Asymmetric AlGaAs/GaAs Quantum Wells in a Dielectric Environment: Electric-Field-Tunable Optical Properties

Asymmetric AlGaAs/GaAs Quantum Wells in a Dielectric Environment: Electric-Field-Tunable Optical Properties J. F. G´omez-Bedoya1and R. L. Restrepo1 Universidad EIA, C. P. 055428 Envigado, Colombia, [email protected] [email protected] Abstract. We present a theoretical investigation of the linear and nonlinear optical properties of asymmetric AlGaAs/GaAs quantum wells embedded in a dielectric medium and subjected to an external electric field. The heterostructure, engineered through stepwise variations in Al concentration, features a four-well potential profile with intrinsic spatial asymmetry. By solving the Schr¨odinger equation within the envelopefunction and effective-mass approximations—including self-energy corrections due to dielectric mismatch—we compute the confined electron states, dipole matrix elements, and optical absorption coefficients. Our results demonstrate that the combined influence of the dielectric environment and electric field dramatically reshapes the confinement potential, leading to Stark shifts, wavefunction relocalization, and field-dependent selection rules. The fundamental 0→1 transition remains spectrally stable in the terahertz range, while higher-order transitions exhibit pronounced blueshifts and can be selectively suppressed at critical field values where dipole matrix elements vanish. These findings highlight the dual tunability of inter-subband optical response through both electric and dielectric degrees of freedom, underscoring the potential of such structures for reconfigurable THz optoelectronic devices. Keywords: Asymmetric quantum wells, Dielectric environment, Stark effect, Nonlinear optical absorption, Terahertz optoelectronics 1 Introduction Scientific research into semiconductor nanostructures (SN), including quantum wells, quantum wires, rings, and quantum dots, necessitates a range of experimental, computational, simulation, theoretical, and numerical application strategies. These strategies are employed to advance the state of the art, with the objective of designing devices that are useful in a variety of cutting-edge quantum technologies [1,2]. Quantum wells (QWs) have emerged as a pivotal class of nanomaterials, exhibiting distinctive optical, electronic, and structural properties, thereby facilitating their application across a wide range of scientific and technological domains. Their tunable bandgap, high quantum yield, and size-dependent emission characteristics render them optimal candidates for 2 J. F. G´omez-Bedoya, R. L. Restrepo optoelectronics, biosensing, and biomedical imaging [3,4,5]. The subsequent review synthesizes recent advances in QW research, grouped into five thematic areas: material synthesis and structural engineering, nonlinear optical properties, optoelectronic applications, dielectric environment effects and biomedical and biosensing applications [6,7]. Advancements in the synthesis and structural control of semiconductor nanostructures (NS) have enabled precise tuning of their properties [8,9]. In their seminal work, the researchers investigated band-edge shifts and magnetic behavior through solvation and doping strategies [10,11]. Theoretical frameworks, including the effective mass approximation and the compact density matrix formalism, were extensively employed to predict and optimize nonlinear responses, as evidenced in works by [12,13]. Employing finite element methods, the model incorporates complex geometries and diffusion effects. These studies underscore the importance of structural engineering in achieving desired optical and electronic outcomes [14,15]. Nonlinear optical phenomena, including third-harmonic generation and refractive index modulation, play a pivotal role in NS-based photonic applications. The effects of these phenomena were examined under quantum plasma and oxide confinement conditions [16,17,18]. The integration of NSs into optoelectronic devices has led to significant improvements in performance, particularly in light-emitting diodes (LEDs), lasers, and photodetectors [19]. In a similar vein, many authors investigated defect control and heavy-metal-free alternatives for QLEDs, emphasizing the pivotal role of surface chemistry and material composition in device efficiency [20,21,22]. The dielectric environment surrounding NSs exerts a pivotal influence on the modulation of their electronic and optical properties. A body of research has emerged in recent years, with studies employing theoretical models to analyze the impact of dielectric matrices and electric fields on energy levels, binding energies, and harmonic generation [23,24,25,26]. Subsequent investigations were conducted into the nonlinear susceptibility and polarizability in various core/shell configurations. These investigations revealed the sensitivity of NSs to geometric and environmental parameters[27,28,29]. Semiconductor nanoestructures have led to a paradigm shift in the fields of biomedical imaging and biosensing, owing to their unique fluorescence characteristics and biocompatibility [30,31]. A recent study investigated the theranostic applications and magnetic properties of doped NSs with the aim of utilizing them for cancer diagnostics. A number of studies have examined the integration of NSs into biosensors, emphasizing their role in enhancing sensitivity and specificity [32,33]. The following paper puts forth a series of models that predict deformation and diffusion in order to mitigate the toxicity of NS and thereby advance their clinical potential [34]. This work presents numerical calculations of the nonlinear optical absorption coefficients in asymmetric AlGaAs/GaAs quantum wells supporting at least four bound electron states, under the combined influence of an external electric field and a surrounding dielectric medium. The energy spectrum and dipole matrix elements are systematically analyzed as functions of the applied field for two Asymmetric Quantum Wells: Dielectric and Field Effects. 3 distinct values of the external permittivity, representing different dielectric environments. The paper is organized as follows: Section 1 provides an introduction contextualized within recent advances in engineered quantum well systems. Section 2 outlines the theoretical model, including the effective-mass approximation, the treatment of dielectric mismatch via self-energy corrections, and the formulation of nonlinear optical absorption. Sections 3 presents the results and discussion, and Section 4 summarizes the main conclusions of the study. 2 Theoretical Framework Fig. 1. Schematic representation of the AlGaAs/GaAs Asymmetric Quantum Well (AQW) used in this work, showing the system of reference, the direction of external electric field, with the coordinates −L/2, Z1, Z2, Z3, Z4, Z5,+L/2 and the potentials V0, V1, V2, V3, V4. We consider asymmetric AlGaAs/GaAs quantum wells embedded in a dielectric medium and subjected to an external electric field. The electronic structure is calculated within the effective-mass and envelope-function approximations, assuming a parabolic conduction-band dispersion near the Γpoint. The model explicitly incorporates the effects of dielectric mismatch at the heterointerfaces, which induce image-charge–mediated self-energy corrections that reshape the confining potential. 4 J. F. G´omez-Bedoya, R. L. Restrepo The single-electron Hamiltonian governing the electronic states is expressed as: H=p2 2m∗+V(z) + e F z +ϕ(z) (1) where m∗is the effective mass of the electron in GaAs (m∗= 0.067m0), assumed constant across the entire structure, V(z) is the confinement potential determined by the aluminum concentration profile, Fis the externally applied electric field, and ϕ(z) is the self-energy potential arising from the dielectric environment. The potential well V(z) is defined as [35]: V(z) =                          V0= 282.8 meV,if −L/2≤z≤Z1, V1= 101.1 meV,if Z1< z ≤Z2, Vbottom = 0.0 meV,if Z2< z ≤Z3, V2= 41.1 meV,if Z3< z ≤Z4, V3= 151.2 meV,if Z4< z ≤Z5, V4= 212.3 meV,if Z5< z ≤+L/2. (2) with L= 60 nm, Z1=−11 nm, Z2=−7 nm, Z3= 3 nm, Z4= 7 nm, and Z5= 15 nm. The dielectric-induced potential ϕ(z) is expressed as [36] ϕ(z) = 1 2L0εin ∞ X k=0 (k+ 1)(εin −εout) (k+ 1)εout +kεin z L02k ,(3) where εin and εout are the dielectric constants inside and outside the quantum well, respectively, and L0denotes half the distance between Z3and Z2, which corresponds to the width of the central quantum well in the heterostructure [37]. The one-dimensional Schr¨odinger equation associated with the Hamiltonian in Eq. 1 is solved numerically using a diagonalization method. This procedure yields the eigenenergies and envelope functions ψn(z) for various asymmetry configurations and applied electric-field strengths. The wave function is expanded in terms of a complete set of orthonormal functions [38]: ψ(z) = ∞ X n=1 Cnφn(z) (4) The complete orthonormal basis set is taken from the analytical solution of the infinite 1D quantum well of width L: φn(z) = r2 Lsin nπz L+nπ 2.(5) Convergence of the calculated energy spectrum is achieved with n= 50 basis functions and a well width of L= 60 nm, consistent with the criteria established in previous studies [35]. Asymmetric Quantum Wells: Dielectric and Field Effects. 5 2.1 Nonlinear Optical Absorption Model The linear and third-order nonlinear optical absorption coefficients are calculated considering inter-subband transitions between confined states. The electric field of the incident radiation is assumed to be linearly polarized along the growth direction (z). The dipole transition matrix element between initial (i) and final (f) states is defined as: Mif =Zψ∗ f(z)z ψi(z)dz. (6) The linear absorption coefficient is given by: α(1)(ω) = ωe2rµ0 ε0εrρ¯hΓif |Mif |2 (Eif −¯hω)2+ (¯hΓif )2,(7) and the third-order nonlinear contribution by: α(3)(ω, I) = −rµ0 ε0εrωI 2nrε0c4ρ¯hΓif |Mif |4 [(Eif −¯hω)2+ (¯hΓif )2]2.(8) The total absorption coefficient is expressed as: α(ω)=α(1)(ω)+α(3)(ω, I),(9) where ρis the electron density, Γif = 1/τif is the damping factor related to the transition lifetime, and Irepresents the optical intensity. 3 Results and Discussion The combined influence of the engineered potential profile, image-charge-induced self-energy corrections, and field-induced Stark shifts leads to a highly tunable nonlinear optical absorption spectrum, as detailed in the following analysis. Fig. 2 shows the confinement potential, energy eigenvalues, and corresponding probability densities for the first four bound electronic states in an AlGaAs/GaAs asymmetric quantum well (AQW), under two distinct conditions: (a) in the absence of external fields, and (b) under an applied static electric field of 40 kV/cm. In the zero-field configuration [Fig. 2(a)], the ground state is strongly localized in the deepest potential well, extending from Z2to Z4. The first excited state primarily occupies the adjacent well (Z4–Z1), while the second and third excited states are distributed across the widest region of the structure, between Z1and Z5. The energy of the third excited state lies close to the top of the right-hand barrier, indicating its proximity to the continuum and weaker confinement. Application of an external electric field of 40 kV/cm [Fig. 2(b)] induces a pronounced tilt in the potential profile—a hallmark of the quantum-confined Stark effect. This field-driven asymmetry effectively deepens the local well between Z2and Z3, causing the ground state to shift toward this region. Likewise, the 6 J. F. G´omez-Bedoya, R. L. Restrepo first and second excited states become more tightly confined within the Z4–Z1 segment. In contrast, the third excited state experiences an upward energy shift and remains localized near the upper edge of the Z5–Z1well. The potential tilt originates from a linear variation in the electrostatic potential across the structure: the energy reference at Z2is reduced to approximately −25 meV, whereas at Z3it rises to about +15 meV, with analogous shifts observed across all barrier and well regions. This linear slope leads to a modest increase in the energies of the higher-lying states—particularly the second and third excited levels. Importantly, the electron probability densities under the applied field exhibit enhanced localization within individual subwells compared to the zero-field case, reflecting the field-induced modification of the effective confinement landscape. Fig. 2. Potential profile, energy levels, and probability density for the first four confined electronic states in an AlGaAs/GaAs AQW, for applied fields of F= 0 (a) and F= 40 kV/cm in (b), without effect of dielectric environment. The results demonstrate a high degree of congruence with those illustrated in Figs. 2(a, b) of the referenced study [35]. The four lowest energy levels, the non-diagonal dipole matrix elements and the total optical absorption coefficients in an AlGaAs/GaAs AQW, are presented in Figs. 3. Fig. 3(a) illustrates the dependence of the energy levels of the first four quantum states on the applied external electric field. In contrast to a symmetric quantum well, the energy levels exhibit an asymmetric response under fields of equal magnitude but opposite polarity (e.g., at ±40 kV/cm). This asymmetry arises from the inherent asymmetry in the potential profile—specifically, the unequal energy barrier heights on either side of the well—which breaks the spatial inversion symmetry. Consequently, the ground and first excited states, whose probability densities are predominantly localized in the central and narrower region of the well, display a negative concavity in their energy-field dependence, Asymmetric Quantum Wells: Dielectric and Field Effects. 7 leading to a decrease in energy with increasing field strength in one direction. In contrast, the third and fourth states, which extend more significantly into the wider regions of the potential, maintain a negative concavity, resulting in an increase in their energies under the same field conditions. This distinct behavior underscores the role of spatial asymmetry in shaping the Stark shift characteristics of confined electronic states. The interplay between quantum well asymmetry and the applied external electric field is clearly reflected in the dipole matrix element curves shown in Fig. 3(b). These curves exhibit a pronounced peak for the transition from the ground state to the first excited state M01, while transitions from the ground state to the second M02 and third M03 excited states yield significantly smaller matrix elements. As previously discussed, this behavior stems from the spatial localization of the electronic wavefunctions: the ground and first excited states are predominantly confined to the central region of the asymmetric well, resulting in strong overlap and hence a large dipole matrix element. In contrast, the second and third excited states extend into the lateral, wider regions of the well, where their probability densities are concentrated away from the center. This spatial separation from the ground state wavefunction reduces the overlap integral6, leading to the observed suppression of the corresponding dipole matrix elements. Fig. 3(c) presents the dependence of the nonlinear optical absorption coefficients on both the incident photon energy and the strength of the applied external electric field. In accordance with the dipole matrix elements discussed earlier, the coefficients corresponding to the transitions α01,α02, and α03 are shown. All absorption peaks exhibit a blueshift as the electric field increases—an effect that is relatively modest for the α01 transition but becomes significantly more pronounced for α02 and α03. To account for the inherently small dipole matrix elements of the higher-order transitions, the vertical scale of the coefficients for the 0 →2 and 0 →3 transitions has been amplified by a factor of 50 relative to α01, enabling clear visualization of their spectral features. The peak positions fall within the mid-infrared region of the electromagnetic spectrum, highlighting the potential of this asymmetric quantum well system for mid-infrared optoelectronic applications. As shown in Figs. 4(a, b), we examine how a dielectric environment characterized by a permittivity lower than that of the quantum well material (εin > εout) influences the confinement potential, the energy spectrum of the first four bound states, and the spatial profiles of their corresponding wave functions. Fig. 4(a), calculated at zero external electric field (F= 0), depicts the modified confinement potential for εout = 3.2, revealing that the dielectric mismatch substantially reshapes the well—enhancing its intrinsic asymmetry and altering its effective depth. When an external electric field is applied (F= 40 kV/cm, as in Fig. 4(b)), the potential profile undergoes a marked tilt, a hallmark signature of the quantum-confined Stark effect. This field-induced distortion further modulates the energy levels and wavefunction localization, underscoring the combined role of dielectric contrast and electric field in tailoring the electronic structure of asymmetric quantum wells. 8 J. F. G´omez-Bedoya, R. L. Restrepo Fig. 3. Energy levels (a), dipole matrix elements (b) in an AlGaAs/GaAs AQW as a function of the electric field, in (c) the total optical absorption coefficients in a function of the photon energy for different values of F, without effect of dielectric environment. The combined influence of the stepped square quantum well structure and the dielectric ambient potential—formulated within the framework of the imagecharge model of classical electromagnetic theory—results in a modified effective confinement potential. This potential features nanoscale quasi-well regions localized between the coordinates Z3–Z2and Z5–Z4, which significantly reshape the spatial distribution of the electron probability density for the four lowest bound states. Consequently, the envelope functions of these states deviate markedly from the characteristic sinusoidal form observed in idealized rectangular wells, instead exhibiting pronounced localization dictated by the fine structure of the engineered potential landscape. Furthermore, the effective bottom of the central quantum well is raised by approximately 35 meV relative to the original reference in Fig. 2(a), a shift attributed to the reduced width of the central region—now confined to roughly 9 nm—induced by the emergence of the adjacent nanonarrow wells (see Figs. 4) . This renormalization of the confinement potential underscores the critical role of both geometric design and dielectric environment in tailoring quantum-state properties in low-dimensional semiconductor heterostructures. In Fig. 4(b), the influence of an externally applied electric field is incorporated, resulting in a pronounced tilting of the confinement potential profile—a direct manifestation of the quantum-confined Stark effect. This field-induced asymmetry lowers the energies of all four bound states and significantly redistributes the spatial probability density of the corresponding electronic wave functions. Markedly, the ground-state wave function shifts toward the left side of the central well, localizing preferentially in the region between Z2and Z3. This displacement arises from the interaction between the electron’s negative charge and the applied electric field, which exerts a force directed opposite to the field vector, thereby driving the electron toward the lower-potential side of the tilted Asymmetric Quantum Wells: Dielectric and Field Effects. 9 well. Such field-driven spatial relocalization underscores the tunability of electronic states in asymmetric quantum well systems and highlights the intimate coupling between external fields and internal quantum confinement. Fig. 4. Potential profile, energy levels, and probability density for the first four confined electronic states in an AlGaAs/GaAs AQW, for applied fields of F= 0 (a) and F= 40 kV/cm in (b), with εin = 12.35 and εout = 3.2. The calculated energy levels and corresponding probability densities demonstrate that both the applied electric field and the dielectric contrast between the well and its surrounding medium exert a strong influence on the spatial localization of the electron wavefunctions. This modulation of wavefunction distribution directly affects the transition energies and dipole matrix elements, thereby shaping the system’s optical response, show in Figs. 5. Now we present in Fig. 5(a) the energy levels of the first four electron states confined within the quantum well, incorporating the influence of the external dielectric environment and plotted as a function of the applied electric field. Although the overall vertical energy range remains comparable to that in Fig. 3(a), the field-dependent behavior of the levels is markedly distinct. Markedly, the ground-state energy (E0) exhibits a monotonically increasing trend as the electric field sweeps from negative to positive values, passing through a minimum near zero field. In contrast, the first excited state (E1) decreases with increasing field strength but displays positive concavity, causing the energy separation E01 =E1−E0to diminish progressively with field magnitude. This reduction in interlevel spacing directly impacts the conduction-band transition energies, leading to a systematic redshift in the corresponding optical transitions. Meanwhile, the second excited state (E2) shows a weak increasing trend with field, while the third excited state (E3) rises more steeply than observed in the absence of dielectric contrast (cf. Fig. 3(a)). These modifications underscore the profound role of dielectric screening in reshaping the Stark response of confined 16 J. F. G´omez-Bedoya, R. L. Restrepo 20. W. Fang, J. Yu, S. Liao, H. Gao, X. Li, B. Liu. Emergence of colloidal quantum well-based optoelectronics (2024). DOI 10.1016/j.xcrp.2024.101936 21. X. Jiang, Z. Fan, L. Luo, L. Wang. Advances and challenges in heavy-metal-free inp quantum dot light-emitting diodes (2022). DOI 10.3390/mi13050709 22. Y. Nandan, M.S. Mehata, Scientific Reports 9(2019). DOI 10.1038/ s41598-018-37676-3 23. H. Song, L. Du, S. Lin, W. Huang, K. Wang, X. Zhou, Y. Zhang, C. Wu, Optics Express 33, 34913 (2025). DOI 10.1364/oe.570147 24. M. Karako¸c, A. Naifar, O. Bayrak, Physica B: Condensed Matter 701 (2025). DOI 10.1016/j.physb.2025.416961 25. N. Zeiri, P. Baser, H.D. Jahromi, N. Yahyaoui, A. Ed-Dahmouny, N. Sfina, C.A. Duque, M. Said. Effects of the size and applied electric field on the photoionization cross-section of elliptical cylindrical cds/zns core-shell quantum dots immersed in various dielectric matrices (2025). DOI 10.1016/j.optlastec.2024.111822 26. S. H´ertilli, N. Yahyaoui, N. Zeiri, P. Baser, M. Said, S. Saadaoui, Results in Physics 56 (2024). DOI 10.1016/j.rinp.2023.107274 27. N. Amin, A.J. Peter, Micro and Nanostructures 185 (2024). DOI 10.1016/j.micrna. 2023.207728 28. A. Ed-Dahmouny, K. El-Bakkari, M. Jaouane, R.V. Hahn, R. Arraoui, A. Fakkahi, H. Azmi, N. Zeiri, M.G. Barseghyan, N. Es-Sbai, A. Sali, M.E. Mora-Ramos, C.A. Duque, Physics Letters, Section A: General, Atomic and Solid State Physics 509 (2024). DOI 10.1016/j.physleta.2024.129502 29. A. Ed-Dahmouny, N. Zeiri, A. Sali, N. Es-Sbai, R. Arraoui, M. Jaouane, A. Fakkahi, K. El-Bakkari, N. Yahyaoui, C.A. Duque, European Physical Journal Plus 139 (2024). DOI 10.1140/epjp/s13360-024-05418-2 30. P. Zhang, M. Jiao, Y. Li, X. Ding, K.J. McHugh, L. Jing. Bright semiconductor quantum dots shed new light on precision nanomedicine for various diseases (2024). DOI 10.1002/smsc.202300081 31. S. Mohanapriya, M. Hariharan, P. Priyadharshini, K. Pushpanathan, Journal of Cluster Science 35, 1255 (2024). DOI 10.1007/s10876-024-02575-3 32. Y. Fu, T. Liu, H. Wang, Z. Wang, L. Hou, J. Jiang, T. Xu. Applications of nanomaterial technology in biosensing (2024). DOI 10.1016/j.jsamd.2024.100694 33. E. Heydari-Bafrooei, A.A. Ensafi, Biosensors and Bioelectronics: X 13 (2023). DOI 10.1016/j.biosx.2022.100245 34. O. Kuzyk, O. Dan’kiv, I. Stolyarchuk, R. Peleshchak, Molecular Crystals and Liquid Crystals (2024). DOI 10.1080/15421406.2024.2358734 35. R.L. Restrepo, J.P. Gonz´alez-Pereira, E. Kasapoglu, M.E. Mora-Ramos, C.A. Duque, Optical Materials 79, 334 (2018). DOI 10.1016/j.optmat.2018.07.019 36. Z. Zeng, E. Paspalakis, C.S. Garoufalis, A.F. Terzis, S. Baskoutas, Journal of Applied Physics 113 (2013). DOI 10.1063/1.4789363 37. Z. Zeng, C.S. Garoufalis, S. Baskoutas, Journal of Nanoelectronics and Optoelectronics 11, 615 (2016). DOI 10.1166/jno.2016.1940 38. J.B. Xia, W.J. Fan, Physical Review B 40(12), 8508 (1989). DOI 10.1103/ PhysRevB.40.8508