An atomistic band anticrossing (BAC) model is developed and used to study “unusual bowing” in energy bandgap and its dependence on the material composition in minority O anion-alloyed ZnS $(ZnS1\u2212xOx)$ and minority S anion-alloyed ZnO $(ZnO1\u2212xSx)$ highly mismatched alloys. For dilute O in $ZnS1\u2212xOx$, it is found that the bandgap decreases as the O composition is increased. A “down-shift” in the conduction band edge (CBE) of host ZnS, which arises from an interaction between the CBE and the localized O defect state, is identified as the root cause. However, the reduction in bandgap as a function of dilute S composition in the $ZnO1\u2212xSx$ alloy follows an “up-shift” in the valence band edge (VBE) of host ZnO, which arises from an interaction between the VBE and the localized S defect state. The BAC model captures the $E+$ and $E\u2212$ splitting in the sub-bands, which are found to be an admixture of the extended CBE (VBE) of ZnS (ZnO) and the localized O (S) state. A fully atomistic 8-band $sp3$-spin tight-binding basis set is used to construct the Hamiltonian for the wurtzite host materials as well as their alloy supercells. For alloy supercells, a strain is computed via the valence force-field formalism using Keating potentials. The O and S energy states are found to be approximately 199 meV below the CBE of ZnS and 190 meV above the VBE of ZnO, respectively. Overall, the calculated energy bandgaps using the BAC model are in good agreement with corrected local density approximation (LDA+U) calculations and experimental results.

## I. INTRODUCTION

The rapid development of electronic, optoelectronic, and renewable energy devices has led to the discovery of new materials and demands detailed investigation of their characteristics. ZnO is an attractive semiconductor material, which is abundant, chemically stable, and nontoxic, and exhibits high radiation hardness.^{1,2} ZnO naturally exists as a *n*-type material and has a wide direct bandgap of about 3.37 eV and an exciton binding energy of 60 meV. It offers great potential for applications in high power electronics, blue and ultraviolet optical emitters, solar cells, and spintronics.^{2} ZnO has high piezoelectric and pyroelectric polarization coefficients that could be used in nanogenerators and enable the increased performance of recently proposed disk-in-wire lighting devices as well as quantum well lasers.^{3–6} The energy bandgap of ZnO can be tuned over a large range varying from 1.5 to 4.5 eV by alloying with other binary compounds such as MgO, CdO, ZnS, and ZnSe.^{7–9} ZnS is also a wide-bandgap semiconducting material having a direct bandgap of approximately 3.8 eV, which could be used for optoelectronic applications such as core-shell nanowire ultraviolet emitters and thin film solar cells.^{10,11} Both ZnO and ZnS possess a stable hexagonal wurtzite crystallographic structure. However, it is challenging to design a *p-n* homojunction in ZnO mainly because the *p*-type junction has unstable conductivity.^{2} The existence of high donor defects such as oxygen vacancies and Zn interstitials has been identified as the main cause of this instability.^{2} It was found that alloying ZnO with S and heavy doping Cu can improve the *p*-type conductivity.^{12}

Cation and anion alloying of ZnO provide more flexibility for engineering the energy bandgap $(EG)$ in room temperature. For example, cation alloyed $Zn1\u2212xMgxO$ and $Zn1\u2212xCdxO$ exhibit a linear variation in $EG$, where the bandgap increases up to 4.5 eV and decreases down to 2.5 eV as Mg and Cd composition increases, respectively.^{7,8} On the other hand, anion alloyed $ZnO1\u2212xSx$ and $ZnO1\u2212xSex$ exhibit an “unusual bowing” and a nonlinear variation in $EG$ ranging from 1.5 eV to 3.4 eV. This has been observed experimentally and explained theoretically using the density functional theory (DFT) and partially self-consistent GW approaches.^{9,12–14} This phenomenon occurs when either the ZnS host is lightly doped with anion O or the ZnO host is lightly doped with anion S. For O in $ZnS1\u2212xOx$, the localized O defect energy state exists below the extended conduction band edge (CBE) of host ZnS and their interaction splits the CBE into two sub-bands, the upper $E+$ band and the lower $E\u2212$ band. On the other hand, for S in $ZnO1\u2212xSx$, the interaction between the extended valence band edge (VBE) of host ZnO and the localized S defect energy state, which is found to be positioned above the VBE of ZnO, leads to a splitting of the host VBE into two main sub-bands, the lower $E+$ band and the upper $E\u2212$ band. These anion alloys are classified as “highly mismatched alloys” (HMAs), where the host materials’ anions are replaced by isovalent atoms with distinct size and sometimes electronegativity.^{9,13} Note that the anion O’s electronegativity is 3.44, while the anion S’s electronegativity is 2.58.^{9,15}

For the electronic band structure of HMAs, the band anticrossing (BAC) model describes the interaction between the localized defect energy state of O (S) and the CBE (VBE) of the host ZnS (ZnO) material. It has been used to explain the pronounced nonlinear variation in $EG$ and the decrease of $EG$ by adding anion N atoms into $Ga1\u2212xInxNyAs1\u2212y$ alloys.^{16} The BAC model was also developed and used using the $sp3s\u2217$ tight-binding as well as the *k ⋅ p* frameworks for a localized N defect energy state interacting with the CBE of GaAs, InAs, GaP, InSb, and GaSb for diluted alloy composition of $GaNxAs1\u2212x$, $InNxAs1\u2212x$, $GaNxP1\u2212x$, $InNxSb1\u2212x$, and $GaNxSb1\u2212x$.^{17–19} Recently, a 10-band tight-binding model has been employed for the study of dilute bismide (Bi) in GaAs, GaP, InGaAs, and GaNAs.^{20–22} As for ZnOS, an analytical model was developed and fitted to experimental data to explain the unusual bowing in the energy bandgap.^{9,13} Also, the method of effective band structure can be used to determine bowing^{23} in alloys. To the best of our knowledge, there has been no report on an atomistic BAC model for ZnOS material system, which could handle realistically sized structures and, at the same time, provide greater accuracy and predictability.

This paper first presents the development of a BAC model based on an atomistic tight-binding framework for highly mismatched wurtzite $ZnO1\u2212xSx$ and $ZnS1\u2212xOx$ alloys. The overall computational methodology, presented in Sec. II, aims at the calculation of the defect energy states and the anticrossing interaction parameter as well as capturing the change in energy bandgap with respect to the composition of minority anions. Section III delineates the quantification of the interaction of localized defect energies with CBE (VBE) of ZnS (ZnO), the resulting band splitting due to the BAC interaction ($E+$ and $E\u2212$ sub-bands)**,** and the characteristic of unusual bowing in the energy bandgap. Finally, conclusions are presented in Sec. IV.

## II. SIMULATION MODEL

The overall computational framework used in this work is incorporated into our in-house QuADS 3-D simulator^{24,25} platform. To construct and diagonalize the atomistic Hamiltonian, the platform employs the open-source NEMO 3-D toolkit.^{26–30} NEMO 3-D computes the single-particle electronic energy states and optical characteristics of several semiconductor structures such as nanocrystals, quantum wells, quantum dots, nanostructures, and bulk materials. Recently, NEMO 3-D has been successfully used to study the electronic and optoelectronic properties of hexagonal wurtzite nanostructures of III-nitride and ZnO/MgO materials for use in disk-in-wire light emitters and core-shell quantum dot solar cells.^{5,31–35}

### A. The atomistic tight-binding approach

The sequence of simulation starts from the geometry construction of the underlying wurtzite lattice. If the material is an alloy, strain computation is enabled, where the atomistic relaxation and the minimum energy of the structure are obtained by means of the valence force-field (VFF) model using Keating potentials.^{36} The electronic bandstructure is computed for binary ZnO and ZnS as well as ZnOS HMAs using 8-band $sp3$-spin tight-binding basis sets. The semiempirical tight-binding parameters used in the study are mainly based on Ref. 37 and are listed in Table I. Note that for the analysis of dilute S in ZnO, we were not required to modify the semiempirical tight-binding parameters of ZnO, as available in Ref. 37, because the bandstructure solver provided a fairly accurate bandgap bowing. On the other hand, we needed to modify some of the tight-binding parameters of ZnO only, when wurtzite ZnS is alloyed by dilute O atoms. These modified parameters are only valid for dilute O less than or equal 4%. The listed tight-binding parameters provide a precise description of the valence bands and a sufficiently accurate representation of the conduction band minima. For a specific cation or anion in the ZnOS HMAs and its four nearest neighbors, the onsite orbital energies in the electronic Hamiltonian are obtained by calculating the weighted average of the binary ZnO and ZnS orbital self-energies including a valence band offset (VBO). The VBO between materials ZnO/ZnS is about 0.84 eV.^{38} Strain causes distortions to both bond lengths and bond angles that influence interatomic interactions.^{26,39} It could be expressed by $dd0\eta (sp\sigma )$, where $d0$ is the equilibrium bond length, *d* is the bond length after distortion, and $\eta (sp\sigma )$ is a scaling parameter.^{26,39}

. | ZnO^{37} (for dilute S in ZnO)
. | ZnO (modified for dilute O in ZnS) . | ZnS^{37}
. |
---|---|---|---|

E(s, a) | −19.046 | −19.046 | −10.634 |

E(p, a) | 4.142 | 4.142 | 1.574 |

E(s, c) | 1.666 | −1.521 | 2.134 |

E(p, c) | 12.368 | 12.368 | 6.626 |

V(s, s) | −6.043 | −2.620 | −4.904 |

V(x, x) | 7.157 | 12.157 | 3.229 |

V(x, y) | 10.578 | 10.578 | 5.168 |

V(sa, pc) | 4.703 | 4.703 | 0.357 |

V(pa, sc) | 8.634 | 12.634 | 6.240 |

λ _{a} | 0.032 | 0.032 | 0.025 |

λ _{c} | 0.085 | 0.085 | 0.085 |

. | ZnO^{37} (for dilute S in ZnO)
. | ZnO (modified for dilute O in ZnS) . | ZnS^{37}
. |
---|---|---|---|

E(s, a) | −19.046 | −19.046 | −10.634 |

E(p, a) | 4.142 | 4.142 | 1.574 |

E(s, c) | 1.666 | −1.521 | 2.134 |

E(p, c) | 12.368 | 12.368 | 6.626 |

V(s, s) | −6.043 | −2.620 | −4.904 |

V(x, x) | 7.157 | 12.157 | 3.229 |

V(x, y) | 10.578 | 10.578 | 5.168 |

V(sa, pc) | 4.703 | 4.703 | 0.357 |

V(pa, sc) | 8.634 | 12.634 | 6.240 |

λ _{a} | 0.032 | 0.032 | 0.025 |

λ _{c} | 0.085 | 0.085 | 0.085 |

### B. Conduction band anticrossing model (CBAC)

The CBAC model determines the effect of the localized defect energy state on the extended CBE of the host material. The CBAC is a two band Hamiltonian, where the computed eigenvalues are the split sub-bands $E+$ and $E\u2212$ of the diluted O in $ZnS1\u2212xOx$ alloy and is given by^{19,20}

Here, $EO$ is the energy of the O defect state and $Ec0$ is the extended CBE of the ZnS host material. The matrix element $VO$ describes the interaction between these energies. *V _{O}* is composition dependent and is given by

^{20}

Here, *β* is the anticrossing interaction parameter and *x* is the material composition. The CBAC analysis starts with constructing the geometry of the $ZnNSN$ host material supercell (N denotes the number of cations and anions in the supercell) and the diluted O in $ZnNSN\u2212LOL$ alloy supercells (L refers to the number of defect atoms). Next, the supercells’ Hamiltonians are constructed using the 8-band $sp3$-spin tight-binding model to compute the electronic band structure for $ZnNSN$ and the diluted O in $ZnNSN\u2212LOL$ supercells. The atom positions are relaxed as described in Sec. II A. The calculated wavefunction of the CBE state $|\Psi c0\u27e9$ for the ZnS host material and the wavefunction of the CBE state $|\Psi c1\u27e9$ for the diluted O in $ZnS1\u2212xOx$ are used to obtain the localized O wavefunction $|\Psi O\u27e9$. Given that $|\Psi c1\u27e9$ is a linear combination of $|\Psi c0\u27e9$ and $|\Psi O\u27e9$, the CBAC model determines $|\Psi O\u27e9$ using^{20}

This step is essential as it determines the localized O defect energy state $EO$ as well as the anticrossing interaction parameter *β*. Here, *β*, which treats the interaction between the localized O defect energy state and the extended CBE of the ZnS host material, is modeled as an inner product incorporating the tight-binding Hamiltonian (*H*) of $ZnS1\u2212xOx$ as follows:

Finally, the values of $EOandVO$ are plugged into Eq. (1) that describes the effect of dilute O composition on the electronic structure of HMAs $ZnS1\u2212xOx$.

### C. Valence band anticrossing model (VBAC)

The VBAC model is implemented to analyze the influence of localized S defect energy state on the extended VBE of the ZnO host material. This model is analogous to the CBAC model and the four-band Hamiltonian of the VBAC is given by^{20}

Here, $EHHandELH$ are the heavy-hole and the light-hole energy states of the VBE for the ZnO host material, respectively. The localized S defect energy state is represented by $ES$. $VS$ represents the interaction between the localized S defect energy state and the extended VBE state of the ZnO host material. *V _{S}* is composition dependent and given by

The eigenvalues of the four-band Hamiltonian are the split degenerate energy states $E\u2212$ and $E+$ of the four-folded degenerate VBE states of the ZnO host. The same procedure as used in the CBAC analysis is followed for the VBAC calculation. The only difference lies in the variables used to carry out the simulation. The geometry is constructed for both the $ZnNON$ supercell and a single-defect S atom in the $ZnNON\u2212LSL$ supercell. Then, the Hamiltonian is constructed and the VBE wavefunctions for both the host material and the HMA alloy are computed. The VBE of host ZnO consists of four-folded degenerate energy states that requires some modifications of Eq. (3) and is given by Eq. (8), as described in Ref. 20. Equation (8) basically computes the wavefunction of the localized S defect wavefunction $|\Psi S,i\u27e9$. As before, the VBE wavefunction $|\Psi v,1,i\u27e9$ of the $ZnO1\u2212xSx$ is a linear combination of the four-folded VBE wavefunctions $|\Psi v,0,i\u27e9$ for the ZnO host material and the localized S defect wavefunction $|\Psi S,i\u27e9$. Accordingly, $|\Psi S,i\u27e9$ is determined as follows:

where *n* denotes the order of the energy level. The localized S defect energy $ES$ and the anticrossing interaction parameter $\beta $ are calculated by

Finally, $ES$ and $VS$ are used in Eq. (6) to calculate the split sub-bands in $ZnSxO1\u2212x$.

### D. Fractional Γ character spectra

The CBE states of ZnS and the VBE states of ZnO are expected to mix in the ZnOS HMAs with the localized O and S defect energy states, respectively. Following the recipe of Ref. 20, to verify the occurrence of the BAC interactions, the fractional Γ character spectra are used to project the CBE and the VBE states into the alloy supercells’ full spectra of levels, which is expressed as

where

Here, *n* and *m* indices are used for the host material and the mixed defect in the alloy state, respectively. The width of the delta function, $\delta (En\u2212E)$, is 2 meV. $g(Em)$ denotes, as needed, the degenerate states of the CBE of ZnS and the four-folded degenerate states of the VBE of ZnO host materials. Note that the unperturbed CBE of ZnO and VBE of ZnS are projected onto the spectrum of the ZnO HMA supercells.

## III. RESULTS AND DISCUSSIONS

### A. Diluted O in $ZnS1\u2212xOx$ supercells

We first analyze the unperturbed ZnS host material to determine its conduction band edge (CBE). Then, a single-defect anion O atom is incorporated randomly replacing an anion S atom. Several supercell sizes are designed, where the number of atoms considered ranges from 5408 to 72 (with a single O defect atom). The main aim is to verify the existence of CBAC interaction and identify the O defect energy as well as the anticrossing interaction parameter. For this analysis, the CBE of ZnS is shifted by 0.84 eV to treat the VBO between ZnO and ZnS. After calculating the electronic structure of the unperturbed ZnS host material and the alloy for each size, the obtained wavefunctions for the CBE of ZnS and the CBE of the alloys with single O defect atom are used to compute Eq. (11), which is the fractional Γ character $G\Gamma (E)$. This is an important step to verify the BAC interaction for these structures. The results are shown in Fig. 1. The CBE of ZnS is projected to the full energy levels spectrum of the alloy for each size. The two degenerate energy states $E+$ and $E\u2212$ are the result of the interaction between localized O defect energy state and the CBE state of ZnS host material. The CBE state of $Zn2704S2704$ host material is plotted as a reference in Fig. 1(a). Figures 1(b)–1(f) refer to the $ZnNSN\u2212LOL$ supercells, where the composition $x=N\u22121$ increases from 5408 atoms with *x* = 0.037% in Fig. 1(b) to 216 atoms with *x* = 0.463% in Fig. 1(f). The CBE of ZnS mixes in the split energy states $E+$ and $E\u2212$ is 90.23% and 4.38%, respectively, in Fig. 1(b). The remaining 5.39% is distributed into other conduction energy bands. For the $E\u2212$ energy state, the percentage of the CBE of the ZnS increases and its quantity become approximately equivalent in each degenerate state as the size of the supercell is reduced. This is observed in Fig. 1(f) where the mixing of CBE decreases to 51.92% for $E+$ energy state and increases to 47.02% at $E\u2212$ energy state. The splitting between $E+$ and $E\u2212$ states increases as the composition *x* increases. It has been verified that the degenerate energy states of the alloy are admixture of the CBE of the unperturbed ZnS host material and the localized O defect energy.

The next step is to obtain the localized O defect energy $EO$ and the anticrossing interaction parameter *β*. Initially, the linear combination is solved using Eq. (3) to find the localized O wavefunction for each supercell size. After that, the localized O wavefunction is used in the calculation of the inner products in Eqs. (4) and (5) for identifying the $EO$ and *β*, respectively. For the 5408 atoms supercell size, the localized O defect energy was found to be 4.441 eV, which is below the CBE of ZnS by 199 meV. This is illustrated in Fig. 2. Here, it is found that as the supercell size increases, the localized defect energy state first decreases and then stabilizes. The trend is similar to Bi in GaP and GaAs host materials as described in Ref. 20. Furthermore, the anticrossing interaction parameter *β* was found to be about 2.57 eV for the 5408 atoms supercell size and it varies in the same fashion with the increase in the supercell size from 72 atoms to 5408 atoms. The difference between the anticrossing interaction parameter *β* of the smallest supercell size (2N = 72 atoms) and the largest supercell size (2N = 5408) is nearly 1.1 eV. The influence of O atom composition in the ZnS supercell is strong mainly due to the difference in electronegativity and size.

### B. Diluted S in $ZnO1\u2212xSx$ supercells

The VBAC analysis is performed constructing several samples of supercells with different sizes from 8100 to 72 atoms including a single S atom that substitutes an anion O atom randomly for the alloy. The Hamiltonians are constructed to compute the electronic structure for these supercells. The generated wavefunctions of the four-folded degenerate VBE states of the ZnO host material and the valence energy levels of the alloy are used to calculate the fractional Γ character $G\Gamma (E)$ for each supercell size. The objective is to verify BAC interaction between the localized S defect energy state and the VBE of ZnO host material and is displayed in Fig. 3. Here, the four-folded degenerate VBE energy states of ZnO are projected to the full energy levels spectrum of the alloy for each supercell size. The VBE of ZnO host material is plotted as a reference in Fig. 3(a). The $ZnNON\u2212LSL$ supercells range from 4032 atoms with *x* = 0.0496% in Fig. 3(b) to 128 atoms with *x* = 1.562% in Fig. 3(f). The four-folded degenerate VBE energy states of ZnO host material are mixed in the split energy states $E\u2212$ and $E+$ as expected from Eq. (6). As the *x* composition increases, the splitting between these two degenerate states rises due to the BAC interaction where the $E+$ decreases and $E\u2212$ increases. In Fig. 3(b), the mixing of four-folded VBE degenerate states in the split energy states $E+$ and $E\u2212$ is 97.6% and 1.64%, respectively. The mixing of VBE reduces to 62.35% for $E+$ and rises to 33.65% for $E\u2212$ as can be seen in Fig. 3(f). This verifies the existence of BAC interaction where the split energies are the admixture of the four-folded degenerate VBE states and the localized S defect energy state. Similar to our observation for the dilute O in ZnS as well as in the case for dilute Bi in GaAs and GaP,^{20} the energy state $E\u2212$ mixes with more band edge of the host material as the size of the supercell is reduced. In other words, the degenerate states $E\u2212$ and $E+$ are constituted of approximately 50% of the defect energy level and the band edge of the host material at small supercell sizes. On the contrary, the majority of the energy state $E+$ is consisted of the band edge of the host material, and the minority of the energy state is defect energy state and vice versa for the energy state $E\u2212$ for larger supercells.

Next, the goal is to find the unknown parameters, namely, the localized S defect energy $ES$ and the anticrossing interaction parameter *β*. To do this, first, the wavefunction of S is obtained by computing the linear combination as expressed in Eq. (8) for each supercell size. Then, the generated wavefunction of S is used in the calculation of the inner product to find $ES$ and *β* as given by Eqs. (9) and (10), respectively. For the 4032 atoms supercell size, the localized S defect energy state is found to be about 190 meV above the VBE of the ZnO host material, whereas the anticrossing interaction parameter *β* is found to be approximately 1.35 eV. It can be seen that the S atom has a smaller anticrossing effect on ZnO host material than the O atom on ZnS host material counterpart. Figure 4 illustrates the variation of the localized S defect energy state and the anticrossing parameter *β* as a function of the supercell size. The localized S defect energy increases as the supercell size rises from 72 to 784 atoms and then it tends to slightly decrease for larger supercell sizes. In addition, the anticrossing interaction parameter establishes a reduction trend as a function of the supercell size similar to the reported dilute Bi in GaAs and GaP.^{20} For the dilute S in ZnO material system, the difference in the anticrossing interaction parameter is about 500 meV between the 72-atom and the 8100-atom supercell sizes, which is smaller by about 50% than the case for dilute O in ZnS material system.

### C. The bandgap of minority alloyed $ZnOS$

The effect on the energy bandgap $(EG)$ in dilute O in $ZnxSx\u22121Ox$ and dilute S in $ZnO1\u2212xSx$ HMAs is illustrated in Figs. 5 and 6, respectively. The energy bandgaps, as calculated using Eqs. (1) and (6), are compared with available experimental and theoretical results and found to be in good agreement. In both cases, $EG$ decreases dramatically when the host material ZnS (ZnO) is lightly doped with anion O (S). At low O composition, the bandgap of $ZnS1\u2212xOx$ decreases from 3.80 eV to approximately 3.16 eV. This reduction is mainly due to a down-shift of the CBE of the $ZnS1\u2212xOx$ material. As the O composition increases beyond 6% (that is, for rich anion O alloys), the CBE starts to decrease linearly. On the other hand, the bandgap of $ZnO1\u2212xSx$ decreases from approximately 3.41 eV to 3.00 eV as S composition is varied from 0% to 10%. The unusual bandgap bowing in this case is due mainly to an up-shift of the VBE of $ZnO1\u2212xSx$ alloy.

## IV. CONCLUSIONS

ZnO and its related alloys have attracted considerable attention for applications in electronic, light-emitting, and energy related devices. ZnO exhibits unusual bandgap bowing when it is doped with anion elements such as in highly mismatched alloys (HMAs) of $ZnO1\u2212xSx$ and $ZnO1\u2212xSex$. In this work, an atomistic band anticrossing (BAC) model is developed to investigate the unusual bowing in energy bandgap (*E*_{G}) in ZnS_{1-x}O_{x} and ZnO_{1-x}S_{x} HMAs and its dependence on the material composition. The presence of the BAC interactions has been quantified using the fractional Γ character. It is found that the split energy states, $E+$ and $E\u2212$, are the admixture of the CBE (VBE) of the ZnS (ZnO) host material and the localized O (S) defect energy states. The predicted defect energy levels for dilute O and dilute S are approximately 199 meV below the CBE of ZnS and approximately 190 meV above the VBE of ZnO, respectively. Furthermore, the anion O atom in the ZnS host material has higher anticrossing interaction parameter than the anion S in the ZnO host material counterpart. The unusual bowing in the lightly alloyed ZnO_{1-x}S_{x} is observed, where the energy bandgap decreases from approximately 3.41 eV to 3.00 eV as the S composition is varied from 0% to 10% and increases from approximately 3.16 eV to 3.80 eV with rich S compositions varying from 96% to 100%.

## ACKNOWLEDGMENTS

This work was financially supported by the U.S. National Science Foundation (Grant No. 1610474). S. M. Alqahtani would like to acknowledge the support from Saudi Arabian Cultural Mission (SACM).

## REFERENCES

_{x}Zn

_{1−x}O as a II–VI widegap semiconductor alloy

_{1−x}S

_{x}and ZnO

_{1−x}Se

_{x}alloys

_{1−x}S

_{x}highly mismatched alloys over the entire composition

_{1−x}S

_{x}thin films deposited by reactive sputtering

_{x}As

_{1−x}and related alloys

_{x}As

_{1–x}/GaAs multiple quantum wells

_{1-x}S

_{x}Alloys

^{3}d

^{5}s* tight-binding simulations