The defect formation energies of transition metals (Cr, Fe, and Ni) doped in the pseudo-H passivated ZnO nanowires and bulk are systematically investigated using first-principles methods. The general chemical trends of the nanowires are similar to those of the bulk. We also show that the formation energy increases as the diameter of the nanowire decreases, indicating that the doping of magnetic ions in the ZnO nanowire becomes more difficult with decreasing diameter. We also systematically calculate the ferromagnetic properties of transition metals doped in the ZnO nanowire and bulk, and find that Cr ions of the nanowire favor ferromagnetic state, which is consistent with the experimental results. We also find that the ferromagnetic coupling state of Cr is more stable in the nanowire than in the bulk, which may lead to a higherTcuseful for the nano-materials design of spintronics.
Keywords:First-principles; Transition-metals; Formation energies; Nanowires; Magnetism
The utilization of the charge freedom degree of electrons in semiconductors to describe the transportation properties and of the spin freedom degree of electrons in magnetic materials to store the information is separated. Until Ohno reports that ferromagnetism can be achieved in (Ga,Mn)As system by introducing magnetic ions Mn into GaAs matrix , which invokes great interests in the diluted magnetic semiconductors (DMSs) because of the utilization of both the charge and spin freedom degrees of electrons in these materials [2,3]. Both theoretical and experimental investigations are performed extensively due to the great potential applications of DMSs in spintronic devices [4-10]. However, one difficulty for the applications is that the Curie temperatures (Tc) of most DMSs materials are below the room temperature. Many attempts have been performed to find high Tc DMSs materials [11-14]. Sato and Katayama-Yoshida  have systematically investigated the ferromagnetic properties of transition metals (TMs) doped in ZnO based DMSs using first principles calculations, and suggested that the Cr-doped ZnO is one candidate for high Tc ferromagnetic DMSs. Nano-scaled DMSs have also been reported recently [12-14]. The Cr-doped single-crystalline nanowires with the stable room-temperature ferromagnetism have been successfully fabricated . However, the origin of the ferromagnetism is still under debate. The Zener model has been proposed to explain the interaction between the TM ions . In addition, wei et al. also proposed a band coupling model based on p-d and d-d repulsions between TM ions and host elements to explain the origin of the ferromagnetism using first-principles band structure calculations [16,17].
Recently, experimental investigations show that the doping is more difficult in the nanosized structures than in the bulk semiconductors, leading to the low solubility of the dopant [18-22]. So far, the origin of this doping difficulty in the nanosized structures is still not clear. Since the concentration of the charge carriers in the nanosized ZnO-based DMSs is important for their applications, it is of great interest to investigate the chemical trends of defect formation and magnetic couplings of TMs doped in the ZnO nanowire theoretically.
In this letter, we systematically calculate the defect formation energies and magnetic coupling properties of transition metals (Cr, Fe, and Ni) doped in the ZnO nanowire and bulk. We find that the two nearest Cr atoms favor ferromagnetic state, which is more stable in the nanowire than in the bulk case. We also find that Cr and Fe can be doped in the ZnO nanowire more easily than Ni due to their lower formation energies, and favor ferromagnetic states.
The paper is organized as follows. In Sect. “Calculational Methods and Details”, we describe our calculational methods and details. In Sect. “Results and Discussion”, we discuss the defect formation energies of transition metals in ZnO nanowires and bulk, and the magnetic coupling properties of the TMs in the two systems. A brief summary of the letter is given in Sect. “Summary”.
Calculational Methods and Details
Our total energy calculations are performed using the density functional theory (DFT) in the generalized gradient approximation (GGA) of PW91 functional  for the exchange correlation potential and the projector augmented wave (PAW) method  as implemented in the Vienna ab initio simulation package . The electron wave function is expanded in plane waves up to a cutoff energy of 300 eV, and all the geometries are fully relaxed until the quantum mechanical forces acting on the atoms become less than 0.01 eV/Å. We used only the Γ point for the points-sampling in the nanowire calculations. The ZnO nanowire of 1.0 nm diameter is generated from the 7 × 7 × 2 supercell of bulk wurtzite (WZ) ZnO along the  direction. The outside of the nanowire within the 7 × 7 × 2 supercell is vacuum space to avoid the interaction between the nanowires in the neighboring supercells. The supercell selected here has a periodical length of 2c, where c is the bulk lattice parameter along the  direction. We use the pseudo-H to passivate the dangling bonds of Zn and O atoms on the surface of the nanowire . Our nanowire supercell for ferromagnetic calculations contains 48 Zn, 48 O, 24 H1.5 and 24 H0.5 atoms, as shown in Fig. 1a.
Figure 1. aThe ZnO nanowire unit-cell andbbulk unit-cell are studied here. Thegrey spheresare Zn,yelloware O,redare H1.5, andblueare H0.5. Thegreen spheresare the Zn sites substituted with Cr atoms or other TM atoms. The geometry is before relaxation
The defect system is modeled by putting one or two defects into supercell mentioned above. To determine the defect formation energy, we calculate the total energy E(α,0) for the supercell containing the relaxed defect α in neutral state, and the total energy E(ZnO) for the host ZnO nanowire in the absence of the defect as well as the total energies of the elemental solids. The defect formation energy ΔHf(α,0) is defined as 
μi is the chemical potential of constituent i relative to element solid with chemical potential μi0[28,29]. The ni’s are the number of Zn and extrinsic defects α. For comparison, we also investigate bulk case doped with the same impurities with 2 × 2 × 2 supercell of WZ containing 16 Zn atoms and 16 O atoms, as shown in Fig. 1b, and the Brillouin zone integration is performed with 4 × 4 × 4 k-meshpoints for the bulk magnetic calculations.
In our previous work, we have systematically studied the ferromagnetic properties of Mn ions doped in the ZnO nanowire in different configurations, and found that the ferromagnetic coupling between the Mn ions exists in the ZnO nanowire with unpassivated surfaces while not in the nanowire with passivated surfaces . In order to exclude the surface states locating in the band gap, which may have effects on the ferromagnetism and are not present in the passivated case, we passivate the nanowire using pseudo-H atoms. Our calculated band gaps are 2.24 and 0.8 eV for the passivated nanowire with the diameter of 1.0 nm and bulk case, respectively.
Results and Discussion
Formation Energies of the Defects in ZnO Nanowires and Bulk
The defect formation energy ΔHf(α,0) of defect α in neutral state determines the dopant solubility in a host at given growth conditions. Generally, high formation energy leads to low solubility. The defect formation energy ΔHf(α,0) of TMs doped in the ZnO nanowires and bulk is calculated using Eq. 1 in the defect rich condition, i.e., μi = 0. The calculated relative formation energy ΔHf(α,0) of the neutral transition metal impurities doped in nanowires and bulk are shown in Fig. 2, indicating that the chemical trends of defect energy are similar in the two cases. For the nanowire, the formation energies for Cr and Fe are lower, leading to the higher doping concentration. For the case of Cr, the formation energy difference between nanowire and bulk is 0.131 eV. The diameter of the nanowire we studied here is 1.0 nm. In order to investigate, the nanosize effect on the formation energy, we also calculate the formation energies of TMs doped in the smallest nanowire with diameter of 0.65 nm (see Fig. 2). The formation energies increase as the size of the nanowire decreases. Therefore, the doping in the nanowires with the smaller diameter becomes more difficult, which is similar to the case of nanocrystals (in our calculation, we ignored the spin–orbit (S–O) coupling interation, thus only the states with the same spin can couple with each other. To the first order, including the S–O coupling will not change the quantity much. Further reference can be found in ). The variations of formation energies could be explained by the shift of the single electron energy levels of 3d relative to the conduction band minimum (CBM) and valence band maximum (VBM) due to the quantum confinement effect.
Figure 2. The calculated relative defect formation energies ΔHf(α,0) of the neutral transition metal impurities in ZnO nanowires of different diameters and bulk case. (All the formation energies take reference to the formation energy of Cr doped in the bulk ZnO)
Magnetic Couplings between the Transition Metals in ZnO Nanowire and Bulk
In order to find whether ferromagnetic (FM) coupling between TM atoms exists in the nanowire with the diameter of 1 nm, we substitute two Zn atoms with two TM atoms with the nearest distance (the two TM atoms are the in-plane nearest neighbors and the plane is perpendicular to thecaxis). Therefore, the transition metal concentration is 4.2%. For comparison, we also study the bulk case with the TMs concentration of 12.5%. Figure 1shows the supercells and the TMs substitutional sites. For each configuration, both ferromagnetic and anti-ferromagnetic (AFM) states are calculated. According to the energy difference ΔEbetween the total energy of AFM and FM states, we can find which state is more favored. The positive (negative) ΔEmeans that the FM (AFM) state is more stable. The calculated results are listed in Table 1. In the nanowire, Cr and Fe favor ferromagnetic states energetically, and Co and Ni prefer anti-ferromagnetic states, whereas the opposite is only true for the cases of Fe and Ni in the bulk.
Table 1. The total energy difference ΔE(ΔE = EAFM−EFM) between AFM and FM states for each configuration is showed in the second column
Based on the band coupling model, Walsh et al. investigated the ferromagnetism of Co-doped ZnO using DFT+ Ud/s to correct the band gap error by applying a Coulomb U on both the s and d orbitals to further raise the CBM level . The correct picture of TMs 3d levels in the nanowire at the Γ point can be obtained due to the enlarged band gap resulting from the quantum confinement effect, even if the gap is not accurate enough. Note that our calculated band gap of the pure passivated nanowire is 2.24 eV. Based on the crystal field theory, the TMs 3d states split into one triply degenerate t2 state and one doubly generate e state in zinc-blende (ZB) structures. The t2 states further split into one doubly degenerate e state and one singly a1 state due to the lower point group C3v in the WZ. The O 2p levels also split into e and a1 states. Figure 3 shows the schematic plot of the Cr 3d levels in majority spin at Γ point in the bulk and nanowire band structures.
Figure 3. The schematic plot for the the Cr 3dlevels in majority spins at Γ point in the bulk and nanowires band structures
Based on the band coupling model, the d–dcoupling of ZnO or GaN based DMSs dominates the ferromagnetism behavior due to the fact that the 3d levels are above the VBM . Figure 4 shows two kinds of magnetic coupling between two Cr atoms in the nanowire. Since the energy levels have included the p–d interaction, the d levels contain the p character, and vice versa. For the FM coupling between the two Cr atoms in Fig. 4a, there is an energy gained because the highest e levels in the majority spin channels are half occupied. In minority spin channels, one e level is pushed up and the other is pushed down by the same amount (denoted as the double exchange Δdd1). Therefore, there is no energy gained due to the empty occupation. For the AFM coupling in Fig. 4b, the e level of one Cr ion with majority spin couples with that of the other Cr ion with the same spin (denoted as the super-exchange Δdd1,2) , and there is also energy gained due to the half occupation of the e levels. Since the energy gained from the double exchange interaction is larger than that from the super-exchange interaction due to the existence of large exchange splitting εdd, the FM states are favored in the Cr-doped ZnO nanowire, which is also true for the Fe doped in ZnO nanowire. However, for the case of Co, the highest occupied e is fully occupied and there is energy gained from the super-exchange interaction, as a result, the AFM state is favored. Note that our calculated results of TMs doped in the nanowire are consistent with the theoretical analysis. To further understand the origin of ferromagnetism of TMs in the ZnO nanowire, we also plot the density of states (DOS) of 3d levels of the FM and AFM states in Fig. 5 for the two Cr and two Co ions, respectively.
Figure 4. The schematic view of the ferromagnetic coupling and anti-ferromagnetic coupling between the two Cr atoms doped in ZnO nanowires. The 3dlevels are showed and the energy levels include thep–dinteraction. The line with thedouble-headed arrowindicates that the two states with the same spin channel couple with each other
Figure 5. The DOS of 3dlevels fora,btwo Cr ions in their FM coupling state,c,dtwo Co ions in their FM coupling state,e,ftwo Cr ions in their AFM coupling state,g,htwo Co ions in their AFM coupling state
Furthermore, we also show that the FM state of Cr is more stable in the nanowire than that in the bulk, which can be interpreted by the fact that compared with the case of Cr doped in the bulk, in the nanowire, thep–drepulsion in the nanowire becomes weaker and the exchange splittingεddincreases due to low concentration of Cr and the VBM moving down. Since the superexchange Δdd1,2term decreases, which favors to the anti-ferromagnetic state, and the FM stabilization energy Δdd1does not significantly change, the ferromagnetic state is more stable in the nanowire. The total energy difference ΔE(ΔE = EAFM−EFM) of 0.275 eV for the Cr-doped nanowire is bigger than that of 0.243 eV for the Cr-doped bulk, suggesting that the former has a higherTc. Since the highTcis crucial to the spintronic devices, our calculated results could be useful in nano-materials design for spintronics.
In summary, we have investigated the formation energies of TMs (Cr, Fe, and Ni) doped in the ZnO nanowires and bulk using first-principles total energy methods. The chemical trends are similar in the two systems. We also find that the formation energies increase as the diameter of the nanowire decreases, indicating that the doping becomes more difficult with decreasing diameter. Finally, it is found that the ferromagnetic state of Cr doped is more stable in nanowires than in the bulk, leading to a higherTcindispensable to the spintronic devices.
This work was supported by the National Basic Research Program of China (973 Program) grant No. G2009CB929300 and the National Natural Science Foundation of China under Grant Nos. 60521001 and 60776061. Part of the CPU time was performed in Supercomputing Center, Chinese Academy of Sciences.
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