1 Introduction

The InxGa1−xN has been considered a material with great potential to achieve high efficiency in tandem solar cells because the energy gap (Eg) can be varied from 0.7 to 3.4 eV [1]. However, even after nearly 20 years of research and significant incremental contributions, only a few reports have shown a small efficiency improvement [2,3,4]. Some reasons for this are the following technological challenges:

The first and the most important challenge is the difficulty to incorporate the adequate Indium concentration at the absorber layer. Typically, when Indium concentration exceeds 20%, the obtained ternary alloy tends to show phase separation and this problem gets worse as the thickness of the InxGa1−xN layer increases [5,6,7]. Fischer et al. [8] have estimated the critical thickness of the stressed InxGa1−xN before phase separation appears, obtaining a thickness of 12 nm for Indium concentrations around 25%, and as the concentration increases the critical thickness is reduced. Although InxGa1−xN has a high absorption coefficient, films with an adequate thickness are difficult to achieve without the phase separation occurring. So, as the concentration of In increases in the ternary, the crystalline quality is reduced [9, 10].

The second challenge is the difficulty to control the p-type conductivity due to the high background electron concentration in InxGa1−xN. Moreover, Mg doping has shown low dopant activation efficiencies around 1% for GaN. As a result, high Mg concentrations near the atomic solubility should be introduced into the films. Thus, under certain synthesis conditions the Mg tends to generate donors that compensate the p-type conductivity [11]. The difficulties for obtaining p-type InxGa1−xN with relatively high In-content have been attributed to the high background electron concentrations caused by the low InxGa1−xN growth temperature which seems to promote the incorporation of donor like defects and impurities [12]. Finally, the third technological challenge is the difficulty to fabricate Ohmic contacts with a low specific contact resistance due to the large work functions ~ 6 eV for p-GaN [13].

For overcoming these technological issues, several approaches have been attempted such as the fabrication of (pin) double hetero-structures p-GaN/InxGa1−xN/n-GaN and solar cells based on InxGa1−xN/GaN multi-quantum wells (MQWs), but poor efficiencies of less than 3.5% have been reported [14,15,16]. In addition, another important aspect worth mentioning is that the use of double hetero-structures such as GaN/InxGa1−xN/GaN will cause the appearance of a potential barrier at each interface, increasing the solar cell series resistance. The main contribution of this work is to establish a practical upper limit efficiency that can be obtained with solar cells based on InxGa1−xN, considering the opto-electronic properties of these materials and the technological difficulties to obtain good material properties and device structures for achieving high efficiencies. These technological difficulties have not been overcome for 20 years and the arguments presented here suggest that they may not be overcome. Therefore, if the maximum expected efficiency for homojunction InxGa1−xN based solar cells does not exceed the efficiencies of more conventional solar cells such as Si or GaAs, then a new approach is needed. For example, using InxGa1−xN nano-particles for making quantum dot solar cells or implementing a graded band-gap absorber layer (varying × gradually) as has been suggested for CdZnTe [17].

2 Solar cell structure

In this work, a homo-junction solar cell structure was chosen, according to the technological difficulties and the experimental advances already reported in the literature, as the most appropriate to be experimentally fabricated. This structure has been experimentally done by Valdueza-Felip et al. [14], with a high In content at the absorber layer (40%) and large hole concentration in the p-type layer. The structure starts with a substrate such as Sapphire, SiC or Si(111) in order to provide an epitaxial seed for III-Nitrides hexagonal phase. Sapphire is chosen because GaN grows on it with a low dislocation density at a low cost in comparison with SiC substrates. Subsequently, an AlN buffer layer is deposited followed by a second GaN buffer layer to reduce the dislocation density to the minimum possible. Then, a n-type InxGa1−xN:Si layer with a very high electron concentration (1019 cm−3) which acts as a back surface field (BSF) should be added. After that, a non-intentionally doped (nid) InxGa1−xN absorber layer should be synthesized. We refer to this (nid) layer because the residual electron concentration is high, and its compensation is a current technological challenge. Finally, the p-type InxGa1−xN emitter layer should be synthesized. According to [14], an InxGa1−xN p-type layer with a hole concentration of 3 × 1018 cm−3 can be achieved. Therefore, the maximum hole concentration that we can consider in this model is 3 × 1018 cm−3 due to the difficulty to obtain higher hole concentrations. In addition, it is important to mention that the p-type layer should be deposited as the top layer because Mg doping degrades the crystalline quality of the material, which in turn may increase its roughness [12, 14, 15]. In other words, if the structure were synthesized over the p-type layer, poor crystalline quality and roughness are expected to spread throughout the structure reducing the crystalline, optical and electrical properties. Moreover, it is worth mentioning that the p-InxGa1−xN layer doping concentration is a key parameter to achieve an adequate depletion layer width. Assuming a free hole concentration of 1018 cm−3 in the p-InxGa1−xN layer and according with Ref. [2], it is reasonable to consider that the residual electron concentration in the InxGa1−xN absorber layer should also be ~ 1018 cm−3. Then, if the hole concentration is less than 1018 cm−3, the depletion region will extend mainly over the emitter layer instead of the base layer, reducing the photo-collection within the (base) depletion region and in turn reducing the efficiency. In addition, if the hole concentration in the p-layer is less than 1018 cm−3, a Schottky barrier at the top contact might be achieved instead of the desired Ohmic contact. So, the hole concentration must be at least 1018 cm−3 to have a reasonable photocurrent generation (in the depletion region) and to avoid the possible formation of a Schottky barrier at the metal contact. The proposed structure is illustrated in Fig. 1. Therefore, considering the present technological issues, the homojunction structure is chosen because it should present low dislocation density, low surface roughness, no barrier potentials at the interfaces as in the case of double heterostructure solar cell (GaN/InxGa1−xN/GaN), an adequate depletion region width, and simple Ohmic contact formation at top of the p-InxGa1−xN emitter layer.

Fig. 1
figure 1

Proposed InxGa1−xN homo-junction solar cell structure for modeling

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3 Modeling

An analytical model was employed to calculate the efficiency that can be achieved with homo-junction InxGa1−xN solar cells. The model includes the In content effect on the photocurrent and the effect of high residual electron background assuming the p-type layers with a free hole concentration of 1018 cm−3. Therefore, the doping concentrations in each layer will be the same with an electron and a hole concentration of 1018 cm−3, respectively. The model includes the voltage dependence of the depletion region (W) and the effect of this on both the illuminated and dark current components. In very thin solar cells, such as the ones to be simulated here, these effects are very important and should be considered using a complete analytical model previously developed by our group [18], as explained below.

First, the bandgap Eg as a function of In content in InxGa1−xN is calculated using a bowing parameter previously established in [13]:

$$E_{g} = {\text{E}}_{\text{g}}^{\text{InN}} + (1 - x){\text{E}}_{\text{g}}^{\text{GaN}} - bx(1 - x)$$
(1)

where the energy band gaps of InN (\({\text{E}}_{\text{g}}^{\text{InN}}\)) and GaN (\({\text{E}}_{\text{g}}^{\text{GaN}}\)) are 0.7 eV and 3.42 eV, respectively. x is the Indium content and b is the bowing parameter (b = 1.43) [1, 5, 19].

The electron affinity (χ) is expressed in Eq. (2) [19, 20].

$$\chi_{{In_{x} Ga_{1 - x} N}} = \chi_{GaN} + 0.7\left( {E_{g}^{\text{GaN}} - E_{g} } \right)$$
(2)

where \(\chi_{GaN} = 4.1\;{\text{eV}}\).

The relative permittivity of the ternary alloy \(\varepsilon_{{In_{x} Ga_{1 - x} N}}\) is expressed in Eq. (3) [20].

$$\varepsilon_{{In_{x} Ga_{1 - x} N}} = x\varepsilon_{InN} + \left( {1 - x} \right)\varepsilon_{GaN}$$
(3)

where (\(\varepsilon_{InN}\)) and (\(\varepsilon_{GaN}\)) are 15.3 and 8.9, respectively.

The effective densities of states in the conduction and valence bands are expressed in Eqs. (4) and (5), respectively [19, 21].

$$N_{C} = (0.9x + 2.3(1 - x))10^{18}$$
(4)
$$N_{V} = (5.3x + 1.8(1 - x))10^{19}$$
(5)

The InxGa1−xN intrinsic carrier concentration is expressed by Eq. (6).

$$n_{i} = \sqrt {N_{V} N_{C} } \exp \left( { - \frac{{E_{g} }}{2kT}} \right)$$
(6)

where k is the Boltzmann constant and T is the absolute temperature. Nc and Nv are the effective density of states in the conduction band and the valence band, respectively.

The electron and hole mobility are expressed as a function of doping concentration in Eq. (7) [19].

$$\mu_{i} (N) = \mu_{\hbox{min} ,i} + \frac{{\mu_{\hbox{max} ,i} + \mu_{\hbox{min} ,i} }}{{1 + \left( {\frac{N}{{N_{g,i} }}} \right)^{{\gamma_{i} }} }}$$
(7)

where i = n for electrons, i = p for holes, and N is the doping concentration, while the parameters μmax,i, μmin,i, Ng,i and γi depends on the type of semiconductor material and their values, as listed in Table 1 [21]. Furthermore, the well-known diffusion coefficients and diffusion lengths are expressed as follows.

$$D_{n} = \frac{kT}{q}\mu_{e} ,\quad D_{p} = \frac{kT}{q}\mu_{h} ,\quad L_{n} = \sqrt {D_{n} \tau_{n} } ,\quad L_{p} = \sqrt {D_{p} \tau_{p} }$$
Table 1 Parameters used for the cell simulation
Full size table

An important parameter which determines the electric and photoelectric properties in a solar cell is the thickness of the space charge region (depletion region) in both the p-type and n-type sides of the junction. The depletion region depends on the doping concentration where high residual electron concentration causes a difficult p-type doping (as mentioned above). The depletion regions for a homo-junction are expressed by Eqs. 8–10 [18, 22].

$$x_{n} = \sqrt {\frac{{2\upvarepsilon_{0}\upvarepsilon_{\text{InGaN}} N_{a} \left( {v_{bi} - V} \right)}}{{qNd\left[ {N_{a} + N_{d} } \right]}}}$$
(8)
$$x_{p} = \sqrt {\frac{{2\upvarepsilon_{0}\upvarepsilon_{\text{InGaN}} N_{d} \left( {v_{bi} - V} \right)}}{{qNa\left[ {N_{a} + N_{d} } \right]}}}$$
(9)
$$W = x_{n} + x_{p}$$
(10)

where \(\upvarepsilon_{0}\) is the permittivity in vacuum, q is the magnitude of the electron charge, vbi is the built-in potential, V is the voltage across the diode, Na and Nd are the acceptor and donor concentrations, respectively. The well-known built-in potential is reported elsewhere [18, 22]. In addition, it is worth mentioning that due to strong absorption coefficient a thin solar cell can be made, and thus, the effect of the voltage in the depletion region thickness should be included. This is because it impacts both the photo-collection within the depletion region and the reverse saturation current due recombination as shown below [18].

The following equations, from 11 to 15, were taken from references [18, 23]. The photocurrent density JL is expressed by the following equation.

$$J_{L} = q\int {Q(\lambda )} F(\lambda )(1 - R(\lambda ))d\lambda$$
(11)

where F(λ) is the density of the incident solar photon flux per unit bandwidth, R(λ) is the device reflectance and Q(λ) is the internal quantum efficiency:

$$Q(\lambda ) = Q_{n} (\lambda ) + Q_{p} (\lambda ) + Q_{dr} (\lambda )$$
(12)

Qn(λ) is the contribution of electrons from the p-side, Qp(λ) is the contribution due to holes from the n-side, and Qdr(λ) is the contribution from the depletion region.

$$Q_{n} = \left( {\frac{{\alpha L_{n} }}{{\alpha^{2} L_{n}^{2} - 1}}} \right)\left( {\frac{{\left( {\frac{{S_{n} \tau_{n} }}{{L_{n} }} + \alpha L_{n} } \right) - e^{{ - \alpha x_{j} }} \left( {\left( {\frac{{S_{n} \tau_{n} }}{{L_{n} }}} \right)\cosh \left( {\frac{{x_{j} }}{{L_{n} }}} \right) + \sinh \left( {\frac{{x_{j} }}{{L_{n} }}} \right)} \right)}}{{\left( {\frac{{S_{n} \tau_{n} }}{{L_{n} }}} \right)\sinh \left( {\frac{{x_{j} }}{{L_{n} }}} \right) + \cosh \left( {\frac{{x_{j} }}{{L_{n} }}} \right)}} - \alpha L_{n} e^{{ - \alpha x_{j} }} } \right)$$
(13)
$$Q_{p} = \left( {\frac{{\alpha L_{p} }}{{\alpha^{2} L_{p}^{2} - 1}}} \right)\left( {e^{{ - \alpha (x_{j} + W)}} } \right)\left( {\alpha L_{p} - \frac{{\left( {\frac{{S_{p} \tau_{p} }}{{L_{p} }}} \right)\left( {\cosh \left( {\frac{H}{{L_{p} }}} \right) - e^{ - \alpha H} } \right) + \sinh \left( {\frac{H}{{L_{p} }}} \right) + \alpha L_{p} e^{ - \alpha H} }}{{\left( {\frac{{S_{p} \tau_{p} }}{{L_{p} }}} \right)\sinh \left( {\frac{H}{{L_{p} }}} \right) + \cosh \left( {\frac{H}{{L_{p} }}} \right)}}} \right)$$
(14)
$$Q_{dr} = e^{{ - \alpha x_{j} }} (1 - e^{ - \alpha W} )$$
(15)

where \(x_{j} = d_{p} - x_{p} ,\;\;H = d_{n} - x_{n}\) and the emitter and absorber layer thickness are \(d_{p}\) and \(d_{n}\), respectively.

The total current density is the difference of the photocurrent density and the dark current density (see Eq. 16). The dark current density is due to the diffusion current in the quasi-neutral regions, in addition to the generation-recombination current at the space-charge region. The well-known saturation current due to diffusion (J0) is given in Ref. [18], and mainly depends on Eg and the diffusion lengths for the n and p type materials. The reverse saturation current due to recombination in the space-charge region is expressed by Eq. (17) below [18, 22, 23], where it is worth noting that if the carrier lifetime decreases, then the saturation current will increase. Previous experimental results [7, 12, 24] (PL, TEM and XRD) have shown that as the In-content increases, the optical and structural properties degrade. Therefore, as the In-content increases, the carrier lifetime should drop. However, in the absence of an expression relating the In content and the carrier lifetime, the carrier lifetime has been considered as constant. Of course, in this case, the reverse saturation current is underestimated for high In concentrations. The parameters used for the simulation are summarized in Table 1.

$$J_{T} \left( V \right) = J_{L} - J_{0} \left( {e^{{\frac{qV}{kT}}} - 1} \right) - J_{{\text{Re} c}} \left( {e^{{\frac{qV}{2kT}}} - 1} \right)$$
(16)
$$J_{Rec} = \frac{{qn_{i} W\left( V \right)}}{{\tau_{p} }}$$
(17)

JL is the illumination current density defined above, J0 is the dark diffusion current density [18], JRec is the dark space-charge recombination current density, ni is the intrinsic carrier concentration and W, which depends on the applied voltage, has been defined before (see Eq. 10). As explained before, JL and both dark current densities depend upon the voltage, and this is an important effect to be considered for very thin cells as is in this case. Usually this effect is not considered for conventional thick solar cells.

4 Results and discussions

In Fig. 2, the calculated output characteristics as a function of the In content for the InxGa1−xN solar cells are shown. As expected, if the Indium content increases, the short circuit current (Jsc) increases due to the reduction of Eg. Nevertheless, the open circuit voltage (Voc) and Fill factor (FF) decrease due to the increase in the reverse saturation current. In accordance to the Shockley–Queisser theory, the optimum Eg is at ~ 1.45 eV which implies that ~ 60% of In should be introduced into InxGa1−xN. However, solar cells with In concentrations greater than 40% have not been reported yet. InxGa1−xN solar cells made with 40% of In have been claimed, but its experimental Eg resulted to be ~ 2.3 eV which is a high band gap for solar cell applications [14]. Figure 2 also shows the efficiency as a function of the In concentration. In concentration of 60% corresponds to Eg of 1.43 eV. Then, according to our calculations, the maximum In0.6Ga0.4N solar cell efficiency is 18.9% which is less than 20.28% predicted for 65% In content in Ref. [20]. The explanation for this difference is due to two reasons, the first one and the most important is because Ref. [20] did not consider the dark current due to recombination at the space-charge region, which becomes a very important component for thin film solar cells. As a result, the estimated efficiency in [20] is not realistic. The second reason for the difference is that the assumed bandgap Eg in [20] is 1.31 eV, but the optimum bandgap should be 1.4–1.5 eV, as confirmed by the above results.

Fig. 2
figure 2

a Fill factor (FF), b open circuit voltage (Voc), c short-circuit current density (Jsc) and d efficiency, as a function of In content in the InxGa1−xN homojunction solar cell

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So, according to the state of the art, InxGa1−xN solar cells with a In content higher than 40% have not been reported yet due to the technological challenges explained in this work. The main challenge is the control of the In incorporation avoiding the phase separation and the reduction of the carrier lifetime due to defects. According to our calculations, the maximum efficiency that can be obtained from an InxGa1−xN homo-junction solar cell (when the reduction of carrier lifetime due the high In content is avoided) is ~ 18.9%. However, the available reports in the literature have shown diode ideality factors and shunt resistances in the range from 2.3 to 27.5 and 8 × 104 to 106 Ohm cm2, respectively for InxGa1−xN layers with an In content no more than 22% [15, 16, 25, 26]. These high ideality factors give evidence of strong recombination at defect traps in InxGa1−xN solar cells. Thus, it is reasonable to expect that the lifetime will be reduced when the Indium content increases. This is because the In incorporation causes stress into InxGa1−xN until the phase separation occurs. This stress might be released in the form of crystal defects resulting in a reduction of the minority carrier lifetime. Therefore, our prediction of the maximum efficiency (18.9%) is an optimistic prediction because we cannot model the effect of reduction of the minority carrier lifetime as the Indium content increases. Thus, experimental measurements for variation of lifetime as a function of the In content are necessary. In order to study the effect of In incorporation in InxGa1−xN on the minority carrier transport properties, Fig. 3 shows the output properties of InxGa1−xN solar cells with an In content of 60% as a function of the minority carrier lifetime. The most important result observed in Fig. 3 is that the efficiency can drop from 18.9 to 3.9% when the minority carrier lifetime is reduced from nanoseconds to picoseconds. Therefore, it is reasonable to claim that we are far from the dreamed high efficiency InxGa1−xN solar cells and the technological issues mentioned above should be overcome, otherwise InxGa1−xN solar cells will always be less efficient than the low-cost Silicon or thin-film solar cells technologies.

Fig. 3
figure 3

In0.6Ga0.4N homojunction solar cell output characteristics as a function of the carrier lifetime

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5 Conclusion

In this work, we have modeled InxGa1−xN based homo-junction solar cells considering realistic electrical transport parameters. In order to get better efficiency estimations, experiments for relating the carrier lifetime and the In content are required. However, the calculations revealed that the maximum efficiency will not be more than 19%. Therefore, due to its high cost and technological challenges, we consider that InxGa1−xN might not become competitive with the most familiar and low-cost silicon cell technologies. In summary, we are far from the dreamed high efficiency InxGa1−xN solar cells. Therefore, a new approach is needed, for example, using InxGa1−xN nano-particles for making quantum dot solar cells or implementing a graded band-gap absorber layer (varying × gradually), as has been suggested for CdZnTe [17].