_{1}

^{*}

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.

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 [_{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. [_{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 [

Roberta De Angelis et al. [_{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 [

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 [

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.

Consider the motion of electron in layer quantum dot of spherical shape (SLQDs) with inner radius R_{1} and outer radiusR_{2}. We consider the transition between both hole and electron states. According to Ref [

Ψ n , l , m e ( h ) ( r , θ , φ , e , p , t ) = π k ( e , p , t ) 2 r [ c 1 ⋅ J ( l + 1 / 2 ) ( k ( e , p , t ) ⋅ r ) + c 2 ⋅ J − ( l + 1 / 2 ) ( k ( e , p , t ) ⋅ r ) ] y l , m ( θ , φ ) (1)

where J ± ( l + 1 / 2 ) 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:

J ( l + 1 / 2 ) ( k ( e , p , t ) ⋅ R 1 ) ⋅ J − ( l + 1 / 2 ) ( k ( e , p , t ) ⋅ R 2 ) − J ( l + 1 / 2 ) ( k ( e , p , t ) ⋅ R 2 ) ⋅ J − ( l + 1 / 2 ) ( k ( e , p , t ) ⋅ R 1 ) = 0 (2)

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

H ( ω , R 1 , R 2 , p , t ) = A v ∑ n , n ′ l , l ′ m , m ′ ( 2 l + 1 ) δ ( ℏ ω − E g ( p , t ) − E n , l , m e − E n ′ , l ′ , m ′ h ) (3)

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

E g ( P , T ) = ( 533 + 7.7 P − 0.276 T 2 T + 83 ) meV (4)

ω is the frequency of the light, The selection rules according to Ref [

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 [

m * ( P , T ) = [ 1 + 15020 E g ( P , T ) + 7510 E g ( P , T ) + 341 ] − 1 m 0 , (6)

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

R 1 ( 2 ) ( P ) = R 1 ( 2 ) ( 0 ) ( 1 − ( S 11 + 2 S 12 ) P ) (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 ( = 1.946 × 10 − 3 kbar − 1 ) and S 12 ( = − 6.855 × 10 − 4 kbar − 1 ) are the elastic constant of the InAs and R 1 ( 2 ) ( 0 ) is the inner (outer) zero-pressure radiuses.

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 = p 2 2 m − e ( V → ⋅ r → ) (8)

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

〈 l ′ , m ′ | V ^ | l , m 〉 = ∫ Y l ′ , m ′ ∗ ( θ , φ ) V ^ ( θ , φ ) Y l , m ( θ , φ ) d Ω (9)

All diagonal elements equal zero [

Δ 1 E l , m = 〈 l , m | V ^ | l , m 〉 = 0. (10)

For the second order correction and according to Ref [

Δ E 2 = ∑ n l I 1 n ′ , l + 1 e 2 E n , l ( 0 ) − E n , l + 1 ( 0 ) ( 1 4 | V x − i V y | 2 a l , m 2 + 1 4 | V x + i V y | 2 a l , − m 2 + ε z 2 b l , m 2 ) + ∑ n l I 2 n ′ , l − 1 e 2 E n , l ( 0 ) − E n , l − 1 ( 0 ) ( 1 4 | V x − i V y | 2 a l − 1 , m − 1 2 + 1 4 | V x + i V y | 2 a l − 1 , m − 1 2 + ε z 2 b l − 1 , m 2 ) (11)

where I 1 n ′ , l + 1 , I 1 n ′ , l − 1 , a l , m and b l , m are according to [

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

P ( ω , R 1 , R 2 , p , t ) = P 0 ⋅ ℏ ω ⋅ H ( ω , R 1 , R 2 , p , t ) ⋅ e ℏ ω − E g K b ⋅ T ⋅ e 0.5 − E g K b ⋅ 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.

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.

_{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 [_{c} which may be attributed to the thermally escape of the carriers to higher energy levels [

The opposite picture appears for photoluminescence peak energy

frequency of incident hole to electron transition 1h → 1e (in meV) at different value of electric field component (e_{x} = 0, e_{y} = 0, e_{z} = 50, 100, 150 V/cm) and fixed values of inner radius (R_{1} = 300 Å), outer radius (R_{2} = 900 Å), pressure (P = 10 k.bar), temperature (T = 4 K) and different azimuthal numbers (l_{c} = 1, 2, 3). From the figure we see that with increasing the electric field component e_{z} there was a red-shift in photoluminescence peak energy (wave length increase) which can be explained according to [_{c}, l_{c}, m_{c}) have a significant effects on photoluminescence peak energy at fixed value of applied electric field (e_{z}) which can be explained as Ref [

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.

· 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.

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

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