Study on the Influence of Plane Strain on Optical Properties of InP Materials through First-Principles Calculations

. This research scrutinizes the impact of plane strain on the optical characteristics of Indium Phosphide (InP) employing first-principles methodology, grounded on the Density Functional Theory (DFT). The findings suggest that the peaks of the spectral response curves of the dielectric function, refractive index, extinction coefficient, optical absorption coefficient, and reflection coefficient of InP, when subjected to plane tension strain, shift towards lower energy of electromagnetic wave frequency in the horizontal coordinate. Concurrently, static quantities such as the dielectric coefficient, refractive coefficient, and reflection coefficient of InP demonstrate an upward trend with an increase in the plane tension strain. Intriguingly, the influence of compressive strain on the photoelectric response of InP manifests contrary behavior to that of the tensile strain.


Introduction
III-V semiconductor materials, known for their superb electrical and optical properties, have garnered significant interest in various fields such as information and communication, semiconductor display and lighting, wearable devices, and aerospace [1]. Recently, indium phosphide (InP) materials have witnessed rapid development in applications such as light-emitting diodes, lasers, high-power radio frequency, photoelectric detection, and solar cells, owing to their remarkable physical attributes such as superior light absorption efficiency, high electron mobility, and exceptional thermal conductivity [2]. However, as the complexity of the structure of InP heterogeneously integrated devices escalates, the lattice constant mismatch between InP and other materials typically induces residual in-plane stress strain within the material film. Furthermore, the variation in the device's operating environment temperature gradient contributes to changes in material stress due to thermal expansion mismatch in the heterogenous film [3]. In certain harsh operational environments characterized by strong electromagnetic and high-intensity mechanical wave pulses, the device experiences pronounced local strains. This lattice strain significantly influences the optical and electrical characteristics of the material. Strain modulation has been shown to alter the electronic structure and optical properties of semiconductor heterojunction materials, thereby stimulating additional theoretical and experimental exploration into device and material property modulation techniques [4,5].
Though the optical properties of InP can be assessed experimentally through various spectroscopic techniques such as Raman spectral analysis and infrared reflectance measurements, there is a noticeable lack of systematic studies related to optical property changes under local planar stress-strain. Consequently, conducting theoretical calculations of the optical properties of InP becomes imperative. This study undertakes first-principles calculation methods to examine the energy band gap, complex permittivity function, complex refractive index, absorption coefficient, and reflection coefficient of InP under plane compressive and tensile strains. Furthermore, the paper discusses and analyzes the variations in optical properties under plane strain conditions.

Computational Models and Methods
This study employs first-principles calculations using the open-source Quantum ESPRESSO software [6,7], predicated on the Density Functional Theory (DFT) [8], plane wave basis set, and pseudopotentials. The computational model for InP is a sphalerite structure encompassing four In atoms and four P atoms, as depicted in Fig. 1. The initial lattice constant of InP is established at 5.8697 Å, aligning well with the experimental data of InP crystals [9]. Within this computational model, the exchange-correlation potential employs the Generalized Gradient Approximation (GGA) [10], and the pseudopotentials utilize the Optimized Norm-Conserving Vanderbilt Pseudopotential (ONCVPSP) as suggested by Hamann [11]. This configuration has demonstrated commendable accuracy in the calculation of the band gap and optical properties of InP. The computational input parameters are established as follows: the plane wave cutoff energy is set at 80 Ry, the total energy and charge density involved in self-consistent and non-self-consistent calculations correspond to 6×6×6 and 8×8×8 for Brillouin zone integration, respectively, and the fast Fourier transform (FFT) grid is fixed at 64×64×64. The convergence accuracies for the self-consistent and nonself-consistent calculations are set at 10E-6 eV/atom and 10E-9 eV/atom, respectively. The computational band gap of the intrinsic InP cell calculated in this study is 1.4406 eV, closely aligning with the experimental band gap value of 1.35 eV for InP [12]. The dielectric spectral response of InP is computed using the epsilon module in the Quantum ESPRESSO software. The computed complex refractive index curve for strainfree InP demonstrates a good fit with experimental values [13], as illustrated in Fig. 2. This study further investigates the influence of plane strain on the optical properties of InP by varying the lattice constants in both directions of the computational model plane, where the compressive strain range spans from -5% to 0% and the tensile strain range from 0% to 5%. The band gap data of InP are extracted from the selfconsistent calculation results, as exhibited in Fig. 3. The band gap of InP is observed to decrease with increasing tensile strain and increase with increasing compressive strain. The real and imaginary parts of the complex dielectric function of InP for different plane strains within the 0-12 eV spectral response energy range were calculated, with the results portrayed in Fig. 4. Upon analyzing the imaginary component ( ε 2), it can be seen that in the frequency energy range of the peak E3 of the imaginary part of the spectral response curve, an increase in compressive strain triggers an overall enhancement of the imaginary part's spectral response, while an increase in tensile strain initiates a general decline in the same spectral response range, as depicted in Figs. 4(a) and (c). However, for energy ranges with electromagnetic wave frequencies exceeding E3 in horizontal coordinates, a rise in compressive strain leads to a reduction in the imaginary spectral response, whereas an increase in tensile strain prompts an overall increase in the imaginary spectral response of that range. Moreover, as the plane compressive strain amplifies and the plane tension strain diminishes, the three peaks marked as E1, E2, and E3 in Fig. 4(a) and Fig. 4(c) demonstrate a gradual decrease in electromagnetic wave frequency energy, as shown in Fig.  5(a). Further analysis suggests that the increase in the band gap, leading to an increase in the optical absorption threshold, triggers a shift of the overall spectral response of the imaginary part towards a more energetic direction.  The static dielectric constant of the intrinsic InP calculated in this study is 8.453, which closely aligns with existing experimental [13] and theoretical results [14]. The variation of the static dielectric constant of InP under plane strain, depicted in Fig. 5(b), shows a progressive increase with the transition of plane strain from compressive to tensile.

Effect of plane strain on the complex refractive index of InP material
Figs. 6(a) and 6(c) depict the real part of the spectral response curves of the complex refractive index of InP material under plane strain. Notably, for different strain conditions of InP material, the refractive index exhibits discernible strain effects in the electromagnetic wave frequency energy range below 3 eV. With an increase in compressive strain, the refractive index decreases gradually, whereas it increases progressively with the rise in tensile strain. This observation suggests that within the calculated electromagnetic wave energy range, the refractive index of InP material is directly proportional to the magnitude of the plane compressive strain and inversely proportional to the magnitude of the plane tensile strain. Moreover, the peak E1 position of the InP refractive index spectral response curve shifts to the right on the horizontal coordinate as the plane compressive strain increases, and shifts to the left as the plane tensile strain increases. Figure 7(a) reveals that as the plane strain applied to the InP material transitions from compressive to tensile strain, the electromagnetic wave frequency energy corresponding to peak E1 of the InP refractive index spectral response curve decreases progressively. Figures 6(b) and 6(d) illustrate the impact of plane strain on the imaginary part of the complex refractive index spectral response curve (i.e., the extinction coefficient) of InP materials. The extinction coefficient spectral response curve of InP showcases significant stress-dependent changes in the approximately 1 eV to 5 eV range, particularly at the low-frequency peak of 1.5 eV and a high-frequency peak of 5 eV, where InP's extinction coefficient exhibits a substantial attenuation phenomenon. As plane strain transitions from -5% compressive strain to 5% tensile strain, the position of the material's extinction coefficient spectral response curve also shifts to the left, as shown in Fig. 7(b). The electromagnetic wave frequency energy corresponding to the peak E2 of the InP extinction coefficient spectral response curve gradually decreases. Further, Fig. 8 presents the variation of the refractive index n(0) of InP at zero frequency under different plane strains. It can be observed that n(0) is inversely proportional to the compressive strain and directly proportional to the tensile strain. This trend is essentially consistent with the static dielectric constant deduced from the real part of the dielectric function.

Effect of plane strain on the absorption and reflection coefficients of InP materials
Figs. 9(a) and 9(c) present the influence of plane strain on the absorption coefficient spectral response curves of InP materials. As compressive strain increases, the peak E1 of the absorption coefficient spectral response curve displays a notable increment, while it declines as tensile strain increases. This finding indicates that compressive strain results in a higher absorption peak of InP material compared to tensile strain, aligning with the analysis from the imaginary part of the dielectric function and bandgap. Additionally, the spectral response curve of InP's absorption coefficient shifts left on the horizontal coordinate axis with an increase in compressive strain, while the curve shifts to the right as tensile strain increases. That is, during the transition from -5% compressive strain to 5% tensile strain, the peak E1 of the absorption coefficient spectral response curve corresponds to a gradual decrease in electromagnetic wave frequency energy, as illustrated in Fig. 9(a). The results indicate that compressive strain leads to higher absorption of electromagnetic wave frequency, thereby endowing the InP material with enhanced absorption capability in a broader photon energy range. The static reflectance at zero frequency from the InP material reflection coefficient spectral response curve (as depicted in Fig. 9(b) and 9(d)) was extracted and its relationship with strain was plotted, as shown in Fig. 10(b). It is observable that the static reflectance R(0) is inversely proportional to the magnitude of the compressive strain and directly proportional to the magnitude of the tensile strain.

Conclusion
The calculation results suggest that with a decrease in the applied plane compressive strain or an increase in the plane tensile strain on the InP material, peak values of the spectral response curves for dielectric function, refractive index, extinction coefficient, light absorption coefficient, and reflection coefficient shift towards lower electromagnetic wave frequency energy along the horizontal axis. Conversely, as the plane strain transitions from compressive to tensile, the static dielectric coefficient, static refractive index, and static reflection coefficient of InP gradually increase. These findings highlight that while the tensile strain can shift the position of the peak frequency response of InP's optoelectronic property towards a smaller electromagnetic wave frequency energy, it also diminishes the maximum peak frequency response of its optoelectronic property. In contrast, the effect of compressive strain on InP's optoelectronic response is the opposite. The insights provided in this study can enhance our understanding of the evolution of performance in photovoltaic and lightemitting devices based on InP. Furthermore, they can foster advancements in the performance modulation technology of InP-based optoelectronic devices within the context of strain engineering.