Nature of charge transport and p-electron ferromagnetism in nitrogen-doped ZrO2: An ab initio perspective

Zirconium dioxide provides an exceptional prototype material for studying the redistribution of the polaron holes and its magnetic coupling with their nearby anions owning to the difference oxygen binding behavior in the monoclinic phase. Here, we perform a comprehensive study of the p-electron magnetism in the nitrogen doped 2 × 2 × 2 monoclinic ZrO2 based on spin-polarized density functional theory. Nitrogen substitutions make the system display half-metallic properties, and the origin of room temperature ferromagnetism ascribes to the p-p coupling interaction between N 2p and the host 2p states. The charge density difference and Mülliken population analyses provide evidences of charge redistributions. Our results reveal that the polaron transfer may alter the magnetic properties and it is greatly facilitated ferromagnetic coupling if the polaron holes are localized around a single anion dopant.

the 2p orbitals of O4 near the Fermi level. It implies that the O4 2p shell should be first fully occupied compare to the O3 2p shell while sufficient holes are injected. It can be anticipated that the redistribution of the induced holes and the interaction with their nearby atoms will have a significant impact on establishing the RTFM.

Results
Structure stability. Let us first consider a Zr 32 O 63 N supercell with a single N substituted at an O site, accompany a formation of a doped hole. After a structural relaxation, the N substituted at O3 site (N O3 ) model system is found to have a lower energy of 65.4 meV than that the N substituted at O4 site (N O4 ) model system. This should be due to the fact that N O3 has a more freedom to relax than the N O4 . In addition, both spin-polarized and non-spin-polarized states are calculated to check the stability of the magnetization. The results reveal that both the model system of N O3 and that of N O4 favor the energy differences of about 387.5 and 293.1 meV between the spinpolarized state and the non-spin-polarized state respectively. The probability of incorporating a defect in a crystal is determined by the formation energy. The calculated formation energy of N O3 (or N O4 ) is 3.92 eV (or 3.99 eV) and 21.56 eV (or 21.49 eV) for the Orich condition and Zr-rich condition, respectively.
Electronic properties. The calculated total density of states (TDOS) and PDOS for the N 2p, the neighboring Zr 4d and O 2p states are shown in Fig. 2. It is found there exist significant hybridization between the N 2p states and O 2p states around the Fermi level (E F ), suggesting a strong p-p coupling interaction between them. Such a strong p-p interaction makes the energy levels near the E F spin split. The partially filled spin-down band split off from the fully occupied spin-up band via Hund exchange, resulting typical FM half-metallic character, which is important for the spintronics applications 37 . Additionally, nitrogen substituting for oxygen introduces some hole states peaked abound 1.0 eV above E F , which are mainly originated from N 2p, O 2p and Zr 4d orbitals. The total magnetic moment of the supercell is 1.00 m B corresponding to the single excess p electron provided by the N dopant. This is consistent with the previous results of N-doped ZnO 15 , TiO 2 14 , In 2 O 3 38 . The moments are mainly localized on the N 2p orbital (,0.86 m B ), with minor contributions of the O (0.18 m B ) and the neighboring Zr atoms is polarized antiferromagnetically with a magnetic moment of 20.04 m B (See the Supplementary Information for more details).
The band structure calculations suggest that the localized holes induced by N dopant mediate the FM interaction. The carriers, i.e. holes, which were introduced by the substitution of N at O sites, reside not only on the N 2p orbitals but also on its near neighboring O atoms. The holes near each anion impurity tend to align parallel to the moment of the impurity ion under the p-d hybridization-like p-p coupling interaction 5,39 . Sufficient spin-polarized holes can mediate the spin alignment of the N dopant atoms, resulting in an indirect strong FM coupling among them. Therefore, the mechanism responsible for the FM coupling in the N-doped ZrO 2 can be classified as the p-p coupling interaction between the impurity 2p states and the host 2p states.
Magnetic coupling. Next, we examine the magnetic coupling of these magnetic moments. To understand this issue, we calculate four nonequivalent structural configurations of Zr 32 O 62 N 2 , which are constructed by replacing two O atoms by N with various N-N distances. Here, the ''dimer'' of N-Zr-N configuration is only considered because we find the magnetic coupling to be negligible small when the N-N pair is well separated. The (i, j) configuration is adopted to denote an N-N pair, in which two O sites are replaced by N at i and j sites, as shown in Fig. 1(c). This amounts to the doping concentration of 3.13%. The magnetic coupling strength can be obtained from the energy difference between FM and AFM states (DE FM-AFM 5 E FM -E AFM ). A positive value means the FM state is favored and vice versa. DE is the relative energy with respect to the (a, b) configuration which has the lowest total energy. The results are summarized in Table 1. It is found that the DE becomes smaller when the N-N distance gets larger, and hence the doped N atoms do not have a tendency of clustering. It is noteworthy that the FM states always exhibit a lower energy than the AFM states except for the (a, d) configuration in which the N-N distance is 2.8 Å with the bond angle ffN-Zr-N near 90u. It is well known that the Curie temperature in the mean field approximation (MFA) T c is calculated as k B T c 5 (2/   35,40,41 . The MFA estimates T c for DMSs with low concentrations within 5,10% errors. The T c is always very lower than the room temperature, due to the short-range magnetic exchange interaction of deep impurity states in wide band-gap semiconductors, percolation of the ferromagnetic coupling is difficult to achieve for small concentrations. However, in the case of (a, b) configuration, the DE FM-AFM value reaches 372.7 meV, larger that of N-doped TiO 2 14 and N-doped In 2 O 3 38 which are known to be RTFM. Therefore, such a strong ferromagnetic interaction implies that the RTFM for Ndoped ZrO 2 is quite feasible considering the simple estimate. Though not shown here, our additional similar calculations for Zr 32 O 62 N 2 with a long N-N distance (7.4 Å ) reveal that the FM and AFM state are degenerate, whereas the total energy is 412.3 meV higher compared to the ground state of (a, b) configuration (See the Supplementary Information for more details). For comparison, let's focus on the largest and shortest N-N distance in the ''dimer'' of N-Zr-N configuration, corresponding to the (d, e) configuration and the ground state of (a, b) configuration, respectively.
For a deeply understanding of the physical origin of the ferromagnetism, the charge density difference of (a, b) and (d, e) configurations in FM states is calculated, as shown in Figs. 3(a) and 3(b), respectively. A weak covalent bonding can be seen between doped N atoms and its nearest-neighboring O atoms, suggesting that there exists a coupling interaction. Furthermore, in (d, e) configuration, the two doped N atoms which both N1 and N2 substituted at O3-type sites gain almost equivalent electrons from their neighboring Zr atoms. However, the situation is quite different in (a, b) configuration. The N4, where N substitutes O at O4-type site, 2p shell is almost completely filled as compared to the N3 where N substitutes O at O3type site, just like the fully filled O 22 2p shell. These observations are consistent with the analysis in Fig. 1(d) that O4 2p shell is more easily to fully occupy than the O3 2p shell. This charge redistribution can also be quantitatively estimated by considering the Mülliken population analysis 42 . The charges of Zr, O and N atoms at different configurations are given in Table 2. To start with, we calculate the number of electrons on each atom in bulk ZrO 2 , which results in charge states of Zr and O atoms of 11.50e and 20.74e, respectively. Note that these values are far from the formal ones corresponding to the ideal ionic situation (Zr 41 (a, b) configuration and the most stable state (either FM or AFM) is used for each configuration. The magnetization energy DE FM-AFM is the energy difference between the FM and AFM states, and M is the total magnetic moment for the FM state  In order to further understand the nature of magnetic coupling in Zr 32 O 62 N 2 , the spin charge density (SCD) distribution of (a, b) and (d, e) configurations in the FM states and the AFM states are shown in Fig. 5. As can be seen, in (d, e) configuration, the p-like shape spin charge resides mainly on both the two N impurities and they have identical spin direction in the FM states while opposite direction in the AFM states. It indicates that there existed a strong FM coupling interaction induced by the N 2p orbitals [see also Fig. 4(b)]. Meanwhile, the lobes of N 2p orbitals are towards the lobes of O 2p orbitals to maximize the overlaps, which manifests that the p-p interaction does extend to the neighboring O ions, similar to the case of single N doping. In great contrast, for (a, b) configuration in the FM states, the main features are that the spin charge is mostly contributed by the N4 dopant and N-N pair coupling interaction nearly vanished [see also Fig. 4(c)]. This can be understood by noticing that the N-N distance is 4.4 Å , almost double of the distance in (d, e) configuration. Such a long distance effectively degenerate the N-N pair coupling interaction.

Discussion
Now we turn to address the question of what stabilizes the ferromagnetic ground state of (a, b) configuration. Spontaneous ferromagnetism can be evaluated according to the ''Stoner Criterion'' 43,44 : D(E F )J . 1, where D(E F ) is the DOS at the Fermi level and J denotes the strength of the exchange interaction. The D(E F ) is proportional to m 3/2 (E VBM -E F ) 1/2 , where m is the effective mass and E VBM is the energy of the valence band maximum (VBM). Due to the stronger N3-Zr than N4-Zr bonds in the (a, b) configuration as indicated by their bond length (N3-Zr,2.05 Å , N4-Zr,2.22 Å ), the N3 spin-up states are occupied and the occupation of the spin-down states will decrease. Thus the spin charge tend to accumulate around the N3 atom, which makes the defect level deeper and wave function more localized, resulting in a larger D(E F ), and a larger spin-exchange splitting J of the band, as can be seen in Figure 4(a). The enhanced  exchange splitting stabilizes the FM state, it can cause the system to be ground state. These results therefore indicate that it will be easier to obtain stable ferromagnetism if the injected holes are not shared by several neighboring anion atoms, but localize around a single anion. A similar situation was observed in the different system of C-doped ZnO 45 .
In summary, our ab initio calculation results indicate that the substitutional N induces spin-split impurity states near the Fermi level and leads to the half-metallic property. The total magnetic moment is found to be proportional to the number of injected holes in the supercell. The magnetic moments in this system are attributed to the strong p-p coupling interaction between the localized N 2p states and the neighboring O 2p states. An analysis from charge density difference and Mülliken population analysis was developed to explain the charge transfer due to the different binding behavior of N substituted at O sites. Our findings pave the way for interpreting further the nature of p-electron magnetism in LE-doped oxide DMSs.
Methods ab initio calculations are performed by using density functional theory (DFT) with the plane-wave pseudopotential (PWP) approach as implemented in the Cambridge Sequential Total Energy Package (CASTEP) 46 . A generalized gradient approximation (GGA) 47 in the form of Perdew-Burke-Ernzerhof (PBE) is used for the exchange correlation functional. The plane-wave cutoff energy is chosen to be 300 eV and a 2 3 2 3 2 Monkhorst-Pack k-point mesh is adopted for integrations over the Brillouin zone. Here, E tot (ZrO 2 :N) is the total energy of the supercell with n oxygen atoms replaced by nitrogen atoms, and E tot (ZrO 2 ) is the total energy of ZrO 2 perfect crystal in the same supercell. m O is the chemical potential of oxygen (m O # m O2 /2), and m N is the chemical potential of nitrogen in nitric oxide (NO) 48 .
For a supercell (Zr m O n N l ) containing mZr atoms, nO and lN atoms, the charge density difference is obtained as follows: where, r(Zr m O n N l ) is the charge density of supercell Zr m O n N l , r(Zr i ), r(O j ) and r(N k ) are the charge density of isolated Zr m , O n and N k in the same supercell, respectively. According to the equation, the charge density difference gives a direct real-space visualization of the local charge redistribution due to the interactions of atoms.
To better understand the magnetic coupling of the Zr 32 O 62 N 2 systems, SCD was used in all investigated systems and defined as: where r:r ð Þ and r;r ð Þ are the spin up and spin down charge densities of the Zr 32 O 62 N 2 system, respectively.