Band‑gap Engineering in InGaNBi Quaternary Alloys: Composition‑Dependent Structural, Electronic, and Strain Properties

Introduction

Wurtzite In_xGa_1−xN alloys and InGaN/GaN quantum wells (QWs) have attracted intense interest for high‑efficiency light‑emitting diodes (LEDs) and laser diodes (LDs) due to their tunable band‑gaps and compatibility with GaN substrates [1–10]. However, the [0001]‑oriented In_xGa_1−xN/GaN QWs suffer from a strong built‑in electric field caused by biaxial compressive strain, which reduces the emission energy and oscillator strength of electron‑hole pairs [11]. In addition, the high density of geometric defects—stacking faults and threading dislocations—serves as non‑radiative recombination centers [12], while electron leakage and Auger recombination contribute to the notorious efficiency droop in InGaN LEDs [13].

In the infrared regime, alloying GaAs with bismuth (Bi) has emerged as an effective strategy to reduce band‑gap energy (E_g) and enhance spin‑orbit (SO) splitting, thereby mitigating Auger recombination [14]. Bi, the largest group‑V element, exerts profound influence on the band structure of III‑V bismide alloys, as demonstrated in AlNBi [15], GaNBi [16,17], GaSbBi [18,19], InPBi [20,21], and InSbBi [22–24]. In these ternaries, Bi incorporation perturbs the valence band (VB) through interaction between Bi impurity states and heavy‑/light‑hole bands, while the large atomic size of Bi induces strain‑related band‑gap modifications [21]. More recently, quaternary bismide systems such as GaAsNBi [25–27], InGaAsBi [28,29], and GaAsPBi [30] have attracted attention for their extended band‑gap tunability. By combining Bi with other III or V elements, one can engineer band‑gaps, SO splitting, and strain simultaneously, expanding the functional design space for optoelectronic devices [25].

Given the potential of InGaNBi to bridge visible and mid‑infrared applications, this study employs first‑principles calculations to elucidate how In and Bi concentrations shape the structural, electronic, and strain characteristics of the alloy. Concentrations are limited to ≤ 50 % In and ≤ 9.375 % Bi to avoid the large lattice mismatch and limited Bi solubility that compromise crystal quality [32,33]. The paper is organized as follows: computational methods are detailed in Section 2; results and discussions of lattice constants, band‑gaps, SO splitting, and strain are presented in Section 3; and key take‑aways are summarized in Section 4.

Methods

All calculations were performed within density‑functional theory (DFT) as implemented in the Vienna Ab‑Initio Simulation Package (VASP) [33,34]. Structural relaxations employed the projector‑augmented wave (PAW) method [35,36] with the Perdew‑Burke‑Ernzerhof (PBE) generalized gradient approximation (GGA) for exchange‑correlation. To correct the systematic underestimation of band‑gaps in PBE, we used the modified Becke‑Johnson (MBJLDA) exchange potential combined with local density approximation (LDA) correlation [38]. Because Bi exhibits strong spin‑orbit coupling (SOC), SOC was explicitly included in all electronic structure calculations.

The valence‑electron configurations were set to 4d¹⁰5s²5p¹ for In, 3d¹⁰4s²4p¹ for Ga, 2s²2p³ for N, and 5d¹⁰6s²6p³ for Bi. Atomic forces were converged to < 0.02 eV/Å, and total energies were converged to 10⁻⁴ eV. A plane‑wave cutoff of 450 eV and a 4 × 4 × 4 Monkhorst‑Pack k‑point mesh were employed to ensure accurate total energies and electronic properties.

Results and Discussion

Structural Properties

We constructed 4 × 2 × 2 supercells (64 atoms) to model 36 distinct compositions of In_yGa_1−yN_1−xBi_x with 0 ≤ x ≤ 0.09375 and 0 ≤ y ≤ 0.5. In each case, In and Bi atoms were distributed as uniformly as possible to minimize clustering effects. The calculated lattice parameters for ternary In_yGa_1−yN and quaternary In_yGa_1−yN_1−xBi_x alloys are plotted in Figure 1. For pristine GaN, we obtained a = 3.211 Å and c = 5.235 Å, which are in excellent agreement with previous theoretical (a = 3.155–3.22 Å, c = 5.144–5.24 Å) and experimental (a = 3.19 Å, c = 5.19 Å) values [39–42]. The lattice constants of In_yGa_1−yN increase almost linearly with y, reaching a = 3.304 Å and c = 5.365 Å for y = 0.25, and a = 3.397 Å and c = 5.509 Å for y = 0.5, consistent with earlier studies [39,40,43,44]. For the quaternary alloys, both a and c exhibit a nearly linear rise with increasing In and Bi content, reflecting the larger ionic radii of In (1.42 Å) and Bi (1.48 Å) relative to Ga (1.22 Å) and N (0.71 Å) [45].

The lattice parameters for a ternary alloys In_yGa_1−yN, with 0≤y≤0.5 and b quaternary alloys In_yGa_1−yN_1−xBi_x, with 0≤x≤0.09375, 0≤y≤0.5. For comparison, we add some other calculations and experimental data from Ref. [39–44] in Fig. 1a. The solid line represents a and dashed line is c

Bond‑length analysis for the representative composition In_0.25Ga_0.75N_0.9375Bi_0.0625 (Fig. 2) reveals that Ga–N bonds shorten to 2.009 Å, whereas In–N bonds lengthen to 2.195 Å, and Ga–Bi and In–Bi bonds reach 2.592 Å and 2.704 Å, respectively. These trends confirm that In and Bi atoms expand the local lattice, consistent with their larger covalent radii. Such lattice distortions, coupled with electronegativity differences, are expected to influence electronic and optical behavior.

Histogram of bond length in In_0.25GaNBi_0.0625. The values in panel indicate the average lengths of the four types of bond

Electronic Properties

Band‑gap energies were computed with the MBJLDA+SOC approach, which has proven reliable for III‑V alloys. Figure 3 shows the dependence of E_g on In content in In_yGa_1−yN, alongside experimental and other theoretical benchmarks (HSE06, mBJ, LMTO‑CPA‑MBJ). The calculated E_g of GaN (3.273 eV) aligns closely with experimental values (3.40–3.50 eV) and previous calculations (3.261–3.33 eV) [39–49]. As y increases from 0 to 0.5, the band‑gap decreases smoothly from 3.273 eV to 1.546 eV, matching both theory and experiment [50,51].

Predicted bandgap energy (E_g, red solid line) as a function of In composition in I_yG_1−yN as well as a fit to the data (black dashed line). Other theoretical [39, 40, 46] and experimental [47–51] results are also plotted

Extending to quaternary In_yGa_1−yN_1−xBi_x alloys, the contour map in Figure 4 demonstrates a pronounced, nonlinear reduction of the band‑gap with both In and Bi content. For Bi up to 9.375 % and In up to 50 %, the band‑gap spans 3.273 to 0.651 eV, corresponding to wavelengths from 0.38 to 1.9 µm—covering the entire visible and mid‑infrared range. Importantly, the inclusion of Bi not only depresses E_g but also enhances the spin‑orbit splitting Δ_SO, reaching 0.220 eV (3.125 % Bi), 0.360 eV (6.25 % Bi), and 0.600 eV (9.375 % Bi). These values remain largely independent of the In fraction, suggesting that Bi is the dominant driver of Δ_SO enhancement. Achieving Δ_SO > E_g in higher‑composition alloys could suppress Auger recombination, a critical bottleneck for high‑efficiency LEDs and LDs.

Contour plot of the bandgap values for In_yGa_1−yN_1−xBi_x alloys, as a function of Bi(x) and In(y) compositions

Projected band structures and total density of states (TDOS) for GaN, In_0.25GaN, and In_0.25GaN_0.96875Bi_0.03125 (Fig. 5) reveal the distinct roles of In and Bi. In incorporation shifts both the conduction band minimum (CBM) and valence band maximum (VBM) toward lower energies, effectively narrowing the gap. In contrast, Bi introduces a defect band just below the Fermi level, strongly hybridizing with the VB edge and further raising the VBM. The TDOS shows a pronounced Bi‑derived peak between −1.0 and −0.5 eV, confirming the formation of localized states that can modulate carrier dynamics.

The projected band structures and their corresponding total density of states (TDOS) of a GaN, b, c In_0.25Ga_0.75N, and d, e In_0.25Ga_0.75N_0.96875Bi_0.03125. The black dashed line represents the Fermi level, which sets to be zero. The relative contributions of In and Bi are highlighted by color: blue (red) corresponds to the state originating from In (Bi)

Strain of InGaNBi on GaN

For practical device integration, the strain state of In_yGa_1−yN_1−xBi_x layers grown on GaN must remain within manageable limits. Figure 6 maps the biaxial strain as a function of In and Bi fractions. At the maximum considered composition (y = 0.5, x = 0.09375), the alloy experiences an 8.5 % compressive strain—substantial but potentially tolerable with appropriate buffer layers. More importantly, for In fractions ≤ 0.0625 and Bi fractions ≤ 0.028, the strain stays below 1 %, indicating that modest alloying can yield a near‑lattice‑matched system suitable for high‑quality epitaxy.

Strain of InGaNBi alloys on GaN substrate at various In (0–0.5) as a function of Bi fraction. Positive values of strain indicate InGaNBi is under compressive strain

Conclusions

First‑principles DFT calculations demonstrate that InGaNBi alloys offer a highly tunable platform for optoelectronic applications across the visible to mid‑infrared spectrum. Lattice constants grow almost linearly with In and Bi content, reflecting the larger atomic sizes of In and Bi. The band‑gap can be engineered from 3.273 eV down to 0.651 eV for Bi up to 9.375 % and In up to 50 %, while spin‑orbit splitting reaches 0.600 eV at the highest Bi fraction—values that can suppress Auger recombination when Δ_SO > E_g. Band‑structure analysis shows that In primarily shifts both CBM and VBM, whereas Bi introduces localized defect states that raise the VBM. Strain analysis indicates that with moderate In (≤ 6.25 %) and Bi (≤ 2.8 %) concentrations, the alloy remains nearly lattice‑matched to GaN, enabling high‑quality epitaxial growth. These findings lay the groundwork for designing high‑efficiency InGaNBi LEDs and laser diodes with tailored emission wavelengths and reduced non‑radiative losses.