Ferromagnetic behavior of native point defects and vacancy-clusters in ZnO studied by first principle calculation

The origin of room temperature ferromagnetism in undoped ZnO is still a question of debate. Experimental and theoretical findings are inconclusive as to the predominant contributor for the magnetic behavior of undoped ZnO. First principle calculation pseudopotential method was used to systematically determine the relaxed atomic geometry, the formation energies and the magnetic properties of the native point defects (vacancies, interstitials and antisites), and vacancy clusters (VZnVO, VZn − 2VO and 2VZn − VO) in ZnO. The results show that ZnO cells consisting of the VZn and the Oi have non-zero magnetic moments, energetically favoring ferromagnetic states and close-to-room-temperature Curie temperatures (294 K). VZn and Oi are also characterized by their low formation energies, in particular in the case of n-type (i.e. Fermi level close to the conduction band minimum) and O-rich conditions. The energy differences between the ferromagnetic state and anti-ferromagnetic state for VZn and Oi are larger than kT at room temperature but still relatively small (∼34 meV). Although VZn and Oi would contribute for the room temperature ferromagnetism, the ferromagnetism states would not be robustly stable for thermal excitation to the anti-ferromagnetic states.


Introduction
Diluted magnetic semiconductor (DMS) has been receiving extensive attention since Munekata et al's first fabrication of (In, Mn)As DMS [1]. These fundamental studies of DMS have been of interest for the development of spintronic devices [2]. The electron spins in spintronic devices are exploited as a further degree of freedom, increasing the efficiency of the device of information storage. A commonly used technique to obtain DMS was to dope magnetic elements into a semiconductor material, such as TiO 2 [3], GaN [4] and ZnO [5]. Other than being the DMS, ZnO is also a multi-functional material suitable for a variety of applications like ultra-violet optoelectronic, transparent electrode, sensors, and photocatalysis [6], which thus attracts extensive focus of research activities. In recent years, Dietl et al [7] studied the Curie temperature T C for different p-type semiconductors, found that room-temperature ferromagnetism (RTFM) can be realized in p-type ZnO doped by Mn. Despite numerous experimental observations of RTFM in transition metal (TM) doped ZnO, it is still uncertain as to what are the relevant origins of the RTFM in doped ZnO, which may be originated from the introduced dopant, the interaction between dopant and intrinsic defect, or second phase, etc. Most of the published work, based on theoretical calculation and experimental verification concluded that intrinsic defects play a crucial role in the magnetic behavior of doped ZnO. For example, Yi et al [8] proposed that RTFM observed in Li-doped ZnO may be associated with the V Zn vacancy based on the results of positron annihilation spectroscopy (PAS) study. They also found that doping with appropriate dopants can lower the formation energy of V Zn . Besides, V O is also critical in one way or another on magnetic property. Hsu et al [9] found a Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. correlation between the enhancement of ferromagnetism and the increase of oxygen vacancies in Co-doped ZnO. By using soft x-ray absorption, x-ray magnetic circular dichroism and first-principle calculations, Herng et al [10] found both Cu impurities and V O were essential to the observed RTFM in Cu doped ZnO and proposed an indirect exchange model as the cause.
RTFM has also been reported in undoped ZnO material with different structures [11][12][13]. Despite of many efforts devoted [14][15][16][17][18], the origin of the observed RTFM in undoped ZnO is still inconclusive. With the comprehensive study based on first-principle calculation and photoluminescence, the observed RTFM in ZnO nano-particles grown by a solution method was attributed to singly ionized oxygen vacancies [19]. Xing et al [19] reported RTFM in ZnO nanowires obtained by using a vapor transport method, and the magnetic property was tunable by adjusting the oxygen deficiency during growth. Using density functional theoretical (DFT) study, Wang et al [20] found that the FM in undoped ZnO could be attributed to V Zn instead of V O , also indicated V Zn prefer to form clusters. Similarly, Chakrabarty and Patterson [21] carried out a DFT study and suggested that the RTFM in undoped ZnO could originate from the isolated V Zn and (V Zn V O )-divacancy. With molecular dynamics and DFT studies, Tietze et al [22] showed the presence of unpaired electrons at the grain boundary and these unpaired electrons were ferromagnetically coupled. Most of the RTFM theoretical studies of native defects in undoped ZnO focused on V Zn , V O , and V Zn V O divacancy. It is no doubt that, under equilibrium condition, V Zn and V O are the predominant defects due to their relatively low formation energies [23]. However, for ZnO samples undergone non-equilibrium process, e.g. electron irradiation [24,25] and ion implantation [26,27], other kinds of intrinsic defects and vacancy clusters having higher formation energies would exist. It is thus also important to understand the magnetic properties of these defects to grip a more comprehensive view for interpreting the experimental data.
Using the density functional theory (DFT) method, we studied the relaxed atomic geometry, the formation energy and magnetic properties for all the native point defects (vacancies, interstitials and antisites) and vacancy clusters (V Zn V O , 2V Zn −V O and 2V O −V Zn ) in ZnO so as to gain the knowledge of their roles in ZnO materials exhibiting RTFM.

Method
All calculations in the present study were performed based on DFT, using the projector augmented wave method (PAW) as implemented in the Vienna ab initio Simulation Package. The exchange-correlation potential was represented by the spin-polarized generalized gradient approximation (SGGA). The plane wave cut-off energy was taken as 400 eV throughout the calculation. All the atoms in the supercell were fully relaxed until the Hellmann-Feynman force converged to less than 0.01 eV A −1 . For the k-space integration, a 3×3×3 k-points grid was used for sampling the irreducible wedge of the Brillouin zone. Test calculation for the selection of cutoff energy and k-points were performed, finding that the results remained unchanged with higher cutoff energy and denser k-points. A homogeneous background-charge was added or removed from the supercell to obtain the different charge states of the defects. The equilibrium defect concentration c depends on the formation energy and is given by where N sites is the number of the possible defect sites in the supercell, E f is the formation energy, and k B is Boltzmann constant. The formation energy can be found by calculating the total energy of the supercell, as given by [28]: q is the charge state of the defects and E f (X q ) is the formation energy of the defect X in the supercell. E tot (X q ) is the total energy obtained from the DFT calculation for defect X with the charge state of q, E tot (ZnO, perfect) is the total energy of the defect-free ZnO supercell, n i is the number of atoms (type i) that have been added (n i <0) or removed (n i >0) from the supercell. μ i is the chemical potential of the element i, E F is the Fermi-level. E v is the valance band maximum of ZnO, which is given as: = - ZnO, .

Structural relaxation
All atoms in the supercell are fully relaxed during the optimized procedure. It is worthy to discuss the relaxation of the vacancies. For the neutral and +1 states of V O , the nearest Zn atoms respectively displaced inward by 12.3% and 0.3% as compared with the equilibrium Zn-O bond length. For the +2 charge states of V O , the neighboring Zn atoms displaced outward by 24%. For V Zn with neutral, −1 and −2 charge states, the relaxations were similar, i.e. 10.6% outward as compared to the equilibrium Zn-O bond length. Vacancy clusters are formed through removing the nearest neighboring atoms in the perfect supercell with the equilibrium Zn-O bond length of 1.98 Å respectively on the a-b basal plane and along the c-axis. For V O −V Zn , both O atoms and Zn atoms neighboring the defect exhibited outward displacement. The most significant outward displacement occurred on the charge state of (V Zn −V O ) +1 . The nearest O atoms displaced outward by 6.2% at the basal plane and 16% along the c-axis, while the surrounding Zn atoms displaced outward by 4% along the c-axis and no obvious displacement at the a-b basal plane. For the 2V O −V Zn and 2V Zn −V O clusters, three nearest atoms were removed in a perfect supercell. The outward displacements for O and Zn atoms around the vacancy clusters are basically similar to that for V O −V Zn , whereas the nearest O atoms along the c-axis had the most significant outward displacement.

Formation energies
The formation energies of the different intrinsic point defects (namely V Zn , V O , Z ni , O i , Zn O , and O Zn ) against the Fermi level positive (measured from the valance band minimum) obtained from the DFT calculation are shown in figure 1. The interstitial defects have two configurations in the wurtzite structure, namely the octahedral coordinated (oct) site and the tetrahedral coordinated (tet) site [23]. The oct-site usually has lower formation energy than that for the tet-site. Therefore, only the low-formation-energy oct-site was studied in the present study. The valence band maximum (VBM) is set to be at 0 eV, whereas the theoretical conduction band minimum (CBM) is indicated by the black dotted line in figure 1. The Fermi-level E F varies from the valence band edge to the experimental conduction band edge, for which the experimental band gap was obtained by

Magnetic properties
The total magnetic moments M total of the unit cells containing the corresponding defects in the different charge states were calculated. All the Zn and O atom sites in the supercell were included to calculate for the total magnetic moments. The resultant total magnetic moments, as well as the corresponding contributions from Zn and O atoms, of the unit cell containing the defects having non-zero total magnetic moments are tabulated in      The iso-surface value is 0.02 e A −3 . The black, red and black balls represent Zn, O and vacancy atoms in supercell respectively. supercell. For the 2V Zn −V O cluster having the neutral charge, the defect staying along the c axis has a slightly larger magnetic moment than that at the a-b plane (namely 1.997 μB and 1.774 μB). For the 2V Zn −V O cluster having the 1− charge, the defect staying along the c axis has a negligibly smaller magnetic moment while compared to that at the a-b plane (namely 0.948 μB and 0.999 μB). It is also noticed that the 2V Zn −V O clusters in neutral charge have the smaller magnetic moment as compared to that in the 1− charge. The Curie temperatures (T C ) were estimated by the mean-field approximation, which was given by:   low, in particular under the O-rich condition for n-type materials. The 2V Zn V O 's in c-axis and ab-plane configurations with 0 and −1 charge states also have non-zero localized magnetic moment and favors for FM state, though the ΔE are relatively small and thus resulting in low Curie temperatures. The largest ΔE and highest Curie temperature among the different configurations and charge states for 2V Zn V O 's is 29 meV and 224 K respectively, which occurs at the c-axis configuration and −1 charge. Thus, 2V Zn V O do not contribute FM effectively at room temperature, while V Zn and O i could be the probable defects responsible for the observed RT FM in ZnO. Khalid et al [33] studied the origin of RTFM in undoped ZnO using a comprehensive approach (SQUID, positron annihilation spectroscopy, x-ray diffraction and first principle calculation) and associated the observed RTFM to V Zn , though there is no experimental report contributing the observed room temperature FM to O i . However, the ΔE for V Zn is still relatively low and its Curie temperature is marginally close to the room temperature. The thermal stability of its RTFM would not be robust against thermal excitation.
The current study shows that (summarized in table 3) the defects possessing non-zero magnetic moment and energetically favoring ferromagnetic state are all acceptors and the magnetic moments are mainly contributed from the O atoms. It is also noticed that the localized magnetic moments are favored if these defects are filled by a hole in a more positive state. Kenmochi et al [34,35] in first principle magnetic studies of MgO, SrO and BaO suggested that FM originated from double exchange could be correlated with hole doping, for which Yamamoto and Katayama [36] reported that hole-doping was favored by co-doping of donor and acceptor.
To discuss for the undertainties of the current results, it would be worthy to bring to the attention that the T C estimated by the mean-field approximation does not involve the interaction range, which would lead to significant errors in the dilute magnetic semiconductor with low defect concentration [37]. Therefore, the realistic T C values could be lower than the ones obtained by the mean-field approximation. Seike et al [38] reported that to induce ferromagnetism under homogenous distribution conditions, 15%-20% doping concentration is required for RTFM [38]. In the present study, the corresponding defect concentration is ∼2.8% for one defect in the unit cell, which is much lower than the threshold. For the practical fabrication of ZnO samples like film grown by pulsed laser deposition, the native defect like the V Zn -related defect has concentration of ∼10 18 cm −3 (~0.01 %) in undoped ZnO [39]. Besides, the assumption of homogeneity is difficult to be confirmed for realistic DMS systems. Nano-scale inhomogeneity like spinodal decomposition cannot be exclude as the cause of the experimental observation of RTFM in the doped ZnO systems [40,41].

Conclusion
In conclusion, first principle calculation was performed to systematically study the relaxed atomic geometry, formation energies and magnetic properties of all the native point defects (vacancies, interstitials and antisites) and the vacancy clusters V Zn V O , V Zn −2V O and 2V Zn −V O in ZnO. Only the unit cells of ZnO containing the V Zn , O i (oct) and 2V Zn −V O in the configurations of a-b plane and c-axis energetically favors for FM state as compared to AFM state and carry non-zero magnetic moments. The energy differences between the FM and AFM states ΔE, and the magnetic moments depends on the charge state and the defect configuration, varying from 7-38 meV and 0.9-2.0 μB per unit cell respectively. V Zn 0 and O i are the two defects having the low formation energies, the largest ΔE and Curie temperautres (both are 38 meV and 294 K respectively). Their magnetic moments are 1.82 μB per unit cell and 1.01 μB per unit cell respectively. These two defects could be the origins of the experimental observed RTFM in ZnO materials, though their FM states are not robustly stable against thermal excitation to the AFM state as their ΔE's are still relatively small. For the case of 2V Zn −V O 's, their ΔE's are even smaller (7-29 meV for different charge states and configurations). The low ΔE leads to the low Curie temperature, which is lower than the room temperature. The formation energies of 2V Zn −2V O 's are larger than those of V Zn and O i 's, indicating their relative lower abundance with the equilibrium situation. 2V Zn −2V O 's are thus not the important contributor to RTFM.