Crystal Growth and the Structure of a New Quaternary Adamantine Cu☐GaGeS₄

Abstract

Single crystals of quaternary adamantine-type Cu☐GaGeS₄ were grown using the chemical vapor transport technique, with iodine as the transport agent. Dark red transparent crystals were grown in a temperature gradient of ΔT = 900–750 °C. Chemical characterization by X-ray fluorescence showed the off-stoichiometric composition of Cu☐GaGeS₄ crystals—in particular, a slight Ge deficiency was observed. By X-ray diffraction, Cu☐GaGeS₄ was found to adopt the chalcopyrite-type structure with the space group $I\overline{4}2d$. Cation distribution in this structure was analyzed by multiple energy anomalous synchrotron X-ray diffraction, and it was found that Cu and vacancies occupied the 4a site, whereas Ga and Ge occupied the 4b site. The band gap energies of several off-stoichiometric Cu☐GaGeS₄ crystals were determined by UV-Vis spectroscopy and ranged from 2.1 to 2.4 eV. A non-linear correlation of the band gap energy with the Ge content of the compound was shown to follow the usual bowing behavior of semiconductor alloys, with a bowing parameter of $b$= −1.45 (0.08).

1. Introduction

“Adamantine”-type compounds, including kesterites, are currently the most promising material for the development of a fully inorganic thin-film photovoltaic technology that is free of critical raw materials and thus provides sustainable solutions. The highest efficiency for fully inorganic thin-film photovoltaic technologies is exhibited by thin-film solar cells based on the ternary chalcopyrite-type compound semiconductor Cu(In,Ga)S₂ (CIGS), with a record power conversion efficiency of 23.6% [1]. This absorber material belongs to the ternary A^IB^IIIX^VI₂ chalcopyrite-type compound, belonging to the adamantine compound family as well. One focus of research in solar energy conversion is hybrid halide perovskites, which result in thin-film devices with a record efficiency of 26.1% [1]. Despite their undisputed high efficiency, hybrid halide perovskites have some drawbacks; materials (and thus solar cell devices) have no long-term stability, and they contain lead, which is a critical toxicity issue. Furthermore, the use of Pb-containing absorber materials is not in line with the EU Directive 2002/95/EC (also known as the RoHS Directive), thereby restricting the use of lead in electrical and electronic equipment [2]. Therefore, the development of compound semiconductors that are free of critical raw materials and are long-term stable is crucial for the transition away from fossil fuels and the move toward a greener energy future. These compound semiconductors are promising materials both in single-junction solar cells and as top absorber layers in tandem cells.

The ternary A^IB^IIIX^VI₂ chalcopyrite compound family, which includes the compound semiconductor Cu(In,Ga)Se₂—used as an absorber layer in high-efficiency thin-film solar cells—can be transformed into two different quaternary adamantines by chemical substitutions by obeying the valence octet rule. Due to added chemical and structural freedom, these quaternary compounds have some novel and exciting properties.

The first possible substitution would be $2B^{III}\leftrightarrow B^{II}+C^{IV}$, resulting in quaternary compound semiconductors A^I₂B^IIC^IVX^VI₄. Such materials have recently attracted considerable attention because of their potential low costs and high-efficiency solar cell absorbers. Thin-film solar cells based on kesterite-type Cu₂ZnSn(S,Se)₄ (CZTSSe) not only show a record power conversion efficiency of 14.9% [1] but also show excellent long-term stability and do not contain toxic or critical components. Due to these advantages, kesterite-based thin-film solar cells are currently the only critical raw material-free and the most promising fully inorganic photovoltaic technologies. Cation alloying for band gap engineering and the extrinsic doping of kesterites to increase the power conversion efficiency of related devices have been performed by several groups. Alloying CZTSe with Ge generates solar cells with an efficiency of 12.3% [3] and alloying CZTSSe with Ag pushes the device’s efficiency to 12.96% [4]. In these Ag-alloyed kesterites, mixed crystals adopt a stannite-type structure, completely blocking Cu/Zn disorder [5]—a possible reason for high V_OC deficits in CZTSSe-based devices. Extrinsic doping with alkali metals has been successfully used in chalcopyrite-type CIGS to achieve high energy-conversion efficiencies [6]. This approach has also been successfully adopted for kesterites. Extrinsic doping with Li can also shift the efficiency of CZTSSe to above 10% [7].

The second possible substitution would be ${A^I+B}^{III}\leftrightarrow C^{IV}+$ ☐, resulting in quaternary semiconductors A^I☐B^IIIC^IVX^VI₄ (the symbol ☐ indicates cation vacancies, i.e., empty cation sites in the crystal structure). These compounds are referred to as defect adamantines [8] and can be considered potential absorber materials for thin-film solar cell applications. However, studies on defect adamantines have made little progress since the first report by Pamplin [8]. Only a few studies on defect adamantine selenides have been published [9,10,11], but less information is available on sulfides [12,13]. Currently, this class of material is undergoing a renaissance because of its potential use in photovoltaic applications. For example, two studies on electronic structures and optical properties of defect adamantines have been performed by Shen et al. [14,15]. These investigations, using first-principle calculations, showed that the values of band gap energies of Cu☐GaSnSe₄ and Cu☐GaGeSe₄ were suitable for photovoltaic applications. Experimental studies on Cu☐GaGeSe₄ thin films [16] have shown that these compounds can be successfully used as absorber layers in thin-film solar cells. Additionally, non-linear optical parameters, such as the non-linear refractive index, have been determined [16]. Thin-film solar cells based on a Cu☐InGeSe₄ absorber showed a power conversion efficiency of 2.38% [17]. The crystal structure of the defect adamantine Cu☐AlGeSe₄ was studied by Quintero et al. [18].

In general, little is known about sulfide adamantines. In this work, a detailed study of the structural and optoelectronic properties of Cu☐GaGeS₄ was performed by preparing single crystals as representative materials.

1.1. Structural Considerations

Adamantines crystallize in tetrahedrally coordinated structures and were derived from diamond-type (space group $Fd\overline{3}m$) and lonsdaleite-type (space group $P\frac{6_3}{m}mc$) crystal structures. Pamplin [8] summarized 249 adamantines derived from these two different crystal structures of carbon (diamond and lonsdaleite). A pre-requisite is not only the tetrahedral coordination of cations by anions and vice versa, but also that four electrons exist per structural site and that the number of cations and anions is equal. In the case where a structural site is a vacancy, such as in A^I☐B^IIIC^IVX^VI₄ compounds, the compounds are referred to as “defect adamantine”.

Ternary compounds of the adamantine family, with general formula A^IB^IIIX^VI_2, crystallize in the chalcopyrite-type crystal structure (space group $I\overline{4}2d$) and belong to the tetragonal crystal system. In this crystal structure, each metal ion is tetrahedrally coordinated with four chalcogen anions and vice versa. Each anion is bonded to two mono-valent and two three-valent cations (Figure 1). In the chalcopyrite-type crystal structure, the A^I cation occupies the Wyckoff position 4a at (0, 0, 0), and the B^III cation occupies the 4b position at (0, 0, 0.5). The anions occupy the Wyckoff position 8d at (x, 0.25, 0.125).

Quaternary compounds of the adamantine family are of A^I₂B^IIC^IVX^VI₄ and A^I☐B^IIIC^IVX^VI₄ types. Figure 1 shows that the distribution of metal ions is important and determines the crystal structure of the quaternary compound. A^I☐B^IIIC^IVX^VI₄ defect adamantines are formed from ternary chalcopyrite-type compounds by doubling the entire formula unit (A^IB^IIIX^VI₂) and replacing one A¹⁺ and one B³⁺ cation with one C⁴⁺ cation. This results in a balanced valence but an unbalanced cation–anion ratio (with respect to the ratio of cation and anion sites in the chalcopyrite-type structure). Accordingly, the structure must be compensated by vacancies (A^I + B^III ↔ vacancy + C^IV).

1.2. Thermodynamic Properties

Limited information is available on the properties of adamantine compounds of interest. Table 1 shows structural information and the melting points and band gap energies of Cu☐GaGeX^VI₄ with X = S, Se.

Table 1. Material properties of the defect adamantines Cu☐GaGeS₄ and Cu☐GaGeSe₄.

.	Cu☐GaGeS4		Cu☐GaGeSe4
	tetragonal [13]	tetragonal $I\overline{4}2d$ [19]	tetragonal $I\overline{4}2d$ [20]	tetragonal $I\overline{4}2d$ [21]
a (Å)	5.334	5.302	5.568	5.5617
c (Å)	10.050	10.212	10.841	10.9238
Tmelting (°C)	1000 [9]		1000 [22]	836 [9]
Tdecomposition (°C)			710 [22]
Eg (eV)	2.73 [23]		1.85 [20]	1.38 [10]

Table 1. Material properties of the defect adamantines Cu☐GaGeS₄ and Cu☐GaGeSe₄.

.	Cu☐GaGeS4		Cu☐GaGeSe4
	tetragonal [13]	tetragonal $I\overline{4}2d$ [19]	tetragonal $I\overline{4}2d$ [20]	tetragonal $I\overline{4}2d$ [21]
a (Å)	5.334	5.302	5.568	5.5617
c (Å)	10.050	10.212	10.841	10.9238
Tmelting (°C)	1000 [9]		1000 [22]	836 [9]
Tdecomposition (°C)			710 [22]
Eg (eV)	2.73 [23]		1.85 [20]	1.38 [10]

Little is known about the phase diagram of CuGaS₂-GeS₂. The phase diagram of Cu₂GeSe₃-Ga₂GeSe₅ [24] is used as a reference point for material synthesis (Figure 2). Very complex phase relations in the class of adamantines are obvious here. The presence of peritectic and eutectic points and phase transitions results in rather difficult conditions for single crystal growth.

The most appropriate single crystal growth method is chemical vapor transport (as introduced by Schäfer [25] and Nitsche [26]). With chemical vapor transport (CVT) enhanced by halogens, crystals can be grown below critical temperatures. Such crystals will grow close to thermodynamic equilibrium.

In this study, we report the growth of Cu☐GaGeS₄ single crystals using the chemical vapor transport technique. Starting with ternary compound CuGaS₂, conditions for growing the single crystals of Cu☐GaGeS₄ are investigated. The evolved material and crystals are characterized with respect to chemical composition, crystal structure, and band gap energy.

2. Materials and Methods

2.1. Crystal Growth

Crystal growth experiments were performed using the chemical vapor transport technique (CVT). The starting elements Cu, Ga, Ge, and S were weighed into a glassy carbon boat, which was placed in a quartz glass ampoule of 28 mm diameter and 200 mm length. A halogen element, such as iodine, was used to enhance transport. The ampoule was evacuated, closed, and placed in a two-zone furnace. The iodine reacted with metals at elevated temperatures to form volatile species. Halogen-containing species were then transported from the hot (900 °C) to the cold (750 °C) part of the ampoule. Transport occurred based on the reaction in Equation (1).

$$\begin{gathered} Cu\left(s\right)+Ga\left(s\right)+Ge\left(s\right)+4S\left(s\right)+{4I}_2\left(s\right)\leftrightarrow CuI\left(g\right)+Ga{I}_3\left(g\right)+Ge{I}_4\left(g\right)+ \\ 2S_2\left(g\right)\leftrightarrow CuGaGe{S}_4\left(s\right)+{4I}_2\left(s\right). \end{gathered}$$

(1)

After cooling to ambient temperatures, ampoules were opened, and gaseous species present in the ampoule were allowed to evaporate.

In chemical vapor transport, growth conditions such as the temperature of the source, temperature gradient, and the concentration of the transport agent are of great importance for crystal growth. All formed phases were analyzed. The results of growth experiments—in this case, occurring phases—showed how important the consideration of the gaseous phase was for transport in a closed system. The main challenge here was to control the composition of the gas phase and optimize the temperature field during growth.

For the growth of CuGaGeS₄ crystals, most conditions were the same for all experiments; only the concentration of the transport agent iodine varied (see Table 2). The exact weights of metals and sulfur were calculated for 5 g of stoichiometric material. A temperature gradient of ΔT = 900 °C–750 °C was maintained for 240 h of growth times.

2.2. Chemical Characterization

Wavelength dispersive X-ray spectroscopy (WDX) was used to determine the composition of present phases; an electron microprobe analysis system was used. In order to obtain reliable results from WDX measurements, the microprobe system was calibrated using NIST elemental standards. A high accuracy of compositional parameters was achieved by averaging over 10 local measured points within one grain and averaging over more than 30 grains of the quaternary phase showing the same compositional values.

Additionally, composition was measured by X-ray fluorescence (XRF) using a Bruker M4 Tornado Micro-XRF spectrometer with Rh excitation beams and two detectors.

2.3. Structural Characterization

X-ray diffraction: Grown crystals were characterized by powder X-ray diffraction (XRD). X-ray powder diffraction data were recorded over a 2θ range of 10–140° with a step size of 0.01313° by means of a BRUKER D8 diffractometer using Cu Kα_1,2 radiation at wavelengths of 1.540598 Å and 1.544426 Å, respectively. Lattice parameters of the material were determined via a LeBail analysis of the diffraction pattern.

Multiple Edge Anomalous Diffraction (MEAD): To clarify cation distribution, anomalous X-ray powder diffraction data (AXRPD) from Cu☐GaGeS₄ were collected at the KMC-2 Diffraction station at the KMC-2 beamline, BESSY II, Berlin, Germany [27], and analyzed as described in the literature [28]. Intensity scans of the 101 Bragg peak in Multiple Edge Anomalous Diffraction (MEAD) analysis were collected at the Κ X-ray absorption edges of the elements Cu, Ga, and Ge. Experimental absorption edges were observed at energies of 8987(1) eV, 10,370(1) eV, and 11,104(1) eV. These values did not deviate significantly from the literature values of 8979 eV (Cu-Κ), 10,367 eV (Ga-Κ), and 11,103 eV (Ge-Κ) [29], with the notable exception of the Cu-Κ edge, which was 8 eV higher than the reference value. Absorption corrections for experimental data were calculated based on the chemical composition of the sample, as determined experimentally by WDX. Full powder diffraction sets within the 2θ-range of 6–132° were collected at energies of 8048 eV (λ = 1.5406 Å, equivalent to Cu Kα₁) and below the absorption edges at 8965 eV, 10,353 eV, and 11,089 eV.

2.4. Optical Characterization

Diffuse Reflectance Spectroscopy (DRS) measurements were carried out in the air and at room temperature using a spectrophotometer equipped with an integrating sphere (Perkin Elmer UV/Vis-spectrometer Lambda 750S). The wavelength range of the measurement was adjusted to 800–1800 nm with a step size of 1 nm. Tauc plots were obtained by plotting (F(R)·hν)² versus photon energies [30]. The linear part of the curve was extrapolated relative to the baseline, and the optical band gap was extracted from the value of the intersection.

Table 2. Chemical composition of off-stoichiometric Cu☐GaGeS₄ crystals determined by X-Ray fluorescence analysis and cation ratios. Iodine concentrations were used for crystal growth. Lattice parameters of crystals were determined using the LeBail analysis of X-ray diffraction data.

Sample No.	Chemical Formular	Cu/(Ga+Ge)	Ge/(Ga+Ge)	Iodine Concentration (mg/cm3)	a (Å)	c (Å)	V (Å3)
To589	Cu0.93☐1.08Ga0.86Ge1.13S4	0.467	0.568	0 *	5.330(1)	10.203(2)	289.856
To557	Cu1.10☐0.90Ga1.09Ge0.91S4	0.550	0.455	4.9	5.315(1)	10.114(2)	285.713
To569	Cu1.18☐0.83Ga1.15Ge0.84S4	0.593	0.422	5.0	5.324(1)	10.187(2)	288.750
To585w	Cu1.22☐0.78Ga1.25Ge0.77S4	0.604	0.381	8.1	5.324(1)	10.184(2)	288.265
To597	Cu1.27☐0.74Ga1.24Ge0.75S4	0.638	0.377	4.7	5.319(1)	10.189(2)	287.308
To555	Cu1.27☐0.75Ga1.22Ge0.76S4	0.641	0.384	4.9	5.318(1)	10.159(2)	287.845
To329	Cu1.27☐0.75Ga1.20Ge0.78S4	0.641	0.394	5.0	5.318(1)	10.178(2)	287.524
To556	Cu1.30☐0.69Ga1.24Ge0.77S4	0.647	0.383	4.9	5.320(1)	10.159(2)	287.650
To581	Cu1.29☐0.72Ga1.23Ge0.76S4	0.648	0.382	5.0	5.323(1)	10.152(2)	288.665
To585	Cu1.30☐0.72Ga1.23Ge0.75S4	0.656	0.379	8.1	5.324(1)	10.184(2)	287.642
To583	Cu1.30☐0.72Ga1.22Ge0.76S4	0.657	0.384	5.0	5.319(1)	10.167(2)	286.942
To531	Cu1.33☐0.68Ga1.27Ge0.72S4	0.668	0.362	5.0	5.323(1)	10.127(2)	289.856
To591	Cu1.35☐0.67Ga1.29Ge0.69S4	0.682	0.348	1.1	5.324(1)	10.226(2)	291.816
To599	Cu1.35☐0.68Ga1.24Ge0.73S4	0.685	0.371	3.1	5.330(1)	10.272(2)	292.484
To580	Cu1.36☐0.66Ga1.28Ge0.70S4	0.687	0.353	5.0	5.362(1)	10.173(2)	291.816
To596	Cu1.47☐0.54Ga1.42Ge0.57S4	0.739	0.286	4.7	5.330(1)	10.272(2)	288.665
To588	Cu1.76☐0.27Ga1.63Ge0.34S4	0.893	0.173	1.1	5.340(1)	10.365(9)	295.564
literature
	CuGaS2 [31]	0.991	0		5.355(1)	10.485(2)	300.668
	CuGaS2 [31]	0.995	0		5.356(1)	10.483(2)	300.723
	GeS2 [32]	0	1		5.68	8.97	287.781

* Solid state reaction without iodine.

Table 2. Chemical composition of off-stoichiometric Cu☐GaGeS₄ crystals determined by X-Ray fluorescence analysis and cation ratios. Iodine concentrations were used for crystal growth. Lattice parameters of crystals were determined using the LeBail analysis of X-ray diffraction data.

Sample No.	Chemical Formular	Cu/(Ga+Ge)	Ge/(Ga+Ge)	Iodine Concentration (mg/cm3)	a (Å)	c (Å)	V (Å3)
To589	Cu0.93☐1.08Ga0.86Ge1.13S4	0.467	0.568	0 *	5.330(1)	10.203(2)	289.856
To557	Cu1.10☐0.90Ga1.09Ge0.91S4	0.550	0.455	4.9	5.315(1)	10.114(2)	285.713
To569	Cu1.18☐0.83Ga1.15Ge0.84S4	0.593	0.422	5.0	5.324(1)	10.187(2)	288.750
To585w	Cu1.22☐0.78Ga1.25Ge0.77S4	0.604	0.381	8.1	5.324(1)	10.184(2)	288.265
To597	Cu1.27☐0.74Ga1.24Ge0.75S4	0.638	0.377	4.7	5.319(1)	10.189(2)	287.308
To555	Cu1.27☐0.75Ga1.22Ge0.76S4	0.641	0.384	4.9	5.318(1)	10.159(2)	287.845
To329	Cu1.27☐0.75Ga1.20Ge0.78S4	0.641	0.394	5.0	5.318(1)	10.178(2)	287.524
To556	Cu1.30☐0.69Ga1.24Ge0.77S4	0.647	0.383	4.9	5.320(1)	10.159(2)	287.650
To581	Cu1.29☐0.72Ga1.23Ge0.76S4	0.648	0.382	5.0	5.323(1)	10.152(2)	288.665
To585	Cu1.30☐0.72Ga1.23Ge0.75S4	0.656	0.379	8.1	5.324(1)	10.184(2)	287.642
To583	Cu1.30☐0.72Ga1.22Ge0.76S4	0.657	0.384	5.0	5.319(1)	10.167(2)	286.942
To531	Cu1.33☐0.68Ga1.27Ge0.72S4	0.668	0.362	5.0	5.323(1)	10.127(2)	289.856
To591	Cu1.35☐0.67Ga1.29Ge0.69S4	0.682	0.348	1.1	5.324(1)	10.226(2)	291.816
To599	Cu1.35☐0.68Ga1.24Ge0.73S4	0.685	0.371	3.1	5.330(1)	10.272(2)	292.484
To580	Cu1.36☐0.66Ga1.28Ge0.70S4	0.687	0.353	5.0	5.362(1)	10.173(2)	291.816
To596	Cu1.47☐0.54Ga1.42Ge0.57S4	0.739	0.286	4.7	5.330(1)	10.272(2)	288.665
To588	Cu1.76☐0.27Ga1.63Ge0.34S4	0.893	0.173	1.1	5.340(1)	10.365(9)	295.564
literature
	CuGaS2 [31]	0.991	0		5.355(1)	10.485(2)	300.668
	CuGaS2 [31]	0.995	0		5.356(1)	10.483(2)	300.723
	GeS2 [32]	0	1		5.68	8.97	287.781

* Solid state reaction without iodine.

3. Results and Discussion

3.1. Crystal Growth

In CVT growth experiments, single crystals of Cu☐GaGeS₄ up to 10 mm in length were obtained, as shown in Figure 3. Crystalline materials had a dark red and orange color. In growth experiments, in addition to transparent red-orange Cu☐GaGeS₄ crystals, other phases also appeared, such as GeS₂ and GaI₃.

3.2. Chemical Composition and Off-Stoichiometry Relations

In analyzing the chemical composition of defect adamantine, chemical analysis of grown crystals by X-ray fluorescence (XRF) revealed that crystals showed Cu/(Ga+Ge) ratios between 0.45 and 0.9 and Ge/(Ga+Ge) ratios between 0.15 and 0.6 (see Table 2). Thus, single crystals showed quite a strong deviation from the stoichiometric composition, which was in accordance with Cu/(Ga+Ge) = Ge/(Ga+Ge) = 0.5. Therefore, defect adamantine Cu☐GaGeS₄ was seen as a compound that was formed within the solid solution between CuGaS₂ and GeS₂, which can be described by (CuGaS₂)_1−x(GeS₂)_x (see Figure 4). The general Cu_2(1−x)☐_2(1−x)Ga_2(1−x)Ge_2xS₄ formula can be applied to describe the off-stoichiometric composition of the material, x = 0.5, resulting in the stoichiometric Cu☐GaGeS₄ composition.

According to this general formula, the chemical compositions for compounds with x = 0 to x = 1 were calculated (calculation 1 and the solid line in Figure 4). Nevertheless, it is well known that CuGaS₂ and CuGaSe₂ exhibit an off-stoichiometric composition by retaining the chalcopyrite-type crystal structure [33,34]. Thus, instead of a stoichiometric end member, off-stoichiometric Cu_2yGa_2(1−y)S_3−2y can be assumed. The chemical compositions for compounds from x = 0 to x = 1, assuming an off-stoichiometric end member with y = 0.51, were calculated (calculation 2 and the dotted line in Figure 4). These calculations showed that, especially within the range of high Cu/(Ga+Ge) values, the experimental chemical composition of Cu☐GaGeS₄ single crystals agreed more with calculation 2, assuming an off-stoichiometric CuGaS₂ end member.

3.3. Crystal Structure of Cu_2(1−x)☐_2(1−x)Ga_2(1−x)Ge_2xS₄ Defect Adamantines

The crystal structure of the CuGaS₂ and GeS₂ end members of the (CuGaS₂)_1−x(GeS₂)_x series were both based on a corner-sharing network of tetrahedra (see Figure 5). CuGaS₂ crystallizes in the chalcopyrite-type structure (space group $I\overline{4}2d$) formed by corner-sharing CuS₄, GaS₄, and GeS₄ tetrahedra [31]. For the network of corner-sharing GeS₄ tetrahedra forming the crystal structure of GeS₂, tetragonal (space group $I\overline{4}2d$), orthorhombic (space group Fdd2), and monoclinic (space group Pc) modifications were reported [32,35]. The difference between these modifications is the degree of the distortion of the GeS₄ tetrahedron. The (CuGaS₂)_1−x(GeS₂)_x series was realized by the substitution of ${Cu}^{+}+{Ga}^{3+}\leftrightarrow {Ge}^{4+}+$ ☐; thus, with increasing Ge content in CuGaS₂, the fraction of vacancies (☐) and thus the fraction of ☐S₄ tetrahedra increased (see Figure 6).

X-ray diffraction data on pulverized crystals were analyzed by LeBail analysis using the chalcopyrite-type structure as the structural model. An exemplarily X-ray diffractogram and corresponding LeBail analysis for a Cu☐GaGeS₄ single crystal (prepared as a powder) is presented in Figure 7.

Tetragonal lattice parameters a and c from different Cu_2(1–x)☐_2(1−x)Ga_2(1−x)Ge_2xS₄ crystals were determined by the LeBail analysis of X-ray diffraction data (see Table 2). For comparisons, the lattice constants of CuGaS₂ and GeS₂ from the literature [31,32] are also provided in Table 2. The unit cell volume correlated linearly with both Cu/(Ga+Ge) (see Figure 8) and Ge/(Ga+Ge) ratios. With increasing ${Cu}^{+}+{Ga}^{3+}\leftrightarrow {Ge}^{4+}+$☐ substitutions in CuGaS₂, the fraction of vacancies increased (Figure 6). In addition, the radius of the incorporated Ge⁴⁺ was smaller than the radius of Cu⁺ and Ga³⁺ (r_Cu¹⁺ = 0.60 Å; r_Ga³⁺ = 0.47 Å; r_Ge⁴⁺ = 0.39 Å [36]). Thus, the unit cell volume decreased with increasing substitutions.

With increasing ${Cu}^{+}+{Ga}^{3+}\leftrightarrow {Ge}^{4+}+$ ☐ substitutions in CuGaS₂, lattice parameters a and c changed in an anisotropic fashion; i.e., the slope of their dependence on cation ratios Cu/(Ga+Ge) and Ge/(Ga+Ge) was different (see Figure 9).

The chalcopyrite-type structure (space group $I\overline{4}2d$) had two different cation sites, Wyckoff positions 4a and 4b (Figure 10), whereas the mono-valent cation occupied the 4a position, and the three-valent cation occupied the 4b position. As Cu⁺, Ga³⁺, and Ge⁴⁺ had the same number of electrons, their X-ray atomic form factors were very similar. Thus, the determination of cation distribution on the two structural sites of the chalcopyrite-type structure by conventional X-ray diffraction was not possible. Published results on the crystal structure of Cu☐GaGeSe₄, which were based on investigations by X-ray diffraction, assumed either Cu and vacancies on 4a and Ga and Ge on 4b positions [20] or Cu and Ga on 4a and Ge and vacancies on 4b sites [21] (see Figure 10). However, it was also possible for Cu and Ge to occupy 4b, in addition to various degrees of cation disorder.

Multiple Edge Anomalous Diffraction (MEAD) using synchrotron X-rays is an established experimental method that distinguishes electronically similar elements during data analysis [28]. The experimental MEAD spectrum was compared to calculated spectra (see Figure 11). As a structural model in calculated spectra, the chalcopyrite-type structure was used, but it was used with three different cation distributions. For calculations, the structural parameters for Cu☐GaGeSe₄ (from Woolley [20]) and the ideal stoichiometric composition were assumed. In addition, a calculated spectrum based on the final refined crystal structure (

Table 3. Results showing the structural refinement of MEAD data for Cu_1.22Ga_1.25Ge_0.77S₄.

Composition (refined): Cu1.152(3) Ga1.554(26) Ge0.446(26) S3.799(13)
$\mathrm{Space}\mathrm{group}:\mathit{I}\overline{4}2\mathit{d}$
Atom	Wyckoff	x	y	z	Biso [Å2]	s.o.f.
Cu	4a	0	0	0	0.830(36)	0.576(2)
Ga	4b	0	0	0.5	0.909(15)	0.777(13)
Ge	4b	0	0	0.5	0.909(15)	0.223(13)
S	8d	0.25518(20)	0.25	0.125	0.804	0.950(3)
S Anisotropic	U11 = 0.010(2), U22 = 0.0029(19), U33 = 0.0179(7), U12 = 0, U13 = 0, U23 = 0.0041(7)
Lattice parameter	a = 5.321383(10) Å		c = 10.18642(3) Å		V = 288.4500(12) Å3
Overall fit indicators (referring to the combined four diffraction patterns):
	Rwp = 0.067		Chi2 = 11.7		Bérar SCOR [37] = 4.42

) was also shown; differences relative to the ideal model were negligible.

It was obvious that the real cation distribution showed that Cu and vacancies occupied the 4a site and Ga and Ge occupied the 4b site.

A subsequent joint Rietveld refinement of the structure using diffraction patterns collected with four different X-ray energies confirmed this result. Due to anomalous scattering, which resulted in a change in the scattering power of chemical elements between individual datasets, the simultaneous independent refinement of all cation site occupation factors was possible. However, this resulted in rather high uncertainties, and the structural model was subsequently simplified by removing atoms with (unphysical) negative occupation factors and limiting total site occupation to full occupation. Note that this was not compulsive; interstitial cations could not be excluded by the methods presented here. The resulting model (Figure 12) was in full agreement with the results from MEAD analyses, with only copper on the 4a position and gallium and germanium on the 4b position, but not copper. Vacancies were found on 4a and 8d anion sites; the 4b Wyckoff site with the highest site occupation factor was assumed to be fully occupied. During the final step, charge neutrality between Cu¹⁺, Ga³⁺, Ge⁴⁺, and S²⁻ was forced; this did not result in a reduction in fit quality and did not significantly affect the refined values of site occupation factors. The results of the refinement are shown in Table 3.

3.4. Band Gap Energy

Band gap energy was determined from diffuse reflectance as measured using UV-VIS spectroscopy. Using the following Kubelka–Munk pseudo-absorption function [38,39]

$$F\left(R\right)=\frac{{(1-R)}^2}{2R},$$

(2)

the band gap energy E_g was determined from the linear slope of function ${\left(F\left(R\right)·h\nu \right)}^2$, assuming a direct band gap for the material studied.

Table 4. Results for off-stoichiometric Cu☐GaGeS₄ crystals: chemical composition and their band gap energies as determined by UV-Vis spectroscopy.

Sample No.	Composition	Band Gap Energy Eg (eV)
To557	Cu1.10☐0.90Ga1.09Ge0.91S4	2.49(5)
To569	Cu1.18☐0.83Ga1.15Ge0.84S4	2.41(5)
To597	Cu1.27☐0.74Ga1.24Ge0.75S4	2.29(5)
To581	Cu1.29☐0.72Ga1.23Ge0.76S4	2.28(5)
To585	Cu1.30☐0.72Ga1.23Ge0.75S4	2.36(5)
To531	Cu1.33☐0.68Ga1.27Ge0.72S4	2.28(5)
To599	Cu1.35☐0.68Ga1.24Ge0.73S4	2.24(5)
To580	Cu1.36☐0.66Ga1.28Ge0.70S4	2.30(5)
To596	Cu1.47☐0.54Ga1.42Ge0.57S4	2.11(5)
To588	Cu1.76☐0.27Ga1.63Ge0.34S4	2.15(5)

summarizes the band gap energies of several off-stoichiometric Cu☐GaGeS₄ crystals and their chemical composition. A non-linear correlation was observed between chemical composition and band gap energy.

The band gap energy of semiconductor alloys is usually described by a quadratic polynomial as a function of the concentration of an alloy component, with the quadratic coefficient referred to as the “bowing parameter”. Accordingly, the band gap energy E_g of alloy (2CuGaS₂)_1−x(Cu☐GaGeS₄)_x is described by

$$E_g\left(x\right)=x{E}_g\left(Cu\mathrm{☐}GaGe{S}_4\right)+\left(1-x\right)E_g\left(CuGa{S}_2\right)-bx\left(1-x\right).$$

(3)

Here, b is the bowing parameter. Figure 13 shows the experimentally determined band gap energy of (2CuGaS₂)_1−x(Cu☐GaGeS₄)_x crystals and band gap bowing according to Equation (3). In the respective fit, Ge content of crystals was selected to represent the x value in Equation (3). The bowing parameter was determined as b = −1.45(0.08) and described the deviation from linearity.

4. Conclusions

Defect adamantines are interesting and promising materials for photovoltaic applications, and are free of critical raw materials. Quaternary adamantines can be derived directly from ternary chalcopyrite A^IB^IIIX^VI₂. In the doubled chemical formula, one A¹⁺ and one B³⁺ are replaced by one C⁴⁺ atom, resulting in cation vacancies containing quaternary compounds with the general formula A^I☐B^IIIC^IVX^VI₄ while maintaining tetrahedral coordination.

Cu☐GaGeS₄ single crystals were grown using the chemical vapor transport technique. Their chemical composition was found to vary and deviate significantly from the stoichiometric composition. Therefore, grown crystals were described as compounds of a solid solution between CuGaS₂ and GeS₂, forming (CuGaS₂)_1−x(GeS₂)_x alloys. The Cu/(Ga+Ge) and Ge/(Ga+Ge) cation ratios described the deviation from the stoichiometric composition (Cu/(Ga+Ge) = Ge/(Ga+Ge) = 0.5). Structural and optoelectronic properties were considerably influenced by these cation ratios and the number of vacancies.

It was shown that Cu☐GaGeS₄ crystallized in the chalcopyrite-type structure (space group $I\overline{4}2d$). The cation distribution in this structure, analyzed using MEAD, was determined to be Cu and vacancies on the Wyckoff position 4a and Ga and Ge were observed on Wyckoff position 4b of the chalcopyrite-type structure.

The band gap energy E_g of off-stoichiometric Cu☐GaGeS₄ crystals varied between 2.1 and 2.4 eV. The non-linear correlation between band gap energy E_g and the chemical composition (Ge content) was described by the usual bowing behavior of semiconductor alloys, with a bowing parameter of b = −1.45(0.08).