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Optical third-harmonic generation of spherical quantum dots under inversely quadratic Hellmann plus inversely quadratic potential

  • Xing Wang ,
  • Xuechao Li
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  • School of Mechanics and Photoelectric Physics, Anhui University of Science and Technology, Huainan 232001, China

Received date: 2024-03-17

  Revised date: 2024-04-13

  Accepted date: 2024-05-16

  Online published: 2024-07-24

Copyright

© 2024 Institute of Theoretical Physics CAS, Chinese Physical Society and IOP Publishing

Abstract

The third-harmonic generation (THG) coefficient for a spherical quantum dot system with inversely quadratic Hellmann plus inversely quadratic potential is investigated theoretically, considering the regulation of quantum size, confinement potential depth and the external environment. The numerical simulation results indicate that the THG coefficient can reach the order of 10−12 m2 V–2, which strongly relies on the tunable factor, with its resonant peak experiencing a redshift or blueshift. Interestingly, the effect of temperature on the THG coefficient in terms of peak location and size is consistent with the quantum dot radius but contrasts with the hydrostatic pressure. Thus, it is crucial to focus on the influence of internal and external parameters on nonlinear optical effects, and to implement the theory in practical experiments and the manufacture of optoelectronic devices.

Cite this article

Xing Wang , Xuechao Li . Optical third-harmonic generation of spherical quantum dots under inversely quadratic Hellmann plus inversely quadratic potential[J]. Communications in Theoretical Physics, 2024 , 76(9) : 095702 . DOI: 10.1088/1572-9494/ad4cdf

1. Introduction

In recent years, due to rapid advances in manufacturing technology, semiconductor structures, especially quantum dots (QDs), have garnered considerable interest [15]. Because of their special structure, semiconductor QDs exhibit unique physical properties not found in traditional materials, such as the quantum local effect [6, 7], quantum tunneling effect [8], surface effect [9], quantum size effect [10, 11] and Coulomb blocking effect [12]. These properties open up a range of applications in photovoltaics, photonics and optoelectronic devices such as light-emitting diodes, electro-optical modulators, far-infrared photodetectors and semiconductor optical amplifiers, all of which hold significant research value [1317].
As is well known, the generation of second harmonics and optical rectification depend heavily on the defects and misalignment in the system, and the third harmonics are not bound by these limitations, making them more versatile for designing optoelectronic devices [1821]. Third harmonics, a sine wave component derived from Fourier decomposition, play a crucial role in laser technology, spectroscopy development and material structure analysis. Third harmonics can be more widely used to design optoelectronic devices [2225]. In recent years, third harmonics have become a hot topic in the field of nonlinear optics. In 2004, Yu et al [26] examined the third harmonic generation (THG) coefficient in cylindrical quantum lines, revealing a substantial increase when factoring in electron–phonon interaction, shifting the curve’s peak to higher energy levels. In 2009, Wang et al [27] reported THG in asymmetric coupled quantum wells, noting that variations in well width and barrier height between potential wells can have a notable impact on these coefficients. In 2012, Li et al [28] delved into the third harmonics in two-dimensional QDs under the polaron condition and found that THG is affected by the applied magnetic field and the size of the system. Despite these advances, a comprehensive study on optical THG of spherical QDs with inversely quadratic Hellmann plus inversely quadratic potential (IQHIQP) remains to be done.
The aim of this paper is to explore THG in spherical QDs with IQHIQP. Compared with THG with the inversely quadratic Hellmann–Kratzer potential [3], our results are one to two orders of magnitude higher. In section 2, the Schrödinger equation will be solved and the specific expression for the THG coefficient will be derived by the Nikiforov–Uvarov function analysis (NUFA) method. In section 3, the influence of internal and external parameters on THG of GaAs/Al0.3Ga0.7As spherical QDs under the IQHIQP is discussed. Eventually, we lay out a brief conclusion in section 4.

2. Model and analysis

In this paper, the constrained potential V(r) is taken as the inversely quadratic Hellmann (IQH) potential Viqh(r) [29] plus the inversely quadratic (IQ) potential Viq(r) [30], which has the following form:
$\begin{eqnarray}{V}_{\mathrm{iqh}}(r)=-\displaystyle \frac{a}{r}+\displaystyle \frac{b}{{r}^{2}}{{\rm{e}}}^{\delta r},{V}_{\mathrm{iq}}(r)=\displaystyle \frac{g}{{r}^{2}}.\end{eqnarray}$
The sum of these potentials (IQHIQP) can be described as
$\begin{eqnarray}V(r)=b{\delta }^{2}-\displaystyle \frac{1}{r}(a+b\delta )+\displaystyle \frac{1}{{r}^{2}}(b+g),\end{eqnarray}$
where δ is the screening parameter and a and b are the strengths of the Coulomb and Yukawa potentials, respectively.
The Schrödinger equation in spherical coordinates can be written as [31, 32]
$\begin{eqnarray}\begin{array}{l}-\displaystyle \frac{{{\hslash }}^{2}}{2m* (P,T)}\left[\displaystyle \frac{1}{{r}^{2}}\displaystyle \frac{\partial }{\partial r}\left({r}^{2}\displaystyle \frac{\partial }{\partial r}\right)+\displaystyle \frac{1}{{r}^{2}\sin \theta }\displaystyle \frac{\partial }{\partial \theta }\left(\sin \theta \displaystyle \frac{\partial }{\partial \theta }\right)\right.\\ \left.+\,\displaystyle \frac{1}{{r}^{2}{\sin }^{2}\theta }\displaystyle \frac{{\partial }^{2}}{\partial {\phi }^{2}}\right]{\rm{\Psi }}(r,\theta ,\phi )+V(r){\rm{\Psi }}(r,\theta ,\phi )\\ =\,E{\rm{\Psi }}(r,\theta ,\phi ),\end{array}\end{eqnarray}$
where m*(P, T) is the effective mass of the electron [33, 34]. Using the common ansatz for the wave function
$\begin{eqnarray}{\rm{\Psi }}(r,\theta ,\phi )=\displaystyle \frac{R(r)}{r}{Y}_{{lm}}(\theta ,\phi )\end{eqnarray}$
in equation (3), we get the following set of equations:
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{{\rm{d}}}^{2}{R}_{{nl}}(r)}{{\rm{d}}{r}^{2}}+\displaystyle \frac{2{m}^{* }(P,T)}{{{\rm{\hslash }}}^{2}}\left[{E}_{{nl}}-V(r)-\displaystyle \frac{l(l+1){{\rm{\hslash }}}^{2}}{2{m}^{* }(P,T)}\displaystyle \frac{1}{{r}^{2}}\right]\\ \qquad \times \,{R}_{{nl}}(r)=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}\displaystyle \frac{{{\rm{d}}}^{2}{{\rm{\Theta }}}_{{ml}}(\theta )}{{\rm{d}}{\theta }^{2}}+\cot \theta \displaystyle \frac{{\rm{d}}{{\rm{\Theta }}}_{{ml}}(\theta )}{{\rm{d}}\theta }\left(l(l+1)-\displaystyle \frac{{m}^{2}}{{\sin }^{2}\theta }\right){{\rm{\Theta }}}_{{ml}}(\theta )=0,\end{eqnarray}$
$\begin{eqnarray}\displaystyle \frac{{{\rm{d}}}^{2}{{\rm{\Phi }}}_{m}(\phi )}{{\rm{d}}{\phi }^{2}}+{m}^{2}{{\rm{\Phi }}}_{m}(\phi )=0,\end{eqnarray}$
where Ylm(θ, φ) = Θml(θm(φ) is referred to as the spherical harmonic function [32], which is the solution to equations (4) and (5), and m2 is the separation constant. Substituting V(r) into equation (5), and by the means of transformation z = r2, the following equation is obtained:
$\begin{eqnarray}\displaystyle \frac{{{\rm{d}}}^{2}R(z)}{{\rm{d}}{z}^{2}}+\displaystyle \frac{1}{2z}\displaystyle \frac{{\rm{d}}R(z)}{{\rm{d}}z}+\displaystyle \frac{1}{4{z}^{2}}\left(-{\beta }_{1}{z}^{2}+{\beta }_{2}z-{\beta }_{3}\right)R(z)=0,\end{eqnarray}$
where R(z) is the radial wave function and
$\begin{eqnarray}{\beta }_{1}=-\displaystyle \frac{2{m}^{* }\left(E-b{\delta }^{2}\right)}{{{\hslash }}^{2}},\end{eqnarray}$
$\begin{eqnarray}{\beta }_{2}=\displaystyle \frac{2{m}^{* }(a+b\delta )}{{{\hslash }}^{2}},\end{eqnarray}$
$\begin{eqnarray}{\beta }_{3}=\displaystyle \frac{2{m}^{* }(b+g)}{{{\hslash }}^{2}}+l(l+1).\end{eqnarray}$
Following the NUFA method [35], the energy eigenvalue equation with the IQHIQP is obtained as
$\begin{eqnarray}{E}_{{nl}}\,=\,b{\delta }^{2}-\displaystyle \frac{{m}^{* }{\left(a+b\delta \right)}^{2}/2{{\hslash }}^{2}}{{\left(n+\tfrac{1}{2}+\sqrt{\tfrac{2{m}^{* }(b+g)}{{{\hslash }}^{2}}+{\left(l+\tfrac{1}{2}\right)}^{2}}\right)}^{2}},\end{eqnarray}$
and the wave function is expressed by the generalized Laguerre polynomial as
$\begin{eqnarray}R(z)={N}_{n}{z}^{(-1+\sqrt{1+4\gamma })/2}{e}^{-\sqrt{\alpha }z}{L}_{n}^{\sqrt{1+4\gamma }}(2\sqrt{\alpha }z).\end{eqnarray}$
The THG coefficient can be obtained using the compact density matrix theory and the iterative method, which is expressed as follows [27, 28]:
$\begin{eqnarray}\begin{array}{l}{\chi }_{3\omega }^{(3)}\,=\,\displaystyle \frac{{{\rm{e}}}^{4}{\sigma }_{v}}{{\varepsilon }_{0}{{\hslash }}^{3}}\\ \,\times \,\displaystyle \frac{{M}_{01}{M}_{12}{M}_{23}{M}_{30}}{\left(\omega -{\omega }_{10}+i{{\rm{\Gamma }}}_{10}\right)\left(2\omega -{\omega }_{20}+{\rm{i}}{{\rm{\Gamma }}}_{20}\right)\left(3\omega -{\omega }_{30}+{\rm{i}}{{\rm{\Gamma }}}_{30}\right)}.\end{array}\end{eqnarray}$
Here ${M}_{{ij}}\,=\,| \left\langle {\psi }_{j}| r\cos \theta | {\psi }_{i}\right\rangle | $ (i, j=0, 1, 2, 3) is the dipole matrix element, ωij = (EiEj)/ is the transition frequency, Γij is the relaxation rate and σv represents electron density.

3. Results and discussions

In this part, the effects of the application of environmental factors and system structure factors on the THG coefficient are analyzed for GaAs/Al0.3Ga0.7As QDs under IQHIQP. The relevant parameters adopted in the calculation process are as follows [3, 27]: Γ10 = Γ20 = Γ30 = 5 × 1012 Hz, ϵ0 = 8.85 × 10−12 F m−1 and σhv = 5 × 1024 m−3.
In the range we studied, we found that in figure 1 E10 surpasses E20/2 and E20/2 exceeds E30/3, failing to meet the three-photon resonance condition (E10 = E20/2 = E30/3), so the THG energy spectrum displays an obvious three-peak structure. Combined with equation (14), it can be concluded that the first peak appears at ω = E30/3, the second peak close to ω = E20/2 and the third peak is positioned near ω = E10, as shown in figures 2(a)–6(a). Physically, the multipeak structure is attributed to the energy level structure of electrons, which affects the nonlinear polarization characteristics, and different electron transitions lead to multiple nonlinear responses with different frequencies. Furthermore, it is evident from figure 1(a) that Eij decreases almost linearly as hydrostatic pressure P increases, aligning with the trend of Eij decreasing as the temperature T is reduced in figure 1(b). However, the rate of change is more pronounced in the former case. Essentially, Eij is closely related to the shift in THG peak position. In other words, in this system, the effect of P on the energy level interval is opposite and more obvious than that of T. From a physical perspective, the slight impact of temperature on THG results from the interplay of energy gap structure and the weak temperature correlation of nonlinear polarizability.
Figure 1. (a) Energy level interval Eij as a function of P when R = 5 nm, V1 = 850 meV, V2 = 350 meV, V3 = 250 meV and T = 150 K. (b) Eij as a function of T when R = 5 nm, V1 = 850 meV, V2 = 350 meV, V3 = 250 meV and P = 10 kbar.
Figure 2(a) shows a functional image of the variation of THG coefficient ${\chi }_{3\omega }^{(3)}$ with incident photon energy ω using P (8, 10 and 12 kbar) under fixed parameters (V1 = 850 meV, V2 = 350 meV, V3 = 250 meV, T = 150 K, R = 5 nm). With increasing P, the peak value of ${\chi }_{3\omega }^{(3)}$ can be clearly observed to decrease, which is because the increase in P promotes an increase in matrix elements. At the same time, it can be seen from equation (14) that ${\chi }_{3\omega }^{(3)}$ is directly proportional to M01M12M23M30. Thus, ${\chi }_{3\omega }^{(3)}$ increases when the product M01M12M23M30 increases. In addition, the peak value shifts to the low-frequency region when P increases, because, as shown in figure 1(a), Eij decreases with increase in P. To further explore the impact of effective mass parameters on the THG coefficient, figure 2(b) illustrates ω as a function of T ranging from 100 to 200 K. Upon zooming into the peak region at the top right of figure 2(b), it becomes evident that, as temperature rises, the THG peak value decreases and shifts towards higher frequencies. This blueshift phenomenon can be explained by the slight increase in Eij due to the increase in T. For physical reasons, an increase in T will promote constraints, which will increase the energy required for transition between subband energy levels. Notably, the trend in peak size and position of THG in figures 2(a) and (b) is opposite, indicating that a higher THG coefficient and the optimal system performance are achievable under certain environmental conditions.
Figure 2. (a) The variation of THG coefficient ${\chi }_{3\omega }^{(3)}$ with incident photon energy ω using P (8, 10 and 12 kbar). (b) The variation of ${\chi }_{3\omega }^{(3)}$ with incident photon energy ω by sing T (100, 150 and 200 K).
In figure 3(a), the THG coefficient ${\chi }_{3\omega }^{(3)}$ is shown as a function of ω regulated by different values of R when V1 = 850 meV, V2 = 350 meV, V3 = 250 meV, P = 8 kbar and T = 150 K. From the general trend of the spectral lines, the greater the structural parameter R, the greater the THG peak value. The origin of this feature is attributed to the increase in M01M12M23M30 with increase in R, which can be observed figure 3(b). Interestingly, from figure 3(b) it can be observed that the smaller R is, the closer M01M12M23M30 is to 0. In the range of 2–4 nm for R, the peak value of the THG coefficient shows a gradual increase, with a widening increment as R extends from 4 to 6 nm, suggesting that the THG coefficient reaches its maximum further away from the reference value. Additionally, as R increases, the resonance peak experiences a redshift due to weakening of the quantum confinement effect which makes it easier for the electron wave function to diffuse, and the energy gap decreases accordingly.
Figure 3. (a) ${\chi }_{3\omega }^{(3)}$ as a function of ω regulated by R (2, 4, 6 nm). (b) M01M12M23M30 as a function of R with P = 8 kbar, V1 = 850 meV, V2 = 350 meV, V3 = 250 meV and T = 150 K.
Figure 4(a) plots a functional image of the variation of the THG coefficient ${\chi }_{3\omega }^{(3)}$ with incident photon energy ω using V1 (800, 850 and 900 meV) under fixed parameters (R = 5 nm, T = 150 K, V2 = 350 meV, V3 =150 meV, P = 8 kbar). It is not difficult to find that the peak value decreases monotonically when V1 increases, which can be explained by the figure 4(b). Here, M01M12M23M30 decreases with increasing V1, and the physical reason for this phenomenon is that the superposition of different electronic states weakens with increase in V1. Additionally, we find that the peak shifts towards the high energy region as V1 increases, attributed to the rise in Eij with increase in V1. From the physical point of view, this phenomenon can be explained by the strengthening of the system’s binding effect when V1 increases.
Figure 4. Variation of ${\chi }_{3\omega }^{(3)}$ with ω for distinct V1 (800, 850, 900 meV). (b) M01M12M23M30 as a function of V1 with P = 8 kbar, R = 5 nm, V2 = 350 meV, V3 = 250 meV and T = 150 K.
In figure 5(a), the THG coefficient ${\chi }_{3\omega }^{(3)}$ is given as a function of ω regulated by distinct V2 when V1 = 850 meV, R = 5 nm, V3 = 250 meV, P = 8 kbar and T = 150 K. The changes in the curve indicate that as V2 increases, the peak value rises, the peak width sharpens and, simultaneously, the peak shifts to the left. This slight redshift phenomenon can be elucidated from figure 5(b). As depicted in figure 5(b), E10, E20/2 and E30/3 all decrease slightly with increasing V2. The physical reason for this phenomenon is that the increase in V2 weakens the quantum confinement of QDs, subsequently reducing the energy level spacing of the confined electrons in the QDs. Consequently, the resonance peak moves in the direction of low energy. Obviously, compared with figure 4(a), it is evident that V1 and V2 have opposite impacts on the energy level interval and matrix elements.
Figure 5. (a) ${\chi }_{3\omega }^{(3)}$ as a function of ω regulated by different values of V2 (300, 350, 400 meV). (b) Eij as a function of V2 when T = 150 K, R = 5 nm, V1 = 850 meV, V3 = 250 meV and P = 10 kbar.
Figure 6(a) depicts the variation of THG coefficient ${\chi }_{3\omega }^{(3)}$ with incident photon energy ω using V3 (200, 250 and 300 meV) under fixed parameters (R = 5 nm, V1 = 850 meV, V2 = 350 meV, T = 150 K, P = 8 kbar). The peak value increases when V3 increases, and at the same time the peak position shifts to the left (i.e. it redshifts), which is similar to the trend of the energy spectrum of figure 5(a) with independent variables. However, the THG redshift in figure 6(a) is more significant. Obviously, comparing figures 5(b) and 6(b), according to the image slope, it can be seen that in the latter Eij decreases faster with increase in the limiting potential parameters, so the THG peak shifts to the left a little more; that is to say, the influence of V3 on the energy level interval is greater than that of V2, making the system more adjustable.
Figure 6. (a) ${\chi }_{3\omega }^{(3)}$ as a function of ω regulated by different values of V3 (200, 250, 300 meV). (b) Eij as a function of V3 when T = 150 K, R = 5 nm, V1 = 850 meV, V2 = 350 meV, and P = 10 kbar.

4. Conclusions

In this paper we studied the effect of various physical factors on the THG coefficient ${\chi }_{3\omega }^{(3)}$. Based on the effective mass m*(P, T), it was applied to the GaAs/Al0.3Ga0.7As QD system with IQHIQP. After a detailed analysis, we summarized some important conclusions: increase in V2 and V3 will make the THG resonance peak position move to the left of the energy axis and the peak value will increase accordingly, but the amplitude changes are different. However, the influence of temperature T on THG resonance peak is opposite to that of P. In addition, ${\chi }_{3\omega }^{(3)}$ is sensitive to the R of the QD system and does not exhibit a straightforward linear pattern. Therefore, the maximum ${\chi }_{3\omega }^{(3)}$ of a QD system can be achieved by adjusting the size of the tuning factor. In summary, the magnitude of the THG coefficient is highly dependent on the magnitude of the tunable factor. Through the theoretical calculation of low-dimensional QDs, we aim for our work to guide experiments and offer insights into the development of a new generation of pressure and temperature sensors, opening up new possibilities for advances in nano-devices.

National Natural Science Foundation of China (Grant Nos. 11674312, 52174161, 51702003, 12174161 and 61775087) and Anhui University of Science and Technology (Grant No. 2023CX2141).

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