Recently, the evolution of the growth techniques such as molecular beam epitaxy and metal-organic chemical vapor deposition combined with the use of the modulation-doped technique made it possible the fabrication of low-dimensional heterostructures such as single and multiple quantum wells, quantum wires, and quantum dots. In these systems, the restriction on the motion of the charge carriers allows us to control the physical properties of the structures. The studies on these systems offer a wide range of potential applications in the development of semiconductor optoelectronic devices[1–5].

GaInNAs/GaAs quantum well (QW) lasers have been attracting significant scientific interest mainly due to their applications in 1.3- or 1.55-μm optical fiber communication[6–12]. These lasers are predominantly based on GaInAsP alloys on the InP substrates, which have a higher temperature sensitivity compared to shorter wavelength lasers that are grown on GaAs substrates. The high-temperature sensitivity is primarily due to Auger recombination and the weak electron confinement resulting from the small conduction band offset in the GaInAsP/InP material system. GaInNAs alloys grown on GaAs substrates have been proposed as a possible alternative to the GaInAsP/InP system for achieving lasers with high-temperature performance[13]. The deeper conduction band well and the larger electron effective mass will provide better confinement for electrons and better match of the valence and conduction band densities of state, which leads to a higher characteristic temperature and higher operating temperature, higher efficiency, and higher output power[6–13].

As known, high-frequency intense laser field (ILF) considerably affects the optical and electronic properties of semiconductors[14–20]. Because when an electronic system is irradiated by ILF, the potential of the system is modified which affects significantly the bound state energy levels, a feature that has been observed in transition energy experiments. The design of new efficient optoelectronic devices depends on the understanding on the basic physics involved in this interaction process. Thus, the effects of a high-frequency ILF on the confining potential and the corresponding bound state energy levels are a very important problem. This problem has been a subject of great interest, and an enormous amount of literature has been devoted to this field[21–27]. However, up to now, to the best of our knowledge, no theoretical studies have been focused on the bound states in Ga_{1 − x}In_{
x
}N_{
y
}As_{1 − y}/GaAs double quantum well (DQW) under the ILF.

The purpose of this work is to investigate the effect of ILF, nitrogen (N), and indium (In) mole fractions on the bound states in Ga_{1 − x}In_{
x
}N_{
y
}As_{1 − y}/GaAsDQW. The paper is organized as follows: in the ‘Theoretical overview’ section, the essential theoretical background is described. The next section is the ‘Results and discussion’ section, and finally, our calculations are given in the ‘Conclusions’ section.

#### Theoretical overview

The method of approach used in the present study is based on non-perturbation theory developed to describe the atomic behavior under intense, high-frequency laser field conditions[

28,

29]. It starts from the space-translated version of the semi-classical Schrödinger equation for a particle moving under the combined forces of potential and a radiation field derived by Kramers in the general context of quantum electrodynamics[

30]. For simplicity, we assume that the radiation field can be represented by a monochromatic plane wave of frequency

*ω*. For linear polarization, the vector potential of the field in the laboratory frame is given by

$\mathbf{A}{(\text{t) = A}}_{0}cos\left(\omega t\right)\widehat{e}$, where

$\widehat{e}$ is the unit vector. By applying the time-dependent translation

$r\phantom{\rule{0.25em}{0ex}}=r\phantom{\rule{0.25em}{0ex}}+\alpha \left(t\right)$, the semi-classical Schrödinger equation in the momentum gauge, describing the interaction dynamics in the laboratory frame of reference, was transformed by Kramers as follows[

30]:

$-\frac{{\hslash}^{2}}{2{m}^{*}}{\nabla}^{2}\phi \left(\mathbf{r},t\right)+V\left(\mathbf{r}+\alpha \left(t\right)\right)\phi \left(\mathbf{r},t\right)=i\hslash \frac{\partial \phi \left(\mathbf{r},t\right)}{\partial t}\text{,}$

(1)

where

*V*(

**r**) is the atomic binding potential, and

$\alpha (\text{t)=}{\alpha}_{0}sin\left(\omega t\right)\widehat{e},\phantom{\rule{2.25em}{0ex}}{\alpha}_{0}=\frac{e{A}_{0}}{{m}^{*}c\omega}$

(2)

represents the quiver motion of a classical electron in the laser field, and

*V*(

**r** +

*α*(

*t*)) is the ‘dressed’ potential energy. In this approximation, the influence of the high-frequency laser field is entirely determined by the ‘dressed potential’

*V*(

**r** +

*α*(

*t*))[

30],

$\phantom{\rule{0.25em}{0ex}}{\alpha}_{0}=\left(\frac{{I}^{1/2}}{{\omega}^{2}}\right)\left(e/{m}^{*}\right){\left(8\pi /c\right)}^{1/2}\text{,}$

(3)

where *e* and *m** are absolute value of the electric charge and effective mass of an electron; *c*, the velocity of the light; *A*_{0}, the amplitude of the vector potential; and *I*, the intensity of ILF.

Following the Floquet approach[

29,

30], the space-translated version of the Schrödinger equation, Equation

1, can be cast in equivalent form of a system of coupled time independent differential equations for the Floquet components of the wave function

*φ*, containing the (in general complex) quasi-energy

*E*. An iteration scheme was developed to solve this; for the zeroth Floquet component

*α*_{0}, the system reduces to the following time-independent Schrödinger equation[

29–

32]:

$\left[-\frac{{\hslash}^{2}}{2{m}^{*}}{\nabla}^{2}+V\left(\mathbf{r},{\alpha}_{0}\right)\right]{\phi}_{0}=E{\phi}_{0}\text{,}$

(4)

where *V*(**r**, *α*_{0}) is the dressed confinement potential which depends on *ω* and *I* only through *α*_{0}[28].

By applying the above-described dressed potential theory to our particular Ga

_{1 - x}In

_{x}N

_{y}As

_{1 - y}/GaAs DQW system, we write down the time-independent Schrödinger equation in one-dimensional case for an electron inside a Ga

_{1 - x}In

_{x}N

_{y}As

_{1 - y}/GaAs DQW (we choose the

*z*-axis along the growth direction) in the presence of an intense high-frequency laser field (the laser-field polarization is along the growth direction), which is given by the following:

$-\frac{{\hslash}^{2}}{2{m}^{*}}\frac{{\partial}^{2}\psi \left(z\right)}{\partial {z}^{2}}+V\left({\alpha}_{0},z\right)\psi \left(z\right)=E\psi \left(z\right)\text{,}$

(5)

where

*ψ*(

*z*) is the wave function, and

*V*(

*α*_{0},

*z*) is the dressed confinement potential which is given by the following expression:

$\begin{array}{l}V\left({\alpha}_{0},z\right)={V}_{0}\left[\theta \left(-{\alpha}_{0}-L/2-z\right)\right]\\ \phantom{\rule{5em}{0ex}}+\frac{{V}_{0}}{\pi}\left[\Theta \left(z+{\alpha}_{0}+L/2\right)\theta \left({\alpha}_{0}-L/2-z\right)\right]\\ \phantom{\rule{5em}{0ex}}\times arccos\left[\frac{z+L/2}{{\alpha}_{0}}\right]+\\ {V}_{0}\left[\theta \left(-{\alpha}_{0}-L/2+z\right)\right]+\frac{{V}_{0}}{\pi}\left[\Theta \left(-z+{\alpha}_{0}+L/2\right)\right.\\ \phantom{\rule{15em}{0ex}}\left.\theta \left({\alpha}_{0}-L/2+z\right)\right]\\ \phantom{\rule{11.2em}{0ex}}\times arccos\left[\frac{L/2-z}{{\alpha}_{0}}\right]+\\ {V}_{0}\left[\Theta \left({\alpha}_{0}+{L}_{b}/2+z\right)-\theta \left(z-{\alpha}_{0}\right)\right]-\\ \frac{{V}_{0}}{\pi}\left[\begin{array}{l}\Theta \left(z+{\alpha}_{0}+{L}_{b}/2\right)\theta \left(-z+{\alpha}_{0}-{L}_{b}/2\right)\\ \phantom{\rule{1.7em}{0ex}}\times arccos\left[\frac{z+{L}_{b}/2}{{\alpha}_{0}}\right]+\\ \Theta \left(-z+{\alpha}_{0}\right)\theta \left(z+{\alpha}_{0}\right)\times arccos\left[\frac{-z}{{\alpha}_{0}}\right]\end{array}\right]+\phantom{\rule{8.25em}{0ex}}\\ {V}_{0}\left[\Theta \left({\alpha}_{0}+{L}_{b}/2-z\right)-\theta \left(-z-{\alpha}_{0}\right)\right]-\\ \frac{{V}_{0}}{\pi}\left[\begin{array}{l}\Theta \left(-z+{\alpha}_{0}+{L}_{b}/2\right)\theta \left(z+{\alpha}_{0}-{L}_{b}/2\right)\\ \phantom{\rule{2.7em}{0ex}}\times arccos\left[\frac{-z+{L}_{b}/2}{{\alpha}_{0}}\right]+\\ \Theta \left(-z+{\alpha}_{0}\right)\theta \left(z+{\alpha}_{0}\right)\times arccos\left[\frac{z}{{\alpha}_{0}}\right]\end{array}\right]\end{array}$

(6)

where *V*_{0} is the conduction band offset at the interface; *L* = *Lw*_{1} + *Lw*_{2} + *L*_{
b
}, *Lw*_{1} = *Lw*_{2}, the well width; *L*_{
b
}, the barrier width; Θ, the Heaviside unit step function which satisfies *Θ*(*z*) = 1 − *θ*(−*z*); and *θ*, the unit step function[33].

To solve the Schrödinger equation in Equation

5, we take as base the eigenfunction of the infinite potential well with

*L*_{
s
} width.

*L*_{
s
} is the well width of the infinite well at the far end of DQW with

*L* width (

*L*_{
s
} > >

*L*), and its value is determined according to the convergence of the energy eigenvalues. These bases are formed as[

34] follows:

${\psi}_{n}\left(z\right)=\sqrt{\frac{2}{{L}_{s}}}cos\left[\frac{n\pi}{{L}_{s}}z-{\delta}_{n}\right]\text{,}$

(7)

where

${\delta}_{n}=\{\begin{array}{l}0\phantom{\rule{1.25em}{0ex}}\text{if}\phantom{\rule{0.25em}{0ex}}n\phantom{\rule{0.25em}{0ex}}\text{is odd,}\\ \frac{\pi}{2}\phantom{\rule{0.75em}{0ex}}\text{if}\phantom{\rule{0.25em}{0ex}}n\phantom{\rule{0.25em}{0ex}}\text{is even,}\end{array}$

and so, the wave function in the

*z*-direction is expanded in a set of basis function as follows:

$\psi \left(z\right)={\displaystyle \sum _{n=1}^{\infty}{c}_{n}{\psi}_{n}\left(z\right)}\text{.}$

(8)

In calculating the wave function *ψ*(*z*), we ensured that the eigenvalues are independent of the chosen infinite potential well width *L*_{
s
} and that the wave functions are localized in the well region of interest. This method, which gives accuracies greater than 0.001 meV, is well controlled, gives the DQW eigenfunctions, and is easily applied to situations of varying potential and effective mass.