Sada Traore^1,2*, Philippe Bernard Himbane², Moustapha Thiame^3
1Laboratory of Semiconductor and Solar Energy, Cheikh Anta Diop University, Dakar, Senegal.
²Laboratory of Electronics, Computer Science, Telecommunications and Renewable Energies, Gaston Berger University, Saint-Louis, Senegal.
³Laboratory of Chemical and Physics of Materials, Assane Seck University, Ziguinchor, Senegal.
DOI: 10.4236/jsemat.2026.161001   PDF    HTML   XML   1 Downloads   16 Views   Citations

Abstract

This paper presents a systematic numerical analysis of the Band Gap Narrowing (BGN) effect in indium gallium nitride (InGaN) solar cells for two indium molar fractions (x = 0.12 and x = 0.28) at T = 300 K. By sweeping the donor doping concentration N from 10¹⁶ to 3 × 10¹⁸ cm^?3, we quantify the reduction of the effective band gap energy (E_geff) and the resulting red-shift of the optical cutoff wavelength (λ_cut). Results show that BGN intensifies non-linearly with doping and is systematically more pronounced for higher indium content. At the highest doping level (3 × 10¹⁸ cm^?3), λ_cut shifts by +154 nm for x = 0.12 and by +378 nm for x = 0.28, extending absorption deep into the visible spectrum. These findings underline the necessity of a careful trade-off between spectral coverage and material quality degradation in the design of high-efficiency InGaN photovoltaic devices.

Keywords

InGaN, Solar Cells, Band Gap Narrowing, Indium Molar Fraction, Cutoff Wavelength, III-Nitride Semiconductors

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1. Introduction

The InGaN alloy (indium-gallium nitride) occupies a strategic position in the field of materials for photovoltaic conversion. Its tunable bandgap spans the spectrum from 0.7 eV (pure InN) to 3.51 eV (pure GaN) through simple variation of the indium molar fraction x. This makes it an ideal material for the fabrication of multi-junction solar cells capable of covering the entire solar spectrum [1]-[4].

However, InGaN-based solar cells face significant physical constraints during fabrication. Doping, which is necessary to form p-n junctions, introduces high charge carrier concentrations that alter the intrinsic electronic properties of the material. One of the most significant effects is Band Gap Narrowing (BGN), a phenomenon in which the effective bandgap narrows due to Coulomb interactions between free carriers and between carriers and ionized impurities [5].

A quantitative understanding of BGN is essential for accurately predicting the spectral response of the solar cell and optimizing its conversion efficiency. Recent numerical studies on InGaN solar cells confirm that doping, active layer thickness, and device architecture simultaneously influence optical absorption, open-circuit voltage, and overall efficiency [6]-[9].

The present study proposes a comparative and systematic analysis of the effect of doping on BGN, the effective bandgap energy E_geff, and the cutoff wavelength λ_cut, for two representative InGaN compositions (x = 0.12 and x = 0.28) at a temperature of 300 K.

2. Theoretical Model

2.1. Bandgap Energy of InGaN

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]:

$E_g^{InGaN}\left(x,T\right)=x\cdot E_g^{InN}\left(T\right)+\left(1-x\right)\cdot E_g^{GaN}\left(T\right)-1.43x\cdot \left(1-x\right)$ (1)

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:

$\{\begin{array}{l}{E}_g^{InN}\left(T\right)=E_g^{InN}\left(0\right)-\frac{{\alpha }_{InN}{T}^2}{T+{\beta }_{InN}}\\ E_g^{GaN}\left(T\right)=E_g^{GaN}\left(0\right)-\frac{{\alpha }_{GaN}{T}^2}{T+{\beta }_{GaN}}\end{array}$

where α and β are material-specific constants. For the alloy, the temperature dependence follows the modified Vegard’s law. $E_g^{InN}$ and $E_g^{GaN}$ are the bandgap energies of InN and GaN, respectively, and b is the bowing parameter.

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

Parameter	Symbol	GaN	InN	Unit	Reference
Bandgap at 0 K	Eg(0)	3.510	0.675	eV	Vurgaftman (2003)
Varshni α	α	9.09 × 10−4	2.45 × 10−4	eV/K	Vurgaftman (2003)
Varshni β	β	830	624	K	Vurgaftman (2003)
Electron effective mass	$m_n^{\ast }$	0.20 m0	0.11 m0	—	Vurgaftman (2003)
Hole effective mass	$m_p^{\ast }$	0.80 m0	0.60 m0	—	Vurgaftman (2003)
Static permittivity	εr	8.9	15.3	—	Levinshtein (2001)
Bowing parameter	b	1.43		eV	Wu et al. (2002)

2.2. Band Gap Narrowing (BGN) - Jain-Roulston Model

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]:

$E_{BGN}^{JR}\left(x,N,T\right)=A\left(x,N\right)\cdot N^{\frac{1}{3}}+B\left(x,N\right)\cdot N^{\frac{1}{4}}+C\left(x,N\right)\cdot \frac{1}{{\lambda }_D\left(x,N,T\right)}$ (2)

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:

$A\left(x,N\right)=\frac{1.83}{r_s^{\ast }\left(x,N\right)\cdot 4\pi {\epsilon }_s\left(x\right)}\cdot \frac{\Lambda }{N_b^{\frac{1}{3}}}$ (3)

$B\left(x,N\right)=\frac{0.95}{r_s^{\ast 3/4}\left(x,N\right)}\cdot \frac{q^2}{4\pi {\epsilon }_s\left(x\right)}$ (4)

$C\left(x,N\right)=\frac{1.57}{r_s^{\ast 3/2}\left(x,N\right)}\cdot \frac{q^2}{4\pi {\epsilon }_s\left(x\right)}\cdot \frac{m_d^{*}\left(x\right)}{m^{*}\left(x\right)}$ (5)

where the reduced Wigner-Seitz radius is given by:

$r_s^{*}\left(x,N\right)={\left(\frac{3}{4\pi N}\right)}^{1/3}\cdot \frac{m^{*}\left(x\right)}{m_0}\cdot \frac{1}{a_B^{*}\left(x\right)}$ (6)

$a_B^{*}\left(x\right)=\frac{4\pi {\epsilon }_s\left(x\right){\hslash }^2}{m^{*}\left(x\right)q^2}$ (7)

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

${\lambda }_D\left(x,N\right)=\sqrt{\frac{{\epsilon }_s\left(x\right)k_{B}T}{q^{2}N}}$ (8)

The material parameters ($m_n^{\ast }$ , $m_p^{\ast }$ , ${\epsilon }_r$ ) are linearly interpolated between the values of GaN and InN as a function of x.

$m^{*}\left(x\right)=x{m}_{InN}^{*}+\left(1-x\right)m_{GaN}^{*}$ , $m^{*}\left(x\right)=x{m}_{InN}^{*}+\left(1-x\right)m_{GaN}^{*}$ et ${\epsilon }_s\left(x\right)=x{\epsilon }_{InN}+\left(1-x\right)x{\epsilon }_{GaN}$

These parameters are given in Table 2.

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

Property	InN	GaN
Band gap (Eg) eV	0.675	3.51
Electron affinity (χ) eV	5.6	4.1
Effective density of states in the conduction band (Nc) (cm−3)	5.1∙1017	2.3∙1018
Effective density of states in the valence band (Nv) (cm−3)	5.3∙1019	4.6∙1019
Electron effective mass (me)	0.11m0	0.2m0
Hole effective mass (mh)	0.65	0.80 m0
Dielectric permittivity (ε)	15.3	8.9

2.3. Validity Range of the Jain-Roulston Model

The Jain-Roulston model is valid for doping concentrations N ≤ 3 × 10¹⁸ cm⁻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.

2.4. Effective Bandgap and Cutoff Wavelength

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:

$E_g^{InGaN}\left(x,N,T\right)=E_g\left(x,T\right)-E_{BGN}^{JR}\left(x,N,T\right){\lambda }_{cut}\left(\text{nm}\right)=\frac{1240}{E_{geff}\left[\text{eV}\right]}$ (9)

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].

3. Results and Discussion

3.1. Summary of Parameters as a Function of Doping

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.

N [cm−3]	Doping Regime	ΔEBGN [meV]	Egeff [eV]	λcut [nm]	Δλ [nm]
No BGN	—	0	2.9524	420.0	—
1016	Low	5.15	2.9473	420.7	+0.7
1017	Low-to-moderate	35.6	2.9168	425.2	+5.2
5 × 1017	Moderate	150.0	2.8024	442.5	+22.5
1018	High	283.7	2.6687	464.7	+44.8
3 × 1018	Very High	792.0	2.1604	574.0	+154

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

N [cm−3]	Doping Regime	ΔEBGN [meV]	Egeff [eV]	λcut [nm]	Δλ [nm]
No BGN	—	0	2.3692	523.4	—
1016	Low	6.24	2.3630	524.8	+1.4
1017	Low-to-moderate	43.30	2.3259	533.2	+9.8
5 × 1017	Moderate	186.5	2.1827	568.1	+44.7
1018	High	240.7	2.1285	582.7	+59.3
3 × 1018	Very High	993.5	1.3757	901.4	+378

3.2. Evolution of Band Gap Narrowing with Doping

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.

3.3. Reduction of the Effective Bandgap Energy

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.

3.4. Shift of the Cutoff Wavelength

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.

3.5. Summary of the Spectral Shift Δλ_cut

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.

Doping Regime	Concentration N [cm−3]	x = 0.12: Δλcut [nm]	x = 0.28: Δλcut [nm]
Low	1016	+0.7	+1.4
Moderate	5 × 1017	+22.5	+44.7
High	1018	+44.7	+59.3
Very High	3 × 1018	+154	+378

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.

3.6. Implications for Photovoltaic Optimization

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].

4. Conclusions

This study presents a systematic numerical analysis of the Band Gap Narrowing (BGN) effect in InGaN solar cells for two Indium molar fractions (x = 0.12 and x = 0.28) at T = 300 K. The main results can be summarized as follows:

• BGN increases in a strongly non-linear manner with doping concentration, with a critical acceleration beyond 10¹⁸ cm⁻³.

• The x = 0.28 composition exhibits a systematically more pronounced BGN, resulting in a greater reduction of Egeff and a larger shift in λ_cut.

• A maximum shift of +378 nm is observed for x = 0.28 at 3 × 10¹⁸ cm⁻³, moving λ_cut to 901.4 nm in the near-infrared region.

• The optimal doping range for photovoltaic applications lies between 10¹⁶ and 5 × 10¹⁷ cm⁻³, providing a good compromise between spectral extension and material quality.

These results constitute an essential quantitative foundation for the design of high-efficiency InGaN solar cells. Future work will incorporate the effects of temperature, the internal piezoelectric electric field inherent to nitrides, and layer morphology on the effective BGN, in connection with recent advances reported for advanced InGaN structures [3] [4] [13].

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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