Synthesis of device-quality GeSn materials with higher Sn compositions is hindered by various factors, such as Sn segregation, clustering, and short-range ordering effects. In the present work, the impact of the clustering of Sn atoms in a GeSn semiconductor alloy was studied by density functional theory using SG15 pseudopotentials in a Synopsys QuantumATK tool, where the thermodynamic stability, effective band structure, indirect and direct bandgaps, and density of states (DOS) were computed to highlight the difference between a cluster-free random GeSn alloy and a GeSn alloy with Sn–Sn clusters. A 54-atom bulk Ge1–xSnx (x = 3.71%–27.77%) supercell was constructed with cluster-free and a first nearest neighbor Sn–Sn clustered GeSn alloy at each composition for this work. Computation using the generalized gradient approximation exchange-correlation functional showed that the thermodynamic stability of GeSn was reduced due to the clustering of Sn, which increased the formation energy of the GeSn alloys by increasing the Hartree potential energy and exchange-correlation energy. Moreover, with the effective band structure of the GeSn material at a Sn composition of ∼22%, both direct (Eg,Γ) and indirect (Eg,L) bandgaps decreased by a large margin of 40.76 and 120.17 meV, respectively, due to Sn–Sn clustering. On the other hand, Eg,Γ and Eg,L decrease is limited to 0.5 and 12.8 meV, respectively, for Sn composition of ∼5.6%. Similar impacts were observed on DOS, in an independent computation without deducing from the electronic band structure, where the width of the forbidden band reduces due to the clustering of Sn atoms in GeSn. Moreover, using the energy bandgaps of GeSn computed with the assumption of it being a random alloy having well-dispersed Sn atoms needs revision by incorporating clustering to align with the experimentally determined bandgap. This necessitates incorporating the effect of Sn atoms clustered together at varying distributions based on experimental characterization techniques such as atom probe tomography or extended x-ray absorption fine structure to substantiate the energy bandgap of the GeSn alloy at a particular composition with precision. Hence, considering the effect of Sn clusters during material characterization, beginning with the accurate energy bandgap characterization of GeSn would help in mitigating the effect of process variations on the performance characteristics of GeSn-based group IV electronic and photonic devices such as varying leakage currents in transistors and photodiodes as well as the deviation from the targeted wavelength of operation in lasers and photodetectors.

To realize highly efficient group IV semiconductor-based optoelectronic devices, significant advancements in the quality of SiGe, SiGeSn, GeSn, and strained Ge (Refs. 1–15) materials are essential. Alloying α-Sn with Ge can make GeSn a direct bandgap semiconductor at 6%–8% Sn composition,1–11 where different Sn alloy compositions were reported for the indirect to direct bandgap transition based on the first-principles calculations.2,16,17 In addition, recent studies predicted a near-zero bandgap GeSn material at Sn composition in the range of 25%–28%.2,16,17 Further, there are also reports of the GeSn material continuing to behave as a semiconductor even at 32% Sn (Ref. 18) based on the ordering of atoms. These developments have added challenges in characterizing GeSn materials at high Sn compositions due to the segregation of Sn (i.e., Sn atoms clustering), creation of point defects, lattice mismatch induced defects and dislocations, etc.19–22 However, there are very few studies reporting the effect of Sn clustering in GeSn materials.23–25 The GeSn alloy has been considered a truly random alloy like SiGe (Ref. 25) while computing the energy bandgaps from the first-principles calculations, and the corresponding compositions are calibrated with experimentally determined energy bandgaps. There is a need to initiate a revision of electronic band structure computation by incorporating the clustering of Sn atoms. For instance, the localized Sn composition in a cluster within the GeSn alloy is reported to be even greater than the average Sn composition using atom probe tomography (APT).26,27 Also reported is increased clustering at compositions greater than 12% using extended x-ray absorption fine structure.28, In the present work, the clustering of Sn atoms in the GeSn alloy is studied by density functional theory (DFT) using the Synopsys QuantumATK tool (i) to compute the effective band structure (both indirect and direct energy bandgaps) and understand the thermodynamic stability of the many-electron system with respect to the formation energy and compute density of states (DOS) for clustered GeSn and (ii) within an integrated tool that facilitates studying atomic-scale effects at material, device, and circuit levels on the same platform.29 The result exhibited a lowering in the indirect-direct energy bandgaps and reduced thermodynamical stability with higher formation energy due to the clustering of Sn atoms in the GeSn alloy. Note that only first nearest neighbor (first-NN) Sn–Sn clusters were modeled in the present work, with the maximum cluster size equaling the localized and average Sn composition in a 54-atom GeSn supercell. For clarity, the distribution of Sn atoms in a cluster as mth-NN (m > 1) was not modeled here. Results from APT have reported ∼14%, ∼21%, and even ∼39% localized Sn composition in a ∼7% average Sn compositional GeSn alloy with peak fraction of Sn atoms as the first-NN.26 Moreover, the Sn–Sn clusters have a Poisson distribution27 (bell-shaped), and the same has been used in the present work. Hence, this work would help to initiate necessary revisions in the electronic band structure computation of the GeSn alloy by incorporating Sn clusters and open up further investigations with intricate cluster characterization techniques to minimize clustering at higher Sn compositions in GeSn semiconductor material systems.

Density functional theory based on the Kohn–Sham (KS)30–34 mathematical formalism was implemented in the present work to study the clustering of Sn atoms with first-NN Sn–Sn clusters in the Ge1–xSnx material system. The generalized gradient approximation (GGA) exchange-correlation functional of Perdew–Burke–Ernzerhof (PBE) (Ref. 35) was applied using the Synopsys QuantumATK (Ref. 36) tool to compute the electronic band structure and the density of states. The KS Hamiltonian is expressed as35,
H ^ K S = 2 2 m 2 + V eff ,
(1)
where Veff is the effective potential of the electronic system and is expressed as35 
V eff [ n ] = V H [ n ] + V X C [ n ] + V ext [ n ] ,
(2)
where n is the electron density, VH is the Hartree potential representing electrostatic interaction between the electrons, VXC is the exchange-correlation potential representing the quantum mechanical nature of the electrons, and Vext represents the electrostatic potential of the ions as well as any externally applied electrostatic fields. The total energy of the many-electron system (ETotal) includes contributions from all these potentials, referred to above, and kinetic energy.
Furthermore, the thermodynamic stability of the GeSn alloy was studied using the formation energy of the material, and lower formation energy indicates better thermodynamic stability.37–39 In the electronic structure calculations, the formation energy has been directly evaluated from the total energy38–41 using the following relation:
E Form = E Total x E Total ( x ) ,
(3)
where EForm is the formation energy of the alloy, ETotal is the total energy of the alloy, and the last term denotes the energy of the source elements. Here, EForm of the GeSn alloy is given as
E Form GeSn = E Total GeSn { E Total ( Ge ) + E Total ( Sn ) } ,
(4)
where the terms ETotal(Ge) and ETotal(Sn) represent the energy associated with the flux coming from the individual Ge and Sn effusion cells, respectively. Moreover, the number of Ge and Sn atoms was kept the same in both the cluster-free and first-NN Sn–Sn clustered GeSn alloys. Hence, the change in the total energy ΔETotal is equivalent to the change in the formation energy ΔEForm. These changes result from the variation in the types of bonds (Ge–Ge, Ge–Sn, and Sn–Sn) between a truly random and a clustered GeSn alloy. The total energy, ETotal, of a many-electron system is expressed in terms of the electron density (n) function as35,
E Total [ n ] = T [ n ] + E H [ n ] + E X C [ n ] + E ext [ n ] ,
(5)
where T[n] is the kinetic energy, EH[n] is the Hartree potential energy, EXC[n] is the exchange-correlation energy, and Eext[n] is the interaction energy due to external potential, Vext. Here, the solution for VH from the Poisson equation,36,
2 V H [ n ] ( r ) = e 2 4 π ε 0 n ( r ) ,
(6)
shows that EH is a functional of the local electron density only and not its gradient as in the GGA functional that is a semi-local approximation for EXC, where it depends on both the local value (n) and the local gradient of the electron density (∇n) as shown below:36,
E X C [ n ] = n ( r ) ε X C ( n ( r ) , n ( r ) ) d r .
(7)
During the DFT computation, the ground state of the electronic system is computed by iteratively minimizing the ETotal of the system at which the system is at its energetically and thermodynamically stable condition.36 

In the present work, a norm-conserving scalar-relativistic (SG15) pseudopotential was used along with the linear combination of atomic orbital (LCAO) basis sets as eigenfunctions of the KS Hamiltonian. These basis sets are internally mapped as fully relativistic (i.e., including the spin–orbit coupling) by solving the Dirac equation of each atom.42,43 The SG15-high accuracy basis sets were used with pseudopotential projector-shift (PPS) parameters for Ge. These PPS parameters enable smoothening of the oscillations in valence electron wavefunctions in the pseudopotential functional, which improves the accuracy of calculations. However, such additional projector parameters for add-on accuracy in pseudopotentials are available only for Si and Ge atoms in the Synopsys QuantumATK framework. It is noted that SG15-Ultra basis sets are more accurate at the cost of computation speed. The QuantumATK tool includes the PPS-PBE parameters for Si and Ge, correcting the energy bandgap obtained from the GGA approximations. This accurately estimates the bandgap corresponding to experimental values.35 Furthermore, the calibration of this DFT computation method with the Ge bandgap, shown in Fig. 1, was performed, giving direct bandgap Eg,Γ = 0.802 eV and indirect bandgap Eg,L= 0.673 eV. Later, this calibrated DFT method was utilized to compute the electronic band structures of different compositional Ge1–xSnx alloys with and without clusters of Sn atoms.

FIG. 1.

Electronic band structure of bulk Ge computed using the SG15 PPS-PBE pseudopotential method with the LCAO (high accuracy) basis set approach, extracting Eg,Γ = 0.802 eV and Eg,L = 0.673 eV from the two-atom primitive bulk configuration.

FIG. 1.

Electronic band structure of bulk Ge computed using the SG15 PPS-PBE pseudopotential method with the LCAO (high accuracy) basis set approach, extracting Eg,Γ = 0.802 eV and Eg,L = 0.673 eV from the two-atom primitive bulk configuration.

Close modal

To investigate Sn clustering in a Ge1–xSnx alloy, a 54-atom bulk supercell configuration was created through a 3 × 3 × 3 repetition of a two-atom bulk configuration. A Monkhorst–Pack k-point grid of 3 × 3 × 3 and the density Hartree mesh cutoff energy of 100 eV were used, where the SG15 pseudopotential with high accuracy basis sets extracts bandgaps that converge to within 10–4 eV. As for the LCAO high accuracy basis sets, for each orbital of Ge and Sn atoms, the radial step size was 0.001 Bohr (i.e., 0.000529 Å) and such a high accuracy basis set led to the total energy convergence to the maximum deviation of 1 meV/atom from ultra-accurate basis set (which itself is 0.1% accurate to the original LCAO basis sets). Moreover, the maximum allowed interaction distance between two orbitals was kept at 20 Å. With a maximum of 100 self-consistent field iterations, the band energies converged to a constant value within 10–4 eV between the consecutive steps, where the self-consistent electron density was found. Cluster-free Ge1–xSnx (x = 3.7%–27.77%) alloy supercells were built with only (i) Ge–Ge and (ii) Ge–Sn bonds in the 54-atom supercell bulk configuration. To study the impact of Sn clustering in the GeSn alloy, the supercell was built with (i) Ge–Ge, (ii) Ge–Sn, and (iii) Sn–Sn bonds with Sn atoms clustered only as first-NNs, as shown in Fig. 2 for x = 22.22%. The present work focused on investigating the effects of Sn clustering over the effective potential, DOS, total energy (deducing the formation energy), and electronic band structure of GeSn. Studies related to the short-range order (SRO) effect observed in GeSn25,44 are outside the scope of this work as neither the random alloy distribution nor the SRO in GeSn depicts the effect of Sn segregation or Sn–Sn clustering as reported.25 Moreover, to compute the effective band structure (EBS) of the Ge1–xSnx alloy (x = 3.71%–27.77%) as a primitive cell configuration, the band structure of each 54-atom supercell was unfolded using the spectral weights of the eigen wavefunctions. The fundamental bandgaps, Eg,Γ and Eg,L, were determined from the folded electronic band structure of the supercell and further confirmed from the respective effective band structures of the Ge1–xSnx alloy. Further, the width of the forbidden gap (noticeably the indirect gap until transition and the direct gap at a higher Sn composition) was also presented from a separately computed DOS for all bands on either side of the valence band maxima and conduction band minima with the Monkhorst–Pack 3 × 3 × 3 k-point grid.

FIG. 2.

Crystal structure of a 54-atom supercell bulk configuration of Ge0.7778Sn0.2222 built in the Synopsys QuantumATK Builder tool, formed with clustered Sn atoms. Schematic drawn using Vesta (Ref. 45).

FIG. 2.

Crystal structure of a 54-atom supercell bulk configuration of Ge0.7778Sn0.2222 built in the Synopsys QuantumATK Builder tool, formed with clustered Sn atoms. Schematic drawn using Vesta (Ref. 45).

Close modal

Effective potential, Veff, is the overall potential experienced by the electrons in a many-electron system, and the contributing factors of Hartree potential VH, exchange-correlation potential VXC, and external potential Vext were as noted in Eq. (2). As denoted in Eq. (1), the KS Hamiltonian considers Veff to compute the total energy of the system. Veff was computed for 11 cluster-free Ge1–xSnx (x = 3.71%–27.77%) and eight first-NN clustered Ge1–xSnx (x = 3.71%–22.22%) alloy compositions as a part of the DFT calculations. Shown in Fig. 3 are three representative compositions of 5.56%, 14.81%, and 22.22%, and it was noted that Veff drops in the coordinate positions of Sn atoms clustering together and has decreased periodicity, vividly observed for a high Sn composition of 22.22% where more Sn–Sn bonds were present. This observation was noted in all the directions at coordinates of clustered Sn atoms, though Veff is shown only for the Z-direction here. Such a variation in the periodicity of Veff leads to the decrease in the direct and indirect energy bandgaps of GeSn (discussed in Sec. III C); however, lack of controllability over Sn–Sn clustering makes it an unintended property of the GeSn material system. Moreover, it leads to variation in the localized Sn composition and the average Sn composition of the material, based on the density of clusters.26–28 Such Sn clustering is widely reported during the synthesis of GeSn material systems,19–22,25,44 where clustering of Sn atoms makes the targeted bandgaps deviate from the design parameters. In turn, first-NN clustering of Sn atoms was selectively investigated for each composition of the GeSn alloy in the present work due to the distributive nature of Sn clustering in epitaxial GeSn from first-NN to fourth-NN Sn clusters observed from APT measurements showing first-NN Sn–Sn as a peak cluster.26 This leads to different regions of a thin film GeSn to exhibit different electronic and optical properties.19–22,25,44

FIG. 3.

Effective potential (Veff) experienced by the electrons in (a) cluster-free GeSn and (b) first-NN Sn–Sn clustered GeSn at Sn compositions of 5.56%, 14.81%, and 22.22%, represented along the Z-direction. For a 22.22% Sn composition, the clustering of Sn atoms decreases the effective potential [see from 8 to 12 Å in (a) and (b)].

FIG. 3.

Effective potential (Veff) experienced by the electrons in (a) cluster-free GeSn and (b) first-NN Sn–Sn clustered GeSn at Sn compositions of 5.56%, 14.81%, and 22.22%, represented along the Z-direction. For a 22.22% Sn composition, the clustering of Sn atoms decreases the effective potential [see from 8 to 12 Å in (a) and (b)].

Close modal

The stability of the GeSn material system during the fabrication remains one of the primary concerns due to Sn segregation or clustering in various process steps.19–22 Here, the thermodynamic stability of the GeSn alloy was studied by computationally arriving at the formation energy comparison between the cluster-free and first-NN Sn–Sn clustered GeSn alloy, as mathematically represented in Sec. II A. As shown in Table I, ETotal of the clustered Ge1–xSnx alloy is higher, and so is the EForm, thereby decreasing the thermodynamic stability of the system. It is imperative to note that, in a five-atom GeSn cluster, SnGe4 is identified as the most stable cluster.24 Moreover, the binding energy of the bonds Ge–Ge > Ge–Sn > Sn–Sn supports better thermodynamic stability24 (higher binding energy correlates to better stability) as also shown by the computation results in the present work. These characteristics were more clearly observed at higher Sn compositions, where the difference in the individual contributing energy terms of EH and EXC continues to increase with increased Sn clustering. The contribution from the potential energy of an electron due to the interaction with other electrons, i.e., EH, is more pronounced. The effect of local electron density n(r) and local gradient of electron density, ∇n(r), due to first-NN clustered Sn–Sn atoms increase the EXC energy as denoted by Eq. (7), even if less than EH as noted from Eq. (6). Certainly, such an increased ETotal (increased EForm) affects the ground state of the system that is used to compute the electron density (n), iteratively, and the electronic band structure and the direct-indirect energy bandgaps of the Ge1–xSnx alloy. With the 54-atom supercell configuration, the unfolded effective band structure as well as the density of states of the first-NN Sn–Sn clustered Ge1–xSnx alloy presented in Sec. III C highlights these effects.

TABLE I.

Total energy difference (ΔETotal), equivalent to formation energy difference (ΔEForm), of first-NN Sn–Sn clustered and cluster-free 54-atom Ge1–xSnx supercell bulk configuration for compositions from 3.7% to 22.22%, with individual energy differences of kinetic (ΔT), electrostatic (ΔEH), and exchange-correlation (ΔEXC) energies. All differences are (clustered) – (cluster-free).

Sn comp. (%)ΔT (eV)ΔEH (eV)ΔEXC (eV)ΔETotal = ΔEForm (eV)
3.7 −0.54 0.77 0.037 0.270 
5.56 −1.03 1.50 0.059 0.535 
7.41 −1.48 2.18 0.086 0.789 
11.11 −2.55 3.73 0.120 1.294 
12.96 −3.05 4.46 0.142 1.544 
14.81 −3.57 5.22 0.140 1.785 
18.52 −4.56 6.61 0.194 2.239 
22.22 −5.58 8.03 0.224 2.671 
Sn comp. (%)ΔT (eV)ΔEH (eV)ΔEXC (eV)ΔETotal = ΔEForm (eV)
3.7 −0.54 0.77 0.037 0.270 
5.56 −1.03 1.50 0.059 0.535 
7.41 −1.48 2.18 0.086 0.789 
11.11 −2.55 3.73 0.120 1.294 
12.96 −3.05 4.46 0.142 1.544 
14.81 −3.57 5.22 0.140 1.785 
18.52 −4.56 6.61 0.194 2.239 
22.22 −5.58 8.03 0.224 2.671 

The electronic band structure of supercells contains hundreds of electronic bands that need unfolding to determine the effective band structure of the primitive cell configuration for the Ge1–xSnx alloy at each composition.46 Here, with first-NN Sn–Sn clusters in the 54-atom GeSn supercell, it becomes even more essential to observe the effective band structure, wherein the eigenfunctions of the wavefunctions in the LCAO basis set approach are assigned weights at each k-point of the supercell, where DFT computation is executed. Figure 4 shows the EBS of cluster-free and clustered GeSn for 5.56% Sn. At a low Sn composition, the reduction in the bandgaps (ΔEg,Γ= 0.5 meV and ΔEg,L= 12.8 meV) is less than at the high Sn compositions such as 22.22% (ΔEg,Γ= 40.76 meV and ΔEg,L = 120.17 meV) shown in Fig. 5, which has Eg, Γcluster-free = 0.17 eV. With clustering at higher Sn compositions, n(r) adds more weightage to the drop in the bandgaps than at a lower Sn composition. This characteristic is observed for clustering from 3.7% to 22.22% and even beyond, though the energy bandgaps shown in Fig. 6 were not computed for clustering beyond 22.22% as Eg,Γclustered becomes 0.05 eV there itself. Moreover, at 27.77% Sn, Eg,Γcluster-free = 0.075 eV. Note that by no means is it claimed in the present work that Ge1–xSnx reaches a 0 eV bandgap at 27.77%, as with the ordering effects reported in the literature18,25,44 even at 32% Sn bandgaps, 0.15 eV has been calculated, and at 28% bandgap crossing, 0 eV has also been reported.2,16,17 These deviations due to ordering effects still exist within the Ge1–xSnx alloy system that is not observed in Si1–xGex material systems or the III–V semiconductors that behave as truly random alloys.25 

FIG. 4.

Effective band structure of the GeSn alloy at a 5.56% Sn composition as (a) cluster-free and (b) first-NN Sn–Sn clustered GeSn. Eg,Γ drops by 0.5 meV and Eg,L by 12.8 meV due to Sn clustering at 5.56%.

FIG. 4.

Effective band structure of the GeSn alloy at a 5.56% Sn composition as (a) cluster-free and (b) first-NN Sn–Sn clustered GeSn. Eg,Γ drops by 0.5 meV and Eg,L by 12.8 meV due to Sn clustering at 5.56%.

Close modal
FIG. 5.

Effective band structure of the GeSn alloy at a 22.22% Sn composition as (a) cluster-free and (b) first-NN Sn–Sn clustered GeSn. Direct bandgap, Eg,Γ, drops by 40.76 meV and indirect bandgap, Eg,L, by 120.17 meV due to Sn clustering at 22.22%, where Eg,Γclustered = 0.05 eV.

FIG. 5.

Effective band structure of the GeSn alloy at a 22.22% Sn composition as (a) cluster-free and (b) first-NN Sn–Sn clustered GeSn. Direct bandgap, Eg,Γ, drops by 40.76 meV and indirect bandgap, Eg,L, by 120.17 meV due to Sn clustering at 22.22%, where Eg,Γclustered = 0.05 eV.

Close modal
FIG. 6.

Decrease in the direct bandgap, Eg, Γ (Δ-symbol), and indirect bandgap, Eg, L (○-symbol), energies due to Sn clustering increases with increasing Sn composition, as observed for cluster-free (light color) and first-NN Sn–Sn clustered (dark color) GeSn.

FIG. 6.

Decrease in the direct bandgap, Eg, Γ (Δ-symbol), and indirect bandgap, Eg, L (○-symbol), energies due to Sn clustering increases with increasing Sn composition, as observed for cluster-free (light color) and first-NN Sn–Sn clustered (dark color) GeSn.

Close modal

This characteristic nature of Sn clustering in the Ge1–xSnx alloy was observed from DOS as well, in a separate computation from band structure calculations, at each k-point using the Monkhorst–Pack k-point grid of 3 × 3 × 3. x = 5.56% and 22.22% are presented in Fig. 7 as representatives of similar observations noted in other compositions. Evident from Fig 7(b) is the rise in DOS and, hence, the reduction in the width of the forbidden gap due to the first-NN Sn–Sn clustering in 22.22% (high composition). Similar characteristics were reported from the bandgaps and effective band structures as noted in Figs. 4–6, further proving the variation in the energy bandgaps at a fixed Sn composition based on the well-dispersed Sn atoms or clustered Sn atoms.

FIG. 7.

Density of states of the Ge1–xSnx alloy at (a) x = 5.56% and (b) x = 22.22%. The width of the forbidden gap decreases with first-NN Sn–Sn clustering in Ge1–xSnx, indicating a decrease in the bandgap. At x = 22.22%, GeSn is fully direct with the observed DOS belonging to the direct band Γ, whereas at x = 5.56%, GeSn is yet to become direct, yet so close that the DOS is apparently from both direct and indirect bands.

FIG. 7.

Density of states of the Ge1–xSnx alloy at (a) x = 5.56% and (b) x = 22.22%. The width of the forbidden gap decreases with first-NN Sn–Sn clustering in Ge1–xSnx, indicating a decrease in the bandgap. At x = 22.22%, GeSn is fully direct with the observed DOS belonging to the direct band Γ, whereas at x = 5.56%, GeSn is yet to become direct, yet so close that the DOS is apparently from both direct and indirect bands.

Close modal

In conclusion, we have shown that the first-NN Sn–Sn clustering in the GeSn alloy has a significant effect on the electronic band structure and thermodynamic stability, thereby modifying the fundamental material properties of direct and indirect energy bandgap in a GeSn alloy. The clustering effect increased the formation energy and reduced the thermodynamic stability of GeSn by increasing both Hartree potential energy and exchange-correlation potential energy. In addition, it reduced both indirect and direct bandgap energies, more significantly at higher Sn compositions. For instance, with first-NN Sn–Sn clustered GeSn at an Sn composition of ∼22%, Eg,Γ and Eg,L bandgaps decreased by 40.76 and 120.17 meV, respectively. In addition, from the separately computed density of states, it was observed that the width of the forbidden bandgap reduced with Sn clustering. This supports the requirement to characterize and identify the clustering in GeSn material systems prior to reporting the accurate direct-indirect energy bandgaps at a particular composition and prevent process variation effects in the device and circuit performance parameters. Hence, it is prudent to initiate revisions in the first-principles calculations for the direct and indirect energy bandgaps of the GeSn alloy with the effect of Sn–Sn clustering incorporated.

The authors thank John K. Ghra (Systems Administrator for ECE, Virginia Tech) for assistance in computational services.

The authors have no conflicts to disclose.

Sengunthar Karthikeyan: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Mantu K Hudait: Investigation (equal); Project administration (equal); Resources (equal); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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