Optical properties of Ag_xCu_1–xI alloy thin films

Over the past years, various ideas for new transparent optoelectronic devices based on wide bandgap semiconductors have been introduced, such as transparent thin-film transistors,¹ transparent electrodes,² electrochromic displays,³ and solar windows.⁴ Regarding the search for a p-type material that combines suitable electrical properties with a high degree of transparency in the visible spectral range, copper halides have emerged as promising candidates in recent years.^5–9 In particular, zincblende γ-CuI combines a large bandgap of about 3.1 eV^5,10 to a large exciton binding energy of 62 meV,¹⁰ with hole carrier densities up to 1 × 10²⁰ cm⁻³ and single crystal hole mobilities up to 44 cm²/V⁻¹ s⁻¹.¹¹ The application potential of p-type CuI was demonstrated in transparent p-n heterojunctions,¹² thin film transistors (TFT),^1,13,14 light emitting diodes (LEDs),^15,16 perovskite solar cells,^17–19 UV-photodetectors (UVPD),²⁰ and thermoelectric devices.⁴ The adjustment of the hole density in CuI thin films, being crucial for active device applications such as UVPD, TFT, and LED,^7,21 was studied for ternary CuBr_xI_1–x alloys.⁷ Moreover, a recent study²² reported the achievement of n-type conductivity in ternary Ag_xCu_1–xI alloys and fabrication of heterostructures based on Ag_xCu_1–xI alloy thin films, making this material system a suitable candidate for future innovative applications in transparent optoelectronics.

Binary AgI is a well-known material with a variety of applications, such as photosensitive material,^23,24 as a solid electrolyte in battery and electrochemical capacitors,^25–29 in artificial weather modification,^30,31 or as a photocatalyst for the purification of dye-containing wastewater.³² However, the majority of publications are focused on the superionic conductor α-phase AgI.^28,33,34 There are only a few reports about the semiconductor zincblende-type γ-phase and wurtzite-type β-phase and, in particular, their optical properties.^35–38 Thus, the studies on the ternary Ag_xCu_1–xI alloy are also mainly limited to the structural phase transition and ionic conductivity properties of the α-phase.^39–43 Although optical transitions of γ- and β-Ag_xCu_1–xI alloy films investigated by using optical absorption spectroscopy were reported by Cardona,⁴⁴ a detailed discussion of the optical transition energies as a function of alloy composition, in combination with the corresponding electronic structure, is still lacking. Moreover, to our knowledge, there are no reports on the temperature dependence of the bandgap of Ag_xCu_1–xI as a function of alloy composition.

In this work, we combine experimental and computational methods to investigate the effect of the chemical composition on the optical transitions at the Γ-point and the electronic band structure of zincblende γ-Ag_xCu_1–xI alloys. A discussion of the origin of effects like the bandgap bowing and the variation of the spin–orbit splitting is provided. The temperature dependence of the E₀ transition between 20 K and 290 K is studied for various alloy compositions.

The investigated Ag_xCu_1–xI thin films were deposited on c-plane oriented Al₂O₃ substrates by a combination of two different deposition techniques: close distance sublimation (CDS)⁴⁵ and solid state reaction (SSR).⁴⁶ In the case of the CDS method, Ag_xCu_1–xI powder pellets with different chemical compositions were used as the source. The deposition temperature was set to 490 °C. Although the CDS method was found to be a suitable technique for the preparation of single-phase zincblende γ-Ag_xCu_1–xI thin films with Cu-rich composition (x $<$ 0.5), Ag-rich thin films exhibit a mixed-phase structure, which contain the wurtzite structure in addition to the zincblende structure. Therefore, single phase zincblende Ag-rich thin films were fabricated by using the SSR technique based on the reaction of AgI thin film, produced by the Bädecker method,⁴⁷ with CuI vapor under CDS conditions. The alloy composition in this case is controlled by the duration of the CDS deposition process. A detailed discussion of the two fabrication methods, as well as the structural properties of the alloyed thin films can be found in Ref. ⁴⁸. Here, we would simply like to point out that all investigated Ag_xCu_1–xI thin films exhibit exclusively γ-phase structure, except for the mixed-phase γ-/β-AgI thin film prepared by the Bädecker method.⁴⁷ The alloy compositions were determined based on the lattice constants applying Vegard’s rule.⁴⁹ 

For temperature-dependent optical measurements, the samples were mounted in a helium-flow cryostat (Janis ST-500). The temperature of the cryostat cold finger was measured with a silicon-diode-based temperature sensor mounted in the vicinity of the samples. A Xenon arc lamp (75 W) was used as a light source for the reflectivity measurements in a spectral range between 2.8 and 3.9 eV. The white light was focused by a long working distance Mitutoyo 50x UV objective (NA = 0.40). The reflected white light was collected confocally, dispersed by using a spectrometer (Horiba Jobin Yvon iHR320) with 600 grooves/mm grating, and detected by a Peltier-cooled CCD camera (Horiba Jobin Yvon Symphony Open STE). The spectral resolution of the setup is ∼0.2 nm, which corresponds to 1.5 meV for photon energies around 3.1 eV.

To increase the accuracy in determining the transition energies of the optical transitions, we numerically calculated the second derivatives of the measured reflectivity spectra by applying the Savitzky–Golay filter algorithm.⁵⁰ The second derivative spectra d²R/dE² were then analyzed by using the transfer matrix technique for a layer stack model consisting of a single semi-infinite layer. The optical response in the vicinity of a single resonance was modeled by using a Lorentzian line shape. The insets in Figs. 1(a) and 1(b) show exemplary experimental data of the CuI thin film along with the model approximation, demonstrating a good agreement with the measured spectra.

All calculations were performed in the framework of density functional theory (DFT), as implemented in the Vienna ab initio simulation package VASP.^51,52 Pseudopotentials produced with the projector augmented wave method⁵³ were applied. The 4s, 3p, and 3d electrons of Cu and the 5s, 4p, and 4d electrons of Ag, as well as the 5s and 5p electrons of I, were explicitly treated as valence electrons. For the plane-wave basis set, a cutoff energy of 640 eV and a k-point grid of 8 × 8 × 8 for the binary and 4 × 4 × 4 for the ordered ternary alloys were used. Both values were obtained to satisfy a convergence criterion of 1 meV/atom regarding the total energy. To limit the computational costs, we decided to focus on the effects of chemical substitution rather than on configurational disorder. Therefore, we performed calculations for ordered alloys consisting of 2 × 2 × 2 primitive supercells (i.e., eight atoms of Ag, Cu and eight I atoms). More information on the structures is provided in the supplementary material (see Fig. S1 in the supplementary material). We consider all possible non-equivalent ways to arrange the Ag and Cu atoms on cation sublattice sites in the supercell. Then, we optimized those geometries by using the PBEsol exchange-correlation functional.⁵⁴ This is a revised Perdew–Burke–Ernzerhof generalized gradient approximation functional (GGA) for solids, which is significantly better than PBE for structural properties.⁵⁵ The relaxation was done until all forces were smaller than 1 meV/Å. Given the small energy differences between the possible configurations at a fixed Ag concentration x, we expect them all to occur in our samples due to their relatively high fabrication temperature of about 490 °C. Therefore, we calculated the electronic properties of interest for all possible configurations and compare the thermal averaged values with the experimental data. Since the Kohn–Sham band structures calculated with just PBEsol drastically underestimate the bandgap, we used the hybrid functional PBE0⁵⁶ with the inclusion of spin–orbit coupling (SOC). This setup describes the bandgap of CuI correctly.⁵⁷ However, given the very high computational cost, we could not include the exciton correction terms directly in our calculations and preferred to account for the exciton binding energy by using a model based on the literature data. In addition, we would like to point out that PBE0 was consistently used for the calculations discussed below unless explicitly stated otherwise.

We show in Fig. 1, as an example, the measured reflectivity spectra of binary CuI and AgI thin films. For the single-phase γ-CuI, the reflectivity spectrum below 3.8 eV is characterized by two optical transitions, in good agreement with the literature results.^44,58,59 The first transition E₀, occurring at 3.06 eV, can be attributed to excitonic transitions at the fundamental bandgap at the Γ-point. Taking into account the exciton binding energy of 62 meV,¹⁰ this yields an experimental bandgap of 3.12 eV. The next higher energy resonance, denoted as E₀ + Δ₀, can also be attributed to excitonic transitions at the Γ-point but involving the split-off valence band instead of the upper valence bands. The deduced spin–orbit splitting for CuI is Δ₀ = 640 meV, which is in reasonable agreement with previously published values for DC-sputtered⁵⁸ and PLD-deposited thin films.⁵⁹ The calculated DFT bandgap of 3.13 eV and the spin–orbit splitting of 670 meV match well with our experimental results.

In the case of AgI [see Fig. 1(b)], the reflectivity spectrum is more complex due to the mixed zincblende and wurtzite phase structure of the polycrystalline sample. The corresponding band structures of AgI in the two different phases are shown in Fig. S2 in the supplementary material. The lowest energy doublet at 2.905 and 2.914 eV is attributed to the E₀ transitions of the zincblende phase, similar to γ-CuI. The observation of the doublet structure instead of a single resonance is caused by the thermal strain that lifts the degeneracy at the valence band maximum (VBM).⁶⁰ We note that in the case of the CuI sample [see Fig. 1(a)], this splitting is not observed experimentally due to the stress relaxation caused by larger film thickness in comparison to AgI^60,61 (1.4 µm compared to 0.7 µm). The effect of thermal strain on the optical transitions is discussed in detail in Sec. IV B. The calculated DFT bandgap of γ-AgI (2.84 eV) slightly underestimates the experimental bandgap value of 2.984 eV considering the exciton binding energy of 79 meV.^62,63 Still, the deviation of the DFT results from the experiment is compatible with the expected accuracy of the PBE0 functional.⁶⁴ 

The additional resonances observed at 2.947 eV, 2.984 eV, and 2.994 eV labeled with $E0wz,1$⁠, $E0wz,2$⁠, and $E0wz,LM$⁠, respectively, are attributed to optical transitions of the wurtzite phase β-AgI.^44,62 The $E0wz,1$ and $E0wz,2$ transitions have the same physical origin as the E₀ transitions in the zincblende crystal, although the degeneracy of the uppermost VBs is removed by the additional crystal field due to anisotropy in the wurtzite structure,^44,65 even in the absence of external perturbations. Here, we observe a VBM splitting of 37 meV at 20 K, which is in good agreement with the literature results^44,62 as well as our calculated value of 46 meV. The $W2LM$ transition at 2.99 eV is attributed to a longitudinal-transverse mixed mode due to the crystal anisotropy.^62,66 Further details on the electronic band structure of β-AgI are presented in Sec. II of the supplementary material.

As for CuI, E₀ + Δ₀ transitions between the lowest conduction band and the split-off valence band are expected at higher energies for both the zincblende and wurtzite structures of AgI.^36,44 However, instead of individual resonances corresponding to the zincblende and wurtzite phases, respectively, a single resonance is observed at 3.75 eV.

We note that although our experimental value $Δ0exp=0.84eV$ is slightly larger compared to the DFT calculations for γ-AgI $(Δ0zb,calc=0.79eV)$⁠, it is still in good agreement with previous reports on γ-AgI.^36,44 However, due to the mixed phase structure of the binary AgI sample, the observed E₀ + Δ₀ resonance and, thus, the value for spin–orbit splitting cannot be unambiguously assigned to a specific phase. Nevertheless, the experimental value is very close to the one-electron spin–orbit splitting of iodine, hinting to only weak contributions of the cation wavefunctions to the valence bands. A detailed discussion of the p-d hybridization for different alloy compositions is given in Sec. IV E. In addition, we refer to the supplementary material for a detailed discussion of the calculated spin–orbit splitting values for γ- and β-AgI.

In conclusion, the calculated energies for the lowest energy transitions at the Γ-point are in good agreement with the experimental data for the binary compounds. In contrast to previous studies^67,68 that rely on GGA or local density approximation (LDA) for the exchange functional, our results with the hybrid functional PBE0 give a very improved description of the absolute value of the bandgap. In the following, we focus the discussion on the bandgap energy as well as the spin–orbit splitting of the zincblende phase Ag_xCu_1–xI alloy as a function of the alloy composition.

As mentioned before, in the zincblende structure, the heavy-hole (hh) and light-hole (lh) valence bands (VBs) degenerate at the Γ-point. However, this degeneracy can be lifted by an external perturbation, such as uniaxial strain.⁶¹ For temperature-dependent measurements, the thermally induced strain resulting from the different thermal expansion coefficients of the thin film and the underlying substrate has to be taken into account.⁶⁰ The effects of such uniaxial strain in CuI and AgI were discussed in detail in Refs. 61, 69, and 70. However, for Ag_xCu_1–xI, no detailed studies about the thermally induced VBM splitting as a function of the alloy composition exist.

Figure 2 shows the VBM splitting $Δhh−lh=E0hh−E0lh$ splitting of the Ag_xCu_1–xI alloy at 20 K as a function of the chemical composition. The assignment of the experimentally observed resonances to the corresponding VBs is based on the assumption that the oscillator strength of the $E0hh$ transitions is significantly larger than that of the $E0lh$ transitions (f_hh/f_lh = 3:1).⁷¹ For the binary CuI, the reported values for the thermal expansion coefficient range between 15 × 10⁻⁶  K⁻¹ and 19.2 × 10⁻⁶ K⁻¹ at room temperature,^70,72,73 whereas the value for the Al₂O₃ substrate is 4.5 × 10⁻⁶ K⁻¹ (see Ref. 74). Such a large difference in the expansion coefficients leads to a tensile strain in Ag-poor samples at low temperatures, resulting in a positive Δ_hh–lh splitting.^60,61,69 The reason that no splitting is observed for our samples with x < 0.4 is due to the increasing thickness with decreasing Ag-content caused by the decreasing deposition rate during the CDS process. Such behavior is in good agreement with previous publications reporting that the thermal stress in CuI films grown on Al₂O₃ starts to relax at 500 nm.⁶⁰ 

Interestingly, in Ag-rich samples, the situation changes significantly so that the $E0hh$ transition, which is energetically higher compared to the $E0lh$ transition in CuI, becomes the energetically lower transition resulting in a negative sign of the Δ_hh–lh splitting. Such behavior is expected for zincblende semiconductors in case of compressive strain. Indeed, the thermal expansion coefficient of AgI was reported to be negative with a value of approximately −4.1 × 10⁻⁶ K⁻¹ at room temperature.⁷⁵ Therefore, due to the positive thermal expansion coefficient of the Al₂O₃ substrate, the Ag-rich thin films are expected to experience compressive strain with decreasing temperature, unlike the tensile strain in the case of Ag-poor thin films. Based on our results, we expect the Δ_hh−lh splitting almost to disappear for alloy compositions around x ≈ 0.6, indicating that the expansion coefficients of the thin film and substrate are identical and thus no strain is present. This is in perfect agreement with dilatometry measurements published by Kumar et al.,⁷⁶ where the authors have shown that Cu_0.25Ag_0.75I has nearly zero thermal expansion. Apart from the fact that such a zero thermal expansion material has potential applications, e.g., for chemical sensing, the thermal expansion coefficient of the Ag_xCu_1–xI alloy can be tuned in a controllable way designing the chemical alloy composition to minimize the thermally induced strain for a variety of substrates, such as silicon, fused silica, and other glass substrates.

Regarding the DFT calculations, when building the 2 × 2 × 2 supercells as written in Sec. III, there are, in principle, different ways to order the Cu and Ag atoms within the crystal. Here, we want to shortly discuss the influence of the different possible configurations of ordered alloys on the results of our calculations. The corresponding values for all the different considered Ag concentrations within the 2 × 2 × 2 supercell are listed in Table I. The energy difference between those different configurations was obtained with PBEsol as the relaxation was performed with that functional. The observed differences are small enough, so we expect a disordered alloy to be formed in the experimental samples due to relatively high deposition temperatures. Interestingly, we find that, for each alloy composition, the results regarding the quantities of interest are almost identical for all possible configurations, in contrast to recent studies on CuBr_xI_1–x alloys.⁵⁹ This indicates that the anion substitution leads to a stronger influence of the corresponding structural disorder on the electronic properties in comparison to the cation substitution.

Since all configurations provide similar values for the properties of interest, we omit the full description of disorder by using other methods such as the quasi-chemical approximation⁷⁷ or cluster expansion,^78,79 and consider the influence of the higher energy configurations by calculating simply the thermal average of values given in Table I using

where m_i is the multiplicity describing how often a configuration appears in the 2 × 2 × 2 supercell, T corresponds to the deposition temperature of 760 K, and Δϵ_i describes the total energy differences per cation–anion pair with respect to the lowest energy configuration listed in Table I. The corresponding variation of the bowing behavior is demonstrated in Fig. S4 in the supplementary material.

As discussed in Sec. IV A, the E₀ transition energy decreases with increasing Ag-content from 3.06 eV for γ-CuI to ∼2.91 eV for γ-AgI. We find that the evolution of the E₀ transition energy does not exhibit a linear dependence on the alloy composition but shows a quadratic bowing behavior. Following the approach presented by Cardona,⁴⁴ we decompose the energetic position of the E₀ peak as a function of alloy composition into a linear and a quadratic term,

where E₀^CuI and E₀^AgI are the transition energies for the binary compounds, x denotes the Ag-content, and b is the bowing parameter. The non-linear contribution to the transition energy as a function of the alloy composition at low temperature is shown in Fig. 3. We determined the bowing parameter of b_exp ≈ 0.54 eV. This result is comparable with previously published values⁴⁴ and is slightly larger than the bowing recently reported for the CuBr_xI_1–x alloy.⁵⁹ In comparison to that experimental value, the bowing parameter obtained from our DFT calculations is b_calc ≈ 0.53 eV (red symbols in Fig. 3). The agreement is significantly improved with respect to previously published DFT calculations, where GGA and LDA functionals were applied.^67,68 We note that although the optical transitions exhibit a strong excitonic character, possible effects of a non-linear change in the exciton binding energy as a function of alloy composition on the observed bandgap bowing⁵⁹ can be almost neglected as discussed in Sec. III in the supplementary material. Therefore, we can compare the experimental bowing parameter with the values obtained from the DFT band structures.

To further discuss the origin of the observed bowing of the E₀ transition, and hence the bandgap energy, we follow the formalism of Bernard and Zunger.⁸⁰ First, the experimentally observed bowing b_exp is split into a parameter b_I that accounts for the effects already included in an ordered alloy and a parameter b_II that accounts for effects due to disorder,

The former part b_I can be further separated into three contributions: volume deformation b_VD, the impact of different chemical electronegativities b_CE, and the effects of full relaxation of the atomic positions within the unit cell of the Ag_xCu_1–xI alloy b_S. The individual contributions can be calculated by comparing the energy transition levels of the binaries with those of the Ag_0.5Cu_0.5I alloy as follows:

where ϵ_CuI and ϵ_AgI denote the energy transition levels of the perfect binary compounds for different lattice constants, and a_CuI, a_AgI, and a_0.5 are the lattice constants of CuI, AgI, and the Ag_0.5Cu_0.5I alloy, respectively. Moreover, u_unrel and u_rel denote the internal atomic degrees of freedom in a perfect undeformed zincblende unit cell and a fully relaxed unit cell, respectively. The total bowing parameter is then obtained by adding the individual effects. The corresponding energy transition levels and the bowing parameters deduced by our calculations are given in Table II. We note that for Ag_0.5Cu_0.5I, we again use thermal averages via Eq. (1).

Interestingly, the contribution of the volume deformation $b̄VD$ almost fully describes the overall bandgap bowing in the Ag_xCu_1–xI alloy. This is quite surprising since the volume deformation of the unit cell to the lattice constant of the Ag_0.5Cu_0.5I alloy within the virtual-crystal approximation (VCA) approach is expected to overestimate the changes in the bond lengths⁸¹ and, thus, the corresponding effect on the bandgap energy, as discussed for III–V alloys.⁸² Indeed, we find that the calculated element-specific bond lengths in a relaxed Ag_xCu_1–xI unit cell are l_Cu–I = 2.59 Å and l_Ag–I = 2.78 Å and are, thus, much closer to their binary values (⁠$lCu−Ibin=2.57Å$ and $lAg−Ibin=2.79Å$⁠) than predicted by the VCA approach (l_Cu∕Ag–I = 2.68 Å) assuming equal bond lengths for different cations (see Fig. S5 in the supplementary material). Therefore, $b̄VD$ should, in fact, strongly overestimate the structural contribution to the bowing behavior and has to be compensated by $b̄S$ in the last step, where the anion position is allowed to relax to its minimum energy configuration. However, we note that $b̄S$ contains not only structural but also electronic effects due to charge redistribution for unequal bond lengths.^80,81 Although $b̄S$ vanishes completely only due to the thermal averaging of the bandgap values obtained for different unit cell configurations, all individual b_S contributions are in the range between −0.05 eV and 0.12 eV. This indicates that the structural and electronic effects almost cancel out in the last structural relaxation step for the Ag_xCu_1–xI alloy. This in turn means that, although charge distribution effects do not play a role in the case of equal bond lengths due to the very similar electronegativity values of copper and silver according to the Pauling scale,⁸³ as indicated by the small value $b̄CE$⁠, the electronic contribution has a major effect on the bandgap bowing in the case of unequal element-specific bond lengths.

To distinguish more clearly between structural and electronic contributions to the bandgap bowing, another two-step approach introduced in Ref. 82 can be used, where the unit cell is deformed directly in the first step to reproduce the individual first neighbor distances found in the Ag_0.5Cu_0.5I alloy. The corresponding computational details illustrating the different impacts of structural and electronic contributions for the lowest energy configuration are treated in Sec. V in the supplementary material. Here, it should only be mentioned that both approaches give very similar results, although the importance of charge redistribution effects is much more evident in the two-step approach.

Finally, we note that our findings are comparable to the bowing caused by cation mixing in (In,Ga)P and (In,Ga)As alloys,⁸¹ but they differ significantly from the situation in the recently reported CuBr_xI_1–x alloy,⁵⁹ where the charge relaxation due to different chemical electronegativities of Br and I plays a major role even in case of equal bond lengths.

In addition to the bandgap bowing, we want to discuss the influence of the alloy composition on the value of the spin–orbit splitting at the Γ-point Δ₀. The experimental values are simply obtained from the energy difference of the peak positions E₀ and E₀ + Δ₀ for the zincblende phase. In the DFT simulations, the spin–orbit splittings are obtained as the energy differences calculated for the bands at the Γ-point. Moreover, in this case, Eq. (1) is used to calculate the thermal averages to account for different atomic configurations. The results for the SOC splitting are summarized in Fig. 4(a). We note that the splitting increases slightly with increasing Ag-content. Although our DFT calculations overestimate the experimental values by (5%–10%), the DFT approach still provides a reasonable description of the impact of Ag incorporation on the SOC splitting energy. As discussed in Sec. IV A, due to the mixed-phase structure of the AgI sample, a unique value for the SOC splitting in the γ-phase cannot be determined.

To investigate the influence of the alloy composition on the spin–orbit splitting more in detail, we examine the effect of the p-d hybridization of the top valence band at the Γ-point, which is typical for all Ag/Cu halides^44,85 and is usually attributed to the contribution from the covalent component in the chemical anion–cation bonding.⁸⁶ Here, we extract the contributions of both metal (Cu/Ag) and halogen atoms (I) to the VB wavefunctions directly from our DFT calculations, considering explicitly the values for the split-off VB and using thermal averages over non-equivalent configurations. We find that the contribution of the Ag/Cu d states to the uppermost VB increases as one moves to lower energies from the VBM [see Figs. 4(c) and 4(d)], indicating that the inclusion of SOC is crucial for extracting these values.

The experimental values are calculated by using a simple empirical model introduced by Cardona,⁴⁴ 

where α represents the contribution of the halogen atom to the VBM wavefunctions at the Γ-point and Δ_halogen (Δ_metal) are contributions of the halogen (metal) atom to the one-electron spin–orbit splittings. For the one-electron spin–orbit splitting energies of Cu, Ag, and I, we use here the values of 0.1, 0.23, and 0.94 eV, respectively.⁴⁴ 

In Fig. 4(b), we compare the experimental and calculated contributions (1 − α) of the metal atoms to the wavefunctions. For binary CuI, we obtain a value of (1 − α) = 0.5, revealing a large contribution of copper atoms to the wavefunctions at the VBM and, thus, a strong degree of p-d hybridization at the Γ-point, which is in good agreement with the literature.^44,86,87 As can be seen in Fig. 4(b), the influence of the metal atoms (Cu/Ag) on the VBM decreases with increasing Ag content, in agreement with our DFT calculations. Interestingly, the contribution of the metal atoms to the VB wavefunction appears to decrease strongly for AgI, which might explain the strong increase in the corresponding spin–orbit splitting as well.

Although the observed composition dependence of the spin–orbit splitting does not exhibit a parabolic behavior like the bandgap energy, it is worth noting that the Δ₀ value for the Ag_0.5Cu_0.5I alloy is smaller than the value that would be expected from a linear interpolation between the two binary compounds. This result is in contrast to the behavior of the CuBr_xI_1–x alloy. Finally, we note that although the observed decrease in the contribution of the metal d orbitals at the VBM with increasing Ag-content [see Fig. 4(b)] should result in increasing bandgap energy,⁴⁴ the observed transition energy E₀ is still smaller in AgI than in CuI. This indicates a strong decrease in the bandgap due to increasing bond lengths, which are l_Cu–I = 2.57 Å and l_Ag–I = 2.79 Å for Cu–I and Ag–I bonds, respectively. This differs from the case of the CuBr_xI_1–x alloy, where the increase in the bandgap energy caused by a decreasing bond length is almost compensated by the effect of increasing p-d hybridization at the VBM.⁵⁹ 

To study the effect of temperature on the E₀ transition energy, temperature-dependent reflectivity measurements were performed between 20 K and 290 K. Figures 5(a) and 5(b) show the evolution of the reflectivity spectra with increasing temperature for binary CuI and AgI with increasing temperature, respectively. The resulting E₀ values are plotted as a function of temperature for different alloy compositions in Fig. 5(c). For CuI, the E₀ resonance shifts monotonically with increasing temperature to lower energies, as previously reported.^10,58 In this case, the observed energy shift of the bandgap between 20 K and room temperature is 25 meV. With increasing Ag concentration, this energy shift decreases approximately linearly to a value of 5 meV for binary AgI [see inset in Fig. 5(c)]. While for alloy compositions with x ≤ 0.76, the transition energy E₀ still remains a monotonic function of temperature, the situation changes considerably for binary AgI. In this case, the temperature-dependent bandgap shift no longer shows a monotonic behavior, but it increases by about 2 meV up to about 100 K before decreasing at higher temperatures.

A commonly used model to describe the dependence of the bandgap on the temperature is the simple Bose–Einstein model,⁸⁸ which describes the interaction with an effective phonon bath at a given energy. Although such a simple model could be, in principle, used for the phenomenological description of the situation of CuI, it cannot explain the non-monotonic behavior of the bandgap energy for binary AgI. Therefore, we extend this model, considering interaction with two different phonon branches, namely, the optical and acoustic phonons, similarly to the approach presented by Serrano et al.⁸⁹ The temperature-dependent transition energy can then be described as follows:

where $E00K$ is the transition energy at 0 K, and α_opt∕ac and k_BΘ_opt∕ac are the coupling strengths and the effective phonon energies of optical and acoustic phonon branches, respectively. For further discussion, we use averaged phonon energies of the optical and acoustic branches obtained considering the corresponding density of states.^90,91 The phonon energies used here as well as the determined coupling strengths obtained by using Eq. (8) are summarized in Table III.

The coupling strength α_opt of the optical phonons has a positive sign in this model, whereas the interaction with the acoustic phonons is described by a negative α_opt. This is in good agreement with previous results for copper halides, where the vibrations of the halogen atoms were assigned mainly to optical phonons, whereas the vibrations of copper atoms were shown to be responsible for the acoustic modes.^89,92,93 However, we note that the values discussed for the coupling strengths include not only the effects of the electron–phonon interaction but also the effects of lattice expansion on the bandgap energy. Here, in particular, for CuI, the contribution of thermal lattice expansion, which has the same sign as the previously discussed contribution of optical phonons, is expected to account for about 50% of the total bandgap shift between 20 K and room temperature.¹⁰ For a detailed discussion of the electron–phonon interaction, it is therefore crucial to explicitly consider the effects of lattice expansion on the bandgap. For this purpose, we use here the recently published values for the temperature-dependent lattice constant, as well as the value of ∂E/∂ ln V = −1.1 eV for the deformation potential of CuI.⁷⁰ By separating the two different effects, as recently discussed in Ref. 10, we obtain a somewhat lower value for the coupling constant of 0.17 meV/K compared to the value of 0.23 meV/K obtained without explicitly considering the thermal expansion effect. The coupling strength for the interaction with acoustic phonons α_ac remains unaffected.

Since the absolute value of the thermal expansion coefficient for AgI is expected to be about an order of magnitude smaller than that of CuI,^70,72,75 the contribution of lattice expansion to the bandgap energy is likely to be rather small in comparison to CuI. However, in addition to the lattice expansion, the deformation potential must also be included in the discussion. Since, to our knowledge, there are no literature values for the deformation potential in AgI, we have performed further DFT calculations for different volumes of the unit cell to simulate the effect of hydrostatic pressure on the bandgap energy for both binary materials (see Fig. S6 in the supplementary material). For the deformation potential ∂E/∂ ln V of CuI, we obtain a value of −1.8 eV, which slightly overestimates the previously mentioned literature value. Interestingly, a positive deformation potential is calculated for AgI with a value of 0.3 eV, which is almost an order of magnitude smaller than for CuI. Taking now into account the temperature-dependent lattice constants of γ-AgI,⁷⁵ the contribution of the thermal expansion to the temperature-dependent energy shift of the bandgap between 10 K and 290 K is estimated to be less than 1 meV. Since our DFT calculations seem to overestimate the absolute values of the deformation potential, at least for CuI, the effect of thermal lattice expansion on the bandgap energy of AgI might be even smaller and thus can be neglected in this case. Therefore, the values obtained for the coupling strengths by using Eq. (8) considering only electron–phonon interaction are listed in Table III. Interestingly, the coupling strengths for optical and acoustic phonons determined for both binary materials are quite similar, with α_opt being slightly larger for CuI compared to AgI. Thus, the observed changes in the temperature dependence of the bandgap energy can be primarily attributed to the decreasing impact of thermal lattice expansion with increasing Ag-content.

We note that without precise knowledge of the deformation potential and temperature-dependent thermal lattice expansion coefficients as a function of the alloy composition, it is difficult to clearly separate the effects of lattice expansion and electron–phonon interaction and, thus, to provide the coupling strengths as a function of alloy composition. Nevertheless, we provide the slope dE₀/dT in the temperature range between 125 K and 290 K as a function of alloy composition in Fig. 5(d) for practical reasons.

We experimentally determined the transition energies E₀ and E₀ + Δ₀ for Ag_xCu_1–xI alloy thin films as a function of the alloy composition. The experimental results are compared with the first-principles band structure calculations performed for different alloy compositions. We show that the E₀ energy reveals a non-linear bowing behavior that can be described by a bowing parameter of b_exp ≈ 0.54 eV. The observed bowing is well described by our DFT calculations for thermal averages of ordered alloys that yield a bowing parameter of 0.53 eV. Different contributions to the bowing parameter, such as volume deformation, different chemical electronegativities, and structural relaxation, are studied following the approach presented by Bernard and Zunger, and comparing it with the approach introduced by Schnohr allows for a better distinction between structural and electronic contributions. We show that the electronic contribution has a major effect on the bandgap bowing in case of unequal element-specific bond lengths. In contrast to that, the effect of charge redistribution at equal bond lengths can be almost neglected due to the similar electronegativities of Cu and Ag, contrary to the case of CuBr_xI_1–x alloys, where this effect plays a major role.⁵⁹ The spin–orbit splitting Δ₀ is found to increase from a value of 640 meV for CuI to ∼790 meV for AgI. The strong increase in the spin–orbit splitting, especially for the binary AgI, was explained in terms of decreasing p-d hybridization of the valence bands. Furthermore, the effect of thermally induced strain on the degenerated uppermost valence bands at low temperatures is investigated. We show that the thermal expansion coefficient of Ag_xCu_1–xI can be matched to that of the Al₂O₃ substrate for alloy compositions around x ≈ 0.6, leading to a stress-free state that can be important for future device applications.

In addition, we investigated the effect of temperature on the E₀ transition for different alloy compositions. The bandgap shift between 20 K and room temperature decreases from 25 meV for CuI to 5 meV for AgI, whereas the bandgap temperature dependence reveals a non-linear behavior for the latter one. We attribute the change in the temperature-dependent bandgap shift almost exclusively to the change in the lattice expansion contribution induced by the changes in the thermal expansion coefficient. We discuss the non-monotonic behavior of the bandgap for AgI in terms of electron–phonon renormalization due to the interaction with optical and acoustic phonons.

Finally, we emphasize that the Ag_xCu_1–xI alloy studied here provides the possibility of developing a material with specific desired properties, including the engineering of the excitonic transition energy and spin–orbit splitting, as well as their temperature dependence and even the targeted control of the thermal expansion coefficient, for use in optoelectronic devices.

See supplementary material for additional information on atomic arrangements for the lowest energy configuration of the compositions discussed in manuscript, band structure calculations of zincblende- and wurtzite-phase AgI, exciton binding energy, and the effect of different ordered configurations on bandgap bowing as well as the calculation of the bowing contributions.

This study was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through FOR 2857 (Project Nos. P02, P04, P06, and P07) under Project No. 403159832. E.K. acknowledges the Leipzig School of Natural Sciences BuildMoNa. M.S. and S.B. acknowledge the Leibniz Supercomputing Center for providing computational resources (Project No. pn68le). Further funding was provided by the Open Access Publishing Fund of Leipzig University supported by the German Research Foundation within the program Open Access Publication Funding.

The authors have no conflicts to disclose.

E.K. and M.S. contributed equally to this work.

Evgeny Krüger: Formal analysis (lead); Investigation (equal); Methodology (equal); Project administration (lead); Validation (equal); Visualization (equal); Writing – original draft (lead). Michael Seifert: Formal analysis (lead); Investigation (equal); Methodology (equal); Project administration (lead); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (supporting); Writing – review & editing (equal). Volker Gottschalch: Resources (equal); Writing – review & editing (equal). Harald Krautscheid: Conceptualization (supporting); Funding acquisition (equal); Project administration (supporting); Resources (equal); Supervision (supporting); Writing – review & editing (equal). Claudia S. Schnohr: Conceptualization (supporting); Funding acquisition (equal); Project administration (supporting); Supervision (supporting); Writing – review & editing (equal). Silvana Botti: Conceptualization (lead); Funding acquisition (equal); Project administration (supporting); Resources (equal); Software (equal); Supervision (lead); Writing – review & editing (supporting). Marius Grundmann: Conceptualization (lead); Funding acquisition (equal); Project administration (supporting); Resources (equal); Supervision (lead); Writing – review & editing (equal). Chris Sturm: Conceptualization (lead); Funding acquisition (equal); Project administration (supporting); Resources (equal); Supervision (lead); Writing – review & editing (equal).

The data that support the findings of this study are available within the article and its supplementary material or are available from the corresponding author upon reasonable request.