Bandgap energy model for GaInNAsSb/GaAs alloys with high N content and strain influence

Bandgap energy of dilute nitride GaInNAsSb/GaAs alloys with N compositions as high as 8% are estimated using a method based on band anti-crossing model used for GaNAs/GaNSb/InNAs/InNSb ternary compounds. The parametrization of the model is defined by fitting with experimental composition and bandgap energy values employing a differential evolution algorithm. The effects of lattice strain on the bandgap energy are taken into account by the model resulting in an accurate prediction of the bandgap energy with an average deviation of only 12 meV compared to the experimental data. The model provides a useful tool for accurate determination of bandgap energies of dilute nitrides, including narrow bandgap, i.e. ~0.7 eV GaInNAsSb alloys, which are becoming increasingly relevant in the development of high-efficiency lattice-matched multijunction solar cells.


A R T I C L E I N F O Communicated by Hajime Asahi
Keywords: A1. Characterization A3. Molecular beam epitaxy B1. Antimonides B1. Nitrides B2. Semiconducting III-V materials A B S T R A C T Bandgap energy of dilute nitride GaInNAsSb/GaAs alloys with N compositions as high as 8% are estimated using a method based on band anti-crossing model used for GaNAs/GaNSb/InNAs/InNSb ternary compounds. The parametrization of the model is defined by fitting with experimental composition and bandgap energy values employing a differential evolution algorithm. The effects of lattice strain on the bandgap energy are taken into account by the model resulting in an accurate prediction of the bandgap energy with an average deviation of only 12 meV compared to the experimental data. The model provides a useful tool for accurate determination of bandgap energies of dilute nitrides, including narrow bandgap, i.e. ~0.7 eV GaInNAsSb alloys, which are becoming increasingly relevant in the development of high-efficiency lattice-matched multijunction solar cells.

Main
Dilute nitrides are a class of compound semiconductors that formed by alloying small concentrations of N to the group V lattice sites in traditional III-V semiconductors [1,2]. Compared to other group V elements, N has a small atomic radius and high electronegativity, which have profound effects on the crystallographic, optical, and electrical properties of the dilute nitrides. Most notably, N incorporation drastically reduces the bandgap energy (E g ) of the host material and simultaneously reduces the lattice constant of the resulting crystal [2,3]. These features have been initially exploited in the development of GaInNAs/GaAs-based laser diodes at 1.3 µm range primary targeting uncooled operation [2,[4][5][6][7]. More recently, the dilute nitrides have had a remarkable impact on the development of lattice-matched multijunction solar cells [8][9][10][11] owing to the fact that GaInNAsSb compounds with E g from 1.4 eV down to 0.7 eV can be grown lattice-matched on GaAs and Ge substrates [2,8]. While the low N-content GaInNAsSb heterostructures with bandgaps of ~1 eV are well studied and the technology is rather mature, GaInNAsSb compounds with high N compositions exhibiting E g around 0.7-0.8 eV require further technological and material science advances to be fully exploited in next-generation multijunction solar cells [12]. These developments are made possible by the use of molecular beam epitaxy (MBE), which enable a high degree of process control for N incorporation and significant reduction of N-related defects that is instrumental for device development [13,14].
Introduction of N to the crystal lattice is associated with formation of localized energy level near the conduction band edge of the host material. In turn, the interaction of these localized N-related resonant states with the conduction band of the host material leads to splitting of the conduction band into E + and Esub-bands, which effectively lowers the E g of the material (i.e. now defined by the transition between Eand valence bands) [15]. The reduction E g in dilute nitride compounds is poorly described by the typical quadratic approximation [16]. Instead, the modification of conduction band structure due to N incorporation can be modelled with a so-called band anti-crossing (BAC) model [17]. Although, the effective use of BAC for GaInNAs has been widely established [18][19][20][21], there is a need for more advanced models for materials with higher N concentrations as well as for alloys with higher complexity incorporating also Sb.
Similar to N, incorporation of Sb also lowers the E g yet this reduction takes place via the valence band structure instead of the conduction band [22]. Indeed, when N and Sb are incorporated simultaneously into GaAs or GaInAs, they affect both the valence and conduction bands [22]. For estimating the E g of GaNAsSb alloys a double band anti-crossing (DBAC) model has been suggested [22]. Moreover, a relatively simple BAC-based model for GaInNAs has been tailored to estimate the E g of GaInNAsSb alloys with N concentrations up to ~6% [23]. The adaptation to GaInNAsSb was based on using interpolated values of E g for GaInAsSb as the host material E g in the BAC-model. This modified BACmodel yielded an excellent fit with the extensive experimental data but it ignored the effects of lattice strain; the model assumed that the deviation arising from residual strain was small within the data set. The modified BAC-model also showed slight drift between the modelled and experimental bandgap energies with higher In and N concentrations. In this paper, we propose an alternative interpolation scheme based on the BAC-model of N containing ternaries (i.e. GaNAs, GaNSb, InNAs and InNSb) for accurately predicting the E g of GaInNAsSb/GaAs alloys, expanding the dataset to include low-bandgap materials, i.e. GaInNAsSb with up to 8 % N compositions. Fig. 1 shows the composition range covered by the proposed model, which is compared with compositions ranges of prior BAC-models used for modelling of dilute nitride alloys.
The modelling of E g for Ga 1-x In x N y As 1-y-z Sb z alloys starts from defining the material composition (x, y, z). As data points for model fitting we used the comprehensive composition and bandgap data presented by Aho et al. [23] together with additional compositions and E g values for GaInNAsSb with higher N concentrations (i.e. bandgap energies down to ~0.7 eV). The compositions for these low-bandgap alloys (samples A-E; summarized in Table 1) were determined using energydispersive x-ray spectroscopy (EDS) measured from the cleaved facets of MBE-grown single-junction GaInNAsSb solar cells. The bandgap energies for these materials were extracted from external quantum efficiency (EQE) measurements made for the solar cells. Additional details on the EDS and EQE measurements for the high N-content solar cells can be found elsewhere [12].
Using the specified compositions, the lattice constant (a) for GaIn-NAsSb is approximated from lattice constants of binaries by using linear Vegard's law: where the subscripts indicate the lattice constants of the binaries. Similar linear approximation is used to estimate the valence band hydrostatic deformation potential (a V ), conduction band hydrostatic deformation potential (a C ), elastic constants c 11 and c 12 , and the shear deformation potential (b ax ) for the GaInNAsSb material from the corresponding values of the binaries. The material parameters of the binaries used in this model are listed in Table 2.
The interpolated material parameters for GaInNAsSb are then used for estimating the influence of lattice strain on the band structure. The influence of hydrostatic strain component on the conduction band (CB) and valence band (VB) is given by [24]: where ε xx is the strain of the GaInNAsSb layer, which is defined by the lattice constants of the substrate (a subs ) and GaInNAsSb as ε xx = (a GaInNAsSb − a subs )/a subs [24]. The influence of the shear component of strain on the band structure is given by [24]: The material parameters in Eqs.
(2)-(4) are now the interpolated material parameters for GaInNAsSb. The hydrostatic strain components from Eqs. (2)-(3) and the shear strain component from Eq. (4) are used for estimating the strain-induced shifts for CB, and light-hole (LH) and heavy-hole (HH) bands [24]: Using these, the total strain-induced shift of the E g is approximated with: Next, the BAC-model is used for estimating the E g of the N-containing ternaries, i.e. GaNAs, GaNSb, InNAs and InNSb corresponding to Egiven by [17,25]: Fig. 1. Comparison of experimental composition ranges used for BACmodelling of GaNAs, GaInNAs, GaInNAsSb and GaNAsSb reported in literature [15,17,22,23] along with the composition range reported in this work.

Table 1
Composition and bandgap data for the high N-content Ga 1-x In x N y As 1-y-z Sb z samples used for the modelling. Values for alloys with lower N compositions are reported in Ref. [23].  Table 2 Material parameters for binaries [16,24].
where E M is the bandgap energy of an unperturbed N-free host material, E N is the energy of the localized N-level near the conduction band edge of the host material, and V NM is the interaction strength between the host material conduction band and the N-level. In this model we have modified the BAC-model such that the strain induced bandgap shift from Eq. (8) is included in the E g of the host material, so E M in Eq. (9) is replaced by (E M + ΔE g ), in a similar fashion as the effect of hydrostatic pressure has been taken into account for GaInNAs by Shan et al. [17] For simplicity, the small effect of strain for other BAC-model parameters has been neglected. The employed bandgap energies of binaries and BAC parameters are listed in Table 3 and Table 4, respectively. The bandgaps of ternaries obtained from Eq. (9) are then used for calculating the E g of quaternary GaNAsSb and InNAsSb. For these alloys a change in the N fraction y affects both the As and Sb fractions, and hence the N fraction should be divided proportionally between the As and Sb fractions. Therefore, the Sb fraction z is replaced by z'=z + yz/(1 − y) [25,27]. Now the bandgaps of GaNAsSb and InNAsSb are approximated in the form of a quadratic function [25]: where C is the quaternary related bowing parameter describing the deviation from a linear interpolation. Finally using the bandgaps of the quaternaries, the E g of the GaInNAsSb alloy is estimated using a similar quadratic formula: where the bowing parameter C is now dependent on the In, N and Sb compositions. The composition dependent empirical bowing parameter for GaInNAsSb is given in a polynomial form: where c 1 -c 7 are fitted parameters. The model was then fitted with the experimental composition and bandgap data by employing differential evolution (DE) algorithm [28]. The parameters used for fitting were the bowing parameters for GaNAsSb and InNAsSb in Eq. (10), and parameters c 1 -c 7 used in Eq. (12) to describe the composition dependency of the bowing parameter. The model parameters obtained from the DE fitting are given in Table 5.
The bandgap energies predicted by the model as function of the measured E g data are plotted in Fig. 2. The datapoints align closely to the ideal line with an average deviation of only 12 meV between the predicted and measured E g values. The maximum deviation within the dataset was 36 meV. In this composition region the calculated E g values do not exhibit any significant systematic drift from the measured E g values with respect to In, N or Sb compositions. This is illustrated by Fig. 3, which shows the ratio between calculated E g and measured E g as function of In, N and Sb compositions. In addition, the data in Fig. 3 shows that all the datapoints predicted by the model are within ±5% from the measured values of E g , further demonstrating the accuracy of the proposed model. The deviation between the predicted and measured E g was also analyzed statistically. Normalized frequency distribution as function of the difference between calculated and measured bandgap values is shown in Fig. 4. The frequency distribution of the deviation within the dataset shows that over 50% of the datapoints are within ±10 meV of the measured bandgap energies and 96% of the datapoints are within ±30 meV of the corresponding experimental E g values. The data in Fig. 4 also indicates that the predicted values tend to be slightly more on the negative side, meaning that the model slightly underestimates the E g of the GaInNAsSb alloys. Yet this deviation is relatively small, which can be seen from the normal distribution fitted for the experimental data points describing the deviation of an expanded dataset. The mean of the fitted distribution is at − 3.5 meV and the standard deviation is only 14.3 meV. Outside the composition range of the samples analyzed in this work (0 ≤ x ≤ 0.17, 0 ≤ y ≤ 0.08, 0 ≤ z ≤ 0.08) the accuracy of the model might deteriorate, and the model parameters might require further refinement.
In conclusion, a novel interpolation method based on the BACmodelling for predicting E g of bulk dilute nitride GaInNAsSb compounds was introduced. The model utilizing the bowing parameters fitted with differential evolution algorithm exhibits an excellent correlation with the experimental E g values in a relatively broad composition range (0 ≤ x ≤ 0.17, 0 ≤ y ≤ 0.08, 0 ≤ z ≤ 0.08), which includes the   Table 5 Fitted values for C GaNAsSb , C InNAsSb and parameters c 1 -c 7 used to calculate the composition dependent bowing parameter of GaInNAsSb.  [23], while the blue crosses represent N-free binary compounds (GaAs, GaSb, InAs and InSb).
high-N containing alloys (up to 8%) especially relevant for development of next-generation multijunction solar cells. Furthermore, the model includes the effect of strain. The average deviation between the estimated and measured E g values was only 12 meV. We consider the model to be a simple and effective tool in epitaxy for accurately predicting bandgaps of GaInNAsSb alloys prior to crystal growth, thus helping in the process for material composition calibration and structure design.

Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.