Spectral features and optical absorption of vertically stacked V-groove quantum wires

The spectral features and the linear, nonlinear, and total optical absorption coefficients of two realistic GaAs/Ga1-xAlxAs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {GaAs/Ga}_{1-x}{\mathrm{Al}}_x{\mathrm{As}}$$\end{document} vertically stacked V-groove quantum wires confining electronic states were theoretically studied. The compact density matrix formalism and the finite element method were used as solution frameworks. The influence of the barrier width, inter-sidewall angle, the crescent thickness of the stack, and the influence of a static electric field on the energy spectrum and the optical coefficients is addressed. Increasing the crescent thickness (inter-sidewall angle) leads to a reduction (non-monotonous increase) of the energy values, while the presence of a static electric field leads to controllable stark-like patterns. In all cases, anti-crossing points are reported. By increasing the crescent thickness, inter-sidewall angle, or barrier width, a non-monotonous blue-/red-shift of the resonant peaks of the optical absorption is observed. In addition, the results in asymmetric/symmetric vertically coupled V-groove quantum wire systems suggest that the quantum tunneling phenomenon can be linked to the enhancement of the total optical absorption due to the possibility of the generation of dipole moments by higher electron mobility. Thus, optical fields oriented along the growth direction enhance the optical absorption. Furthermore, the absorption coefficient maximum is drastically modified by varying the inter-sidewall angle, and even the evolution tendency can be inverted due to the anti-crossing points in the energy spectra. Since the obtained resonant peaks are located below 20 meV in the spectra, the analyzed system could be interesting for exploring and designing new Terahertz devices.


Introduction
Pioneer investigations on (Al,Ga)As/GaAs crystal growth on non-planar substrates showed the possibility of producing lateral variations in the physical properties of semiconductor heterostructures by varying the thickness of the deposited materials [1,2]. Later improvements in the growth technology allowed the fabrication of high-quality one-dimensional semiconductor heterostructures such as the T-shaped [3,4] and sidewall quantum wires [5], quantum ridges [6,7], and the V-groove quantum wires [8,9] (VGQWs). In particular, vertically stacked VGQWs are ideal systems to study tunneling phenomena and the transition of electronic states from one-dimensional to two-dimensional spaces [10]. Within the last decade, from V-grooved (001) GaAs substrates, it has been possible to grow high-quality site-controlled InGaAsN VGQWs via modulated-flux metal-organic vapor phase epitaxy [11], or low-pressure metal-organic chemical vapor deposition [12,13]. The broad set of technological applications for nano-electronics based in VGQWs structures, ranging from VGQW lasers [8,14,15], pMOSFETs [16], 1.3 µm wavelength emitters [11], to VGQW AlGaAs/GaAs multilayers for complex optical devices [17], is still stimulating theoretical [18][19][20] and experimental [20,21] investigations.
After an exhaustive review of the state of the art in the growth of epitaxial VGQWs [11, 14-17, 20, 22-24], it is feasible to define the range of values of some relevant geometric parameters that characterize their typical size/shape aspect. Through cross-sectional transmission electron microscopy [20,23,25], the cross-section geometry of GaAs/AlGaAs VGQWs can be characterized. The results reveal that the angle between the VGQW sidewalls typically ranges from 75 • to 120 • and the VGQW's crescent (thickest zone of the wire) measures about 3-22 nm thick. The size of the pinch-offs, that is, the characteristic constrictions that slightly uncouple the sidewall quantum wells and the crescent, is between 1 and 5 nm, while the coupled quantum wells are between 2 and 10 nm width. The lateral distance from the pinch-offs to the crescent varies between 10 and 58 nm, and the vertical distance between V-grooves (which can be defined through the barrier's width between two consecutive crescents) is typically between 2 and 55 nm. a e-mail: eugenio.giraldo@eia.edu.co (corresponding author) b e-mail: jlpalaci@pascualbravo.edu.co c e-mail: guillermo.miranda@eia.edu.co d e-mail: marlonfulla@yahoo.com These data are relevant since the systematical variation of the crescent's thickness, size and position of the pinch-offs, intersidewall angles, quantum well widths, and VGQW array's pitch can lead to fashionable tailoring of the electronic states [10], optical properties [10,21], quantum magnetotransport [26], and carrier transfer phenomena [27]. In this regard, these findings have stimulated several theoretical explorations related to electron [28,29], hole [29] and impurity [30,31] states calculations in single VGQWs by using an effective potential scheme. This effective lateral potential simplifies the treatment and the decoupling of the two-dimensional system Hamiltonian into two one-dimensional Hamiltonians due to the non-trivial nature of the saw-tooth surface of the V-grooved heterostructure [30]. Some authors have explored algebraic approximations of trigonometric or hyperbolic functions in order to provide approximate analytical expressions for the electronic energy spectrum in quantum wires [32] and have studied the mobility in V-shaped quantum well structures [33]. Recently, other authors have made important computational efforts to solve the single VGQW problem [19] by using a finite differences method. More recently, an excellent approximation to realistic VGQWs [34,35] by using the finite element method was reported allowing to calculate electronic states [34] and the corresponding nonlinear optical generation [35]. Nevertheless, at this stage, theoretical studies related to vertically stacked V-grooves are really scarce [36]. In this article, we study some spectral features of two vertically stacked GaAs/Ga 1−x Al x As VGQWs (with the realistic size/shape described in [34,35]) such as the energy level ordering as a function of the VGQW geometry, and the linear, nonlinear, and total optical absorption. These findings can potentially give some hints to optimally tune the optical response of realistic v-groove superlattices or foster the study of the fundamental physics underlying the development of new optical devices [37][38][39].

V-groove stack's geometry
The analyzed semiconductor heterostructure consists of two vertically stacked epitaxial GaAs VGQWs. In the present work, it is assumed that the same ternary compound (Ga 1−x Al x As) as substrate (at the bottom), barrier (between the VGQWs), and cap (at the top) as shown in Fig. 1. The physical system under consideration corresponds to a conduction-band electron confined into the coupled VGQWs of GaAs due to the energy gap mismatch among the substrate-lower VGQW-barrier and barrier-upper VGQW-cap. An economical and computationally efficient parametrization of a wide variety of VGQW cross-sections extracted from experimental reports [11, 14-17, 20, 22-24] is the one defined by the five-parameter curves [35] representing the GaAs quasi-one-dimensional spaces in Fig. 1 (dark patterned zones): Here, the index i 1, 2 denotes the lower and upper VGQW, respectively, while (x, y) are the dimensionless rectangular coordinates on the cross-sectional plane. The set of parameters in Eqs. (1b) and (1a), α i , β i , γ i , and h 0i control for the i-th VGQW, its inter-sidewall angle, the crescent's thickness (and also to a lesser extent, the quantum well's width), the crescent's curvature in the y 2i (x) contour, and the inter-curve distance (between y 1i (x) and y 2i (x)), respectively. The fifth geometrical parameter L b denotes the barrier width between the VGQWs and allows to control the stack's pitch. Consequently, the equations of the upper VGQW correspond to the same as the lower one plus a vertical displacement equals to L b + y 0 , where y 0 y 11 (0) − y 21 (0). The Fig. 1 Graphical representation of two realistic and vertically stacked GaAs VGQWs surrounded by Ga 1−x Al x As. The blue curves delimiting the lower VGQW's contour correspond to Eqs. (1a) and (1b). The system is irradiated by an optical field E opt (t) with Poynting vector S(t) and under the presence of a static electric field F where ψ( r ⊥ ) and E ⊥ denote the eigenfunctions and eigenenergies, respectively, while r ⊥ corresponds to the electron's vector position on the VGQW's cross-section. Since it is necessary to take into account the effective mass mismatch at the heterojunctions (due to the position-dependent aluminum concentration x Al ), the dimensionless kinetic energy operator is given by the Hermitian operator corresponding to the first term in Eq. (2). The parameter x Al takes the value Al x (0) outside (inside) of each VGQW. The second term is related to the energy interaction with the tilted static electric field F which forms an angle φ with the horizontal axis, and the third term is associated with the confinement potential produced by the energy gap mismatches at VGQW-barrier interfaces. From the dimensioning calculations, naturally arises the μ(x Al ) function defined as the ratio between the position-dependent electron's effective mass in Ga 1−x Al x As and in GaAs, that is, μ(x Al ) m * (x Al )/m * (0). Consequently, with this definition, it can be established [34] the effective Bohr radius a * 0 , effective Rydberg Ry * , and the renormalized electric field strength η as units of length, energy, and electric field, respectively, which are defined as a * 0 2 (4π )/m * (0)e 2 , Ry * m * (0)e 4 /2 2 (4π ) 2 , and η eFa * 0 /Ry * . In the present work, we have implemented the relations extensively discussed by [40] describing the electron's effective mass m * (x Al ) in Ga 1−x Al x As and the confinement potential in the GaAs/Ga 1−x Al x As heterojunction, both dependent on the aluminum concentration. The electron's effective mass can be written as follows: being m 0 the free-electron mass at rest, 2 (x Al ) C 2 1 + C 2 2 x Al is the square of the interband matrix elements that defines the coupling between the s states of the c 6 conduction band and the hybrid sp-valence states from the v 5 band [40]. On the other hand, E g (x Al ) E c 6 (x Al ) − E v 8 (x Al ) is the fundamental energy gap and 0 (x Al ) E v 8 (x Al ) − E v 7 (x Al ) is the split-off valence gap whose linear fit can be written as 0 (x Al ) C 0 1 + C 0 2 x Al [40]. In addition, the effects of the remote-band on the electron's effective mass are included in the δ m (x Al ) term, which is properly fit to the experimental data by a parabolic relation of the form Al [40]. Moreover, the energy gap at the point of the conduction band can be fit as [40]: In Table 1, the fitting parameters list related to the physical quantities described above is summarized. With the energy gap fit to the experimental data, consequently, the dimensionless confinement potential in Eq. (2) can be calculated asV ( r ⊥ , x Al ) V ( r ⊥ , x Al )/Ry * , where: inside the VGQWs (dark patterned zones in Fig. (1)), V eff conf (x Al ) otherwise.
is the effective potential confinement that takes into account the energy gap mismatch through the band off-set alignment factor Q c 0.67 [35].
With all the previous elements composing the system Hamiltonian, the eigenvalue problem on the cross-sectional planê can be firstly solved [34,35] with the aim to solve the three-dimensional problem as will be shown later. The cross-sectional problem can be alternatively obtained [41] from the minimization of the functional: Subsequently, the VGQW's cross-section domain can be divided into n elements to evaluate the action as a discrete sum of action integrals F i in this way: F n i 1 F i [42,43]. Thus, the eigenfunction can be written as a linear combination of interpolant functions P j , in this form, M−1 j 0 j P j (where j are variational parameters and M denotes the number of vertices [42]). By defining H jk as a matrix element of the representation of the [Ĥ ⊥ − E ⊥ ] operator made in the P j basis, the total action can be rewritten as: Thereby, by differentiating the total action with respect to the functions * j , it is possible to obtain a secular equation with k unknown coefficients [41,43]. Once these unknown coefficients are obtained by standard diagonalization matrix methods [41,43] (such as those integrated into the FreeFem++ software [34,44]), the eigenfunctions can be built up, and their corresponding eigenvalues can be straightforwardly obtained.

Electronic spectrum and optical absorption
After obtaining the eigenfunctions, ψ( r ⊥ ) and their corresponding eigenvalues E ⊥ of the cross-sectional problem, the dimensionless band structure can be calculated as: z , being k z the dimensionless electron's wave vector in the z-direction (this is a consequence of the separability of the electron's motion in the xy plane and z-direction). We will focus our analysis on the discrete part of the energy spectrum; that is, we will recalibrate the energy with respect to the free-electron energy in the z-direction.
The electronic transitions and the related optical absorption phenomena can be studied within the widely used density matrix formalism [45][46][47] suitable for semiconductor nanostructures. In this present work, we calculate the optical absorption coefficients of a single electron confined in a vertically stacked VGQW system when it is irradiated by a classical optical field E opt (t) of frequency ω and optical intensity I, propagating in the positive z-direction. The optical field is linearly polarized and oriented by the angle ϕ measured from the positive x-axis (see Fig. 1). According to the used formalism, the evolution of the density matrix operatorρ(t) is governed by the master equation [48]: whereĤ denotes the unperturbed Hamiltonian (Eq. 2),μ is the dipole moment operator,ˆ is the phenomenological operator linked to the carriers lifetimes, andρ (0) is the non-perturbed density operator. From the dielectric susceptibility definition in connection with a standard perturbative expansion procedure [48][49][50], the linear and nonlinear optical absorption coefficients are written as: Here, I and 0 1/T with T is the reciprocal relaxation time. The physical parameters e and 0 correspond to the elementary charge and free space dielectric permittivity. On the other hand, R , n r , and μ denote the GaAs real part of the permittivity ( R n 2 r 0 ), the refractive index, and magnetic permeability, respectively. Finally, the total absorption is calculated as the sum of the linear of nonlinear contributions as: In our calculations, the states |1 and |2 that define the transition energy E 21 in Eqs. (9) and (10) correspond to the ground and first excited states.

Validation of results
Since the proposed model has a certain high degree of complexity, we have previously compared some numerical results obtained in a limit case with the ones obtained by Harrison [51,52], who calculated the low-lying energy levels of a single electron trapped in a post-etch overgrowth quantum wire with a square cross-section of side L. In Fig. 2a, the red diamonds correspond to the low-lying energy levels as a function of L calculated by Harrison [51,52]. His solution procedure considers the electron's effective confinement potential as a superposition of perpendicular independent finite one-dimensional potential wells, giving rise to a two-dimensional space (quantum wire's cross-section) with zero confinement potential inside of a square of side L and surrounded by a confinement potential 2V at the corners and V elsewhere (see the inset of Fig. 2a). In our case, we have taken the VGQW stack model to the limit case of two quantum wires with a square-like cross-section of side L and separated a large distance L b (barrier width between VGQWs) in which the quantum uncoupling between them can be guaranteed (see Fig. 2b as reference). To achieve this configuration, we considered two VGQWs with the same set of geometrical parameters: The potential value V used by Harrison [51,52] is obtained from our model as the aluminum concentration is set to x Al 0.2. With this set of parameter values, we calculated the low-lying energy levels by using two solution schemes: The standard diagonalization matrix method (DDM) (blue circles) and the finite element method (FEM) (solid lines). In Fig. 2a for Harrison's system, the label E a denotes the energy of the state (1, 1) whose value corresponds to the sum of the ground state energies obtained in each independent one-dimensional quantum well. On the other hand, E b denotes the energy of the degenerated states (1, 2) and (2, 1) whose value corresponds to the sum of the ground and first excited state energies in each quantum well, while E c is the energy of the non-degenerated state (2, 2) equal to the sum of the first excited state energies. In contrast, in our single-electron system confined in vertically stacked quantum wires (very Fig. 2 a Low-lying energy levels of a single electron confined in a square cross-sectional quantum wire obtained by Harrison [51,52] (red diamonds) and the ones obtained in the present work for a single electron confined in two vertically stacked VGQWs with a square-like cross-section via DDM (blue circles) and FEM (solid lines). b Probability densities of some low-lying states of the electron confined in the vertically stacked square-like cross-sectional VGQWs (obtained by FEM) separated) with a square-like cross-section, the label E a refers to the ground state energy, which is doubly degenerated due to the double-quantum wire nature of the system (a similar behavior is observed in uncoupled double finite quantum wells [53]). In this regard, E b denotes the first excited energy (fourfold degenerated), and E c is the second excited energy (eight-fold degenerated). We empirically found that the barrier width L b that allows decoupling the vertically stacked quantum wires as a function of the side of the square L within the range 50 − 300 Å is L b (Å) 266 − 3 7 L(Å). With this relation, we observed that the electronic states behave with similar features to those observed in one-dimensional double quantum wells in the uncoupled regime [53]. In addition, as will be shown below, good results in reasonable computing times were obtained.
A glance at Fig. 2a allows to conclude that there is an excellent agreement between our E a , E b , and E c energies via DDM and FEM, with those found in [51] within the range of L between 120 to 300 Å. As expected, for these relatively large values of L, it is possible to separate the vertically stacked VGQWs at a point that can be decoupled; thus, the corresponding energy levels coincide with those of a single quantum square of side L. For L values less than 120 Å, the energy E c calculated with both DDM and FEM schemes differs from the results by Harrison. This behavior is because, for small values of L, an overflow in some states is observed. The state associated with energy E c calculated by [51] does not overflow since the electron is stronger confined due to the higher values of the confinement potential at the corners of the square.
Similarly, we also obtained that the states related to the energy E b overflows (differing from Harrison's value) near L 90 Å due to the reasons above. This assertion is supported by Fig. 2b in which the probability densities for the |1 , |2 , and |7 states (first, second, and third column, respectively) are shown for two values of square's side, L 60 Å (first row) and L 200 Å (second row). At L 200 Å, the square's side is large enough to confine the excited state |7 (corresponding to the E c energy) characterized by the location of 4 bumps at the upper VGQW of the stack. In addition, the probability density of the two states with energy E a (|1 and |2 ) is exchanged between the lower and upper VGQWs similarly, such as in the symmetric double quantum well problem in the uncoupled regime [53]. In contrast, from Fig. 2b at L 60 Å, the state |7 corresponds to an overflowed state, and therefore the energy value does not match the one obtained by Harrison. However, as expected, the most low-lying states |1 and |2 are confined and exhibit the same behavior, such as at L 200 Å.
From this analysis, three important conclusions are obtained: (1) the comparative analysis shows a qualitative correspondence of our system with another in a limit case in a wide range of L parameter values. (2) In the region where there are differences with the cited author, the calculation by two different methods (DMM and FEM) has concluded the appearance of behavior in the square cross-sectional quantum wire system that under the considerations of the other researchers would not be visualized. (3) This comparative analysis constitutes a validation test of the proposed model, and from this point, it is possible to proceed reliably to study the electronic properties of the vertically stacked VGQWs.

Spectral features
In this subsection, in order to study the incidence of the most relevant geometrical parameters on the single-electron energy spectrum in vertically stacked VGQWs and based on the experimental reports of the state of the art in growth of epitaxial VGQWs, the following parameters as standard configuration will be used throughout this work: α i 90 • , β i 0.5 a * 0 , γ i 1 a * 0 , h 0i 0, L p l i L p r i 2.46 a * 0 , R major i 0.8a * 0 , R minori 70% of the inter-curve distance, and x Al 0.3. In Fig. 3a, b, the first six low-lying energy levels as a function of the barrier width L b for the two different values of β 0.5 a * 0 and β 0.75 a * 0 are respectively shown. As can be seen in both cases, by increasing the barrier width, the interaction between the electronic states localized in each VGQW is decreased, and the system tends to behave as two independent VGQWs. Thus, for large values of L b , the energy levels become closer, leading to degenerated energy levels. These energy values are expected to be close to the ones of a single electron confined in a single VGQW. To corroborate this assertion, the energy levels of a single-electron VGQW under the conditions stated above were calculated by using the scheme detailed in Refs. [34,35] and are shown as orange points in the figures. The closeness between the single VGQW and vertically stacked VGQWs energies indirectly reinforces the validation of the model. In addition, by reducing the barrier width, the energy levels splitting occurs in both cases with (without) the presence of anti-crossing points with β 0.5 a * 0 (β 0.75 a * 0 ), and the energy values are higher for the a) case in contrast to b) case. These two last facts are related to the stronger confinement provided by the structure with the smaller β value [34].
In Fig. 4a, b, the first six low-lying energy levels as a function of the inter-sidewall angle α are shown. As can be seen in Fig. 4a, at a small barrier width of L b 0.4 a * 0 and by increasing the inter-sidewall angle, all the energy values increase, which is an indication that the system tends to be more unstable; reaching a maximum value near α 140 • . At a larger value of barrier width L b 0.6 a * 0 (Fig. 4b), the evolution is similar; however, the slope of the curves is smaller. This fact shows the incidence of the inter-spacing between VGQWs, which allows controlling the degree of coupling between the electronic states localized in each VGQW. It is also important to point out that the evolution of the energy spectrum with the α parameter for vertically stacked VGQWs significantly differs from the single VGQW case, exhibiting the latter a monotonous evolution [34]. The contrasting behavior of the E i Vs. α curves for vertically stacked VGQWs characterized by a non-monotonous evolution with the presence of anti-crossing points lies in the tunneling phenomenon that can be seen in Fig. 4c. In this figure, the probability densities of the first three low-lying levels corresponding to the configuration of Fig. 4a for the three different α values 80 • , 84 • , and 90 • (first, second, and third columns, respectively) are shown. From Fig. 4a, it is evident the presence of an anti-crossing point very close to α 84 • between states |2 and |3 . Consequently, the probability densities of these two states in Fig. 4c qualitatively shows the typical interchange of their identities as the α parameter is varied from 80 • to 90 • (close to the vicinity of the anti-crossing point).
In Fig. 5a, the first six low-lying energy levels as a function of the β parameter (that controls the thicknesses of both VGQWs) are displayed. As expected, all the energy values decreased by increasing the β parameter (in accordance to single VGQWs [34]

(c) (d)
but with the presence of a few anti-crossing points (between sates |2 and |3 near β 0.48 a * 0 for instance). On the other hand, in Fig. 5b, the first six low-lying energy levels for two VGQWs vertically stacked with different crescent thicknesses (asymmetric configuration) are displayed. In this case, the lower VGQW's β parameter (β 1 ) is fixed to 0.5 a * 0 , and the energy is plotted as a function of β 2 (the parameter that controls the upper VGQW's crescent thickness). A glance at Fig. 5b reveals the same general trend as the previous configuration since by increasing the crescent thickness of the upper VGQW, the energy values decrease, and the electronic states tend to be more stable. Nevertheless, in this case, since the Hamiltonian contains the varying β 2 parameter and a fixed parameter β 1 , a less monotonous evolution of the energy with the appearance of more anti-crossing points (in contrast to the symmetric case of Fig. 5a) can be seen.
In Fig. 6a, the first six low-lying energy levels as a function of the electric field strength F and oriented toward the y-axis (φ 90 • ) are shown. As can be seen, the energy levels evolution corresponds to a Stark-like spectrum with the presence of the typical anti-crossing points. However, in this case, some geometrical parameter values were carefully selected to achieve that the anti-crossing points between states |1 and |2 , |3 and |4 , and |5 and |6 , take place at (approximately) the same electric field strength value F 5.8 kV/cm. This situation is achieved in an asymmetric configuration of two vertically stacked VGQWs with different crescent thicknesses (with β 1 0.7 a * 0 and β 1 0.8 a * 0 ) and a barrier width of L b 0.9 a * 0 . In Fig. 6b the corresponding probability densities of the first six low-lying levels at F 4 kV/cm (first column) and F 7 kV/cm (second column) are shown. In this figure, the interchange of the identities between the two states involved in an anti-crossing point (analogously as in Fig. 4c) due in this case to the increasing electric field strength can be seen. In order to objectively determine the F value in which the occurrence of an anti-crossing between two states |i and | j takes place is through the calculation of the overlap integral of the moduli [54] defined as : |i| | | j| ∞ −∞ |ψ * i ||ψ j |dV . In Fig. 6c, the overlap integrals of the moduli as a function of the electric field strength for the configuration corresponding to Fig. 6a are shown. As can be seen, due to the orthonormal property of the eigenstates, the numerical overlap integrals of the moduli correspond to delta-like profiles with peaks (maxima values) placed at the F value in which the interchange of identities between the states involved in the integral (anti-crossing point) takes place. In addition, the integral takes small values far from the anti-crossing point. Consequently, from Fig. 6c it can be concluded that the three anti-crossing points seen in Fig. 6a take place very close to F 5.8 kV/cm. Furthermore, if one of the configuration parameters is varied (barrier width, for instance), the situation can significantly change; for example, in Fig. 6d, the overlap integrals of the moduli as a function of F but with a barrier width of L b 0.6 a * 0 are displayed. Clearly, the three anti-crossing points are placed at different F values in contrast with the previous configuration. From this subsection, it can be concluded that the spectral features are strongly dependent on the vertically stacked VGQWs and external fields. Secondly, the importance of determining the energy level ordering lies in the possibility of revealing general behaviors of some optical quantities (such as the optical absorption) from the calculation of resonance energies (or resonance frequencies). For instance, a configuration in which an anti-crossing point between two transition states takes place could not be desired, as will be discussed in the next subsection.

Optical absorption
The study of the linear and nonlinear optical coefficients of the two vertically stacked VGQWs as a function of the most relevant geometrical parameters and the external field is addressed in this section. For the heterojunction under analysis, the physical parameter values are [3,49]: ρ v 3.0 × 10 22 m −3 , 0 1/T 0 1/(0.2 ps), and n r 3.2. In this subsection, the standard conditions for the illumination process (optical intensity and optical field polarization angle) are I 5 × 10 8 W/m 2 and ϕ 45 • . In Fig. 7, the linear (dashed lines), nonlinear (dotted lines), and total (solid lines) optical coefficients as a function of the incident photon energy for different values of the α (Fig. 7a) and β (Fig. 7b) parameters are shown. In Fig. 7a, a non-monotonous shift of the resonant peak by varying the α parameter can be observed. From α 70 • to 90 • , a blue-shift is observed. Moreover, from the latter value to α 110 • , a red-shift of the absorption spectrum occurs. This fact is in accordance with the transition energy E 21 as a function of α within the range [60 • , 160 • ] plotted in the inset of the figure. The inset clearly shows that the resonant peak will be blue-shifted, starting from α 60 • to a value near 84 • . From this point, a red-shift is expected by increasing the α parameter. An important finding that can be seen in the inset is that the vertically stacked VGQWs are optically active within the THz band (below energies of 20 meV [35]). Analogously in Fig. 7b, a non-monotonous shift of the resonant peak by varying the β parameter within the range [0.4 a * 0 , 0.75 a * 0 ] can be seen. This left-to-right and right-to-left shifting is explained in a similar fashion by the inset, which shows that the resonant peak can be blue-shifted up to the maximum energy of 12 meV for a β value of 0.48 a * 0 . In addition, as can be seen, by varying the β parameter within the range of [0.3 a * 0 , 1.5 a * 0 ], the system is optically active within the THz band. Besides, Fig. 7 shows two important facts. In first instance, the maxima in the insets of Fig. 7a, b correspond to the anti-crossing points between states |2 and |3 discussed in Figs. 4a and 5a, respectively. The avoided crossing between the states |2 and |3 leads to broadening the inter-level spacing between states |1 and |2 and consequently, the transition energy is maximized at the anti-crossing points as shown in the insets. On the other hand, the smaller values of α and β parameters appearing in Fig. 7 (70 • and 0.4 a  *   0 ) are linked to the higher values of the total optical absorption (406 and 420 cm −1 , respectively).
In Fig. 8, the linear (dashed lines), nonlinear (dotted lines), and total (solid lines) optical coefficients as a function of the incident photon energy for different values of the barrier width L b (Fig. 8a) and the optical field polarization angle ϕ (Fig. 8b) are displayed. In Fig. 8a, a red-shift of the resonant peak by increasing the L b parameter within the interval [0.4 a * 0 , 0.9 a * 0 ] can be seen. However, the transition energy E 21 as a function of L b (plotted in the inset) in the interval [0.2 a * 0 , 1.2 a * 0 ] shows that the resonant peak shifting is non-monotonous (as discussed in Fig. 7) and follows a blue-shift from 0.2 a * 0 to the maximum point (near L b 0.36 a * 0 ) and then a red-shift. Also, the inset reveals an optical activity within the THz band. Moreover, the general trend is that the larger the barrier width, the smaller the absorption coefficient. The spectra from Fig. 8b reveal that the variation of the polarization angle ϕ does not affect the resonant peak positioning; however, it can drastically change the absorption coefficient. When the optical field is oriented along the x-axis (ϕ 0 • ), the α T coefficient takes very small values since the electron mobility is strongly restricted along this direction and, therefore, the electron-optical field coupling is weak. Nevertheless, by tilting the optical field, α T is increased, reaching large values of the order of 600 cm −1 when the polarization vector is oriented along the y-axis. In this case, the electron mobility is high due to the possibility of the electron to tunnel between the vertically stacked VGQWs.  Fig. 9 a, b Optical absorption spectra for different electric field (F) values, c, e low-lying energy levels and transition energy E 21 as a function of F, d, f modulus of the dipole matrix element M 12 as a function of F, and g, h ground and first excited state probability densities for two different F values for a symmetric (asymmetric) vertically stacked VGQW system In Fig. 9a, b the total absorption coefficient as a function of the incident photon energy for different values of electric field strength F are shown. In order to study the electric field effect on the VGQW-to-VGQW interaction, its orientation angle and the optical field polarization angle have been set to φ 90 • and ϕ 90 • , respectively. In Fig. 9a, a symmetric configuration of VGQWs with the same crescent thickness (β 0.5 a * 0 ) is analyzed. As can be seen, the electric field applied does not significantly shift the resonant peak (located near 11.8 meV) when varied from 0 to 5 kV/cm. In addition, at zero electric field, the absorption coefficient value is approximately 480 cm −1 and slightly decreases with increasing the electric field (at 5 kV/cm, the α T coefficient is reduced to only 3%). These facts are in agreement with the results in Fig. 9c, d and g. In Fig. 9c, the first three low-lying energy levels and the transition energy E 21 as a function of F for the symmetric case are plotted. As can be seen, the inter-level spacing between the ground state and first excited state energies is not drastically modified by the presence of the electric field within the analyzed range. Therefore, the transition energy (blue line) slightly increases, which explains the slight blue-shift of the absorption spectra from Fig. 9a. In Fig. 9d, the dipole matrix element modulus |M 21 | (whose square is in a direct relation with the linear and nonlinear optical coefficients) as a function of F for the symmetric case is plotted. The general trend of this quantity with the increasing field F is slightly decreasing and consequently reducing the optical absorption coefficient. This effect can be understood from the idea of classical physics that the electron under the presence of a static field F oriented along the positive y-axis undergoes an electric force directed downward, limiting the electron mobility and thus affecting the coupling between the electron and the optical field. The considerable value of the α T coefficient is also linked to the possibility of producing significant dipole moments. This fact is due to the electron's cloud is spread over the VGQW stack (as a result of the quantum tunneling), which reveals large electron mobility with the selected geometrical parameters as seen in Fig. 9g.
In contrast, in Fig. 9b, an asymmetric configuration of VGQWs with different crescent thicknesses (β 1 0.6 a * 0 and β 2 0.5 a * 0 for the lower and upper VGQW, respectively) is analyzed. In this case, at zero electric field, the optical absorption is 275 cm −1 being reduced by approximately 43% in comparison with the symmetric case. Additionally, by increasing the electric field from 0 Eur. Phys. J. Plus (2022) 137:1039 to 4.8 kV/cm, an approximate reduction of 20% in the α T parameter is obtained. This fact contrasts with the variation of 3% in the symmetric case). Furthermore, a more noticeable blue-shift of the resonant peak is calculated. For example, the resonant peak is shifted from 15.5 to 19.5 meV by increasing F from zero to 4.8 kV/cm. These behaviors are supported by the results of Fig. 9e and f, in which the first three low-lying energy levels (and the related transition energy E 21 ) and the dipole matrix element modulus |M 21 | as a function of F are plotted, respectively. In Fig. 9e, the increase in the transition energy with the electric field explains the blue-shift of the spectra in Fig. 9b. Also, the transition energy undergoes a change in the slope near F 4.8 kV/cm due to the anti-crossing point between states |2 and |3 at this value (see red and green curves). In Fig. 9f, the decrease in the dipole element with the electric field within the range 0 − 4.8 kV/cm explains the reduction of α T in Fig. 9b. In addition, a drastic decrease in |M 21 | is observed just at the anti-crossing point. This fact is due to the electron's cloud of the state |2 , which is located in the upper VGQW for F ≤ 4.8 kV/cm (see the upper right panel in Fig. 9h at zero electric field for instance) is shifted downwards to the lower VGQW due to the electric force. If the electric field is still increased, it is expected to have stronger confinement of the electron's cloud in the lower VGQW (see, the lower right panel in Fig. 9h for F 6 kV/cm). Thus affecting the mobility and almost setting to zero the dipole moment (as seen in Fig. 9f for F ≥ 6 kV/cm).
Finally, two general aspects of this analysis can be mentioned. Firstly, the effect produced by the lower VGQW with a thicker crescent can be seen comparing the upper left panels of Fig. 9g and h. Since β 1 > β 2 , the electron's cloud of the ground state is located within the wider VGQW, and the quantum tunneling between VGQWs Fig. 9g is no longer noticeable. As a consequence, the electron mobility is smaller for the asymmetric case, and it explains the reduction of the α T coefficient from Fig. 9a, b. Secondly, in both configurations, symmetric and asymmetric vertically stacked VGQWs, a blue-shift of the spectra was obtained, which is in agreement with the results obtained by Campi and Alibert [55] in double quantum well structures.
In Fig. 10, the absorption coefficient maximum (α max ), that is, the absorption coefficient peak value, for the standard symmetric VGQWs configuration as a function of the optical field polarization angle ϕ for several values of the barrier width L b (Fig. 10a) and the inter-sidewall angle α (Fig. 10c) is shown. A glance at the figures allows us to conclude that α max is strongly dependent on L b , α, and ϕ. In Fig. 10a, there are two critical points ϕ 0 • (or ϕ 180 • ) and ϕ 90 • in which regardless the barrier width, the absorption coefficient maximum takes a near-zero and a maximum value, respectively. The fact that α max takes a maximum value at ϕ 90 • is in agreement with the discussion of the results of Fig. 8b. In addition, regardless of the ϕ value, the greater the barrier width, the smaller the absorption coefficient maximum. This fact is in agreement with the results of Fig. 9 since, by increasing the barrier width, the quantum tunneling is reduced and also the electron mobility, thus affecting the generation of dipole moments. As a consequence, the absorption coefficient maximum is smaller. This assertion is supported by Fig. 10b in which the modulus of the dipole moment matrix element M 21 as a function of the polarization angle ϕ and for the four different values of inter-sidewall angle α 60 • , 83 • , 85 • , and 90 • is plotted. Since the configurations of Fig. 10a are plotted with the same standard value of inter-sidewall angle α 90 • , the pink dashed line in Fig. 10b must be taken as a reference. As can be seen, the profile of the curves of Fig. 10a follows the same behavior as the pink dashed line in Fig. 10b, reaching a maximum of the absorption peak if the polarization angle is ϕ 90 • and a very small value if ϕ 0 • . However, Fig. 10b suggests an inversion of this behavior if the inter-sidewall angle is reduced to 83 • (or even 60 • ). Since for these values, the maximum, and minimum of |M 21 | are reached at ϕ 0 • and ϕ 90 • , respectively.
To see this effect, it can be seen that the α max profiles of Fig. 10c are similar to those of |M 21 | in Fig. 10b for the inter-sidewall angles α 60 • , 83 • , 85 • , and 90 • . Furthermore, in Fig. 10c, the absorption coefficient maximum has been plotted for other α values (70 • , 80 • , 100 • , and 110 • ). These results show that starting at ϕ 0 • with α 60 • the absorption coefficient maximum is 430 cm −1 , and monotonically increases up to 745 cm −1 with α 83 • . But from α 85 • , the situation is inverted, the maximum reached with this inter-sidewall angle is 540 cm −1 at ϕ 90 • and by increasing α to 110 • , then the absorption coefficient maximum is reduced to 330 cm −1 (also at ϕ 90 • ). The drastic change of α max at α 84 • is due to the anti-crossing point discussed in Fig. 4c. The situation can be qualitatively understood as follows. From Fig. 4c, the probability densities of the states |1 and |2 show for α ≤ 84 • that they can favor the generation of significant dipole moment elements |M 21 | along the x-axis (when ϕ 0 • ) since there exist for the |2 state a fourfold electron cloud distribution far from the y-axis. On the contrary, if ϕ 90 • , the |M 21 | values are small because in the integration process over the y-axis, the |2 probability density gives null values (see the black zones). However, for α > 84 • the behavior is inverted since the nature of the states |2 and |3 are exchanged (see the probability densities for α 84 • and α 90 • as a reference). This inversion leads to that for α values above 84 • the generation of dipole moments along the y-axis is favored (giving appreciable α max values at ϕ 90 • ) while due to the parity conditions of the integrand of the elements M 21 , the α max gives small values at ϕ 0 • . This inversion effect due to the anti-crossing point is not noticeable for example, in terms such as the modulus of the difference of the dipole matrix elements M 22 and M 11 (|M 22 − M 11 |) appearing in Eq. (10) and which is plotted as a function of ϕ in Fig. 10d. Here, since |M 22 − M 11 | does not involve off-diagonal elements, then when an anti-crossing point takes place, its presence is not seen due to a lack of correlation between different states. This last non-trivial phenomenon reveals, in summary, that the optical properties of a system of vertically coupled VGQWs can be drastically modified and even inverted (as in the anti-crossing points) by varying their geometrical parameters.
Lastly, the optical absorption coefficients as a function of the incident photon energy for different values of the aluminum concentration in the barriers x Al 0.15, 0.25, 0.3, and 0.35 are displayed in Fig. 11a. Within the intervals [0.15,0.25] and [0.25,0.35], a blue-shift and a red-shift of the resonant peak can be seen, respectively. This behavior follows the transition energy evolution (displayed in the inset) as a function of x Al , which increases from 0.15, reaching a maximum at 0.25 and then decreasing for greater aluminum concentration values. This fact is due to the anti-crossing between states |2 and |3 (similarly as those discussed in Figs. 7 and 8a) that takes place at x Al 0.25. This phenomenon can be seen in Fig. 11b, which shows the probability densities of the two first states for the concentration values displayed in Fig. 11a. Clearly, from x Al 0.25 to 0.3, the aspect of density probability of the |2 state is modified as a consequence of the role exchange in the anti-crossing point (as explained in Fig. 4). It is important to notice two remarkable aspects. In the first instance, the optical response lies in the THz range by varying x Al within the interval [0. 15, 0.35]. Secondly, the greater the x Al value, the smaller the absorption peak value. This effect was expected since large aluminum concentration values produce stronger quantum confinement. Thus, the electron mobility and the optical peak values are reduced.

Conclusions
In this work, the spectral features and the linear, nonlinear, and total optical absorption coefficients of two realistic GaAs/Ga 1−x Al x As vertically stacked VGQWs confining single-electron states were theoretically studied. The effects of the variations of geometrical parameters such as the barrier width, inter-sidewall angle, crescent thickness, and the influence of a static electric field on the level ordering and optical coefficients are analyzed. The results show consistency between the energy levels of a stack of two VGQWs with large barrier width and those of a single VGQW, which can be understood as quality proof of the obtained results. Some general trends of the electron states in two vertically coupled and identical VGQWs can be remarked. Increasing the crescent thickness leads to a reduction of the energy values, while an increase in the inter-sidewall angle produces a non-monotonous increase in the energy significantly differing for the single VGQW case. In both cases, anti-crossing points in the spectra are seen, and in which clearly, the VGQW inter-spacing plays a key role since it controls the degree of coupling between them. In addition, the presence of a static electric field leads to stark-like patterns, which can be more controllable in contrast to the single VGQW case [34]. Furthermore, the compact density matrix formalism predicts a non-monotonous blue-/red-shift of the resonant peaks of the optical absorption by increasing the crescent thickness, inter-sidewall angle, or the barrier width, which suggests a strong dependency between the VGQW stack geometry and its spectral and optical properties. As a general trend, the model predicts that optical fields oriented along the growth direction enhance the optical absorption. Finally, asymmetric vertically coupled VGQWs systems in the presence of electric fields can significantly modify the optical properties compared to symmetric systems, especially in the quantum coupling regime, due to the modification of the quantum tunneling as similarly occurs in analog heterostructures such as the asymmetric double quantum wells [51]. In vertically coupled VGQWs systems, the obtained absorption coefficient maximum can be drastically modified by varying the inter-sidewall angle, and even the evolution tendency be inverted due to the anti-crossing points in the energy spectra. Considering the VGQW stack optical response lies in the THz range, they could be suitable for exploring and designing novel technological applications.