External Parameters Affecting on the Photoluminescence of InAs Spherical Layer Quantum Dot ()

Marwan Zuhair Elias^{}

Biophysics Department, Sciences College, Mosul University, Mosul, Iraq.

**DOI: **10.4236/jamp.2021.910155
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Biophysics Department, Sciences College, Mosul University, Mosul, Iraq.

Spherical layer quantum dots (SLQDs) attract a great deal of importance, and have various optoelectronics applications due to their outstanding optical and electrical properties. The photoluminescence (PL) and the electroluminescence (EL) spectra of InAs (SLQDs) were investigated theoretically under the presence of external parameters (pressure, temperature, electric field). Existing of both the temperature and the applied electric field lead to a significant decrease in photoluminescence peak energy (red-shift), while an increase existed in presence of applied hydrostatic pressure (blue-shift). Also with increasing the quantum azimuthal number the photoluminescence peak energy increase. In addition, we found no effect on the band shape of the luminescence as a result of existing such parameters. The study indicates the importance of such parameters as fitting parameters for photoluminescence spectra.

Keywords

Layer Quantum Dots, Photoluminescence, Electroluminescence, Pressure-Temperature Dependence

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Elias, M. (2021) External Parameters Affecting on the Photoluminescence of InAs Spherical Layer Quantum Dot. *Journal of Applied Mathematics and Physics*, **9**, 2439-2446. doi: 10.4236/jamp.2021.910155.

1. Introduction

Studying the electronic devices using photoluminescence is a powerful technique to extract valuable information about semiconductor sample. The promising objects of quantum dots (QDs) are due to their wide range of applications, ranging from bio-labeling, photodetection, light emitting diode and solar cells. In the case of applications, numerous parameters may affect the photoluminescence (PL) of QDs [1] . The authors of [2] investigate the photoluminescence properties of core-shell InP/ZnS quantum dots under temperature from 80 - 300 K with four different sizes by 405 nm continuous excitation light, the authors found that there was a red-shift in photoluminescence peak with increasing temperature which attributed to the interaction of longitudinal acoustic phonons with excitons. In Ref. [3] , the authors studied the quantum dots photoluminescence of perovskite CsPbBr_{3} enhanced by nanorods plasmonic Au. They found that the quantum dots film photoluminescence intensity of Au nanorods/CsPbBr_{3} exhibits an enhancement of 2-fold compared with pristine CsPbBr_{3} quantum dots film. Kyun Geun Ung *et al*. [4] studied the perovskite quantum dots (PQDs) and reported an effective strategy for highly luminescent of (PQDs) by a simple doping. In Ref [5] , the fabrication of hybrid nanoflowers graphene quantum dots (NFGQDs)/ZnO was investigated. The structure showed strong photoluminescence emissions were more efficient integration of NFGQDs exist as a result of the large ZnO surface area. Neslihan Ayarcı Kuruoğlu *et al*. [6] investigate the Carrier Conduction Mechanism over InAs/InP Quantum Dashes and InAs/GaAs Quantum Dots Based p-i-n Laser Heterostructures. The charge transfer characteristics of the long wavelength semiconductor laser structures, containing quantum dot layers (QDs), were investigated by means of temperature dependent current-voltage and electroluminescence measurements over InAs/InP, and InAs/GaAs based p-i-n structures, they found that the peak value of emitted laser light for InAs/InP QDashes and InAs/GaAs QDs occurred in 1.55 μm and 1.3 μm, respectively. The authors of [7] propose a synthesis method to synthesize CsPbCl_{3}:Mn quantum dots using a single-step ultrasonic. The quantum dots synthesized improved the photoluminescence properties and exhibit both blue and orange Mn emissions. In Ref [8] , the authors demonstrated the synthesis of scalable carbon quantum dots by simply heating or microwave using a mixture of ethylenediamine (EDA) and citric acid (CA) in presence of glycerol (Gly) as solvent of high boiling point. The CQDs showed UV absorption at about 346 nm and a broad spectrum of photoluminescence with quantum yield of 49.9%.

Roberta De Angelis *et al*. [9] investigated the Chemical Sensitivity of Luminescent Epitaxial Surface InP Quantum Dots, where a broad-band near-infrared photoluminescence ranging from 750 to 865 nm was reported from the surface InP quantum dots grown by gas source molecular beam epitaxy on In_{0.48}Ga_{0.52}P buffer layer lattice matched to GaAs substrate. When the quantum dots exposed to vapors of different chemical solvents with the highest sensitivity for alcohol (methanol and ethanol) vapours a reversible luminescence intensity enhancement has been observed. The authors found that the luminescent behavior depends on the solvent type and concentration. In addition to that, they proved that the solvent vapor has no effect on the peak energy and band shape of the luminescence.

The authors of [10] studied the mechanisms of radiative recombination of large InAs/GaAs quantum dots, the large InAs/GaAs quantum dots optical properties were investigated by low-temperature photoluminescence versus excitation-power density. They concluded that to have more than one exciton by dots the probability must be considered, and we must revise the usual equation to correctly describe the origin of the recombination and must include other factors as relaxation time, radiative recombination rate, scattering, and others.

The investigation of layer quantum dots (SLQDs), in which motion of radial charge carriers limited on inner and outer radii are of great interest and allow flexible manipulations of the optical absorption of layer quantum structures [11] .

In our present work, the effects of external parameters on the photoluminescence of (SLQDs) were investigated theoretically. We studied the ability of using electric field intensity, temperature and pressure, as fitting parameters on such characteristics.

2. Theory

2.1. Pressure-Temperature Effects

Consider the motion of electron in layer quantum dot of spherical shape (SLQDs) with inner radius *R*_{1} and outer radius*R*_{2}. We consider the transition between both hole and electron states. According to Ref [11] and considering the effect of both temperature and pressure. The wave functions of (SLQD) are.

$\begin{array}{l}{\Psi}_{n,l,m}^{e\left(h\right)}\left(r,\theta ,\phi ,e,p,t\right)\\ =\sqrt{\frac{\pi k\left(e,p,t\right)}{2r}}\left[{c}_{1}\cdot {J}_{\left(l+1/2\right)}\left(k\left(e,p,t\right)\cdot r\right)+{c}_{2}\cdot {J}_{-\left(l+1/2\right)}\left(k\left(e,p,t\right)\cdot r\right)\right]{y}_{l,m}\left(\theta ,\phi \right)\end{array}$ (1)

where
${J}_{\pm \left(l+1/2\right)}$ are the Bessel functions, *l* and *m* are the azimuthal and the magnetic quantum numbers respectively, *C*_{1(2)} are the constants of normalization. The wave function (1) should satisfy the boundary conditions (wave function vanishes outside the layer) the electron energy spectrum determined from the transcendent equation:

$\begin{array}{l}{J}_{\left(l+1/2\right)}\left(k\left(e,p,t\right)\cdot {R}_{1}\right)\cdot {J}_{-\left(l+1/2\right)}\left(k\left(e,p,t\right)\cdot {R}_{2}\right)\\ \text{\hspace{0.05em}}-{J}_{\left(l+1/2\right)}\left(k\left(e,p,t\right)\cdot {R}_{2}\right)\cdot {J}_{-\left(l+1/2\right)}\left(k\left(e,p,t\right)\cdot {R}_{1}\right)=0\end{array}$ (2)

The absorption coefficient for the transition between the holes and the electron states of conduction band are:

$H\left(\omega ,{R}_{1},{R}_{2},p,t\right)={A}_{v}{\displaystyle {\sum}_{\begin{array}{l}n,{n}^{\prime}\\ l,{l}^{\prime}\\ m,{m}^{\prime}\end{array}}\left(2l+1\right)\delta \left(\hslash \omega -{E}_{g}\left(p,t\right)-{E}_{n,l,m}^{e}-{E}_{{n}^{\prime},{l}^{\prime},{m}^{\prime}}^{h}\right)}$ (3)

where ${A}_{v}$ coefficient proportional to the square of the modulus of the matrix elements, $n=1,2,3$, $m=0,\pm 1,\pm 2$ and $l=0,1,2$, ${E}_{g}\left(p,t\right)$ is the bulk InAs band gap given by:

${E}_{g}\left(P,T\right)=\left(533+7.7P-\frac{0.276{T}^{2}}{T+83}\right)\text{\hspace{0.17em}}\text{meV}$ (4)

*ω* is the frequency of the light, The selection rules according to Ref [12] will be,

${n}^{e}={n}^{h},{m}^{e}=-{m}^{h},{l}^{e}={l}^{h}$ (5)

For the effective mass of the electron in presence of pressure-temperature effect will be [13] .

${m}^{*}\left(P,T\right)={\left[1+\frac{15020}{{E}_{g}\left(P,T\right)}+\frac{7510}{{E}_{g}\left(P,T\right)+341}\right]}^{-1}{m}_{0},$ (6)

The fractional change in the volume of the spherical layer quantum dot is given by:

${R}_{1\left(2\right)}\left(P\right)={R}_{1\left(2\right)}\left(0\right)\left(1-\left({S}_{11}+2{S}_{12}\right)P\right)$ (7)

where *P* is the pressure in (k.bar), *T* is the temperature in (K), and *m*_{0} is the free electron mass.
${S}_{11}\left(=1.946\times {10}^{-3}\text{\hspace{0.05em}}\text{\hspace{0.05em}}{\text{kbar}}^{-1}\right)$ and
${S}_{12}\left(=-6.855\times {10}^{-4}\text{\hspace{0.05em}}\text{\hspace{0.05em}}{\text{kbar}}^{-1}\right)$ are the elastic constant of the InAs and
${R}_{1\left(2\right)}\left(0\right)$ is the inner (outer) zero-pressure radiuses.

2.2. Presence of Electric Field

Now let an electric field (*V*) be applied uniformly on the system as perturbation. Then the corresponding energy in frames of perturbation theory has the form:

$E=\frac{{p}^{2}}{2m}-e\left(\stackrel{\to}{V}\cdot \stackrel{\to}{r}\right)$ (8)

Corrections of energy in presence of the electric field *V*(*r*) were calculated using the following integrals:

$\langle {l}^{\prime},{m}^{\prime}|\stackrel{^}{V}|l,m\rangle ={\displaystyle \int {Y}_{{l}^{\prime},{m}^{\prime}}^{\ast}\left(\theta ,\phi \right)\stackrel{^}{V}\left(\theta ,\phi \right){Y}_{l,m}\left(\theta ,\phi \right)\text{d}\Omega}$ (9)

All diagonal elements equal zero [12] , so the first-order correction of the perturbation theory will be zero:

${\Delta}_{1}{E}_{l,m}=\langle l,m|\stackrel{^}{V}|l,m\rangle =0.$ (10)

For the second order correction and according to Ref [14] we can write:

$\begin{array}{l}\Delta {E}_{2}={\displaystyle \underset{{{\displaystyle n}}^{l}}{\sum}\frac{{I}_{1}^{{n}^{\prime},l+1}{e}^{2}}{{E}_{n,l}^{\left(0\right)}-{E}_{n,l+1}^{\left(0\right)}}}\left(\frac{1}{4}{\left|{V}_{x}-i{V}_{y}\right|}^{2}{a}_{l,m}^{2}+\frac{1}{4}{\left|{V}_{x}+i{V}_{y}\right|}^{2}{a}_{l,-m}^{2}+{\epsilon}_{z}^{2}{b}_{l,m}^{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\displaystyle \underset{{{\displaystyle n}}^{l}}{\sum}\frac{{I}_{2}^{{n}^{\prime},l-1}{e}^{2}}{{E}_{n,l}^{\left(0\right)}-{E}_{n,l-1}^{\left(0\right)}}}\left(\frac{1}{4}{\left|{V}_{x}-i{V}_{y}\right|}^{2}{a}_{l-1,m-1}^{2}+\frac{1}{4}{\left|{V}_{x}+i{V}_{y}\right|}^{2}{a}_{l-1,m-1}^{2}+{\epsilon}_{z}^{2}{b}_{l-1,m}^{2}\right)\end{array}$ (11)

where ${I}_{1}^{{n}^{\prime},l+1}$, ${I}_{1}^{{n}^{\prime},l-1}$, ${a}_{l,m}$ and ${b}_{l,m}$ are according to [14] .

The photoluminescence spectra for the cases under consideration are calculated using the relation [15] :

$P\left(\omega ,{R}_{1},{R}_{2},p,t\right)={P}_{0}\cdot \hslash \omega \cdot H\left(\omega ,{R}_{1},{R}_{2},p,t\right)\cdot {\text{e}}^{\frac{\hslash \omega -{E}_{g}}{{K}_{b}\cdot T}}\cdot {\text{e}}^{\frac{0.5-{E}_{g}}{{K}_{b}\cdot T}}$ (12)

where *P*_{0} is a coefficient proportional to the square of the modulus of the matrix elements

All calculations were made numerically by the finite element method using mathematica 5.

3. Results and Discussions

Now, let us proceed to the results and discussions. Numerical calculations have been performed for the photoluminescence of (SLQDs). All needed parameters interring our numerical calculations were taken as a function of pressure and temperature for InAs. Let us consider the photoluminescence of (SLQDs) neglecting the interaction between the hole and electron within the framework of the regime of strong size quantization.

Figure 1 (left) represents the photoluminescence (in arbitrary units) versus the frequency of incident hole to electron transition 1h → 1e (in meV) at different value of temperatures (*T* = 100 K, 200 K, 300 K, 400 K), fixed values of inner radius (*R*_{1} = 300 Å), outer radius (*R*_{2} = 900 Å), pressure (*P* = 10 k∙bar) and different azimuthal numbers (*l** _{c}* = 1, 2, 3). From the figures, there was a red-shift in photoluminescence peak energy and a decrease in intensity with increasing temperature which attributed to the interaction of excitons with longitudinal acoustic phonons [2] . In addition to that, we notice that the increase in temperature has no effect on the band shape of the luminescence [9] . Also we see the photoluminescence peak energy increase with increasing the quantum azimuthal number

The opposite picture appears for photoluminescence peak energy Figure 1 (right) where there was a blue-shift in photoluminescence peak energy with increasing pressure at fixed value of temperature (*T* = 4 K) which may be attributed to the energy dependencies on the applied pressure, where a change in the energy gap of InAs as a result of changing the applied pressure will exist lead to changing the Coulomb interaction energy as a result of an internal huge strain of the dot of InAs, which will change the electrons and holes quantization energies [12] . In addition to that Ref [13] showed that the interband emission energy increases when the hydrostatic pressure increases.

Figure 2 represents the photoluminescence (in arbitrary units) versus the

Figure 1. Photoluminescence (in arbitrary units) of InAs (SLQDs) versus the frequency of incident hole to electron transition 1 h → 1e (in meV) at fixed values of inner and outer radiuses (*R*_{1} & *R*_{2}) and different azimuthal numbers (*l** _{c}* = 1, 2, 3) for two cases (left) fixed value of pressure (

Figure 2. Represent the photoluminescence (in arbitrary units) versus the frequency of incident hole to electron transition 1h → 1e (in meV) at different value of electric field component (*e _{x}* = 0,

frequency of incident hole to electron transition 1h → 1e (in meV) at different value of electric field component (*e _{x}* = 0,

4. Conclusion

The photoluminescence spectra of SLQDs were investigated theoretically under the presence of external parameters (pressure, temperature, electric field), a red-shift in photoluminescence peak was observed as a result of existing both temperature and the applied electric field while a blue-shift in photoluminescence peak was observed in presence of applied hydrostatic pressure. In addition to that the quantum azimuthal number has a significant effect on photoluminescence peak energy. The study reveals the importance of such parameters in photoluminescence spectra as fitting parameters.

Highlights

· A red-shift in PL, EL peaks energy was observed with increasing both the temperature and the applied electric field while a blue-shift was observed in PL peak with increasing the hydrostatic pressure.

· The photoluminescence peak energy increase with increasing the quantum azimuthal number.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

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