# Quantum Confined Stark Effect on the Linear and Nonlinear Optical Properties of SiGe/Si Semi Oblate and Prolate Quantum Dots Grown in Si Wetting Layer

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theory and Model

## 3. Results and Discussion

^{−3}, the electron density, ${n}_{r}(\eta =0.3)=3.55$ represents the refractive index of the QD, $\epsilon (\eta =0.3)=13.05$ is the static dielectric constant, ${\Gamma}_{\mathrm{if}}=0.38$ ps

^{−1}is the inverse of the relaxation time material and $I=2\times {10}^{7}$ W/m

^{2}is the intensity of the incident electromagnetic field [59,67]. The effective mass of conductivity ${m}_{SiGe}^{*}=0.26{m}_{0}$, where ${m}_{0}$ is the mass of free electrons.

#### 3.1. With Lateral Electric Field (In x-Direction)

^{2}/V

^{2}) and for the SiGe case are near to the GaAs/AlAs case (${10}^{-14}$ m

^{2}/V

^{2}) as reported by Reference [72].

#### 3.2. With Electric Field (In z-Direction) Perpendicular to Wl

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Variation of four low lying energy levels as a function of the (

**a**) oblate QD height (with R = 7 nm) and (

**b**) prolate QD R (with h = 7 nm) without electric field i.e., $F=0$ kV/cm for electron and hole states.

**Figure 3.**Ground and excited state energies as a function of the (

**a**) oblate QD height h and (

**b**) prolate QD radius R for different values of parallel electric field $F=20$, 30 and 40 kV/cm.

**Figure 4.**The DM elements (

**a**) ${M}_{12}$, (

**b**) ${M}_{23}$, (

**c**) ${M}_{34}$, (

**d**) ${M}_{31}$, and (

**e**) ${M}_{41}$ as a function of oblate QD height h for R = 7 nm and for different values of lateral electric field $F=20,\text{}30$ and 40 kV/cm respectively.

**Figure 5.**The DM elements (

**a**) ${M}_{12}$, (

**b**) ${M}_{23}$, (

**c**) ${M}_{34}$, (

**d**) ${M}_{31}$, and (

**e**) ${M}_{41}$ as a function of prolate QD Radius R for h = 7 nm and for different values of the lateral electric field $F=20,\text{}30$ and 40 kV/cm respectively.

**Figure 6.**Linear, non-linear and total absorption coefficients as a function of the photon energy for (

**a**) oblate height R = 4 nm with radius R = 7 nm at WL thickness 0.5 nm (

**b**) prolate QD with R = 4 nm, h = 7 nm with WL thickness = 0.5 nm, (

**c**) oblate QD with R = 4 nm, R = 7 nm with WL thickness = 1 nm, and (

**d**) prolate QD with R = 4 nm, h = 7 nm with WL thickness = 1 nm for different values of the lateral electric field $F=20,\text{}30$ and 40 kV/cm.

**Figure 7.**Linear, non-linear and total refractive index changes as a function of the photon energy for (

**a**) oblate height R = 4 nm with radius R = 7 nm at WL thickness 0.5 nm (

**b**) prolate QD with R = 4 nm, h = 7 nm with WL thickness = 0.5 nm, (

**c**) oblate QD with R = 4 nm, R = 7 nm with WL thickness = 1 nm, and (

**d**) prolate QD with R = 4 nm, h = 7 nm with WL thickness = 1 nm for different values of the lateral electric field $F=20,\text{}30$ and 40 kV/cm.

**Figure 8.**Second harmonic generation as a function of the photon energy for (

**a**) oblate height R = 4 nm with radius R = 7 nm at WL thickness 0.5 nm (

**b**) prolate QD with R = 4 nm, h = 7 nm with WL thickness = 0.5 nm for different values of the lateral electric field $F=20,\text{}30$ and 40 kV/cm.

**Figure 9.**Third harmonic generation as a function of the photon energy for (

**a**) oblate height R = 4 nm with radius R = 7 nm at WL thickness 0.5 nm (

**b**) prolate QD with R = 4 nm, h = 7 nm with WL thickness = 0.5 nm for different values of lateral electric field $F=20,\text{}30$ and 40 kV/cm.

**Figure 10.**Second harmonic generation as a function of the photon energy for (

**a**) oblate height R = 4 nm with radius R = 7 nm at WL thickness 1 nm (

**b**) prolate QD with R = 4 nm, h = 7 nm with WL thickness = 1 nm for different values of lateral electric field $F=20,\text{}30$ and 40 kV/cm.

**Figure 11.**Third harmonic generation as a function of the photon energy for (

**a**) oblate height R = 4 nm with radius R = 7 nm at WL thickness 1 nm (

**b**) prolate QD with R = 4 nm, h = 7 nm with WL thickness = 1 nm for different values of lateral electric field $F=20,\text{}30$ and 40 kV/cm.

**Figure 12.**Ground and excited state energies as a function of the (

**a**) oblate QD height (with R = 7 nm) and (

**b**) prolate QD R (with h = 7 nm) for different values of the perpendicular electric field at $F=5,\phantom{\rule{0.277778em}{0ex}}10$ and 15 kV/cm.

**Figure 13.**The DM elements (

**a**) ${M}_{12}$, (

**b**) ${M}_{23}$, (

**c**) ${M}_{34}$, (

**d**) ${M}_{31}$, and (

**e**) ${M}_{41}$ as a function of oblate QD height h with R = 7 nm for different values of the perpendicular electric field $F=5,\phantom{\rule{0.277778em}{0ex}}10$ and $15\phantom{\rule{0.277778em}{0ex}}$ kV/cm respectively.

**Figure 14.**The DM elements (

**a**) ${M}_{12}$, (

**b**) ${M}_{23}$, (

**c**) ${M}_{34}$, (

**d**) ${M}_{31}$, and (

**e**) ${M}_{41}$ as a function of prolate QD radius R with h = 7 nm for different values of the perpendicular electric field $F=5,\phantom{\rule{0.277778em}{0ex}}10$ and 15 kV/cm respectively.

**Figure 15.**Linear, non-linear and total absorption coefficients as a function of the photon energy for (

**a**) oblate height h = 5 nm with radius R = 7 nm at WL thickness 0.5 nm and 1 nm (

**b**) prolate QD with R = 5 nm, h = 7 nm with WL = 0.5 nm and 1 nm, at four perpendicular electric field strengths of $F=0,\phantom{\rule{0.277778em}{0ex}}5,\phantom{\rule{0.277778em}{0ex}}10$ and 15 kV/cm from left to right respectively.

**Figure 16.**Linear, non-linear and total refractive index changes as a function of the photon energy for (

**a**) oblate height h = 5 nm with radius R = 7 nm at WL thickness 0.5 nm and 1 nm (

**b**) prolate QD with R = 5 nm, h = 7 nm with WL = 0.5 nm and 1 nm, at four perpendicular electric field strengths of $F=0,\phantom{\rule{0.277778em}{0ex}}5,\phantom{\rule{0.277778em}{0ex}}10$ and 15 kV/cm from left to right respectively.

**Figure 17.**Second harmonic generation as a function of the photon energy for (

**a**) oblate height h = 5 nm with radius R = 7 nm at WL thickness 0.5 nm (

**b**) prolate QD with $R=5$ nm, h = 7 nm with WL thickness = 0.5 nm for different perpendicular electric field strengths from F = 0 to 15 kV/cm.

**Figure 18.**Third harmonic generation as a function of the photon energy for (

**a**) oblate height h = 5 nm with radius R = 7 nm at WL thickness 0.5 nm (

**b**) prolate QD with R = 5 nm, h = 7 nm with WL thickness = 0.5 nm for different perpendicular electric field strengths from $F=0$ to 15 kV/cm.

**Figure 19.**Second harmonic generation as a function of the photon energy for (

**a**) oblate height h = 5 nm with radius R = 7 nm at WL thickness 1 nm (

**b**) prolate QD with R = 5 nm, h = 7 nm with WL thickness = 1 nm for different perpendicular electric field strengths from $F=0\phantom{\rule{0.277778em}{0ex}}$ to 15 kV/cm.

**Figure 20.**Third harmonic generation as a function of the photon energy for (

**a**) oblate height h = 5 nm with radius R = 7 nm at WL thickness 1 nm (

**b**) prolate QD with R = 5 nm, h = 7 nm with WL thickness = 1 nm for different perpendicular electric field strengths from $F=0\phantom{\rule{0.277778em}{0ex}}$ to 15 kV/cm.

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**MDPI and ACS Style**

Varsha; Kria, M.; Hamdaoui, J.E.; Pérez, L.M.; Prasad, V.; El-Yadri, M.; Laroze, D.; Feddi, E.M.
Quantum Confined Stark Effect on the Linear and Nonlinear Optical Properties of SiGe/Si Semi Oblate and Prolate Quantum Dots Grown in Si Wetting Layer. *Nanomaterials* **2021**, *11*, 1513.
https://doi.org/10.3390/nano11061513

**AMA Style**

Varsha, Kria M, Hamdaoui JE, Pérez LM, Prasad V, El-Yadri M, Laroze D, Feddi EM.
Quantum Confined Stark Effect on the Linear and Nonlinear Optical Properties of SiGe/Si Semi Oblate and Prolate Quantum Dots Grown in Si Wetting Layer. *Nanomaterials*. 2021; 11(6):1513.
https://doi.org/10.3390/nano11061513

**Chicago/Turabian Style**

Varsha, Mohamed Kria, Jawad El Hamdaoui, Laura M. Pérez, Vinod Prasad, Mohamed El-Yadri, David Laroze, and El Mustapha Feddi.
2021. "Quantum Confined Stark Effect on the Linear and Nonlinear Optical Properties of SiGe/Si Semi Oblate and Prolate Quantum Dots Grown in Si Wetting Layer" *Nanomaterials* 11, no. 6: 1513.
https://doi.org/10.3390/nano11061513