1. Introduction

Optical isotropy and anisotropy, fundamental to photonics and materials science, have consistently garnered significant attention, especially their in-depth exploration at the nanostructure. Optical isotropy, characterized by uniform properties such as absorption and refractive index regardless of the direction of incident light, has been pivotal in developing applications requiring consistent optical performances [1–3]. Conversely, optical anisotropy, notable for its direction-dependent properties, catalyzes novel dimensions in light manipulation, leading to breakthroughs in a variety of domains, including multifunctional optoelectronic devices [4], multichannel colorimetric sensors [5], and polarization-based bioimaging [6]. Undoubtedly, rigorous investigation into the directional features of optical behavior in nanostructure has been crucial not only for developing nanodevices with enhanced precision and efficiency, but also for establishing novel frameworks in understanding and manipulation of light-matter interactions at a fundamental level.

Recent advancements have heralded a groundbreaking shift in adaptive optics [7] and intelligent photonic systems [8], marked by the advent of technologies enabling reversible transition between optical isotropy and anisotropy. Exploration has bifurcated into two investigative paths over the past few decades. One delves into the transition from optical isotropy to anisotropy by destroying the original electronic state. Recently, Yuan et al. [9] innovatively combined an anisotropic dielectric SiP₂ with isotropic 1L-MoS₂, resulting in a SiP₂/MoS₂ heterointerface, and they successfully altered the electronic states of 1L-MoS₂ from isotropy to anisotropy, which can be tuned back using SiP₂ grid voltage. Contrastingly, the transition from optical anisotropy to isotropy poses a formidable challenge, necessitating the precise manipulation of the degeneracy in energy level and wavefunction symmetry in order to reverse the initial mechanisms that resulted in optical anisotropy. Experimentally, Mandel et al. [8] have developed a smart superparamagnetic microrods that combine luminescence and high reflectivity, resulting in switchable isotropic and anisotropic optical properties modulated by the applied wavelength of light. However, the primary emphasis of these studies has been on the innovation and transformation of material properties.

Core-shell nanostructures [10,11], defined as heterostructures with a core of distinct material encased in a shell layer, have recently gained significant attention for their promising applications in optical and electronic devices [12–15]. The core shift, induced by different growth rates of the shell layers during the synthesis process, leads to the formation of both coaxial [16–18] and off-centered [19–21] core-shell structures. In the case of coaxial types, the electron states are generally characterized by degenerate energy levels in excited states, accompanied by wavefunctions displaying pronounced circular symmetry. Conversely, a tiny core shift can disrupt the degeneracy and symmetry of electronic states, leading to pronounced polarization sensitivity in optical absorption. For example, Amthong et al. [22,23] have demonstrated that a minimal shift distance of less than 1 nm can generate perfectly polarization-dependent optical absorption coefficients (OAC). However, despite the remarkable flexibility in configuring core-shell structures, achieving central inversion symmetry remains a significant challenge due to the inherent difficulty in controlling core shift. To date, researches have extensively explored the reversible transformation of core-shell nanostructures from isotropic to anisotropic optical absorption, yet investigations into the converse transition — from anisotropic to isotropic optical absorption — are notably lacking. Addressing this deficiency is vital for augmenting the optical versatility of core-shell nanostructures and broadening their applicability in intelligent optical devices, underscoring the urgency of bridging this research void.

Electro-optic modulators, governed by electric field modulation, are now extensively deployed for applications such as spectrally shifted enhancement [24], interleaver with high extinction ratios [25], and the augmentation of nonlinear susceptibilities [26]. Despite these advancements, the industry lacks electro-optic modulators capable of modulating optical absorption from an anisotropic to an isotropic state. In our previous investigations, we identified that an electric field induces “anti-crossing behavior of energy levels” (ACE) [27], significantly enhancing the reinstatement of energy level degeneracy [28–31]. This process effectively concentrates the discrete energy levels, enabling the development of electronic structures characterized by isotropic optical responses. Recent experimental studies have highlighted the critical role of ACE in gauging degeneracy within coupled quantum systems [32,33]. Hence, electro-optic modulators based on the ACE mechanism of core-shell nanostructures, are expected to facilitate the reversible transformation from anisotropic to isotropic optical absorption.

In this work, we propose, for the first time, a transition from anisotropic to isotropic optical absorption in off-centered square nanowires (OSN), achieved by an electric field. Initially, the finite difference method is employed to determine the electric field necessary for inducing ACE. Subsequently, the dependence of OAC on the rotating angle of the system and photon energy is investigated with and without external electric field. We discover that the onset of anti-crossing closely aligns the energy structures of off-centered and coaxial nanowires, neutralizing polarization sensitivity and enhancing isotropic optical absorption for the mid-infrared (mid-IR) wavelengths longer than 10 μm. However, as the core shift increases, the challenge of reinstating optical isotropy correspondingly escalates. Remarkably, applying the electric field in a direction opposite to the core's off-center displacement reveals new concentrations of transition energies, observed as non-ACE phenomena, which facilitates diverse isotropic optical absorption in the mid-IR wavelengths. These insights offer theoretical support for a switchable mechanism that controls the polarization sensitivity of mid-IR wavelengths in various mediums.

2. Theory

As schematically depicted in Fig. 1(a), the system we investigated comprises a nanowire with a square cross-section. It features a core of AlGaAs and a shell of GaAs, whose side lengths are denoted as L_s and L_c, respectively. The centroid of the square shell is located at the origin (x = 0, y = 0), whereas that of the square core, denoted by coordinate (x_c, y_c), moves along the white deviation line at angles of d = 0°, d = 30°, and d = 45°, with the x-axis, as shown in Fig. 1(b). In our work, the incident light, linearly polarized along the x-axis, propagates along the z-direction. The anisotropic and isotropic optical absorption are detected by rotating the system around the origin. The rotation angle θ, defined as the angle between the y-axis and the y′-axis, quantifies the rotation of the incident light relative to the x-axis as reflected in Fig. 1(c). Based on the effective mass approximation, the Hamiltonian, describing the lateral motion of electrons confined by the square core-shell confinement under an electric field, is given as:

 

Fig. 1. (a) Schematic representation of a square core–shell nanowire under electric field, where AlGaAs and GaAs are respectively core and shell materials. (b) Schematic diagram of the core shift along three kinds of directions with d = 0°, 30°, and 45° studied in this work. (c) A cross-section of the system where orange and blue solid lines represent the core and shell edges respectively. The cross sign expresses the centroid of the core square. After rotating the system through an angle θ clockwise, the core and shell edges become those represented by orange and blue dash lines.

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The Schrödinger equation $H\psi _n^{\theta = 0}({x,y} )= {E_n}\psi _n^{\theta = 0}({x,y} )$ in this system is solved using the 2D finite difference method by partitioning the xy plane into a uniform spatial mesh [34–36]. As depicted in Fig. 2, the xy plane is discretized into a square grid with width $h = {L_s}/({N - 1} )$, where N–1 is the number of intervals along the x and y directions. Boundary conditions dictate zero electron wavefunctions at the grid’s perimeter (blue line). Such discretization allows the Schrödinger equation in a discrete form as:

 

Fig. 2. A square grid with width h. Orange and blue lines represent the core and shell edges respectively. The centroid of the shell square is located at the origin (x = 0, y = 0).

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Based on the compact density matrix and the iterative method, the total OAC for arbitrary rotating angles θ is defined by ${\alpha ^\theta }(\omega )= \alpha _1^\theta (\omega )+ \alpha _3^\theta ({\omega ,I} )$ [22,23], where the linear and third-order nonlinear OAC are, respectively, given by

Here, μ is the system permeability, ε₀ is the vacuum permittivity, ε_r is the relative permittivity, σ_v is the electron density in the system, Γ is the relaxation rate, ω is the angular frequency of incident photon, c is the speed of light, ${n_r} = \sqrt {{\varepsilon _r}} $ is the refractive index of the system, and I is the optical intensity of the incident electromagnetic wave that leads to the intersubband optical transition. $M_{fi}^\theta = \psi _f^\theta \textrm{|}x\textrm{|}\psi _i^\theta $ represents the dipole moment matrix element for x-polarized light, and E_fi = E_f–E_i signifies the transition energy between the initial state $\psi _i^\theta $ and final state $\psi _f^\theta $. E_fi and $M_{fi}^\theta $ play a crucial role in analyzing and understanding the characteristics of the OAC. E_fi serves as effective approximations of the incident photon energy where ${\alpha ^\theta }(\omega )$ reaches its peak, while the maximum values of ${\alpha ^\theta }(\omega )$ are significantly influenced by $M_{fi}^\theta $. The parameters used in our calculations are as follows [23]: μ= 4π×10⁻⁷H/m, ɛ₀= 8.85 × 10⁻¹² Fm^-1, σ_ν= 3.0 × 10²²m^-3, ɛ_r = 12.53, Γ = 1/0.14 ps^-1, I = 0.05 MW/cm², V₀ = 342.95 meV, m₁ = 0.067m₀, m₂ = 0.1002 m₀ (m₀ is the free electron mass), L_c = 4 nm, and L_s = 12 nm. The Al concentration x in the core material Al_xGa_1-xAs is 0.4.

Prior to delving into our work, our numerical procedures is validated by examining the Hamiltonian with V₀ = 0, which aligns with the scenario of a particle confined within a two-dimensional square box [37]. The eigenvalues derived from our methods are found to be consistent with the exact eigenenergies $E = {\hbar ^2}{\pi ^2}({n_x^2 + n_y^2} )/2{m^\ast }L_s^2$.

3. Results and discussions

This work concentrates on the optical transitions among the first four states (n = 1, 2, 3, and 4). It is worth noting that when the centroids of the core are located at the origin (x_c = y_c = 0), the first excited electron states of the core–shell structure emerge as double degenerate states. Figure 3(a) shows the total OAC ${\alpha ^\theta }(\omega )$ of the coaxial core as a function of incident photon energy ħω and rotating angle θ for the system. It is important to recognize that ${\alpha ^\theta }(\omega )$ adheres to the relation ${\alpha ^\theta }(\omega )= {\alpha ^{\theta + \pi }}(\omega )$, thus the values in the range of $\pi < \theta \le 2\pi $ are omitted here. Figure 3(a) clearly illustrates that the relative maximums of ${\alpha ^\theta }(\omega )$ occur approximately at transition wavelengths about 22.8 μm and 13.8 μm, which correspond to the transition energies E₂₁ or E₃₁ and E₄₂ or E₄₃, respectively. The forbidden and allowed transitions are determined by the dipole moment matrix element $M_{fi}^\theta = \psi _f^\theta \textrm{|}x\textrm{|}\psi _i^\theta $. If $M_{fi}^\theta $ is zero (nonzero), the i→f transition is forbidden (allowed). Hence, the forbidden 1→4 transition leads to the nullification of ${\alpha ^\theta }(\omega )$ at ħω = E₄₁. Notably, ${\alpha ^\theta }(\omega )$ is hardly dependent on the rotating angle θ, suggesting near-optical isotropy. Meanwhile, the core shifts disrupt the degeneracy of first excited electron states, leading to a split into two distinct energy levels (E₂ and E₃), as seen in Fig. 3(b). Further, Fig. 3(b) demonstrates that the separation between E₂ and E₃ intensifies with the increase in x_c or d. The energy spitting results in the expansion of the total absorption spectrum to encompass six distinct transition energies.

 

Fig. 3. (a) The total OAC ${\alpha ^\theta }(\omega )$ of the coaxial core as a function of incident photon energy ħω and rotating angle θ for the system. The inset shows the allowed and forbidden transition energies. (b) The first four energy levels (E₁, E₂, E₃, and E₄) as functions of a shift distance x_c of the square core with different values of d, d = 0°, d = 30° and d = 45°.

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In Fig. 4, ${\alpha ^\theta }(\omega )$ as functions of incident photon energy and rotating angle θ are presented with different values of x_c (x_c = 0.4, x_c = 0.8, and x_c = 1.6 nm) and d (d = 0°, d = 30°, and d = 45°). Colored arrows indicate the transition energies E_fi. The dominant peaks of ${\alpha ^\theta }(\omega )$ are governed by photon energies of E₂₁, E₃₁, E₄₂ and E₄₃. As increments in x_c and d, there is a gradual separation between the transition energies E_fi, with E₂₁, E₃₁ and E₄₂ exhibiting a blue shift, and E₄₃ showing a red shift. Meanwhile, the anisotropic features of ${\alpha ^\theta }(\omega )$ become more pronounced with the increase of x_c and d. The transitions 1→2, 1→4, 2→3, and 3→4 are forbidden at θ = 0 when the core moves along the direction of d = 0°, while the transitions 1→2, 2→3, and 2→4 are similarly precluded at θ = π/4 corresponding to the core’s movement along the direction of d = 45°. Specially, the transitions between the first four states are all allowed for every value of θ. As a consequence, the relative maxima of θ-dependent ${\alpha ^\theta }(\omega )$ approximately occur at ħω = E₃₁ and E₄₂ for θ = 0, and at ħω = E₂₁ and E₄₃ for θ = π/2 in case of d = 0°. The maxima shift to ħω = E₃₁ and E₄₃ for θ = π/4, and at ħω = E₂₁ and E₄₂ for θ = 3π/4 when d = 45°. For d = 30°, the maxima are observed at ħω = E₄₂, E₃₁, E₄₃ and E₂₁ corresponding to the values of θ = 0, π/6, π/3, and 2π/3, respectively. Moreover, a substantial core shift at d = 0° and x_c = 1.6 nm induces broadband anisotropic absorption spanning 14.8 to 21 μm in the long-wavelength IR range. This anisotropic property is attributed to $M_{fi}^\theta $, whose values are significantly affected by the symmetry of the initial state $\psi _i^\theta $ and final state $\psi _f^\theta $. The increase in core shift distance gradually distorts the properties of the wavefunction, transforming it from circular symmetry into functions that are odd, even, or neither. A detailed discussion of this phenomenon will follow later.

 

Fig. 4. The total OAC ${\alpha ^\theta }(\omega )$ as a function of incident photon energy ħω and rotating angle θ for the system with different values of x_c (x_c = 0.4, x_c = 0.8, and x_c = 1.6 nm) and d (d = 0°, d = 30°, and d = 45°). Colored arrows represent transition energies E_fi.

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Building on the insights highlighted earlier, the distortion of the circular symmetry in wavefunction and the splitting of the first excited state contribute to the emergence of anisotropic optical absorption. The restoration of anisotropic optical absorption hinges on systematically rectifying the previous distortions. Given the critical role of ACE in the rectification of splitting electron states, it becomes imperative to investigate the value of electric field capable of inducing ACE in OSN. The energy levels of E₂ and E₃ as functions of F and φ with different values of x_c (x_c = 0.4, x_c = 0.8, and x_c = 1.6 nm) and d (d = 0°, d = 30°, and d = 45°) are shown in Fig. 5. From Fig. 5(a) to (f) (x_c ≤ 0.8 nm), a distinct ACE region is evident in each OSN, characterized by the intersection of the blue and red lines. Conversely, ACE is absent in Fig. 5(g) to (i) (x_c = 1.6 nm); however, E₂ and E₃ demonstrate similar rates of change in response to variations in F and φ. To facilitate the discussion, representative angles φ have been selected as 0°, 40°, 45°, 25°, 45°, 45°, 60°, 345° and 45°, respectively, for Fig. 5(a) through Fig. 5(i). In these cases, the transition energies E_fi as functions of the electric field F with different values of x_c (x_c = 0.4, x_c = 0.8, and x_c = 1.6 nm) and d (d = 0°, d = 30°, and d = 45°) are depicted in Fig. 6(a) to (c). Figure 6(a) to (c) vividly illustrates that as F increases, ACE leads to the merging of the four initially dominant transition energies (E₂₁, E₃₁, E₄₂ and E₄₃) into approximately two distinct transition energies (E₂₁ ≈ E₃₁ and E₄₂ ≈ E₄₃) as shown in the region of color band, which closely resembles the energy structure observed in coaxial core shell nanostructure. Unexpectedly, despite the absence of ACE when x_c = 1.6 nm, tuning the electric field felicitously can still induce two dominant transition energies, characterized by E₂₁ ≈ E₄₃ and E₃₁ ≈ E₄₂. Analogous behaviors are observed at x_c = 0.4 and 0.8 nm with d = 45°, as illustrated in Fig. 6(d). Figure 6(d) reveals that, if φ = 225°, inversely aligned with the core's off-center displacement, two concentrated region of transition energies (E₂₁ ≈ E₄₃ and E₃₁ ≈ E₄₂) are discernible, provided the amplitude of the electric field is selected appropriately. These new concentrations of transition energies appear to stem from the relative stability among the energy levels. Specially, Within the concentrated region of transition energies depicted in Fig. 6, a specific electric field F has been chosen for detailed discussion.

 

Fig. 5. The energy levels of E₂ and E₃ as functions of F and φ with different values of x_c (x_c = 0.4, x_c = 0.8, and x_c = 1.6 nm) and d (d = 0°, d = 30°, and d = 45°).

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Fig. 6. The transition energies (E_fi = E_f–E_i) as functions of F with different values of d: (a-c) d = 0°, d = 30° and d = 45°. (d) d = 45° and φ = 225°. The color band represents the concentrated region of transition energies. The inset shows the allowed and forbidden transition energies.

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The ${\alpha ^\theta }(\omega )$ of the OSN as a function of incident photon energy ħω and rotating angle θ are depicted in Fig. 7 with different values of x_c (x_c = 0.4, x_c = 0.8, and x_c = 1.6 nm) and d (d = 0°, d = 30°, and d = 45°), while the selected parameters F and φ are presented. Figure 7(a) to (g) depict two distinct isotropic optical absorption bands (E₂₁ ≈ E₃₁ and E₄₂ ≈ E₄₃). Notably, both Fig. 7(h) and (i) also exhibit two significant absorption bands (E₂₁ ≈ E₄₃ and E₃₁ ≈ E₄₂), even in the presence of a more pronounced discrepancy in the absorption intensity of incident light under different polarization directions. It suggests that as the core deviates further, the task of restoring the optical isotropy becomes progressively more challenging. The transition wavelengths corresponding to the two absorption bands fall within the mid-IR range, as labeled in the Fig. 7, which exhibit a minor blue shift with increasing x_c. It is noteworthy that despite each of the two isotropic optical absorption bands comprising two transition energies that are closely aligned, only one transition predominates when subjected to incident light with varying polarization directions. For instance, In Fig. 6(a), when θ = 0, d = 0°, and x_c = 0.4 nm, the forbidden 1→2, 1→4, 2→3 and 3→4 transition result in the vanishing of ${\alpha ^\theta }(\omega )$ at ħω = E₂₁ and ħω = E₄₃, accompanied with the government at ħω = E₃₁ and ħω = E₄₂, as seen in Fig. 7(a). Subsequently, as θ increases, 1→2 and 3→4 transition progressively assume a dominant role. Similarly, the inset of Fig. 6(c) indicates the transitions 1→2, 2→3 and 2→4 are forbidden at θ = π/4 and d = 45°, which leads to pronounced optical absorption in E₃₁ and E₄₃.

 

Fig. 7. The total OAC ${\alpha ^\theta }(\omega )$ as a function of incident photon energy ħω and rotating angle θ for the system with different values of x_c (x_c = 0.4, x_c = 0.8, and x_c = 1.6 nm) and d (d = 0°, d = 30°, and d = 45°). Colored arrows represent transition energies E_fi. The values of two dominant transition wavelengths are labeled.

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The allowed and forbidden transitions are determined by the nonzero and zero component of $M_{fi}^\theta = \langle\psi _f^\theta \textrm{|}x\textrm{|}\psi _i^\theta \rangle$, which depends on the parity of $\psi _f^\theta $ and $\psi _i^\theta $. Figure 8 illustrates the contour lines of the first four wavefunction in different values of x_c (x_c = 0.4, x_c = 0.8, and x_c = 1.6 nm) and d (d = 0°, d = 30°, and d = 45°) with and without electric field, whose values are consistent with those in Fig. 7. Considering the integration rule of odd and even functions, the zero component of $M_{fi}^\theta = \langle\psi _f^\theta \textrm{|}x\textrm{|}\psi _i^\theta \rangle$ can be concluded as the following two cases: (i) when $\psi _i^\theta $ and $\psi _f^\theta $ present opposite parity with respect to reflection along the θ + 90° axis. (ii) when $\psi _i^\theta $ and $\psi _f^\theta $ have the same parity for the reflection of the θ axis. Specifically, the first scenario, in Fig. 8, is exemplified by $M_{21}^{\theta = 0}$, $M_{41}^{\theta = 0}$, $M_{32}^{\theta = 0}$, and $M_{43}^{\theta = 0}$ at d = 0°, accompanied with $M_{21}^{\theta = \pi /4}$, $M_{32}^{\theta = \pi /4}$, and $M_{42}^{\theta = \pi /4}$ at d = 45°, except in the case of x_c = 0.8 nm under an electric field. Furthermore, comparative analyses reveal that specific modulations of the electric field significantly enhance the symmetry of the wavefunction.

 

Fig. 8. Th contour lines of the first four wavefunction in different values of x_c (x_c = 0.4, x_c = 0.8, and x_c = 1.6 nm) and d (d = 0°, d = 30°, and d = 45°) with and without electric field. The value of electric field is the same as that in Fig. 7.

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Finally, since the quasi-isotropic optical absorption band depicted in Fig. 7(g) to (i) can be attributed to the new concentrations of transition energies stemming from non-ACE, we have conducted exploratory research to ascertain the feasibility of using this mechanism to re-establish isotropic optical absorption. As representatives, Fig. 9 reveals the ${\alpha ^\theta }(\omega )$ of the OSN as a function of incident photon energy ħω and rotating angle θ with x_c = 0.4 nm and x_c = 0.8 at d = 45° and φ = 225°. It can be clearly observed that two isotropic optical absorption peaks appear in both OSN. In addition, compared with Fig. 7(c) and (f), Fig. 9(a) and (b) distinctly reveal the appearance of new isotropic absorption peaks at wavelengths of 9.9 μm and 10.2 μm, respectively. To the best of our knowledge, this is the first time that the transition from anisotropic to isotropic OAC in the OSN has been achieved. However, the physical mechanism underlying the repair mode of non-ACE requires further exploration in subsequent studies, which will pave the way for a more comprehensive understanding of this process.

 

Fig. 9. The total OAC ${\alpha ^\theta }(\omega )$ as a function of incident photon energy ħω and rotating angle θ for the system with x_c = 0.4 nm, and x_c = 0.8 at d = 45°. Colored arrows represent transition energies E_fi. The values of two dominant transition wavelengths are labeled.

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4. Conclusion

In summary, the transition from anisotropic to isotropic OAC in the off-centered core-shell AlGaAs/GaAs square nanowires are researched numerically. The finite difference approach is employed to solve the Schrodinger equation. The results indicate that an off-centered core disrupts the degeneracy of quantum states, leading to anisotropic OAC. Notably, the maximum OAC, characterized by the rotating angle θ, is intricately linked to the offset direction d. The anisotropic OAC are elucidated by allowed and forbidden optical transitions, governed by odd and even symmetries of the wavefunctions. To the best of our knowledge, this is the first instance where, by judiciously adjusting the electric field, we have effectively concentrated transition energies, either ACE or non-ACE, to align the energy structures in both eccentric and concentric nanowires, which significantly neutralizes polarization sensitivity and enhances isotropic optical absorption in the mid-IR wavelengths. This work provides innovative approaches to significantly advance the switchable polarization sensitivity and extend the operational spectrum in next-generation optical devices.