Electronic structure and ferromagnetic properties of Zn vacancies in ZnO screw dislocations: First-principles calculations

ZnO has gained a renewed interest during the last decade, due to its many technological applications, particularly in short wavelength optoelectronics, since it has a large band gap of 3.36 eV at 300 K¹ and a large exciton binding energy of 59 meV.² The existence of native defects is inevitable in the preparation of ZnO devices, and it is significant to understand their properties. Among the native defects, linear defects such as dislocations, often occur in ZnO heteroepitaxial growth and gain a renewed interest during the last years. Recent many experimental researches have shown that dislocations can drive the epitaxial growth of ZnO, particularly for one-dimensional nanowire.^3–6 Moreover, dislocations also effect on the luminescence properties. In ZnO epitaxial layers grown on c-plane p-GaN templates, the intensity of the defect related visible luminescence band was found to strongly depend on the threading edge dislocation density, suggesting a relationship between this band and the dislocation.⁷ All of the above experimental researches show the dislocations play a key role for the structure and properties of ZnO.

Since ZnO is a compound semiconductor, many Zn-Zn, O-O and dangling bonds occur around the core of the dislocations, which have different impact on the electronic structure of the dislocations. Thus, to investigate the electronic properties of the dislocations in ZnO, an important step is the construction of the atomic detailed configurations. High resolution Transmission Electron Microscopy can realize the direct observation of dislocations at the atomic level,⁸ but its observation is a plane-view. Based on the 2D view, it is impossible to determinate the spatial atomic structures of dislocations, especially for the case of screw dislocations. To the best of our knowledge, no detailed spatial configurations of ZnO dislocations have been reported so far. As ZnO is a wurtzite semiconductor, its dislocation configuration can be drawn from other wurtzite compounds. Recently, Belabbas et al. spatially resolved the threading screw dislocations along [0001] and [11$2¯$0] in GaN and investigated its energetics, stability and mobility.⁹ Based on the methods and the models in these works, it is possible to construct the supercells of ZnO dislocations for the electronic structure calculations.

In addition, dislocations generally contain a large amount of strain, which can favor the formation of Zn vacancies (V_Zn). Since V_Zn can serve as the origin of d⁰ ferromagnetism,^10–12 dislocations should also effect on the ferromagnetism in ZnO. Therefore, in this paper, first-principle calculation is applied to study the atomic and electronic structure of ZnO dislocations, and the effect of dislocations on d⁰ ferromagnetism.

The calculations were performed in the framework of DFT with the projected augmented wave method,¹³ using the Vienna ab initio simulation package (VASP).¹⁴ For the exchange-correlation functional, a spin-polarized generalized gradient approximation (GGA)¹⁵ in the form of Perdew-Burke-Ernzerhof (PBE) was employed. A Hubbard U term was also considered in the description of the electron-electron interaction in the 3d orbital.¹⁶ In GGA+U, the Coulomb parameter U was set to 12.8 and 4.0 eV for Zn and O, respectively. Both of Zn and O had the value of J = 0.0 eV. The calculated values were 3.36, 4.00 and -8.8 eV, for the band gap, the formation energy and the position of the Zn-3d bands, respectively. The corresponding experimental values were 3.37, 3.6¹⁷ and -8.8 ∼ -7.5 eV.¹⁸ The parameters could substantially reproduce the experimental results. The plane-wave energy cutoff was set to 400 eV and the Γ-point centered 1×1×7 and 1×1×9 k-point mesh were used for atomic relaxation and electronic structure analysis, respectively.

In a supercell, periodic boundary conditions are applied in the all three direction of space. However, dislocations only have a structural periodicity along its line direction, which makes it difficult to build a supercell with dislocations. To solve the problem, according to the method proposed by Belabbas,^9,19 the supercell for screw dislocations along [0001] (dislocation A) contained 200 atoms, and its three supercell vectors were set as: a=10×e₁, b=3×e₁+3×e₂+0.5×e₃, c=e₃, where e₁=a[11$2¯$0], e₂=$a3[101¯0]$ and e₃=c[0001], (a and c are relaxed equilibrium lattice parameters of ZnO). In the supercell, two dislocations with opposite Burgers vectors were included and separated by about 13.0 Å. Similarly, for screw dislocations along [11$2¯$0] (dislocation B), the three vectors of the supercell were set as: a=8×e₁, b=4×e₁+4×e₂+0.5×e₃, c=e₃, where, e₁=$a3[101¯0]$⁠, e₂=c[0001] and e₃=a[11$2¯$0]. The supercell contained 192 atoms and two dislocations with a distance of 19.6 Å. The initial atomic positions in supercell were generated by imposing the displacement field given by isotropic linear elasticity theory.⁹ Generally, the screw dislocations in each supercell have several core configurations, by changing the position of the dislocation center. In the supercell with dislocation A, there are two possible dislocation centers, including S0 and S1, as shown in Ref. 9. However, after relaxing, the configuration S0 is unstable and transforms to the configuration S1 spontaneously. Similar phenomenon also happens in the supercell for dislocation B, where four dislocation centers are possible, containing G, M1, M2 and S, as shown in Ref. 19. The configuration S is most stable and reasonable one. Therefore, in the following research, we only investigate the two configurations.

The formation energy of a Zn vacancy in charge state q is defined as^20,21

where $E(VZnq)$ is the total energy for the studied supercell with V_Zn in charge state q and E(host) is the total energy of the same supercell without V_Zn. n_Zn is the number of the removed Zn atoms (n_Zn > 0). E_Fermi is the Fermi level, and set to 0 eV in this calculation. E_VBM is the energy of valence band maximum in host supercell. μ_Zn and E(Zn) indicate the chemical potential and cohesive energy of Zn. Since the formation energy of ZnO is -4.00 eV, μ_Zn has the same value of -4.00 eV under O-rich growth condition.

The relaxed configurations are showed in Fig. 1. For clarity, in the side view, we only present half of supercells perpendicular to dislocation direct and two periods of supercells along dislocation direct. As expected, the main mismatch happens in the channel of dislocations. The calculated formation energy is 4.51 and 1.51 eV/Å, for dislocation A and B, respectively. The difference in energy can be attributed to the magnitude of the Burgers vector, and the values are 4.91 and 3.00 Å for the two dislocations, respectively. Generally, the magnitude of the Burgers vector presents the lattice distortion resulting from a dislocation in a crystal lattice. To describe the distortion in the two dislocations, the bond lengths of the atoms (from site 01 to 10) around the dislocation A have been shown in Table I. Many bond lengths are longer than the value in bulk ZnO, besides site 01 and site 02. The two sites are 3-fold coordination ones in the screw dislocation A. The low coordination number shortens the bond lengths of the atoms. In Table II, most of the bond lengths are close to that in the bulk ZnO. The coordination states of the atoms are significant in determining the bond lengths. All of the atoms around the dislocation B keep 4-fold coordination, which also results in low formation energy.

Fig. 2 shows the total density of states (DOS) of the two supercells and the bulk ZnO. Compared with bulk ZnO, the dislocation A has an occupied level in the forbidden band, which is caused by the O-O bond between the two O atoms at the 3-fold coordination sites. The O-O bond length is only 2.60 Å, and the interaction between the two O atoms becomes possible. In fact, the energy level arises from the antibonding state of p-p coupling in the O-O bond, and similar result has been reported in ZnO twin grain boundary.²² For the dislocations B, it almost has a similar DOS to bulk ZnO. The result is also consistent with the bonding situation in the dislocations. To investigate the effect of the dislocations on the ferromagnetic properties in ZnO, we also calculated the dislocations in different charge states, but ferromagnetism (FM) was still not found in ZnO. Thus, it can be concluded that the screw dislocations of stoichiometric ZnO cannot cause the generation of FM.

For the origin of FM in ZnO, especially for d⁰ FM, many researchers reported that the presence of V_Zn was responsible for the observed room temperature FM in polycrystalline ZnO.^{10,12,23–25} Since every screw dislocation has a channel with special bonding situation, it may favor the aggregation of impurities and native defects. In order to research the FM property of the ZnO dislocations, 10 and 12 unequivalent sites for V_Zn in the dislocation A and B were calculated, respectively, and all of the positions have been labeled, as shown in Fig. 1. Fig. 3 is the formation energies of V_Zn under O-rich condition. For comparison, the formation energy of V_Zn in bulk ZnO is also listed, with a value of 2.12 eV. The value is lower than 3.7 eV, the extrapolated LDA+U result,²¹ but it is close to other LDA and GGA+U results.^26,27 As shown in Fig. 3, the formation energies in all the sites of dislocation A are negative, and the formation of V_Zn is spontaneous. In the dislocation B, although all of the formation energies are positive, the values are still lower than that of bulk ZnO. V_Zn can effectively release the strain in the dislocations, and the formation of V_Zn is energetically favorable. In addition, the previous calculations show that the stoichiometric screw dislocations in ZnO have very high formation energies, and the formation of the screw dislocations is almost impossible. However, the screw dislocations, especially the dislocations along [0001], were often found in many experimental reports.^28,29 In our opinion, the observed screw dislocations should be nonstoichiometric, and vacancies contribute to stabilize the dislocations. On the basis of our calculation, one V_Zn can lead to 0.37 eV drop in energy.

Substitution of two Zn atoms by two vacancies allows for investigation into the coupling between the two vacancies and calculation of the relative energies of FM and antiferromagnetic (AFM) orderings. Several configurations with two vacancies were built and discussed. The previous calculation shows the FM ordering can be found with the presence of V_Zn and V_Zn is most stable at site 02 in dislocation A. Therefore, we only discuss the case of V_Zn pairs with V_Zn at site 02. The calculated results are presented in Table III, including E_f under O-rich condition and magnetic moment of different V_Zn pair configurations, and the separation distance (D) between the vacancies. 02-05 V_Zn pair is the most stable one and has the lowest E_f with a value of -2.26 eV under O-rich condition. The formation energies of the V_Zn pairs are independent of D, but the total magnetic moment is affected by D. Large D induces the decrease of the magnetic moment.

To achieve the FM coupling originating from the V_Zn pairs located in the dislocation, the AFM states should be inhibited. Therefore, we calculated the energy difference (ΔE) between FM and AFM states for some V_Zn pairs, which is defined as ΔE = E(FM) - E(AFM). A negative value of ∆E means that FM is favored over AFM and vice versa. Considering the formation energy, 02-01 and 02-05 V_Zn pairs were chosen as objects of study. In addition, the dislocations are linear defects and easy to trap carriers. Some reports showed spin polarization, caused by cation vacancies in ZnO, could be influenced by the localized holes.^30,31 Thus, ΔE of V_Zn pairs in different charge states was also discussed. As shown in Table IV, all of ∆E have negative values, and FM coupling between Zn vacancies is possible. Although the FM coupling becomes weak with the increasing D, it can be stabilized by holes. Fig. 4 is DOS and spin density distribution of the two V_Zn pairs. It is obvious that the origin of FM is the 2p orbitals of the O atoms around V_Zn. Different from V_Zn in bulk and nanowire ZnO,^30,32 the spin density distribution of the two V_Zn pairs has a typical character of antibonding state, which originates from p-p coupling between O atoms. Similar phenomenon has been reported when V_Zn appears in twin grain boundary of ZnO, where O-O bonds arise from the special boundary structure.²² For 02-01 and 02-05 V_Zn pairs, the average O-O bond length of O atoms around V_Zn is 2.41 and 2.53 Å, respectively, shorter than 2.85 Å in bulk ZnO. As a result, the p-p coupling between O atoms is possible.

Macroscopic FM ordering requires a high density of vacancies to obtain magnetic percolation, which should perhaps be difficult to achieve in equilibrium conditions.²⁷ However, the above calculation results has confirmed that dislocations can favor the formation of V_Zn. Due to the periodic structural character of dislocations, they can be seen as a 1D continuous line and a mass of V_Zn aggregate in the line. The spin-polarized O atoms around V_Zn bond with each other, and the macroscopic FM ordering could be realized. In fact, dislocations serve as traps to capture Zn vacancies and mediate them. Therefore, dislocations are potential origin of d⁰ ferromagnetism in ZnO.

In summary, we presented the configurations of two screw dislocations. The dislocations with different orientations have different bonding states. The dislocation along [0001] has O-O bonds in its core, which induce an occupied energy level in forbidden band. For the dislocation along [11$2¯$0], all the atoms are 4-fold coordination and there is almost no change in the electronic structure. The dislocations can favor the formation of Zn vacancies, due to the stress relief. It is concluded that many screw dislocations may be nonstoichiometric, and vacancies contribute to stabilize the dislocations. Moreover, dislocations are potential origin of d⁰ ferromagnetism in ZnO. The V_Zn pairs can be stabilized in the dislocation along [0001] and their FM ordering is energetically the preferred state. Due to the periodic structural character of dislocations, the macroscopic FM ordering is expected.

The work is supported by the National Natural Science Foundation of China under Grant No’s. 11364009, and Natural Science Foundation of Guangxi Province (No. 2014GXNSFFA118004). The authors would like to thank Dr. I. Belabbas for his help in building the dislocation models.