Elsevier

Current Applied Physics

Volume 25, May 2021, Pages 1-11
Current Applied Physics

Wetting layer and size effects on the nonlinear optical properties of semi oblate and prolate Si0.7Ge0.3/Si quantum dots

https://doi.org/10.1016/j.cap.2021.02.004Get rights and content

Highlights

  • SiGe prolate and oblate quantum dots.

  • Influence of the geometrical parameters and wetting layer on energy spectrum.

  • Nonlinear optical properties.

Abstract

Semi oblate and semi prolate are among the most probable self-organized nanostructures shapes. The optoelectronic properties of such nanostructures are not just manipulated with the height and lateral size but also with the wetting layer element. The practical interest of derivatives of germanium and silicon has a great important role in optoelectronic devices. This study is a contribution to the analysis of linear and nonlinear optical properties of Si0.7Ge0.3/Si. In the framework of the effective mass approximation, we solve numerically the Schrödinger equation relative to one particle confined in Si0.7Ge0.3/Si semi prolate and semi oblate quantum dots by using the finite element method and by taking into consideration the effect of the wetting layer. The energy spectrum of the lowest states and the dipolar matrix for the fourth allowed transitions are determined and discussed. We also calculate the detailed optical properties, including absorption coefficients, refractive index changes, second and third harmonic generation as a function of the quantum dot sizes. We found that with the change in the size of prolate and oblate quantum dots, there is a shift in the resonance peaks for the absorption coefficient and refractive index. It is due to the modification in the energy levels with changing size. The study proves a redshift in the second harmonic generation and third harmonic generation coefficients with an increase in the height/radius of the oblate/prolate quantum dot, respectively. We also demonstrated the variation of wavefunction inside the quantum dot with the change in wetting layer thickness.

Graphical abstract

The wavefunction for the four low lying states for different values of parametres.

Image 1
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Introduction

Due to their promising optical and electronic properties, quantum dots (QDs), also labeled as artificial atoms, have attracted notable attention from theorists and experimenters [1,2]. These nanostructures are widely used in several optoelectronic devices in different fields space (LEDs, Lasers, quantum information, biomarkers, among others) [[3], [4], [5], [6]]. In QDs, the quantum effects induced by both size and shape alter the energy spectra of charge carriers and control their optical and electronic properties. Thus, the choice of confinement and geometrical factors are crucial tools to design the most attractive optoelectronic devices [7,8]. The control of QD's optoelectronic properties known as bandgap engineering is supported by advancements in nanofabrication techniques that allow the design of different types of nanostructures of different shapes [[9], [10], [11], [12]]. Historically, Si, Ge and their combinations Si1−ηGeη (where η is the concentration of Ge) have a special place in the field of semiconductor devices and applied physics [13]. Various components demonstrate the importance of SiGe material in field effect transistor (FET) applications [14], complementary metal-oxide-semiconductor (CMOS), quantum well MOSFET [15], heterojunction bipolar transistor (HBT) [16], photodetectors and modulators [17,18] and tunneling devices [19,20]. The SiGe alloy is known for its high density, superior optical properties, and higher dielectric constant compared to Si or Ge alone. For more details about its enormous applications, we refer the reader to some theoretical and experimental works [[21], [22], [23], [24]].

It is worth mentioning that from the experimental point of view, during the growth of the SiGe material, QDs arise at the wetting layer. Thus, the well-known growth techniques (Starsky Kratsanov or Metalo-Organic Chemical Vapor Deposition) lead to the formation of nonhomogenous islands with a large dispersion of the sizes and shapes of SiGe QDs [[25], [26], [27], [28], [29]]. Indeed, due to the lattices mismatch of different materials and increasing the wetting layer, the strain energy increases rapidly, and dislocated islands formation can occur to stabilize the structure. The process leads to an inhomogeneous dispersion of size and shape [25]. The forms self-organized dots tend to have different non-symmetrical shapes. The WL considered as a fine layer under the plane of QD islands has a large influence on the optoelectronic properties of QDs. To study this type of structures, authors have suggested semi-spherical, dome-shaped, lens, disk shapes [[30], [31], [32], [33], [34], [35]]. Nevertheless, in such a growth process, we cant avoid the deformation of shape in the course of fabrication of pyramidal, prolate and oblate shapes are most acceptable are more realistic model compared to the spherical or cylindrical shapes. This consideration of shape influences the entire set of properties of these QDs.

Different methods have been used to study their optoelectronic properties. For example, Dvoyan et al. [36,37] have used the variational method to determine the impurity states in a weakly prolate and oblate ellipsoidal microcrystal under the effect of a magnetic field. They have also analyzed the optical properties as a function of the sizes, electric and magnetic fields. Dujardin et al. [38] studied the excitonic binding energy in prolate and oblate spheroidal QDs. Using the variational method, the authors found that the binding of the spherical case is minimal. It increases when the deformation is accentuated and explained that the bandgap is also tuned with the help of the shape of QD. Using the matrix diagonalization method, energy eigenvalues of an electron confined in ellipsoidal and semi-ellipsoidal QDs, in presence and absence of hydrogenic impurity, under an external electric field has been studied by Razaei et al. [39]. The authors have calculated the electron's low-lying states as a function of the electric field strength, the dot size, and its geometry. Baira et al. [40] have investigated the dome and pyramidal of GeSn/Ge. They have determined the optical transitions for integrated optics on Si substrate using the finite element method. Their study found that bigger QDs with higher aspect ratios are likely to have higher oscillator strength and longer radiative lifetime. Their results demonstrate that this QD system could be fascinating for CMOS compatible light emitters and detectors operating in the mid-IR range. They also evaluated the intersubband optical properties for GeSn pyramidal QD with the wetting layer embedded in the Ge matrix under the vertical electric field [41]. Recently, Sabaeian and Shahzadeh have developed several interesting studies concerning pyramidal and dome forms with different effects [[42], [43], [44], [45]]. They have analyzed the effects of wetting layer thickness and quantum dot (QD) shape on S state and P state of strained pyramid QDs. They have demonstrated an overall redshift by increasing the WL. The same authors have analyzed the effect of size and wetting layer on subband electronic envelop functions, eigenenergies, linear and nonlinear absorption coefficients, and refractive indices of dome-shaped InAs/GaAs. They have also determined the linear, and nonlinear optical properties regarding P-to-S transition in pyramid-shaped GaAs quantum dots (QDs) coupled to wetting layer (WL) in an Al0.3Ga0.7As matrix. Sabaeian and Riyahi [46] have investigated THz wave emission and absorption coefficients in truncated pyramidal-shaped InAs/GaAs quantum dots in the presence of a vertical magnetic field. Shahzadesh et al. [47] have analyzed the optical properties of InAs/GaAs oblate and semi-spheroid-shaped quantum dots coupled to WL. In this study, the perturbation method has been used combined with the exact solution for the single-band effective mass Schrödinger equation. They have shown that the electric dipole moment of the prolate QDs is more prominent with an increase in the base length than oblate QDs, with an increase in the height.

On the other hand, the nonlinear optical properties in quantum structures can be associated with the intrasubband and/or intersubband transitions. In the last years, researchers have focused on the linear and nonlinear optical properties of quantum dots: Mastering these properties requires understanding their behavior concerning the different parameters such as the size, shape, and the effect of the external medium. Many studies have treated optical non-linearity in quantum dots under different external perturbations: magnetic field, electric field, pressure, and temperature. The whole of these investigations shows that the nonlinear optical properties are susceptible to several factors: the nature of the material, the shape of the QD, and the external perturbation. It is crucial to point out that unlike the third order, the second-order response only occurs in systems that exhibit asymmetry as in the quantum medium, characterized by allowed transitions between states having the same parities. For more details we refer the reader to some interesting references [[48], [49], [50], [51], [52], [53], [54], [55], [56], [57], [58]]. Indeed, in these studies, the absorption coefficient, refractive index changes, second and third harmonic generations are the most investigated properties for different types of confinement: quantum wells, quantum disk, quantum wire, core/shell, conical and pyramidal dots, etc. However, to our knowledge and despite their importance, there is no study concerning the nonlinear optical properties induced by the low first state transitions of SiGe with the wetting layer's effect. Thus, in this study, we will try to analyze the behavior of different optical properties: absorption coefficient, refractive index changes, second and third harmonic generation.

In the present manuscript, we focus on the lower four states and intraband transitions between these states in two configurations semi-prolate and semi-oblate QDs of Si0.7Ge0.3/Si material and for wetting layer surrounded by Si matrix. It is interesting to note that the optical transitions can be controlled by the height (h) and the radius (R) of the QDs. We determine the eigenenergies, the wave functions, and the dipole matrix elements as a function of the thickness, h, and R. These results are used to analyze the linear and non-linear optical AC, RIC, SHG, and THG. The article is arranged as follows: In Section 2, we give a detailed theory and model for the system with the formulation we used, followed by detailed results and thorough discussion in Section 3. Finally, a brief conclusion is illustrated for the findings of the results in Section 4.

Section snippets

Theory and model

Let us consider semi oblate and semi prolate Si0.7Ge0.3/Si QDs deposited on a thick SiGe wetting layer (WL) surrounded by Si matrix with dimension L3 where L = 40 nm. The sketch of the system is provided in Fig. 1. The system can be described by the Schrödinger equation:where ψ(r) and E are the wave function and the energy level respectively. For a given region here, the effective mass of the electron mi* depends on the position such that:mi*(r)=mSiGe*inside the core and the WLmSi*otherwise 

Results and discussion

The parameters used in the numerical calculations for Si0.7Ge0.3/Si QD are as follows: σ = 2.8 × 1022m−3, nr = 3.55, ϵ = 13.05, Γfi = 0.38 ps−1 and I = 2 × 107W/m2 [66,67]. Here the width of WL is taken as 0.5 nm.

From the numerical point of view, Eq. (1) is solved with Finite Element Method using COMSOL Multiphysics software [68]. The Schr ödinger equation was established in the form of the general partial differential equation. As it is shown in Fig. 1a, we describe the delimitation of

Conclusion

We have studied the energy states, the transition matrix elements between different states of the oblate and the prolate shaped Si0.7Ge0.3/Si QDs with Si0.7Ge0.3 wetting layer (WL), which is surrounded by the Si matrix. We found that the low-lying states' energy, which is important in the present study, decreases with the increasing height in oblate QD and radius in prolate QD. Also presented are the wave functions for these low lying states for both dot shapes, which present a clear picture of

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

LMP and DL acknowledge partial financial support from FONDECYT 1180905. DL acknowledges partial financial support from Centers of Excellence with BASAL/CONICYT financing, Grant AFB180001, CEDENNA.

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