Effect of Changing and Combining Trivalent Metals in the Structural and Electronic Properties of Cu-Based Crystal Delafossite Materials

Abstract

Cu-based ternary oxides with delafossite structure have received considerable attention in recent years for their versatility in a wide range of applications, among which is the possibility to use them in heterostructure solar cells as hole transport layers, due to their promising behavior as p-type conducting oxides. Ab initio calculations have been performed with density functional theory to investigate the role of the trivalent metal within the CuMO₂ structure and the dependence of structural and electronic properties with the species (M = Al, Ga, In, Fe, Cr, Co, Sc, Y) occupying the site of the metal. Generalized Gradient Approximation also including a Hubbard term and nonlocal Heyd–Scuseria–Enzerhof screened hybrid functional schemes were tested and their results were compared. Excellent agreement with experimental lattice parameters and measured gaps have been found. The use of hybrid functionals in HSE approximation considerably improves the bandgaps when compared with the experimental results but takes considerable time to converge, hence the need to explore less demanding methodologies. Trends in the geometry as well as in the electronic properties are discussed, and the effect of mixing different metals (CuM_xN_1−xO₂, M, N being the aforementioned elements) in the geometry and electronic properties of these delafossite materials is investigated. Due to the high cost of HSE calculations, especially when supercells are needed to model several x concentrations, statistical models and techniques based on machine learning have also been explored to predict HSE bandgap values from GGA and structural information.

1. Introduction

The discovery of p-type transparent conductivity of CuAlO₂ films with delafossite structure [1] was a milestone in the race for finding p-type transparent conducting oxides (TCOs). These kinds of materials, among which electronic doped ZnO and SnO₂ are perhaps the most representative [2,3,4,5], have some very unusual physical properties; they are transparent, similar to glass, but conductive almost like a metal. They are widely used in a variety of electronic applications, from panel displays and touch screens to solar cell applications. In the design of heterostructure solar cells, they are mainly used as electron (ETL) and hole (HTL) transport layers. Problems arise with the fact that most TCOs are n-type conductors, and finding p-type conductors with wide bandgap and good conductivity has been elusive.

It has been discussed [3,4,6] that for semiconductor oxides, the mobility of doping holes at the top of the valence band, typically filled by O-$2p$ bands, tends to be very low due to the strong localization of these levels. Possible solutions to achieve higher mobility of holes would be the use of materials for which the valence band edge is not O-$2p$ but cations with d electrons at the top of the band or modifying the valence band edge by mixing these orbitals with appropriate cations that have energy-filled levels comparable to those of the O-$2p$. Allegedly, this would be effective in reducing the strong Coulomb interaction of oxygens and therefore in delocalizing positive holes. The choice of a monovalent copper Cu⁺ as a candidate, with filled $3d^{10}$ orbitals in a closed shell after giving its $4s^1$ electron, and energy of the Cu-$3d$ levels compared to those of the O-$2p$ leads to the idea of Cu oxides as good candidates to overcome the localization of holes via covalent bonding between the Cu⁺ and the O⁻² ions. However, the simplest Cu oxide, CuO₂, has a rather small bandgap and it is not transparent in visible light, despite not having a considerably large conductivity.

Therefore, the discovery of p-type conductivity in CuAlO₂ triggered an increased interest in the experimental and theoretical characterization of this compound [3,7,8,9]. CuAlO₂ was followed by the family of CuM^III−AO₂ metals, in which the M^III−A cation is a metal from the group III-A in the periodic table; CuGaO₂ was found to also have p-type conductivity and CuInO₂ was found to exhibit the ability to be both n- and p-type doped [2,5,10,11,12,13,14,15]. Also, CuScO₂ [16,17] and CuYO₂ [18,19] have been studied, as well as CuCrO₂, which has been doped with Mg, exhibiting the highest conductivity but not good transparency [20,21,22,23,24]. To date, numerous Cu ternary oxides with delafossite structure CuMO₂ (M = B, Al, Ga, In, Fe, Cr, Sc, Y, etc.), and even some with general formula AMO₂ (where A = Ag, Cu), have been reported with p-type conductivity as high as ${10}^2$ S/cm and high optical transparency (50–85%), depending not only on their chemical compositions but also grain sizes or film deposition methods [4,25,26,27,28,29,30,31,32]. Theoretically, special attention has been received by the Fe-delafossite for the difficulties in its calculation [33,34,35,36].

The effect of doping with divalent metals in the delafossite compounds has also been investigated. Other studies of mixed structures are common for different kinds of applications and materials [37,38] and are useful not only to predict properties in novel materials but also for the understanding of underlying mechanisms of the different properties of interest withing given materials. The race to achieve good transparency and conductivity has included the study of Mg-doped delafossites [4,21,22,24,29], Ni- or Ca-doped delafossites [20,39], as well as doping with another trivalent metal in structures with high percentages of substitution [27,40].

In this paper, we aim to study a variety of Cu-based delafossite semiconductors as promising materials for photovoltaic applications in the design of heterostructure solar cells, as hole transport layers (HTLs). We focus on the CuMO₂ structure, with M = Al, Ga, In, Co, Sc, Y, Fe, Cr. Geometrical and electronic characterization were performed using the density functional theory for these materials, as described in Section 2: Methodology. The main geometrical characteristics of the two phases in which these delafossites can be found are discussed in Section 3: Structural characterization. A discussion about bandgap values, density of states, and band structure can be found in Section 4: Electronic Structure. Then, the effects in the geometry and bandgap when one of the trivalent metals within the oxides is substituted are tested. Section 5 shows results for CuM_xN_1−xO₂ structures, with N, M being the same species previously mentioned, and $x=1/3$. Different relative concentrations for N, M require larger supercells, making the more accurate calculations using hybrid exchange correlation functionals such as HSE prohibitive. Thus, a discussion on statistical models and the use of techniques based on machine learning to describe the correlation between the bandgap values described with the GGA and HSE functionals is presented in Section 6. Finally, some concluding remarks highlighting the fundamental properties of these materials are given.

2. Methodology

Since theoretical simulations have proven to be useful for the understanding of the properties of systems at the atomic level, first-principles calculations were performed to gain an insight into the structural and electronic properties of these Cu-based ternary oxides with delafossite structures.

For this purpose, the atomic positions were optimized and the electronic properties were computed within the density functional theory (DFT) approach, based on the framework of the generalized Kohn–Sham scheme [41,42] in combination with the projector augmented wave (PAW) method [43] for the description of the core electrons and the generalized gradient approximation (GGA) within the Perdew–Burke–Ernzerhof (PBE) formalism [44] for the exchange and correlation term, as implemented in the Vienna ab initio simulation package (VASP) [45,46,47]. It is well-known that GGA typically underestimates gap values, especially when localized orbitals such as metal d or f states are present. Hence, for a more accurate electronic description, the Heyd–Scuseria–Ernzerhof hybrid functional with the modified fraction of screened short-range Hartree–Fock exchange (HSE06) [48,49,50] was also used to calculate the density of states and band energies. Hybrid functional calculation results in a more accurate description of electronic properties for some systems compared to simple DFT calculations with a generalized gradient approximation for the exchange and correlation term in the Kohn–Sham scheme. The higher accuracy, however, is reached at the cost of a higher computational time needed to achieve convergence.

In the search for an alternative method to hold acceptable accuracy with less demanding computational resources, calculations using DFT corrected for on-site Coulomb interactions (DFT + U) [51] were also performed. The inclusion of the so-called Hubbard term (+U) provides an approximate correction for strongly correlated d or f electrons that are usually inaccurately described with standard GGA functionals. The choice of the U parameter is typically an ad hoc choice, sometimes with empirical information. In the present study, the U value has been chosen such that the valence band description in GGA + U reproduces the shape of the band in the density of state curve obtained with HSE06 calculations, i.e., the value of U for which the position of peaks in the valence band edge is the same as in the HSE simulations. Even when the HSE gap values are not reproduced, having a correctly described valence band allows further calculations with a reasonable computational time cost.

Table 1. Different U values used for the on-site Coulomb interaction scheme (DFT + U).

Element	U (eV)
Cu	6.0
Co	5.2
Cr	5.0
Fe	4.0

shows the U values used in this work.

To model the tetragonal phase of CuMO₂, a unit cell with 12 atoms (Cu₃M₃O₆) was used, while for the hexagonal cell, a smaller cell with 8 atoms (Cu₂M₂O₄) was needed. To model the CuM_xN_1−xO₂ mixed structures when $x=1/3$, the same 12-atom cells were used. However, for the machine learning techniques tested, a small number of calculations were run for which $x=0.5$ and $x=1/6$, and for these structures, 1 × 2 × 1 supercells, containing 24 atoms, were built.

The electronic wave functions were expanded in a plane wave basis setup to a kinetic energy cutoff of 450 eV, and the atomic positions were optimized using the conjugate gradient method up until the forces on each atom were less than 0.01 eV${\AA }^{-1}$ and the energy convergence less than ${10}^{-8}$ eV for the relaxation and GGA calculations, and less than ${10}^{-5}$ eV for the calculations with the hybrid functional. The structures with the heaviest atoms—the delafossites, including In and Y—were optimized, including spin–orbit coupling (SOC) [52], but the $E_{SOC}$ falls under $0.1$ eV, leading to the conclusion that the effect of spin–orbit coupling does not need to be taken into account with these structures. For the Brillouin zone integration, a 12 × 12 × 6 Monkhorst–Pack scheme k-point mesh was used [53] both for the optimization and electronic structure calculation.

As mentioned before, due to the high cost of running HSE calculations for each of these structures to obtain accurate bandgap values, we also explored the use of statistical models and more complex techniques based on machine learning to describe the correlation between the bandgap values described with the GGA functional as a first approximation, and those obtained with the HSE functional over already relaxed structures.

The applicability of this approach lies not only in the significant reduction in computational cost when calculating the bandgap values of these materials. It would also allow us to extend the study to new configurations with this delafossite structure which, due to their different relative concentration of elements occupying cationic positions, may require larger supercells for modeling. This would make the cost of HSE calculations practically prohibitive due to the greater number of atoms, but through the implementation of this approach, a simple GGA calculation would be enough to infer a meaningful value of the bandgap for a given configuration.

Several models and combinations of chemical and structural information to describe the compounds have been tested, searching for the best description of the structure–property correlation in this group of materials. In the present manuscript, only the main metrics of the best-performing pair of descriptors (set of features or parameters that better describe the system in an appropriate way for its digestion by the statistical model) and models are shown, together with some remarkable comments, in the corresponding section.

3. Structural Characterization

The delafossite structure of these Cu-based ternary oxides can be appreciated fairly in Figure 1. It shows the atom positions and geometrical structure of the converged unit cell for the CuAlO₂ delafossite, as a prototype of the structure. The aforementioned delafossite structure consists of close-packed alternate layers of Cu atoms linearly coordinated with O atoms, forming O-Cu-O dumbbells parallel to the z axis, and M-centered octahedra (M = Al, Ga, In, Co, Sc, Y, Cr, Fe) parallel to the xy plane. The oxygens on the O-Cu-O dumbbells coordinate to three metal ions each, while metal ions coordinate to eight oxygens, forming the mentioned octahedra. The stacking can occur following an ABCABC pattern, thus forming a trigonal system with a rhombohedral Bravais lattice unit cell with space group $R\overline{3}m$ (symmetry number 166), as shown in Figure 1a, or with an ABAB stacking pattern, thus showing a hexagonal unit cell with $P{6}_3/mmc$ (no. 194) space group, as shown in Figure 1b.

3.1. Rhombohedral Structures

The structural parameters of the unit cells for the $R\overline{3}m$ phase were calculated and

Table 2. Lattice parameters a and c for the studied delafossite in the rhombohedral structure (R3). The relaxed values from our simulations are compared to previous experimental results and the deviation from experimental values is expressed in %. When more than one experimental reference is shown, the deviation written in the Table is the one with the highest absolute value.

		a (Å)
	Relaxed value	Exp. references	Deviation from exp.
CuAlO2	2.87	2.85 [54], 2.86 [55,56]	+0.7%
CuGaO2	2.97	2.97 [54], 2.98 [55]	−0.3%
CuInO2	3.28	3.29 [57]	−0.3%
CuCoO2	2.86	2.85 [55]	+0.35%
CuScO2	3.19	3.22 [17]	−0.93%
CuYO2	3.47	3.53 [54]	−1.7%
CuCrO2	2.98	2.97 [24], 2.98 [55]	+0.34%
CuFeO2	3.02	3.04 [36,55]	−0.7%
		c (Å)
	Relaxed value	Exp. references	Deviation
CuAlO2	17.05	16.94 [55,56], 16.95 [54]	+0.65%
CuGaO2	17.22	17.15 [55], 17.17 [54]	+0.41%
CuInO2	17.62	17.39 [57]	+1.3%
CuCoO2	16.96	16.92 [55]	+0.24%
CuScO2	17.21	17.10 [17]	+0.64%
CuYO2	17.31	17.13 [54]	+1.05%
CuCrO2	17.10	17.09 [24,55]	+0.06%
CuFeO2	17.34	17.17 [36,55]	+0.99%

summarizes the optimized lattice parameters for all the studied structures, with experimentally measured values for some of the compounds and the deviation percentage that our simulated value represents with respect with the measured one.

Our simulations are in good agreement with experimentally measured values for the cells, with deviations that fall under 2% for almost all the compounds, except for the $CuY{O}_2$, which exhibits the highest deviation for both a and c parameters. Particularly good agreement between experimental values and our simulations was obtained for $Ga$-, $Co$-, and $Cr$- delafossites.

The role of the trivalent metal ion in the structural and electronic properties of these materials has been discussed [16,19,28,58]. The Cu-Cu distance (equal to the a lattice constant, since Cu atoms are in the corners of the unit cell) has been pointed out as playing a relevant role for p-type mobility [58] with holes hopping between Cu ions. On the other hand, the size of the metal ion seems decisive for the a constant. Note, in Table 2, the trends followed by the a constant for CuAlO₂, CuGaO₂, CuInO₂, CuScO₂, and CuYO₂, for example. It is strictly true for CuAlO₂, CuGaO₂, and CuInO₂, as well as for CuScO₂ and CuYO₂ in separate groups. Put together, with CuInO₂ and CuScO₂, the trend to be expected is not clear; however, both delafossites show similar lattice constants, while In³⁺ and Sc³⁺ ions also exhibit similar sizes according to [59]. The same can be said for CuCrO₂ and CuFeO₂ if we look for the Fe³⁺ size in the high-spin configuration. An unexpectedly low value of a compared to what could be intuitively expected is the one for CuCoO₂, which seems not to be an error of the simulation, since the experimental value [55] is very similar. However, it is consistent with the fact that the electronic structure calculated for this delafossite shows no spin polarization, indicating the Co³⁺ is in its low-spin configuration, and again the ion size of Co³⁺ in the low-spin configuration, as reported in [59], is very similar to the size of Al³⁺, similar to the a lattice constant.

Therefore, the size of the metal ion, which is decisive to the a lattice constant, should play an important role in the conductivity of the delafossites, especially those with p-type conductivity. This might be the explanation behind the fact that undoped CuAlO₂ has higher conductivity than both undoped CuScO₂ and CuYO₂ [16,28]. More discussion on the effect of the M cation size and a lattice constant on the conductivity can be found in [4,29] for some Mg-mixed Cu-based delafossites. It is of course not the only factor that contributes, with the covalency of the M-O bond being another important factor, as discussed in [19].

3.2. Hexagonal Structures

As mentioned before, in the hexagonal cells, the same layers of O-Cu-O dumbbells and M-centered octahedral are observed, but with an ABAB stacking pattern. It is expected, then, to find the lattice constant a to be practically identical between both phases, while the lattice constant c is expected to have a significantly lower value. The structure parameters found in our calculations, with some experimental values for comparison, are shown in

Table 3. Lattice parameters a and c for the studied delafossite in the hexagonal structure (2H). The relaxed values from our simulations are compared to previous experimental results and the deviation from experimental values is expressed in %. When more than one experimental reference is shown, the deviation in the Table is the one with the highest absolute value.

		a (Å)
	Relaxed value	Exp. references	Deviation from exp.
CuAlO2	2.84	2.86 [54]	−0.7%
CuGaO2	2.90
CuInO2	3.20
CuCoO2	2.83
CuScO2	3.13	3.22 [17,54]	−2.8%
CuYO2	3.39	3.52 [60]	−3.7%
CuCrO2	2.93	2.97 [61]	−1.3%
CuFeO2	2.97
		c (Å)
	Relaxed value	Exp. references	Deviation
CuAlO2	11.34	11.31 [54]	+0.27%
CuGaO2	11.45
CuInO2	11.83
CuCoO2	11.298
CuScO2	11.52	11.41 [17,54]	+0.96%
CuYO2	11.66	11.42 [60]	+2.1%
CuCrO2	11.42	11.40 [61]	+0.18%
CuFeO2	11.60

.

The mentioned difference in c lattice constant between both phases is caused by the fact that a smaller distance is enough to represent the structure, because of the stacking sequence. Hence, important distances within the unit cell, such as the distance between two Cu neighbors in the $xy$ plane (equal to the a lattice constant), the distance between Cu-O atoms in the O-Cu-O dumbbells, or the distance from cornered O to centered M in the octahedra are practically the same, as can be appreciated in

Table 4. Comparative values for important interatomic distances within 3R and 2H phases of delafossite structure.

		Cu-Cu (Å)	Cu-O (Å)	M-O (Å)
CuAlO2	3R	2.87	1.87	1.92
	2H	2.84	1.87	1.91
CuGaO2	3R	2.97	1.85	1.995
	2H	2.90	1.83	1.97
CuInO2	3R	3.28	1.84	2.19
	2H	3.20	1.84	2.16
CuCoO2	3R	2.86	1.85	1.92
	2H	2.83	1.85	1.91
CuScO2	3R	3.19	1.84	2.11
	2H	3.13	1.83	2.09
CuYO2	3R	3.47	1.83	2.27
	2H	3.39	1.82	2.24
CuCrO2	3R	2.98	1.85	1.99
	2H	2.93	1.84	1.97
CuFeO2	3R	3.02	1.83	2.04
	2H	2.97	1.83	2.02

.

Once more, the trend shown in the Cu-Cu distances (equal to the a lattice constant) is in agreement with the size of the trivalent metal ion. The same goes for the M-O distances, as expected. The O-Cu-O dumbbell distances, however, seem to be practically the same for all the structures, varying in a very short range of no more than $0.1\AA$.

The relative stability of the two different phases is shown in

Table 5. Difference in basic state energy of rhombohedral vs. hexagonal structures for the studied delafossites E_3R-E_2H.

	CuAlO2	CuGaO2	CuInO2	CuCoO2	CuScO2	CuYO2	CuCrO2	CuFeO2
$\Delta$E0 (eV)	−0.26	−0.22	−0.21	−0.24	−0.298	−0.30	−0.29	−0.26

. A negative $\Delta$ means the first structure has a lower energy, i.e., is the most stable one: the rhombohedral structure, in all our cases. However, the energy difference for all compounds falls under 0.3 eV, meaning both structures are relatively equally stable, and must be found indistinctly in nature [62].

4. Electronic Structure

4.1. CuAlO₂, CuGaO₂ and CuInO₂

CuAlO₂ is probably the most extensively studied delafossite, with numerous theoretical and experimental studies. Some experimental works have reported a minimal value for the indirect bandgap in CuAlO₂ of 1.8 eV [3], very much deviated from the 3.4 eV we have obtained. This small value has been called into question [5,7], suggesting this highly underestimated measured bandgap could be related to some sort of defect. Several theoretical calculations (

Table 6. Comparison of calculated bandgaps with previously reported bandgaps obtained with different simulation schemes.

Our simulations
GGA	1.8
HSE06	3.4
Theoretical References
GGA [5]	2.1
sX-LDA [12]	2.8
HSE03 [14]	3.1
G0W0 [14]	3.1
HSE06 [14]	3.6
B3LYP [5]	3.9

) give rise to a considerably larger indirect bandgap, supporting this consideration. Values range from 2.8 eV with an sX-LDA, which is expected to underestimate the gaps due to the LDA contribution, to 3.9 eV, the highest reported. In particular, three hybrid calculated values—including ours—seem to agree in a smaller range of values from 3.6 to 3.9 eV.

A detailed description of the electronic structure of the CuAlO₂ is shown in Figure 2a: HSE06 total and projected Density of States (DoS) as well as the band diagram with projected element contribution. While total DoS is shown in a black curve, element projected DoS is represented with color curves following the colors given to atomic species in Figure 1: red for Cu contributions, green for O contributions, and magenta for the contribution of the trivalent metal, Al in this case. In the bands diagram, every dot is a weighted mixture of these colors, where the projected contribution of each element to each point has been used as the weight.

The curves from DoS graphics are in excellent agreement with previous x-ray spectrum measures for CuAlO₂ [7]. The valence band edge shows a relatively weak—compared to its immediate neighbor—but well-defined peak dominated by Cu-3d states. This peak shows the red curve almost under the black one, which means the Cu contribution is dominant to the DoS, and the same can be appreciated by the almost pure red dots in the bands at the L-K line. A second and much stronger peak is observed next to it, also dominated by Cu-3d contributions mixed with O-2p states. This region’s progressive fading from red to green in the band diagram accounts for the Cu-O mix. As the energies go deep into the band, a third broader and weaker region is observed, in which the O-2p contributions overcome the Cu presence. The dominant and wide region of green dots in the band suggests a wide dispersion of oxygen through the valence band. To find Al contributions, it is needed to look at the bottom of the band, where Al projected DoS is almost equal to Cu projected DoS in a fourth region, which corresponds in the band diagram to bands in which predominantly green dots turn to a darker color due to the magenta contribution.

The conduction band, on the other hand, starts with contributions mainly from Cu atoms and exhibits Al contributions near 10 eV. The valence band maximum (VBM) is located at the L point. The conduction band minimum is located at the $\Gamma$ point and is constituted by Cu-3d states mixed with O-2p states. As CBM and VBM are located at different points of the Brillouin zone, the gap of the material is indirect.

For the CuGaO₂ (Figure 2b) and CuInO₂ (Figure 2c) delafossites, the electronic description is topographically quite similar, i.e., Cu dominated VB edge, the rest of the band showing the presence of Cu-3d mixed with O-2p states, with little to no contribution from the trivalent metal and O states, showing a wide dispersion in the region. A comparison of HSE06 calculated gaps for these three ternary oxides of group III-A and experimentally reported values can be found in

Table 7. Calculated bandgaps and experimental references for group III-A delafossites.

	CuAlO2	CuGaO2	CuInO2
Indirect gap	3.44	2.42	1.60
Direct gap, $\Gamma$	4.85	3.74	2.54
Direct gap, F	4.03	3.87	4.34
Direct gap, L	4.06	3.93	4.39
Experimental reference	3.5 [3]	3.6 [11]	3.9 [10]
	3.60–3.94 [8]

.

Some general trends can be highlighted:

• The three of them show an indirect fundamental bandgap.
• Also, for the three of them, VBM is located at L point, while CBM is located at $\Gamma$ point. The fundamental direct bandgap, i.e., the smallest direct gap, is located at $\Gamma$ point for CuGaO₂ and CuInO₂ but at L point in CuAlO₂.
• The fundamental bandgap, i.e., the indirect one, decreases from CuAlO₂ to CuGaO₂ and to CuInO₂. This trend is coherent with what is expected for group III-A containing semiconductors, but not with what has been experimentally measured.
• The value of the bandgap at $\Gamma$ point also decreases from CuAlO₂ to CuGaO₂ and to CuInO₂. This has been pointed out as the fundamental direct bandgap in previous reports [2] where matrix elements for direct transitions between band edge states have been calculated, obtaining that transitions at $\Gamma$ are forbidden, therefore explaining the inconsistency mentioned before between experimentally measured optical gaps and the fundamental gaps obtained in the calculations.
• The bandgap at L point, in fact, slightly decreases from CuAlO₂ to CuGaO₂ but then increases to CuInO₂. This aligns with the experimental result. The same trend is observed at F point, and in general in the F-L line where band edges are closest to each other. The optical bandgap that is being measured must be related to this transition at the L point.
• The width of the valence band decreases from CuAlO₂ to CuGaO₂ and to CuInO₂, which is coherent with experimental measurements [13].
• The contribution of M^III−A states at the edge of the valence band is almost negligible for the three delafossites but becomes increasingly present at the bottom of the band from CuAlO₂ to CuGaO₂ and to CuInO₂, as can be observed in the coloring of the dots in the graphics. Actually, in CuInO₂, the contribution of In states begins to be appreciable around −5 eV, in the region dominated by O states.
• The presence of states from the trivalent metal becomes increasingly important at lower energies, as well as in the conduction band, as seen in projected DoS from CuAlO₂ to CuGaO₂ and to CuInO₂.
• The CBM at $\Gamma$ is located at a lower energy from CuAlO₂ to CuGaO₂ and to CuInO₂.

The calculated band structure for CuGaO₂ with GGA, GGA + U, and HSE06 formalisms along the high-symmetry $\Gamma$-F-L-$\Gamma$ points of the Brillouin zone is shown in Figure 3. It can be fairly appreciated that the increasing accuracy in calculations induces a vertical upward shift of the conduction band with respect to the Fermi Energy—which has been set equal to the Valence Band Maximum (VBM) and set to zero—with the consequent increase in bandgap value. However, despite this upward shift, the form of the conduction and valence bands at the edge are not appreciably modified. As a consequence, the Conduction Band Minimum (CBM) is located at $\Gamma$ point in all three calculations, as well as the smallest direct gap, represented by arrows in the diagrams. The VBM—represented, as the CBM, by bigger dots in the pictures—is also located at the same point despite the approximation used: L point. Therefore, the fundamental bandgap is indirect in all three simulations. In summary, at least for the CuM^III−AO₂ delafossites, increasing accuracy does not change the band’s description qualitatively.

However, some quantitative changes do happen when increasing accuracy. We previously mentioned the upward shift of the conduction band. The width of the valence band is also increased, deepening the lower states with the increase in accuracy from GGA to HSE, though the GGA + U shows a smaller bandwidth. This can be appreciated better by comparing the DoS and projected DoS for the three levels of calculation in Figure 4. However, it is possible to observe that despite the method, the valence band is dominated by Cu and O contributions, while the Ga contribution relies on the bottom of the band. In conclusion, despite quantitative differences, even at the GGA level, the electronic description of the material is qualitatively quite good. The same can be said for CuAlO₂ and CuInO₂.

Electronic Properties in Hexagonal Structures

The electronic properties of the hexagonal structures are substantially the same as in their rhombohedral counterparts. Indirect HSE bandgap values differ less than $1\%$ for these structures, and the density of states follows very similar curves, as can be appreciated in Figure 5 for the group III-A ternary oxides previously analyzed.

Therefore, at least for applications depending only on electronic properties, the presence of either rhombohedral or hexagonal delafossite structures in a sample is indistinct. These similarities in electrical properties are quite an expected output due to the ones found in the geometry analyses of these two faces within the delafossite structure. If distances between atoms and angles of coordination are almost the same, bonds and electrical properties are expected to behave almost the same. This result has been previously discussed for CuAlO₂ [18] as well as for CuCoO₂ [62].

4.2. CuCoO₂

Despite what was discussed in the previous section for the delafossites with metals of group III-A, in the description of delafossite oxides in which the valence electrons of the metal M are d electrons, the GGA methodology fails not just quantitatively. This is the case for the CuCoO₂, CuScO₂, CuYO₂, CuCrO₂, and CuFeO₂ structures. Taking CuCoO₂ to exemplify, Figure 6 shows the bands in the GGA (Figure 6a), GGA + U (Figure 6b), and HSE (Figure 6c) approximations. In the GGA, the bandgap is almost negligible, which means the energies of excited states are strongly underestimated. The CuFeO₂ shows the most inaccurate GGA calculation, leading to the band description of a conductor, i.e., zero gap.

However, more notably, there is a considerable difference in the form of the valence bands from GGA compared to GGA + U and HSE. Looking at the projected density of states in Figure 7, the problem in GGA calculation is the erroneous high energies of Co-$3d$ states at the top of the band. The inclusion of the Hubbard term that matches the HSE density of states description corrects the location of these Co-$3d$ states, moving them deeper into the band, and therefore, the GGA + U and HSE are qualitatively very much in agreement; however, the values of the excited states remain underestimated in GGA + U.

Figure 8 shows the band diagram and projected DoS for the CuCoO₂ delafossite obtained with HSE functionals. Spin-polarized calculations were performed, but up/down configurations produced the same result, thus confirming the Co³⁺ ion is in its low state and there are no unpaired electrons in the material. The VBM is at the F point, while the CBM is somewhere in the $L-\Gamma$ line. Therefore, the fundamental bandgap is indirect and has a value of $E_{gap}=2.34$ eV. The smallest direct bandgap is also located in the $L-\Gamma$ line, near the L point, and has a value of $3.40$ eV, consistent with measured optical bandgaps in the range 3.4–3.9 [62,63].

The electronic structure shows a valence band edge dominated by Cu states. Like in the previously described delafossites, the first peak (from 0 to −1 eV) is energetically smaller than its immediate neighbor and is Cu-dominated. However, the second and higher peak, situated from −1 to −3 eV, exhibits a strong presence of Co-$3d$ states, comparable to the Cu-$3d$ contribution, even overcoming at some point the Cu contribution. From −3 eV and deeper, the main contribution is due to very delocalized O-$2p$ states, mixed with Co contributions instead of Cu, as was the case for the M^III−A delafossites. On the other hand, the conduction band edge shows mainly Co-$3d$ contributions mixed with a small O-$2p$ presence and an almost negligible Cu presence in the first peak observed in the DoS diagram, approximately from 2.35 to 4 eV. The second and slightly smallest peak—from 4.2 to 6 eV—shows equal Co, Cu, and O contribution to the density of states.

4.3. CuScO₂ and CuYO₂

The delafossite compounds formed with Sc and Y in the place of the trivalent metal show a very notable difference compared with all other studied delafossites in this paper. As can be appreciated in Figure 9, these two band diagrams show that CuScO₂ and CuYO₂ have a direct bandgap at F point, as has been previously reported [12,64]. On the other hand, as this transition at F is symmetry-allowed, a good agreement between the calculated direct bandgap and the experimental measured optical gap is expected. Our 3.5 eV is only 0.2 eV overestimated to the measured 3.3 eV gap for CuScO₂ [65], while our 3.9 eV calculated bandgap for CuYO₂ is also slightly overestimated to the measured 3.5 eV for this delafossite [66].

An important difference in the electronic structure when compared with the previously described delafossites (CuM^III−AO₂ and CuCoO₂) lies in the bottom of the conduction band. While the CuM^III−AO₂ structures showed the edge of the conduction band filled with Cu-d states hybridized with O-p states and the CuCoO₂ conduction band edge was filled with Co-d states also hybridized with O-p states, these two CuM^III−BO₂ exhibit a conduction band with an important and almost exclusive contribution of Sc/Y-d states, as can be appreciated in the almost pure magenta coloring of the dots in the conduction band of the band diagrams in Figure 9. In CuScO₂, the conduction band has no definite and separated peaks, but a wide band with mainly Sc-d states, followed by a smaller contribution of Cu-d states mixed with O-$2p$ with a high dispersion, as can also be seen in the curves of DoS on the right side of Figure 9a. In CuYO₂, the conduction band edge shows a first peak with equal contribution of Y-d states with the mixture of Cu-d and O-p states, as shown by almost overlapping curves from 4 to 5 eV on the right side of Figure 9b, followed by a wide band, as in the case of Sc delafossite, with almost exclusively Y-d contribution.

4.4. CuCrO₂ and CuFeO₂

For the CuCrO₂ and CuFeO₂, as both Fe—in its high-spin configuration—and Cr atoms present unpaired electrons, spin-polarized calculations were performed. In Figure 10 HSE DoS are represented for these materials, in which positive curves account for Spin-Up contributions to the DoS and negative curves for the Spin-Down.

The electronic structure of CuCrO₂ can be appreciated in Figure 10a. It is similar to the ones observed in previous delafossite structures and quite similar in both Up/Down configurations. The edge of the valence band belongs to the Up configuration. It presents a first peak—from 0 to −1 eV—of mainly Cu contributions, separated from the second and higher peak—from −1 to −3.5 eV—which exhibits mostly Cr contributions, with mixed Cu-O states. On the downside, the first peak is almost identical to its corresponding Up configuration, but the second one has almost no Cr contribution, showing an important Cu presence and a relatively small O presence. From −3.5 eV and deeper into the band, both sides show a wide dispersion of O states, being almost the only contribution in the down configuration, while the up configuration also shows Cr and smaller but appreciable Cu contributions. On the other hand, the conduction band starts growing both in the Up and Down configuration with mainly Cr contributions, mixed in the Up configuration with O states, but being almost the only contribution on the downside.

A different scenario is found by inspecting the electronic structure of CuFeO₂ in Figure 10b. The valence band edge is very similar in energy for both Up and Down configurations, and exhibits mainly Cu contributions, as all described structures before. This dominant Cu contribution remains through a wide range of energy into the band—reaching −3 eV in the Down configuration and −3.5 eV in the Up part—with a very discrete contribution of Fe states. From this point and deeper, on the downside, the valence band DoS is almost solely O contributions, while in the Up configuration, a strong Fe contribution appears around −7 eV. This is coherent with previously reported DoS [34,35]. The conduction band starts growing on the downside with a strong contribution of Fe states, from 1.3 to 2 eV, and a second Fe-dominant peak from 2.4 to 3.7 eV. The Down-spin states that pair with the 5 unpaired d-electrons of Fe are responsible for the small gap found in this delafossite, the smallest within the studied delafossites in this paper.

5. Mixed Structures

To explore the effects of the substitution of one of the trivalent metals in the delafossite compounds, we have studied the substitution of one of the trivalent metals (Al, Ga, In, Co, Sc, Y, Cr, Fe), having as possible substitutes the same set of metals. Changes in geometry, gap value, and electronic structure have been analyzed in CuM_xN_1−xO₂ structures, where $x=1/3$. In this section, as for the pure delafossites, structural results arise from GGA calculations while bandgaps were obtained in HSE simulations.

The inclusion of ions with a larger radius in a delafossite with a smaller M metal is expected to increase both a and c parameters, as well as the cell volume. However, in general, we have observed that while the parameter a increases with the increase in a larger metal in the composition, it is not the case for parameter c. This can be fairly appreciated in Figure 11 for a selection of these mixed structures.

As the amount of Ga increases in Figure 11a, the parameter a also increases, suggesting a linear tendency. The same can be said for the amount of In in Figure 11b, In in Figure 11c, Sc in Figure 11d, Fe in Figure 11e, or Cr in Figure 11f. For every mix, we have calculated only four structures, and a rigorous fit can not be performed with such a small amount of data points.

The case is completely different for the cell parameter c. It can be appreciated in the inset plots in Figure 11. In Figure 11a–c, this parameter also seems to increase with the increase in the amount of larger metal. In these three examples, the substituted metal has a very similar electronic composition as the one to be substituted, since Al, Ga, In belong to the same group in the periodic table. But in all other cases, it is fairly appreciated that there is no clear dependency of c on the amount of substituted metal, since the dispersion of the four points is notable. This behavior was also found previously [40] for CuFe_xGa_1−xO₂. The referred study concluded that the c parameter is almost independent of the concentration of the substituted cation. A possible explanation is that the strong repulsion of the M cations in the $ab$ plane affects the MO₆ octahedra, rearranging the oxygen atom positions. As the M cation becomes larger, the M–O distance increases, but the O–O contact distance varies only slightly, and therefore, there is little impact on the c cell parameter.

The complete set of values for the geometry of these structures can be found in

Table 8. Lattice constant a and c (in Å) for CuM_1/3N_2/3O₂ delafossite structures obtained with GGA functional. In the diagonal, the values of the pure delafossites previously discussed are shown.

a(Å) c(Å)	-Al2/3	-Ga2/3	-In2/3	-Co2/3	-Sc2/3	-Y2/3	-Cr2/3	-Fe2/3
Al1/3-	2.87	2.96	3.19	2.86	3.11	3.35	2.96	2.96
	17.05	17.24	17.50	17.01	17.20	17.22	17.16	17.33
Ga1/3-	2.91	2.97	3.24	2.91	3.15	3.39	2.97	3.02
	17.17	17.22	17.49	17.04	17.22	17.199	17.13	17.34
In1/3-	2.998	3.09	3.28	3.00	3.25	3.48	3.13	3.10
	17.40	17.47	17.62	17.19	17.29	17.23	17.27	17.40
Co1/3-	2.87	2.96	3.20	2.87	3.12	3.37	2.96	2.96
	17.04	17.16	17.33	16.96	17.05	16.98	17.08	17.26
Sc1/3-	2.97	3.06	3.29	2.98	3.19	3.44	3.07	3.098
	17.20	17.29	17.40	17.02	17.21	17.15	17.12	17.26
Y1/3-	3.09	3.27	3.42	3.12	3.33	3.47	3.099	3.24
	17.31	17.31	17.35	16.98	17.17	17.31	17.16	17.18
Cr1/3-	2.91	2.96	3.24	2.90	3.15	3.39	2.98	3.02
	17.15	17.18	17.36	17.05	17.12	17.06	17.10	17.28
Fe1/3-	2.898	2.97	3.26	2.90	3.17	3.42	3.01	3.02
	17.24	17.29	17.42	17.15	17.19	17.10	17.21	17.34

.

Bandgap values have been calculated and the DoS structures analyzed. In general, it is impossible to find any clear trend in the behavior of the bandgap values, as was true for the geometry. An exception to this is the CuGa_xAl_1−xO₂ case. As can be appreciated in Figure 12a, the gaps seem to follow a linear decrease with the increase in Ga concentration. This is not surprising, since in both CuAlO₂ and CuGaO₂ electronic structures, the states of the trivalent metal are very deep in the valence band, and very high in energy in the conduction band. All the interaction near the gap is due to Cu atoms and the tightness of the bond shall only be determined by its Cu-Cu distance. As mentioned before, the structure increases linearly in a and c for these delafossites, thus causing the electrons to feel a softer potential and be less tightly bonded to the Cu atoms. A second exception is found by inspecting every delafossite that includes Fe in the substitution. The Fe-states appear so energetically low in the conduction band that the inclusion of any Fe atom causes the bandgap to exhibit very similar values, despite which was the host delafossite (Figure 12b).

However, apart from these two mentioned examples, there is no clear trend in the gap values that can be analyzed. The complete set of HSE values for these structures can be found in

Table 9. Indirect bandgap values (in eV) with HSE functionals for CuM_1/3N_2/3O₂ delafossite structures. Arrows indicate whether the gap belongs to up/down configurations in spin-polarized calculations for structures containing Cr and Fe atoms.

Bandgap (eV)	-Al2/3	-Ga2/3	-In2/3	-Co2/3	-Sc2/3	-Y2/3	-Cr2/3	-Fe2/3
Al1/3-	3.39	2.74	1.90	2.45	3.38	2.90	3.17↑	1.49↓
Ga1/3-	2.96	2.38	1.88	2.38	2.48	1.72	2.96↓	1.56↓
In1/3-	2.04	1.99	1.73	2.17	2.16	1.96	2.18↑	1.66↓
Co1/3-	2.58	2.47	1.95	2.31	2.06	1.53	2.34↑	1.52↓
Sc1/3-	3.17	2.48	1.83	2.28	3.45	3.64	2.99↑	1.55↓
Y1/3-	2.65	2.36	1.80	2.11	3.54	3.82	2.87↑	1.59↓
Cr1/3-	3.29↓	2.67↓	1.82↓	2.32↑	3.02↑	2.66↑	2.94↑	1.57↓
Fe1/3-	1.53↓	1.66↓	1.75↓	1.59↓	1.76↓	1.65↓	1.74↓	1.31↓

.

6. Statistical Models and Machine Learning

The computational effort to calculate HSE bandgaps can be, as has been said, demanding in terms of time and memory needed. This is especially true for structures with relative compositions M/N different than 1/3 (or 2/3). Therefore, we discuss the use of statistical models and techniques based on machine learning to predict HSE bandgaps given the structural information of the material and its GGA calculated bandgap. A step of ab initio calculation is still needed, but the cost of a calculation with a GGA functional is far less expensive than those using HSE functionals.

To use machine learning techniques or other statistical models, a numerical way of fully describing structures is needed. These numerical representations are the descriptors, which must be invariant to symmetry operations, complete and unique, meaning able to distinguish between any two different structures [67]. The descriptor for which the best performance has been obtained in our current work is the sine matrix [68], consisting of an ansatz that mimics the periodicity and the basic features of the elements in the Ewald sum matrix for a periodic system, using a sine function of the crystal coordinates of the atoms.

Apart from the sine matrix description, some other parameters of chemical nature have been added to the descriptor: the covalent radius, electronegativity, and ionization potential of the metal cations occupying M, N positions. Additionally, given the aim of this approach, the GGA results have been added as well as an extra feature of the descriptor. The featurization of the structures was carried out with the Python libraries matminer [69] and smact [70,71,72]. This featurization was performed for structures with 24 atoms (2 × 1 × 1 rhombohedral supercells), the ones that will be used to test results for new unbalanced configurations, but 12-atom structures (the cells used to model all results in previous sections) were processed as well, in order to check the consistency of results. We ensured that identical models were trained separately for both sets of structures of 12- and 24-atom unit cells, and the average deviation of bandgap results between both was 56 meV, a very small value in terms of bandgaps of semiconductors in the range between approximately 2 and 4 eV. This result suggests that scaling up to even larger unit cells should also yield consistent results.

Regarding the model, because of the limited number of structures considered relevant to describe this family of compounds with delafossite structure, high-complexity models like neural networks, which require large training sets, were automatically discarded even for initial evaluation. Beyond those, other smaller models have been tested, like simple regularized and non-regularized linear regressors, support vector regressors, and gradient-boosting regressors based on decision trees. For all of them, a proper search of optimal hyperparameters has been performed. The best results were achieved with a LASSO regressor (linear regressor with L1 regularization) with an alpha parameter for the regularization of 0.01.

For the evaluation of the models and descriptors, a standard train–test splitting was implemented within a leave-one-group-out cross-validation scheme [73], using the 64 structures of 12 atoms described in Section 5. The size of the test is equal to the square root of the total number of compounds in all the cases, with the training set made up of the remaining compounds. Groups for cross-validation were chosen so the balance between training and test sets was always maintained, meaning that the number of compounds in the set with any of the different metal cations studied was always the same. Following that scheme, average and best results between the series of folds in the cross-validation are here presented in

Table 10. Performance metrics comparing statistically predicted and DFT calculated bandgaps for the different sets of structures in eV.

	Average of Folds		Best Fold
	MAE	${\epsilon }_{\mathit{max}}$	MAE	${\epsilon }_{\mathit{max}}$
Whole set	0.162	0.330	0.078	0.165
Set without Fe	0.130	0.288	0.077	0.166

, which shows main performance evaluation metrics like the mean absolute error (MAE) and maximum error (${\epsilon }_{max}$) between the predicted and DFT calculated results. According to common procedures of model evaluation, the errors presented in the table have been calculated only for the test sets. The results for the best fold are also presented in Figure 13a. Given that all the Fe compounds present a GGA bandgap of 0, an analogous study was performed without including these compounds. Results for the sets without Fe compounds are also presented in the table, for comparison purposes, as it is clearly seen how the performance metrics improve filtering them.

Average results show that this method yields approximate results, which certainly should not be taken as definitive nor exact. However, the aim of this process is not to obtain the most accurate bandgap values but to obtain results within an acceptable range of accuracy that allow us to preselect which compositions would be the most suitable for our purposes.

With the choice of the descriptor and the model already defined, the same process has been followed to evaluate the performance of new structures, with the same crystal structure and same set of species in N, M positions but with relative concentration ratios of metal cations not used for the training of the model. A set of eight new structures (CuAl_1/6Ga_5/6O₂, CuAl_1/2Ga_1/2O₂, CuAl_5/6Ga_1/6O₂, CuAl_5/6In_1/6O₂), CuAl_5/6Cr_1/6O₂, CuAl_1/6Co_5/6O₂, CuGa_1/2In_1/2O₂, CuSc_5/6Y_1/6O₂) have been used as the test set, using the sixty-four original compounds as the training set in this case. These results, with the corresponding performance metrics (MAE, ${\epsilon }_{max}$ and r2), are presented in Figure 13b. Slightly higher errors are obtained in this case, compared with those obtained for the original compounds; however, they are very similar, considering that we are predicting concentration ratios not used to train the model. These results, with errors that fall within a tolerance margin of +/−0.2 eV, could still be of perfect use for preselecting compounds based on this property with high levels of confidence.

These results prove that this machine learning-based approach could be extremely useful to further explore this family of compounds efficiently and affordably, especially considering the titanic computational cost of running HSE calculations for several of these compounds that requires a description with large unit cells and the very inaccurate results obtained with GGA functionals. Indeed, a similar approach could be used as well to predict other properties of interest to sketch a map of properties to efficiently select the most promising compounds within the compositional landscape of this family of compounds, given those properties can be directly inferred from DFT calculations like the ones performed in this work.

7. Conclusions

A study with several CuMO₂ compounds with delafossite structure has been carried out, using DFT and different levels of accuracy in the representation of the exchange correlation functional. The structural characterization for both reported phases has been compared to experimental results, finding excellent agreement. For the electronic properties, it has been found that even GGA simulations yield qualitatively acceptable results for the compounds with a metal from group III-A occupying the M site, but fail to represent other compounds. For the delafossites with the rest of the studied elements, simulations with GGA + U were found to be able to qualitatively represent the electronic properties of the materials, with a proper selection of the U parameter. All of the materials studied exhibited indirect bandgaps except for the ones containing Sc and Y on the M site. Mixed structures were also analyzed, and machine learning techniques were tested and showed the ability to infer HSE bandgap values with a reasonable tolerance margin, even for structures with a larger number of atoms than those calculated with DFT.