The bandgap energy of the In_xGa_1−xN alloy as a function of the indium molar fraction x is given by the modified Vegard’s law including the Varshni correction [10]:

where b= 1.43 eV is the bowing parameter [1]. The Varshni parameters from Vurgaftman & Meyer (2003) are given in Table 1.

Temperature Dependence: Varshni’s Law

For the binary compounds, the following expression is generally used:

where α and β are material-specific constants. For the alloy, the temperature dependence follows the modified Vegard’s law. E g I n N and E g G a N are the bandgap energies of InN and GaN, respectively, and b is the bowing parameter.

Table 1. Varshni parameters and material properties [10]-[12].

Band Gap Narrowing results from many-body interactions in heavily doped semiconductors: carrier-carrier exchange interactions, correlation effects, and ion-carrier interactions. For an n-type semiconductor, the bandgap reduction ΔE_BGN is modeled according to the Jain-Roulston formalism [5]:

where A, B, and C are material constants that depend on the composition x, and N is the doping concentration in cm⁻³. where N denotes the donor concentration (N_D). Throughout this work, complete donor ionization at T = 300 K is assumed, so that the free electron concentration is approximated by n ≈ N_D.

These coefficients are determined from the Jain-Roulston relation:

where the reduced Wigner-Seitz radius is given by:

where λ_D is the Debye length given by the following equation:

The material parameters ( m n ∗ , m p ∗ , ε r ) are linearly interpolated between the values of GaN and InN as a function ofx.

m * ( x ) = x m I n N * + ( 1 − x ) m G a N * , m * ( x ) = x m I n N * + ( 1 − x ) m G a N * et ε s ( x ) = x ε I n N + ( 1 − x ) x ε G a N

These parameters are given in Table 2.

Table 2. Some values of the electrical properties of GaN and InN at 300 K.

The Jain-Roulston model is valid for doping concentrations N≤ 3 × 10¹⁸cm⁻³. Beyond this limit, two physical effects not included in the model become significant: Although the Jain–Roulston model was originally developed for conventional semiconductors such as Si, Ge and GaAs, it has been widely employed as a first-order approximation for evaluating many-body interaction effects in heavily doped III-V semiconductors. The predicted BGN magnitudes obtained in this study remain within the order of magnitude reported for heavily doped GaN and InGaN materials in the literature, supporting the applicability of the model for comparative analysis.

Burstein-Moss effect: Band filling by degenerate carriers leads to an apparent increase in the optical bandgap, which partially compensates the Band Gap Narrowing (BGN).

Carrier degeneracy: The Fermi level penetrates into the conduction band, invalidating the classical Boltzmann statistics used in the derivation of the Jain-Roulston model.

The cutoff wavelength λ_cut corresponds to the minimum photon energy required to excite an electron across the effective bandgap. The relations for the effective bandgap energy and the cutoff wavelength are given by the following equation:

A red shift of λ_cut (i.e., an increase) indicates the possibility of absorbing photons of lower energy. This can increase the short-circuit current density Jsc but may potentially degrade the open-circuit voltage Voc, as shown by several recent numerical optimization studies on single-junction InGaN structures, thin polar layers, and intermediate-band architectures [6]-[8][13].

Table 3 and Table 4 present the calculated values of bandgap narrowing (ΔE_BGN), effective bandgap energy (E_geff), and cutoff wavelength (λ_cut) for the two compositions investigated at the different doping levels considered, ranging from the absence of BGN to the heavily doped regime (3 × 10¹⁸ cm⁻³).

Table 3. Physical parameters for x = 0.12 (In_0.12Ga_0.88N), T = 300 K.

Table 4. Physical parameters for x = 0.28 (In_0.28Ga_0.72N), T = 300 K.

Figure 1 illustrates the evolution of ΔEBGN_{BGN}BGN as a function of the doping concentration for the two Indium mole fractions. A strongly non-linear increase in the BGN is observed: negligible at low doping levels (<5.15 meV at 10¹⁶ cm⁻³ for x = 0.12), it becomes critical beyond 10¹⁸ cm⁻³, reaching 792.0 meV and 993.5 meV for x = 0.12 and x = 0.28, respectively, at 3 × 10¹⁸ cm⁻³.

In general, the composition x = 0.28 exhibits a larger BGN than x = 0.12, particularly at low and very high doping levels. However, an exception is observed at N = 10¹⁸ cm⁻³, where the calculated BGN for x = 0.12 (283.7 meV) slightly exceeds that of x = 0.28 (240.7 meV). This deviation may arise from the interplay between effective mass and dielectric screening effects in the Jain–Roulston formulation.

Figure 2 shows the evolution of Egeff with doping. For x = 0.12, Egeff decreases from 2.9524 eV (without BGN) to 2.1604 eV at 3 × 10¹⁸ cm⁻³, representing a reduction of 26.8%. For x = 0.28, the drop is even more pronounced: from 2.3692 eV to 1.3757 eV, corresponding to a reduction of 41.9%.

These significant reductions in Egeff can disrupt band alignment in multi-junction structures and alter the ideal diode quality factor, directly impacting the fill factor FF and the conversion efficiency η of the solar cell [4][8].

Figure 1.Variation of band gap narrowing ΔE_BGN as a function of doping concentration N for x = 0.12 and x = 0.28 at T = 300 K.

Figure 2. Effective bandgap energy E_geff as a function of doping concentration N for the two InGaN compositions.

Figure 3 shows the evolution of λ_cut with doping. The red shift is moderate at low doping levels but becomes spectacular in the high doping regime. For x = 0.28, λ_cut increases from 523.4 nm to 901.4 nm, crossing the visible-infrared boundary of the solar spectrum. This significantly broadens the spectral absorption window.

Table 5 and Figure 4 summarize the shifts in Δλ_cut relative to the reference case (without BGN) for each doping regime. The comparison between the two compositions shows that x = 0.28 consistently produces a larger shift, with an amplification factor of approximately 2 to 2.5 in the moderate and high doping regimes, reaching a factor greater than 2.4 in the very high doping regime.

Figure 3.Cutoff wavelength λ_cut as a function of doping concentration N. The redshift is more pronounced for x = 0.28.

Table 5. Shift Δλ_cut of the cutoff wavelength according to the doping regime.

Δλ_cut calculated with respect to the reference case without BGN for each composition.

Figure 4. Spectral shift Δλ_cut by doping regime for x = 0.12 and x = 0.28. The nonlinearity is particularly pronounced in the very heavy regime.

The analysis shows that doping in the range of 10¹⁶ - 5 × 10¹⁷ cm⁻³ enables a moderate spectral extension (Δλ_cut < 45 nm for both compositions) while keeping the effective bandgap energy close to the nominal values. This regime therefore represents an optimal doping window for taking advantage of BGN without severely compromising the material’s electronic properties, in line with several recent studies on InGaN cell optimization [6]-[9].

Beyond 10¹⁸ cm⁻³, the non-linear increase in BGN induces an excessive reduction in Egeff, which is likely to lower the open-circuit voltage Voc and degrade the overall efficiency of the solar cell, despite the broadening of the absorption window. An optimal compromise must therefore be determined according to the incident solar spectrum and the specific cell architecture, whether single-junction, tandem, or intermediate-band devices [4][8][9][13].