Stoner Ferromagnetism in Hole-Doped CuMIIIAO2 with MIIIA = Al, Ga, and In

Using density functional theory calculations, we examine the effect of hole doping on the magnetic and electronic properties of CuMIIIAO2, with MIIIA = Al, Ga, and In. CuMIIIAO2 nonmagnetic semiconductors switch to ferromagnetic half-metals upon hole doping. For CuAlO2, the nonmagnetic-to-ferromagnetic transition occurs for hole densities of ∼7 × 1019/cm3. Ferromagnetism arises from an exchange splitting of the electronic states at the valence band edge, and it can be attributed to the high-lying Cu-d states. Hole doping induced by cation vacancies and substitutional divalent dopants is also investigated. Interestingly, both vacancies and nonmagnetic divalent dopants result in the emergence of ferromagnetism.


INTRODUCTION
Ferromagnets such as iron, cobalt, and nickel, are typically metals, whereas materials with both ferromagnetic and semiconducting properties are scarce. Ferromagnetic semiconductors are attractive candidates for future spintronic devices, owing to their easy integration into the existing semiconductor devices. 1 Over the past few decades, diluted magnetic semiconductors (DMS), i.e., nonmagnetic semiconductors doped with magnetic ions, have triggered intense interest. 2−5 Magnetic moments mainly arise from partially filled 3d and 4f orbitals of transition metals and rare earth metals, and examples of DMS include (Ga,Mn)As, (Zn,Cr)Se, and (Pb,Eu)Te. Although Ga 1−x Mn x As has been one of the most widely studied DMS, its low Curie temperature hinders its utilization in practical applications. Contrary to GaAs-based DMS, room-temperature ferromagnetism has been observed in diluted magnetic oxides such as TiO 2 , ZnO, and SnO. 6−10 Intriguingly, high-temperature ferromagnetism in the absence of magnetic ions has been additionally reported. 11−21 For instance, Kenmochi et al. found that carbon-doped oxides such as MgO, SrO, BaO, and CaO are ferromagnetic without any transition-metal impurities, and based on the mean-field approximation, Curie temperatures higher than room temperature have been predicted. 19−21 Two types of diluted magnetic semiconductors with different properties have been widely studied. First, for DMS like Mn-doped GaSb with localized majority d-states deep in the valence band, ferromagnetism arises from the Zener's p−d exchange mechanism, resulting in holes in the majority pvalence band. Such interaction is relatively weak and long-range, and the Curie temperature scales linearly with the impurity concentration. On the other hand, for DMS like Mndoped GaN with impurity bands within the band gap, ferromagnetism arises from the Zener's double-exchange mechanism. The magnetic couple is strong and short-ranged, and the Curie temperature scales proportionally to the square root of the impurity concentration. 22,23 Doping of CuAlO 2 with magnetic ions such as Mn, Fe, and Co, to form oxide-based DMS, has been intensively investigated. 24−27 For instance, first-principles calculations revealed that (Cu,Fe)AlO 2 and (Cu,Co)AlO 2 are attractive candidates as high-Tc ferromagnets. Overall, CuAlO 2 is an attractive candidate for various applications owing to its p-type conductivity (without intentional doping) and its visible light transparency. 28−30 In general, the design of p-type transparent conductive oxides (TCO) is found to be difficult, and the difficulty arises from the nature of the VB edge, dominated by localized O-2p states. 31,32 Holes are typically trapped at O sites and cannot migrate through the oxide, leading to poor p-type behavior. To overcome this limitation, modulation of the VB edge through hybridization of the O-2p orbitals with metal orbitals has been proposed. 31 The discovery of p-type conductivity in CuAlO 2 has paved the way to the discovery of other p-type TCO like CuGaO 2 and CuInO 2 .
In this paper, using density functional theory (DFT) calculations, we examine the effect of hole doping on the magnetic properties of CuM IIIA O 2 with M IIIA = Al, Ga, and In, having CuAlO 2 as a prototype material. Despite that these materials are intrinsically nonmagnetic, stable ferromagnetic phases appear for a wide range of hole densities. In addition the possibility of inducing hole doping and also ferromagnetism by cation vacancies and substitutional divalent dopants is investigated. Contrary to previous studies, we focus on doping with nonmagnetic ions. Magnetic ions typically lead to localized magnetic moments instead of ferromagnetic phases, and materials with localized magnetic moments are likely to present low spin polarization of the charge carriers, hindering their utilization in future spintronic devices. In our work, we find that nonmagnetic divalent dopants result in the emergence of ferromagnetism, and ferromagnetism can be attributed to the high-lying Cu-d states, dominating over the O-p states at the VB edge.

METHODS AND COMPUTATIONAL DETAILS
We perform spin-polarized DFT calculations using the Vienna Ab initio Simulation Package (VASP). 33,34 For the exchange correlation functional, the generalized gradient approximation introduced by Perdew, Burke, and Ernzerhof (PBE) is used. 35 The electron−ion interaction is described by the projector augmented wave (PAW) method. 36 For the unit cells, both the atomic positions and the volume are optimized, through the conjugate gradient method, with 10 −8 eV energy convergence criteria and 10 −2 eV/Å force convergence criteria. The kinetic energy cutoff is set to 600 eV, and the Brillouin zone is sampled by a 16 × 16 × 16 k-point grid for the atomic relaxations, whereas a 24 × 24 × 24 k-point grid is used for the density of states (DOS) calculations.
To achieve hole doping, we change the number of electrons of the unit cell and we add a compensating opposite charge background. Since the magnetic instabilities sensitively depend on the DOS near the Fermi level, a very dense grid of 48 × 48 × 48 k-points was used for the calculations of the magnetic moments. Hybrid functional calculations were also performed, using the Heyd−Scuseria− Ernzerhof (HSE) functional and the standard range-separation and mixing (0.25 Fock exchange) parameters. 37 Due to computational issues, for the hybrid functional calculations, an 8 × 8 × 8 k-point grid was adopted.
Finally, for the defective systems, we used supercells consisting of 192 atoms in total. The Brillouin zone was sampled by a 2 × 2 × 2 kpoint grid for the atomic relaxations, whereas a 4 × 4 × 4 k-point grid was used for the DOS calculations.  . Two-dimensional band structure of CuAlO 2 unit cell, and three-dimensional band structure for the topmost valence band and the bottommost conduction band. The three-dimensional band structure is obtained using the VASPKIT code. 56 The energies refer to the valence band maximum. The high-symmetry k-points are Γ (0, 0, 0), F (0.5, 0.5, 0), L (0.5, 0, 0), and Z (0.5, 0.5, 0.5). The topmost VB is highlighted in red.
and out-of-plane lattice constants are a = b = 2.880 and c = 17.119 Å, in excellent agreement with experimental observations. 38 A summary of the structural parameters of CuM IIIA O 2 with M IIIA = Al, Ga, and In, in the delafossite structures is given in Table S1 in the Supporting Information. As shown in Figure  1, CuAlO 2 is an indirect band gap semiconductor with E g = 1.79 eV, in agreement with previously reported DFT-PBE calculations. 39 Using HSE calculations, a gap opening of about 1.8 eV is found. The valence band (VB) top and the conduction band (CB) bottom are located at the F and Γ points, respectively, and the VB top is mainly derived from Cud orbitals along with a smaller contribution of O-p orbitals, as observed in Figure 2. Notably, the dispersion of the topmost VB looks like a slightly asymmetric inverted Mexican hat, where the band extrema are located at the F and L points, and the energy difference between the F and L points is 85 meV. This peculiar VB dispersion leads to a sharp high peak of the DOS near the VB edge. By shifting the Fermi level close to this peak, the Stoner criterion is fulfilled and the system becomes ferromagnetic. 40−47 Hence, hole-doped CuAlO 2 could be a ferromagnetic material.
To verify this assumption, we consider charged systems. Upon hole doping, a very slight shortening of the Cu−O bond lengths is observed, whereas the Al−O bond lengths present the opposite trend (see Supporting Information Figure S1). The magnetic moments and the spin polarization energies as a function of the injected holes are computed. The spin polarization energies are obtained through the equation E SP = E NM − E FM , where E NM and E FM are the energies of the nonmagnetic and ferromagnetic phases, respectively. Using this definition, positive spin polarization energies indicate stable ferromagnetic phases. As shown in Figure 3, a ferromagnetic state spontaneously emerges for hole densities of ∼7 × 10 19 / cm 3 . The magnetic moment per carrier rapidly reaches the saturation value of 1 μ B , and it remains unchanged for hole densities up to ∼1 × 10 22 /cm 3 . Further increase of the hole density leads to the reduction of the magnetic moment. On the other hand, the spin polarization energy per carrier strongly depends on the doping level. At the relatively low doping level, it increases by increasing the density. For p = 6 × 10 21 /cm 3 , it reaches the maximum value of ∼46 meV/carrier, which is larger compared to the corresponding values in similar investigations, 44,45 whereas for larger hole densities, it gradually decreases until it becomes zero. Contrary to the magnetic moment per carrier, no plateau region is observed in the spin polarization energy per carrier.
Owing to the self-interaction error, semilocal functionals tend to overdelocalize electrons, and for systems with d electrons, this can be particularly problematic. Hartree−Fock and semilocal functionals typically present an opposite selfinteraction error. Thus, hybrid functionals, which use a fraction of the Hartree−Fock and a fraction of the semilocal functional, tend to reduce the self-interaction error. 48 In our work, hybrid functional calculations have been additionally performed for the hole-doped systems. Notably, using HSE calculations, the peculiar Mexican-hat-like valence band dispersion is retained and the energy difference between the F and L points is 91 meV, i.e., slightly larger compared to the PBE calculations. The magnetic properties of CuAlO 2 at different hole densities in the range of ∼10 19 −10 22 /cm 3 are examined, and similar to our PBE calculations, hole-doped CuAlO 2 is found to be ferromagnetic. It is worth noting that the so-called pseudopotential self-interaction method (pseudo-SIC) has been widely used to eliminate self-interaction errors in standard DFT calculations. 49−51 For instance, the selfinteraction-corrected electronic structures of ZnO-based DMS have been reported. Compared to those computed using standard DFT, a larger band gap and a qualitatively different description of the transition-metal d states have been observed.
As a next step, we compute the electronic band structures and the spin density plots of the charged systems. The spin density plots are obtained by the spin-up and spin-down charge density differences. As shown in Figure 4, an exchange splitting of the electronic states near the Fermi level is observed at p = 5.0 × 10 20 /cm 3 . The energy difference between the spin-up and spin-down valence band maxima significantly increases, by increasing the hole density. On the other hand, we observe a shift of the Fermi level toward the lower energies. This results in a half-metallic behavior, i.e., the spin-up and spin-down states present semiconducting and metallic nature, respectively, allowing a fully polarized spin transport. The magnetization mainly arises from Cu-d orbitals along with O-p orbitals. For instance, for p = 1 × 10 21 /cm 3 , the magnetization of Cu, Al, and O atoms is 86, 0, and 14% of the total magnetization, and very similar results are found for other hole densities. These results are consistent with the observation that the magnetic behavior is originated from the VB edge, which mainly consists of Cu-d states with a smaller contribution of Op states.
Finally, we perform calculations to estimate the Curie temperature of CuAlO 2 at various hole densities. Our calculations are based on the mean-field evaluation of the magnetic moment by minimizing the electronic free energy of the system at specific temperatures. 52 To introduce the finite temperature, we modify the smearing factor (σ = k B T) of the Fermi−Dirac distribution. At the relatively low doping level, the Curie temperature is below room temperature. By increasing the hole density, the Curie temperature increases, and at the high doping level, it becomes much larger than room temperature (see Supporting Information Figure S2). It should be noted that the Curie temperatures can be significantly overestimated using this approach, and the real values can be lower compared to the predicted values by as much as a factor of 2. 53 Overall, our findings revealed that CuAlO 2 could be used for future spintronic devices like the spin field effect transistors (spin FETs) and the magnetic tunnel junctions (MTJs). We found that the injection of holes in CuAlO 2 could control/ manipulate its electronic and magnetic properties. Control of magnetism through an applied voltage offers various advantages like ultralow power dissipation and reversibility, and electrically controlled ferromagnetism is highly promising for achieving optimal spintronic devices.
Besides CuAlO 2 , other group-IIIA Cu-based oxides are examined, namely, CuGaO 2 and CuInO 2 . Both CuGaO 2 and CuInO 2 present an asymmetric Mexican-hat-like VB dis-  Furthermore, we examine if the corresponding chalcogenides CuAlY 2 , where Y = S, Se, and Te, in the delafossite structures, are also promising for realizing a ferromagnetic phase upon hole doping. First, we introduce a measure for the hybridization strength between the projected orbital l of atom α and the projected orbital l′ of atom α′, using the normalized cross-correlation of DOS g(E) In the entire VB region, the hybridization strengths between Cu-d and Y-p orbitals are found to be 0.50, 0.50, and 0.41 for CuAlS 2 , CuAlSe 2 , and CuAlTe 2 , respectively, whereas the corresponding value for CuAlO 2 is only 0.21. In addition, contrary to the oxides where the VB edge is dominated by Cud states, for the sulfides and selenides, a strong hybridization of Cu-d and Y-p states is observed, whereas for tellurides, the VB edge is dominated by Te-p states, as shown in Figure 2. Thus, going from oxygen to tellurium compounds, the Cu-d character of the VB edge is found to decrease. The different nature of the VB edge in chalcogenides compared to the oxides leads to the absence of the peculiar Mexican-hat-like VB dispersion and the subsequent absence of the sharp high peak of the DOS at the VB edge. Thus, CuAlY 2 , with Y = S, Se, and Te is expected to be nonmagnetic upon hole doping. To verify this assumption, additional calculations were performed and we found that hole doping in the range of 1 × 10 19 −1 × 10 22 / cm 3 results in no magnetization in CuAlY 2 . AgM IIIA O 2 , where M IIIA = Al, Ga, and In, as well as AuAlO 2 are also discussed. The hybridization strengths between Ag-d and O-p orbitals, in the entire VB region, are 0.83, 0.79, and 0.78 for M IIIA = Al, Ga, and In, respectively, whereas the corresponding values for their Cu-based counterparts are only 0.21, 0.22, and 0.20. In addition, going from Cu-to Ag-based compounds, the metal-d character of the VB edge significantly decreases, leading to a less sharp peak of the DOS at the VB edge, as shown in Figure 2. Accordingly, for Ag-based compounds, the asymmetry in the Mexican-hat-like VB dispersion significantly increases and the energy differences between the band extremum located at the F and L points are 258, 428, and 340 meV for AgAlO 2 , AgGaO 2 , and AgInO 2 , respectively. This should lead to a much less promising behavior of hole-doped AgM IIIA O 2 compounds compared to CuM IIIA O 2 . Indeed, our calculations reveal that for AgM IIIA O 2 , no ferromagnetism is observed for hole densities up to ∼1 × 10 22 /cm 3 .
Interestingly, AuAlO 2 presents an intermediate behavior between CuAlO 2 and AgAlO 2 . The hybridization strength between Au-d and O-p orbitals in the entire VB region is 0.76, whereas the VB edge has a weaker (stronger) d-character compared to CuAlO 2 (AgAlO 2 ), as shown in Figure 2. Furthermore, for AuAlO 2 , the asymmetry in the Mexican-hat VB is larger (smaller) compared to CuAlO 2 (AgAlO 2 ) and the energy difference between the F and L points is 204 meV. We find that AuAlO 2 becomes ferromagnetic for only a narrow range of hole densities, i.e., ∼5 × 10 21 < p < 9 × 10 21 /cm 3 , and for the optimal hole density, the magnetic moment is less than 1 μ B /carrier. A summary of the key electronic-and magneticrelated characteristics for all studied systems is shown in Table   1. A measure of the d-state strength in the VB edge is introduced using the formula 2) and for this measure, we use an integration region of 0.3 eV below the VB top.

Hole
Doping Induced by Intrinsic Defects in CuAlO 2 . As a next step, we examine the possibility of inducing hole doping and also ferromagnetism by intrinsic defects. In our study, we focus on cation vacancies (V Cu and V Al ) in the neutral charge state. Regarding the structural properties, by removing a Cu atom, two undercoordinated O atoms are found. These atoms exhibit a slight outward relaxation by 2.1%, leading to shorter Al−O bond lengths of 1.852 Å. Accordingly, Cu atoms surrounding the missing lattice site display an outward relaxation of ∼1.3%. Contrary to Cu vacancy, which causes only a minor distortion of the lattice, Al vacancy induces a larger deformation. In particular, threefoldcoordinated O atoms, neighboring the vacant site, relax outward by 10.5%, resulting in shorter Cu−O bond lengths of 1.805 Å.
We compute the defect formation energies through the equation E for = E tot (CuAlO 2 :V A ) − E tot (CuAlO 2 ) + μ A , where E tot (CuAlO 2 :V A ) and E tot (CuAlO 2 ) are the energies of the defective and defect-free structures, respectively, whereas μ A is the chemical potential of the removed cation atom, which refers to the corresponding solid structure. Cu vacancy formation energy is found to be 0.9 eV, whereas the formation energy of Al vacancy is as high as 8.4 eV, in agreement with previously reported theoretical calculations. 54 When a Cu atom is removed, one hole is introduced in the system, whereas the removal of an Al atom corresponds to three induced holes in the system. The corresponding hole densities are about 5 × 10 20 and 15 × 10 20 /cm 3 for Cu-and Aldeficient systems, respectively. Cu vacancy leads to a total magnetization of 1 μ B , whereas Al vacancy results in a total In addition, the total Cu, Al, and O magnetizations are found to be ∼0.9, 0.0, and 0.1 μ B , respectively. At the same hole concentration, for the holeinjected CuAlO 2 , the Cu, Al, and O magnetizations are also 90, 0, and 10% of the total magnetization, respectively. Next, the relaxation effect on the magnetic properties is investigated. In agreement with the previous results, the unrelaxed defective structure has a total magnetic moment of 1 μ B and the distribution of the local magnetic moments is very similar. Figure 5 shows the spin density plots for both vacancy defects.
The spin polarization energies of Cu-and Al-deficient systems are found to be ∼7 and 93 meV, respectively, and the positive values reveal that the ferromagnetic phases are favorable compared to the nonmagnetic phases. Test calculations for singly negatively charged Cu vacancies and triply negatively charged Al vacancies are also performed. The charged defective systems are found to be nonmagnetic, confirming that the induced holes are responsible for the emergence of ferromagnetism.
Regarding the electronic properties, Figure 6 shows the DOS of the defective CuAlO 2 . For both cation vacancies, no deep gap states are observed. Due to the induced holes, the Fermi level is shifted within the VB, leading to an asymmetry between the spin-up and spin-down states, and both systems are found to be half-metals. Obviously, the Fermi-level shift for the Aldeficient system is larger compared to the Cu-deficient one. 3 where E tot (CuAlO 2 :X) and E tot (CuAlO 2 ) are the energies of the doped and undoped structures, whereas μ Al and μ X are the chemical potentials of Al and dopant atoms, respectively, which refer to the corresponding solid structures. The formation energies are found to be 1.7, 2.8, 4.8, and 6.7 eV for Mg-, Ca-, Zn-, and Cddoped systems, respectively.  Taking Mg-doped CuAlO 2 as an example, the total magnetic moment is found to be 1 μ B , i.e., equivalent to the number of holes. The 32 Cu atoms, which sandwich the dopantcontaining Al plane, present local magnetic moments of about 0.02−0.04μ B , whereas the remaining 16 Cu atoms present negligible magnetic moments. Magnetization is found to be less localized in plane, compared to the out-of-plane direction. The total Cu and O magnetizations are found to be ∼ 0.9 and 0.1 μ B , respectively, whereas the dopant atom and all Al atoms have zero magnetic moments. Remarkably, our findings are different from those reported in similar investigations, where the magnetic moments are found to be strongly localized in the impurity atoms. 44 Our theoretical calculations are also in good agreement with recent experimental observations, where CuAl 1−x Mg x O 2 with x = 0.05 is found to be ferromagnetic at room temperature. 55 Similar results are also obtained for the magnetic moments for the other group-IIA-and group-IIB-doped systems, and their spin density plots are shown in Figure 7. For all doped systems, positive spin polarization energies are found and their values are 7.2, 6.3, 7.3, and 7.2 meV for Mg, Ca, Zn, and Cd dopants, respectively. Concerning the electronic properties, no deep gap states are observed, and an obvious shift of the Fermi level within the VB along with half-metallicity is observed, as shown in Figure 8.

CONCLUSIONS
Using density functional theory calculations, we examined the electronic and magnetic properties of CuM IIIA O 2 with M IIIA = Al, Ga, and In, upon hole doping. Our simulations revealed a nonmagnetic-to-ferromagnetic transition for the hole-doped systems. Ferromagnetism arises from an exchange splitting of the electronic states at the valence band edge, and it can be attributed to the high-lying Cu-d states dominating over the Op states. Besides ferromagnetism, half-metallicity has been additionally observed, i.e., the spin-up and spin-down states present semiconducting and metallic behavior, respectively, allowing a fully polarized spin transport. The possibility of inducing hole doping and a subsequent ferromagnetic order by cation vacancies was also investigated. We found that Cu  vacancy presents a formation energy lower than 1 eV and leads to a total magnetization of 1 μ B . The local magnetic moments of the nearest-neighboring Cu sites are ∼0.05 μ B , whereas all other Cu sites present nonzero magnetic moments up to 0.04 μ B . Besides cation vacancies, group-IIA and group-IIB atoms replacing Al atoms were examined. Interestingly, these nonmagnetic divalent dopants also result in the emergence of ferromagnetism.