1 Introduction

Low-dimensional semiconductor heterostructures, such as quantum wells (QWs), quantum wires (QWRs), and quantum dots (QDs), are widely used in today's electronics and optoelectronics technologies because they enable the fabrication of efficient and fast circuit elements [1,2,3,4,5]. Some applications for these structures include the production of fast transistors such as modulation-doped field effect transistors (MODFET) [6] or heterojunction bipolar transistor (HBT) [7], infrared lasers [8], efficient solar cells [9], waveguides [10], DBR mirror production [11], and so on. The effective confinement property and the fact that impurity states change their optical, and transport properties are important factors in examining their electronic, optical, and magnetic properties both experimentally [12,13,14,15,16] and theoretically [17,18,19,20]. The confinement potentials of low-dimensional heterostructures can be manipulated effectively by using varied methods such as changing externally applied fields (electric, magnetic, laser, etc.) or changing the structural parameters (well width, barrier width, etc.) one by one or together. Manipulating the confinement potential affects the electronic structure of the system and therefore the transitions between intersubband energy levels. Variations in these transitions result in changes in linear and nonlinear optical properties. These changes in optical properties enable major and important advances in optoelectronics. Researchers working on this subject have focused on the investigation of the second-order nonlinear optical properties like nonlinear optical rectification (NOR), second harmonic generation (SHG), and third harmonic generation (THG) of various semiconductor heterostructures with different geometric structures and conditions to design materials with improved nonlinear optical properties and have carried out many studies. Martínez-Orozco et al. [21] reported how the NOR, SHG, and THG changed under the magnetic field’s effect for double δ-doped GaAs QWs. They showed that the configuration they worked on was suitable for photodetectors that want to be produced in the far infrared/THz region. In their theoretical study, Restrepo et al. [22] examined the linear and nonlinear optical properties in AlxGa1−xAs/GaAs asymmetric multiple-step quantum wells under the effects of electric, magnetic, and non-resonant intense laser fields. They found that variations in applied external fields significantly influenced the peak positions and magnitude of the SHG and THG coefficients. Sayrac et al. [23] studied the InP/InGaAs triple quantum well structure to observe the effects of structural variables and different external perturbations on NOR, SHG, and THG. Besides, Giraldo-Tobón et al. [24] calculated the NOR, SHG, and THG coefficients in a two-dimensional elliptical quantum dot. Alaydin et al. [25] studied the honeycomb quantum well-wire (HQWW) structure by altering intense laser fields to optimize NOR, SHG, and THG coefficients under the effects of constant electric and magnetic fields. They proved that HQWW could be used for harmonic generation-based optical devices. In another work, Pourmand et al. [26] investigated the optical rectification (OR), second, and third harmonic generations (SHG and THG) of a GaAs quantum ring. They showed that both the amplitude and position of OR, SHG, and THG peaks exhibit changes depending on external fields and inner radius.

Within this study, we investigated the impacts of external fields such as electric, magnetic, and intense laser fields on the THG coefficient for GaAs/AlGaAs Manning-like double quantum well structure. Besides, we analyzed how THG coefficients are affected by changes in some structural parameters such as well’s depth and width. To achieve the aforementioned goals, we used the diagonalization and compact density matrix methods to determine the energy levels with wave functions and the mathematical description of THG coefficients.. The article is organized as follows: Sect. 2 provides a detailed explanation of the problem and its solution, Sect. 3 contains all of the work's results and comments, and Sect. 4 presents the conclusions.

2 Theory

Firstly, we will present the mathematical calculations of the electron subband energy levels together with their corresponding wave functions of the GaAs/AlGaAs Manning-like double quantum well heterostructure which is given in Fig. 1. After these calculations, the derivation of the equation of THG coefficient is given accordingly in detail. The directions of the applied external electric, magnetic, and intense laser fields have been chosen as parallel to the growth direction \(\left( {{\varvec{F}} = F\hat{z}} \right)\), perpendicular to the growth direction \(\left( {{\varvec{B}} = B\hat{x}} \right)\), and oriented along the \(z\)-axis respectively.

Fig. 1
figure 1

The confinement potential’s profiles and the probability densities of the four lowest states for \(V_{0} = 228\;{\text{meV}}\) and \(L_{wt} = 100\) Å in the presence of the a electric field \(\left( {B = 0, \alpha_{0} = 0} \right)\), b magnetic field \(\left( {F = 0, \alpha_{0} = 0} \right)\), and c intense laser field \(\left( {F = 0, B = 0} \right)\)

Full size image

Within the effective mass and parabolic band approximations, the total Hamiltonian expression for a confined electron in this structure with the contributions of the external fields mentioned above can be written as [27]

$$ {\mathbb{H}}_{0} = \frac{1}{{2m^{*} }}\left[ {{\varvec{p}} + \frac{e}{c}{\varvec{A}}\left( {\varvec{r}} \right)} \right]^{2} + {\mathbb{V}}\left( {z,\alpha_{0} } \right) + eFz $$
(1)

In the given expression, \(m^{*}\) and \(e\) indicate the electron’s effective mass and charge values, respectively. \(c\) shows the light’s speed and \({\varvec{p}}\) is the momentum operator. The magnetic vector potential is given by \({\varvec{A}}\left( {\varvec{r}} \right) = \left( {0, - Bz,0} \right)\). The magnitude of electric field is \(F\). The laser-dressed confinement potential is expressed by \({\mathbb{V}}\left( {z,\alpha_{0} } \right)\) in the Hamiltonian and is found out by calculating the integral value of the following expression [28]

$$ {\mathbb{V}}\left( {z,\alpha_{0} } \right) = \frac{{\mathbb{W}}}{2\pi }\mathop \int \limits_{0}^{{\frac{2\pi }{{\mathbb{W}}}}} {\mathbb{V}}\left( {z + \alpha_{0} \sin {\mathbb{W}}t} \right){\text{d}}t $$
(2)

Here \(\alpha_{0}\), which is directly proportional to the laser field’s strength and inversely proportional to the square of the laser field’s non-resonant frequency, is the laser dress parameter \(\left( {\frac{{eF_{0} }}{{m^{*} {\mathbb{W}}}}} \right)\). When the intense laser field equals zero, the confinement potential of the Manning-like double quantum well is denoted by \({\mathbb{V}}\left( z \right)\) and defined as

$$ {\mathbb{V}}\left( z \right) = V_{0} \left[ {{\mathcal{V}}_{1} \sec h^{4} \left( {z/L_{wt} } \right) - {\mathcal{V}}_{2} \sec h^{2} \left( {z/L_{wt} } \right)} \right] $$
(3)

\({\mathcal{V}}_{1}\) and \({\mathcal{V}}_{2}\) are the structural parameters, which determine the depths of both wells separately. Another parameter is \(L_{wt}\) and it sets wells’ widths. \(V_{0}\) is the barrier’s height.

The electron wave function belonging the system can be given as [29]

$$ \psi \left( {\vec{r}} \right) = e^{{\left( {ik_{ \bot } \vec{r}} \right)}} {\Psi }\left( z \right) $$
(4)

where \(k_{ \bot }\) has \(k_{x}\) and \(k_{y}\) components. \({\Psi }\left( z \right)\) satisfies the Schrödinger equation given in one dimension below [30]

$$ \left[ { - \frac{{\hbar^{2} }}{{2m^{*} }}\frac{{d^{2} }}{{dz^{2} }} + \frac{{e^{2} B^{2} }}{{2m^{*} c^{2} }}\left( {z + \frac{{c\hbar k_{x} }}{eB}} \right)^{2} + {\mathbb{V}}\left( {z,\alpha_{0} } \right) + eFz} \right]{\Psi }\left( z \right) = \left[ {E_{z} - \frac{{\hbar^{2} k_{y}^{2} }}{{2m^{*} }}} \right]{\Psi }\left( z \right) $$
(5)

When \(k_{ \bot } = 0\), Eq. (5) becomes the following

$$ \left[ { - \frac{{\hbar^{2} }}{{2m^{*} }}\frac{{d^{2} }}{{dz^{2} }} + \frac{{e^{2} B^{2} }}{{2m^{*} c^{2} }}z^{2} + {\mathbb{V}}\left( {z,\alpha_{0} } \right) + eFz} \right]{\Psi }\left( z \right) = E_{z} {\Psi }\left( z \right) $$
(6)

The energy eigenvalues and wave functions are expressed by \(E_{z}\) and \({\Psi }\left( z \right)\) for each level. According to the diagonalization method, the single electron wave function \({\Psi }\left( z \right) \) is expressed in terms of the complete set of orthonormal functions,

$$ {\Psi }\left( z \right) = \sqrt {\frac{2}{{L_{\infty } }}} \mathop \sum \limits_{n} C_{n} {\text{sin}}\left( {\frac{n\pi z}{{L_{\infty } }} + \frac{n\pi }{2}} \right) $$
(7)

The width of the well \(L_{\infty }\) (1000 Å) is quite large compared to the width of the examined structure. While \(C_{n}\) corresponds to expansion coefficient, \(n\) represents a positive integer. The calculations that we performed so far provided us with the necessary information about the transitions between sub-bands and the dipole matrix elements related to the sub-bands. Afterward, we used the compact density matrix method [21, 31,32,33] to find out the mathematical expression of third harmonic generation (THG) coefficients and assumed that the system was excited by an electromagnetic field with frequency \(\omega\) \( \left( {E\left( t \right) = \check{E} e^{{i\omega t}} + {\check{E}^{*} } e^{{ - i\omega t}} } \right) \). \(E\left( t \right)\) is applied with polarization along the heterostructure's growth direction. Time-evolution of one-electron density matrix is obtained by using time dependent Van Neumann differential equation given as follows [32]

$$ \frac{{\partial \hat{\rho }_{ij} }}{\partial t} = \frac{1}{i\hbar }\left[ {\hat{H}_{0} - \hat{M}E\left( t \right),\hat{\rho }} \right]_{ij} - {\Gamma }_{ij} \left( {\hat{\rho } - \hat{\rho }^{\left( 0 \right)} } \right)_{ij} $$
(8)

where \(\hat{\rho }\) and \(\hat{\rho }^{\left( 0 \right)}\) are the density matrix of single electron state and the unperturbed density matrix operator, respectively. While \(\hat{H}_{0}\) is the Hamiltonian of the system in the case of \(E\left( t \right) = 0\), the electric dipole moment operator is given as \(\hat{M}\). \({\Gamma }_{ij}\) is the relaxation rate which causes the damping processes. We obtained the solution of Eq. (8) by using iterative method,

$$ \hat{\rho }\left( t \right) = \mathop \sum \limits_{n = 0}^{\infty } \hat{\rho }^{\left( n \right)} \left( t \right) $$
(9)

with

$$ \frac{{\partial \hat{\rho }_{ij}^{n + 1} }}{\partial t} = \frac{1}{i\hbar }\left\{ {\left[ {\hat{H}_{0} ,\hat{\rho }^{n + 1} } \right]_{ij} - i\hbar {\Gamma }_{ij} \hat{\rho }_{ij}^{n + 1} } \right\} - \frac{1}{i\hbar }\left[ {\hat{M}, \hat{\rho }^{n} } \right]_{ij} E\left( t \right) $$
(10)

The electronic polarization of the system caused by the optical field \(E\left( t \right)\) can be given as follows [34],

$$ P\left( t \right) = \varepsilon _{0} \chi _{\omega }^{{\left( 1 \right)}} \check{E}e^{{i\omega t}} + \varepsilon _{0} \chi _{0}^{{\left( 2 \right)}} \check{E}^{2} + \varepsilon _{0} \chi _{{2\omega }}^{{\left( 2 \right)}} \check{E}^{2} e^{{2i\omega t}} + \varepsilon _{0} \chi _{\omega }^{{\left( 3 \right)}} \check{E}^{2} \check{E}e^{{i\omega t}} + \varepsilon _{0} \chi _{{3\omega }}^{{\left( 3 \right)}} \check{E}^{3} e^{{3i\omega t}} + cc $$
(11)

where the free space’s permittivity is expressed by \(\varepsilon_{0}\). \(\chi_{\omega }^{\left( 1 \right)}\), \(\chi_{0}^{\left( 2 \right)}\), \(\chi_{2\omega }^{\left( 2 \right)}\), \(\chi_{\omega }^{\left( 3 \right)}\) and \(\chi_{3\omega }^{\left( 3 \right)}\) are the linear, optical rectification, second harmonic generation, third-order, and third harmonic generation susceptibilities, respectively. \(\chi_{3\omega }^{\left( 3 \right)}\) is THG susceptibility and is expressed as,

$$ \chi_{3\omega }^{\left( 3 \right)} = \frac{{e^{4} \rho_{v} }}{{\varepsilon_{0} \hbar^{3} }}\frac{{M_{01} M_{12} M_{23} M_{30} }}{{\left( {\omega - \omega_{10} - i\Gamma_{3} } \right)\left( {2\omega - \omega_{20} /2 - i\Gamma_{3} } \right)\left( {3\omega - \omega_{30} /3 - i\Gamma_{3} } \right)}} $$
(12)

In the expression obtained above \(\rho_{v}\), \(\varepsilon_{0}\) and \(\hbar\) specify the values of the electrons’ concentration in three dimensions, vacuum’s permittivity, and reduced Planck constant, respectively. The dipole matrix elements are defined as \(M_{ij}\) \(\left( { = \left\langle {\Psi_{i} } \right|z\left| {\Psi_{j} } \right\rangle , \;\left( {i, j = 0, 1, 2, 3} \right)} \right)\). \(\omega_{ij} = \left( {E_{i} - E_{j} } \right)/\hbar\) gives the transition frequency between the \(i{\text{th}}\) and \(j{\text{th}}\) sub-bands. The last term we will define is the relaxation rate \({\Gamma }_{3} = 1/T_{3}\) concerned with the transition lifetime of electrons.

Numerical values ​​of the physical parameters used in this research are; \(m^{*} = 0.067m_{0}\) (\(m_{0}\) is the mass of free electron), \(\hbar = 1.056 \times 10^{ - 34} \;{\text{Js}}\), \(\varepsilon = 12.58\), \(\varepsilon_{0} = 8.854 \times 10^{ - 12}\), \(\rho_{\upsilon } = 3 \times 10^{22} \;m^{ - 3}\),\( e = 1.602 \times 10^{ - 19} \;{\text{C}}\), \(\Gamma_{3} = 7.0\) THz. The structural parameters \({\mathcal{V}}_{1}\) and \({\mathcal{V}}_{2}\) are chosen as \(6\) and \(9\) throughout the study.

3 Results and discussion

In this section, the numerical results and analysis for the GaAs/AlGaAs Manning-like double quantum well are presented. We first got the four lowest subband levels under the influence of external fields (electric, magnetic, and intense laser) separately for the confined electron in the structure as shown in Fig. 1. Figure 1 shows variations in confining potential and the probability density for the envelope wave functions of the four lowest subband states under the effect applied external field in order of electric, magnetic, and intense laser for \(V_{0} = 228 \;{\text{meV}}\) and \(L_{wt} = 100\) Å. In Fig. 1a, we took the magnetic and laser fields as zero \(\left( {B = 0, \alpha_{0} = 0} \right)\) and changed the electric field \(\left( F \right)\) value to \(0\) and \(30\;{\text{kV/cm}}\). When \(F\) equals zero, the probability densities are almost symmetrical with respect to the center of the heterostructure. While the ground state and the first excited states have two peaks, the second and the third excited states have three and four peaks, respectively. In the case of an electric field of magnitude \(30\;{\text{kV/cm}}\) being applied to the structure, the confinement potential tilts, and the left side of the quantum well structure becomes deeper than the right side. Therefore, charge carriers accumulate toward the deeper potential zone. In this situation, the symmetries in the electron’s localizations for the ground and the first excited states disappear. The localization of the electron increases in the left well and the right well for the ground and the first excited states, respectively. Despite these, there is no significant change in probability densities for the second and third excited states.

In the absence of the electric and laser fields \(\left( {F = 0, \alpha_{0} = 0} \right)\), we can see the effects of the applied magnetic field \(\left( B \right)\) in Fig. 1b, which is perpendicular to the growth direction, where the values of \(B\) are assigned as \(0\) and \(12 T\). While \(B = 0\), as mentioned before, the probability densities are almost symmetrical with respect to the center of the heterostructure. Including the effect of the magnetic field in the calculations means that a parabolic term \(\left( {z^{2} } \right)\) will be added to the Schrödinger equation. Due to this term, the upper part of the confinement potential takes a parabolic shape as shown in Fig. 1b.

While there is no visible change in the well parts of the structure, there are very slight increases in the probability densities of the four lowest sub-bands. We attempt to observe the impacts of the intense laser field \(\left( {\alpha_{0} } \right)\) on the probability densities and structure in Fig. 1c when the electric and magnetic fields are both zero \(\left( {F = 0, B = 0} \right)\). With the application of an intense laser field with a magnitude of 40 Å, while there was no change in the upper parts of the structure, the well regions of ​​the structure have effectively changed and became a single well. Instead of the symmetric two peaks present in the case of \(\alpha_{0} = 0\), the ground state features a single, bigger peak at the center of the structure in this instance. When we examine the probability densities of the first, second, and third excited levels, we can say that their symmetry and peak numbers do not change, but there are increases in the probability densities.

In cases where there is no external field effect, the transition energies between the ground state and the first three excited states as a function of the heterostructure’s depth \((V_{0} )\) and width \(\left( {L_{wt} } \right)\) are given in Fig. 2a and b, respectively. When we examine Fig. 2a, we can see that with increasing the structure’s depth, while the value of transition energy from the ground state to the first excited state decreases, the transition energies between the ground state and the second and third excited states show an increase. The increased amounts here are more evident. According to Fig. 2b, increasing the value of \(L_{wt}\) causes all transition energy values ​​to decrease prominently.

Fig. 2
figure 2

Under zero external field, the transition energies between the ground state and the first, second, and third excited states as a function of the a depth of the heterostructure \(\left( {V_{0} } \right)\) and b width of the heterostructure \(\left( {L_{wt} } \right)\)

Full size image

The effects of the depth and width of the structure on the change of the dipole moment matrix element used in the calculation of the THG coefficient are given in Fig. 3a, b under the conditions used in Fig. 2. The dipole moment matrix element decreases with the increase in structure's depth, as shown in Fig. 3a. However, it shows a rapid increase with increasing well width, as shown in Fig. 3b.

Fig. 3
figure 3

The variations of the dipole moment matrix element necessary for computing THG coefficient as a function of the a depth of the heterostructure \(\left( {V_{0} } \right)\) and b width of the heterostructure \(\left( {L_{wt} } \right)\)

Full size image

After we obtain changes in the dipole moment matrix element and the transition energies for different values of \(V_{0}\) and \(L_{wt}\), we examine the THG coefficient’s variations as a function of incident photon energy for varied \(V_{0}\) and \(L_{wt}\) values under the condition that there is no external field effect in Fig. 4. In Fig. 4a, we take the values of the depth of the heterostructure \((V_{0} )\) as \(180\;{\text{meV}}\), \(200 \;{\text{meV}}\) and \(228 \;{\text{meV}}\). The given figure shows three resonant peaks of the THG coefficient. While the peaks that state the main and secondary peaks corresponding to \((E_{3} - E_{0} )/3\) and \((E_{2} - E_{0} )/2\) , respectively, are visible, the peak relating to \(\left( {E_{1} - E_{0} } \right)\) is so small and obscure compared to the others. By examining the main and secondary peaks, we can easily state that both peaks exhibit shifting behavior to the high-energy region. The increases in \(\left( {E_{2} - E_{0} } \right)/2\) and \(\left( {E_{3} - E_{0} } \right)/3\) expressed in Fig. 2a are consistent with this result. With this, the dipole matrix element behavior given in Fig. 3a also explains the decrease in the amplitudes of peaks here. We can make a similar explanation for Fig. 4b, where the \(L_{wt}\) value is changed to 90 Å, 100 Å, and 110 Å. The amplitudes of the main and secondary peaks increase and have red shift, consistent with the increase in the dipole matrix element (Fig. 3b) and the decreases in \(\left( {E_{2} - E_{0} } \right)/2\) and \(\left( {E_{3} - E_{0} } \right)/3\) (Fig. 2b).

Fig. 4
figure 4

The variations of third harmonic generation (THG) coefficient as a function of incident photon energy for different values of the a depth of the heterostructure \(\left( {V_{0} } \right)\) and b width of the heterostructure \(\left( {L_{wt} } \right)\)

Full size image

In Fig. 5, we introduce all external fields' effects affecting the heterostructure on the variations of the transition energies between the ground and first three excited states separately. If we examine all three figures, we can say that in the case where the other two external fields are zero and only the change of one external field occurs, the increase in the applied field's magnitude generates an increment in the intersubband transition energies. The difference between the figures is that while \(\left( {E_{1} - E_{0} } \right)\), \(\left( {E_{2} - E_{0} } \right)/2\) and \(\left( {E_{3} - E_{0} } \right)/3\) show approximately the same behavior with the change of the magnetic field (Fig. 5b), on the other hand, \(\left( {E_{1} - E_{0} } \right)\) shows a faster increase compared to \(\left( {E_{2} - E_{0} } \right)/2\) and \(\left( {E_{3} - E_{0} } \right)/3\) in the changes of the electric and laser fields (Fig. 5a, c).

Fig. 5
figure 5

The changes of transition energies between the ground state and the first three excited state as a function of the a electric field under zero magnetic and intense laser fields, b magnetic field under zero electric and intense laser fields, and c intense laser field under zero electric and magnetic fields

Full size image

In Fig. 6, the change that occurs in the dipole moment matrix element with the change of only one external field (the other two external fields are zero) is examined, respectively, under the conditions used in Fig. 5. While there is a sharp decrease in the value of the dipole moment matrix element with the increase in the electric and laser fields (Fig. 6a, c), there is a smaller change in the value of the matrix element with the increase in the magnetic field (Fig. 6b).

Fig. 6
figure 6

The variations of the dipole moment matrix element as as a function of the a electric field under zero magnetic and intense laser fields, b magnetic field under zero electric and intense laser fields, and c intense laser field under zero electric and magnetic fields

Full size image

In the last part of the study, we analyze how just a single external field change affects the behaviors of the main and secondary peaks (related to \(\left( {E_{3} - E_{0} } \right)/3\) and \(\left( {E_{2} - E_{0} } \right)/2\)) and the amplitudes of these peaks in the THG coefficient (Fig. 7). The peak corresponding to \(\left( {E_{1} - E_{0} } \right)\) is so small. By taking the other two external field values ​​to zero, we change the electric field’s value to \(0,\;15\;{\text{kV/cm}},\) \(20\;{\text{kV/cm}},\;30\;{\text{kV/cm}}\) (Fig. 7a), the magnetic field’s value to \(0, 4\;{\text{T}}, 8\;{\text{T}}, 12\;{\text{T}}\) (Fig. 7b) and the intense laser field value to 0, 10 Å, 20 Å, 40 Å (Fig. 7c), respectively. Firstly, we can see that, changes for each of the three applied field (Fig. 7a–c), as the results, we obtained in Fig. 5a–c in parallel with the changes of transition energies between the ground state and the first three excited states (\((E_{1} - E_{0} )\), \((E_{2} - E_{0} )/2\) and \((E_{3} - E_{0} )/3\)), there is a shift in the peaks to the high-energy region. In addition, as supported by the data in Fig. 6, the amplitude values ​​show a more significant decrease for the changes in electric and laser fields (Fig. 7a, c) compared to the change in the magnetic field (Fig. 7b).

Fig. 7
figure 7

The variations of third harmonic generation (THG) coefficient as a function of incident photon energy for different values of the external a electric field \(\left( F \right)\), b magnetic field \(\left( B \right)\), and c intense laser field \(\left( {\alpha_{0} } \right)\)

Full size image

In addition to obtained results, we can say the following; if we compare our work with other studies on the Manning-like double quantum well structure [35, 36], the structure is asymmetric due to the different choice of \({\mathcal{V}}_{1}\) and \({\mathcal{V}}_{2}\) parameters in Eq. (3). No energy degeneration was observed in our asymmetric structure. Besides this, applied external fields has not showed opposing effects on the THG coefficient, as in the double quantum well studies of Altuntas [37], Durmuslar et al. [38]. The external fields (electric, magnetic, and intense laser) that we applied to the structure separately had the same effect on the THG coefficients, and a blue shift was obtained for each three external effect.

4 Conclusions

We theoretically studied the changes occurring in the third harmonic generation (THG) coefficient of GaAs/AlGaAs Manning-like double quantum well structure according to the applied external fields’ variations, such as electric, magnetic, and intense laser. The situations in which external fields are applied are covered individually. In addition, how some structural parameters (such as depth and width of the well) changes affect the THG coefficient are investigated. We used the diagonalization and the compact density matrix methods to obtain subband energy levels with corresponding wave functions and the mathematical form of the THG coefficient, respectively. The results of our investigation can be summarized as follows: The resonance peaks of the THG coefficient have a blue shift with an increase in the values of each of the three external fields (electric, magnetic, and intense laser). Moreover, while the resonance peaks of the THG coefficient shift to the high-energy region with an increment in the depth of the structure, increasing the width of the structure causes the shifting of the resonance peaks to the low-energy region. We think that the results we found in this search will make significant contributions to current experimental device designs and applications.