[Paper Review] Linear and nonlinear optical properties in spherical quantum dots: Inversely quadratic Hellmann potential

Levente Máthé, Chibueze P. Onyenegecha+4 more|arXiv (Cornell University)|Dec 16, 2020

Quantum and electron transport phenomena40 references112 citations

TL;DR

This study investigates linear and nonlinear optical properties in spherical quantum dots confined by an inversely quadratic Hellmann (IQH) potential. Using the Nikiforov-Uvarov method, analytical eigenenergies and wave functions are derived, and the density matrix formalism is applied to compute absorption coefficients and refractive index changes. The key finding is that optical intensity induces destructive interference between one- and three-photon processes, leading to a dip and even negative total absorption coefficient at high intensities, while peak positions remain unchanged.

ABSTRACT

The aims of the reported work are to provide new insights into the quantum dot optical properties confined in an inverse of a quadratic Hellmann potential. The Schr\"odinger equation is solved using the Nikiforov-Uvarov (NU) method, in order to obtain the analytical expressions of the eigenenergies and the eigenfunctions. The linear together with the third-order nonlinear changes in absorption coefficients and refractive index are investigated using the density matrix formalism. The absorption coefficients and the refractive index changes are strong related to the structure parameters and the optical intensity.

Motivation & Objective

• To investigate linear and nonlinear optical properties in spherical quantum dots under an inversely quadratic Hellmann (IQH) potential.
• To derive analytical expressions for eigenenergies and eigenfunctions using the Nikiforov-Uvarov (NU) method.
• To analyze the dependence of absorption coefficients and refractive index changes on quantum dot size, potential height, and optical intensity.
• To explore the role of destructive interference between one- and three-photon absorption processes in shaping the total absorption coefficient.
• To determine how structural parameters and optical intensity influence the resonant peak positions and amplitudes of optical responses.

Proposed method

• The Schrödinger equation for an electron in a spherical quantum dot is solved using the effective mass approximation in spherical coordinates.
• The inversely quadratic Hellmann potential is defined as $ V(r) = -\frac{A}{r} + \frac{B}{r^2} $, with $ A = V_0 R_0 $, $ B = V_0 R_0^2 $, and $ R_0 = 1/\alpha $, where $ \alpha $ controls the dot radius.
• The Nikiforov-Uvarov (NU) method is applied to solve the radial Schrödinger equation analytically, yielding eigenenergies and radial wave functions in terms of Laguerre and Jacobi polynomials.
• The density matrix formalism is used to compute the linear and third-order nonlinear absorption coefficients and refractive index changes as functions of photon energy and optical intensity.
• The total absorption coefficient is calculated as the sum of linear and third-order nonlinear contributions: $ \alpha(\omega) = \alpha^{(1)}(\omega) + \alpha^{(3)}(\omega) $, with interference effects considered.
• The refractive index change is decomposed into linear and third-order nonlinear components, with the total change dependent on intensity via the nonlinear term.

Experimental results

Research questions

• RQ1 How do the eigenenergies and wave functions of a quantum dot confined by the IQH potential depend on the dot radius and potential height?
• RQ2 How does the optical intensity influence the interference between one- and three-photon absorption processes in the total absorption coefficient?
• RQ3 What is the effect of quantum dot size on the peak positions and amplitudes of absorption and refractive index changes?
• RQ4 How does the potential height affect the magnitude and energy shift of the refractive index change peaks?
• RQ5 Does the optical intensity shift the resonant frequency of the total absorption coefficient, or only alter its amplitude and sign?

Key findings

• The total absorption coefficient reaches zero when the magnitudes of the linear and third-order nonlinear absorption coefficients are equal, which occurs at an optical intensity of $ I \approx 5.25 \times 10^9 \, \text{W/m}^2 $.
• At higher intensities ($ I > 5.25 \times 10^9 \, \text{W/m}^2 $), the total absorption coefficient becomes negative due to destructive interference between one- and three-photon processes.
• The resonant peak position of the total absorption coefficient remains unchanged with increasing optical intensity, indicating no shift in the resonant frequency.
• Increasing the quantum dot radius from 3 nm to 5 nm causes a red shift in the refractive index change peaks due to a reduced energy splitting between the ground and first excited states.
• Increasing the potential height from 112.23 meV to 336.69 meV induces a blue shift in the refractive index change peaks, as the energy gap increases.
• The magnitude of the refractive index change peaks increases with dot radius due to enhanced dipole matrix element $ M_{21} $, while it decreases with increasing potential height due to reduced $ M_{21} $.