Ti_Ga–V_N complexes in GaN: a new prospect of carrier mediated ferromagnetism

Abstract

First principle investigations exploring the effects of nitrogen vacancies on ferromagnetism in Ti doped wurtzite GaN are reported. The presence of nitrogen vacancies demonstrated no noticeable effect in the case of pure GaN but exhibited ferromagnetism in the case of Ti doped GaN. The magnetic moment however ceased upon doubling the concentration of dopant and vacancies in the host which points towards possible antiferromagnetic coupling. The conventional double exchange ordering observed in the case of the vacancy-added Ti:GaN switched to a carrier mediated exchange for the Ti_Ga–V_N complex in Ti:GaN. For Ti doped GaN, the energy difference calculated with and without N vacancies is found to be relatively smaller than that of other 3d transition metal (Cr, Mn, Fe, Co, Ni, Cu) doped GaN. The results calculated for different configurations to explore the effects of nitrogen vacancies on the electronic and magnetic properties of Ti:GaN are discussed in detail.

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1. Introduction

Diluted magnetic semiconductors (DMSs) gained central attention in material science because of their potential for applications in future appliances. These materials are realized by introducing magnetic dopants into conventional semiconductors and have attracted intense interest due to their prospective use in developing novel magneto-electronic, magneto-optical and specially spintronic devices that utilize both the charge and spin character of electrons to create new functionalities.^1–3 After Ohno’s discovery of carrier induced ferromagnetism in the initial DMS material Mn:GaAs, which exhibited a critical temperature of 110 K, numerous experimental and theoretical studies appeared reporting room temperature magnetic ordering in II–VI, III–V, and IV–VI DMSs.⁴ Dietl predicted that wide bandgap semiconductors were good candidates for obtaining room temperature ferromagnetism and voted in favor of GaN:Mn.⁵ Owing to the development of sophisticated growth techniques, the scope of material science is expanding and the searching curve of suitable DMS materials is at its peak.

Although GaN has been extensively studied for its exceptional electronic and optoelectronic properties, its potential for use in magnetic random access memory (MRAM), spin light emitting diodes (spin LEDs), spin field effect transistors (FETs), spin based quantum dots, magnetic tunnel junctions and other storage elements has opened new research areas.⁶ The doping of the GaN matrix with transition metals (TMs), bearing partial occupancy of d-orbitals, is an effective method to realize nitride based DMSs.^7–10 The induced ferromagnetism roots from a magnetic exchange interaction as explained by several models including double exchange, p–d exchange, super-exchange, bound magnetic polaron and carrier mediated interaction. The carrier mediated interaction is held by the interaction between the magnetic impurities and propagating charge carriers in the host. These carriers are generated as a result of native defects in the form of vacancies, interstitials and antisites that may control many aspects of the DMSs, particularly FM, by significant variations in their carrier densities.¹¹

In the early 1990s, vacancies, the most prevalent and inevitable native point defects, found in both the cationic and anionic sublattices of bulk GaN, were examined in numerous theoretical studies.^7–13 Anionic nitrogen vacancies typically behave as shallow donors and are reported to be a dominant defect in GaN showing its n-type conductivity¹⁴ while cationic gallium vacancies act as acceptors¹⁵ and have been found to induce a local magnetic moment.¹⁶ Pratibha Dev et al. suggested that a neutral cation vacancy in GaN led to the strong localization of defect states favoring spontaneous spin polarization and the formation of a local magnetic moment. The extended tails of defect wave functions, on the other hand, mediate surprisingly long-range magnetic interactions between the defect-induced moments.¹⁷ Zhihua Xiong et al. also proposed that ferromagnetism in GaN is due to the Ga vacancy whereas the N vacancy induces no magnetism.¹⁸ It was found that the room-temperature ferromagnetism of undoped GaN nanoparticles¹⁹ originated from the ferromagnetic spin coupling of the nitrogen dangling bonds associated with the surface Ga-vacancies through bond spin polarization and was effective even when the vacancy separation was as long as 8 Å. Later, in 2010, in GaN thin films and nanowires, ferromagnetism driven by cation vacancy was studied by Anlong Kuanga et al. who suggested that magnetic moments mainly came from the unpaired 2p electrons at the nearest-neighbor N atoms of the Ga vacancy.²⁰ In 2011, the experimental evidence of Ga-vacancy induced room temperature ferromagnetic behavior in GaN films was reported, using plasma-assisted molecular beam epitaxy, and was believed to originate from the polarization of the unpaired 2p electrons of N surrounding the Ga vacancy.²¹ Earlier investigations mainly focused on the fact that considerable amounts of native defects only argued for cationic Ga sites in as-grown GaN^17–21 but recently, a theoretical study²² established that the nitrogen vacancy should also serve as a dominant defect in GaN. Recently, it has been indicated that a Curie temperature of 150 K at a density lower than the concentration value of 1.28 × 10²¹ cm⁻³ disabled V_Ga from inducing room temperature ferromagnetism.²³ Instead of the frequently studied cationic sublattice vacancies, in the field of defect induced ferromagnetism in semiconductors, the existence of another type of anionic sublattice vacancy induced magnetism is still controversial as a cationic nitrogen vacancy in pure GaN causes no spin polarization but induces a magnetic moment when introduced in the presence of an impurity.¹⁸

There have been many materials studied up to now which indicate this novel way of ferromagnetism assisted by introducing anionic vacancies in doped semiconductors as in (Be, Mg)-GaN (2003),²⁴ magnetic ion-TiO₂ (2007),²⁵ Ti-ZnO,²⁶ Mn-GaN films,²⁷ Mn-SrTiO₃,²⁸ and Co-Dy₂O₃.²⁹ Recently, the effects of nitrogen vacancies on 3d-transition metals including Cr, Mn, Fe, Co, Ni and Cu in GaN³⁰ were studied. However, the literature still lacks the Ti-doped GaN study in the context of anionic defect induced ferromagnetism, although it may prove very important in prospect as Ti could favorably replace Ga in the Ti³⁺ form, and would be ultimately left with one unpaired electron thus offering a high flexibility to study the effects of the N vacancy in magnetic interaction variation. Ti, for being an important nonmagnetic dopant responsible for inducing favorable ferromagnetism, has been studied in the host environment of ZnO,^31,32 AlN,³³ AlP,³⁴ and GaN.³⁵ Motivated by the growing interest in innovative defect-impurity state induced ferromagnetism, a detailed study in the framework of the anionic nitrogen vacancy effects on Ti-GaN is presented here.

2. Computational details

The first-principle calculations based on density functional theory were performed on pure and Ti-doped bulk wurtzite GaN using the supercell approach implemented in the Amsterdam density functional (ADF) BAND package.³⁵ All calculations were carried out on 32 atom supercells.

The first part of the calculations was devoted to pure GaN in the form of (a) defect free GaN with the unit cell formula Ga₁₆N₁₆ and (b) with a nitrogen vacancy defect denoted by the unit cell formula Ga₁₆V_NN₁₅. The second part of the calculations deals with the doping of Ti onto the cationic sites of GaN in the form of Ti_Ga with and without a nitrogen vacancy represented by the unit cell formulae Ga₁₅Ti₁V_NN₁₅ and Ga₁₅Ti₁N₁₆, respectively, where V_N symbolizes the absence of single nitrogen atoms on the anionic site of the matrix. The calculations were also carried out for two Ti atom doped GaN without vacancies in the form of supercell Ti₂Ga₁₄N₁₆ (Ga₁₄N₁₆ + 2Ti_Ga) and with two vacancies in the form of supercell Ti₂Ga₁₄N₁₄ (Ga₁₅N₁₅ + 2V_N + 2Ti_Ga).

The calculations were performed using generalized gradient approximation (GGA) based on the Perdew–Burke–Ernzerhof (PBE) functional as it was found to be very accurate to theoretically investigate the electric and magnetic properties of DMSs.^36–39 However there exists an inevitable limitation of the GGA functional to properly describe the magnetic properties of TM doped compounds, therefore, to improve the calculation, we included on-site d–d Coulomb interactions by adding the Hubbard U parameter to GGA. Hence this work also includes the comparative study of Ti-doped GaN using GGA and GGA + U. The value of Hubbard U was considered as U = 4.5 eV and J = 0.5 eV in agreement with the literature.^40–43 For self-consistent field (SCF) convergence, the basis set of linear combination of atomic orbitals (LCAO) with triple zeta double polarization (LCAO + Tz2P) was utilized. All the configurations of wurtzite GaN were fully relaxed and geometrically optimized, using the experimental lattice constants of a = 3.189 Å and c = 5.185 Å and internal parameter 0.377 as starting parameters.⁴⁴ The cutoff energy was found to be 1.00 × 10⁻⁴ eV. The Brillouin zone integration was performed by using a Γ-centered k-point grid with a 2 × 2 × 2 Monkhorst–Pack mesh.

TMs preferably substitute a cationic substitutional site in compound semiconductors as it offers a relatively low formation energy, hence we considered the dopant in the form of Ti_Ga.⁴⁵ The vacancy is introduced at the anionic N site, denoted V_N, which usually introduces shallow donor levels in materials.^46,47 The orbitals from the electronic configurations used in the calculations were Ga [4s², 3d¹⁰, 4p¹], N [2s², 2p²] and Ti [4s², 3d²] and were taken as the valence while the rest of the inner electrons were kept frozen.

3. Results and discussions

The densities of states (DOS) calculated for a series of geometrically optimized structures of GaN, in pure and Ti doped forms, were discussed in detail to explore the potentials of the material for carrier mediated ferromagnetism. The following sections describe our findings to explain the effects of nitrogen vacancies on the structural, electronic and magnetic properties on Ti doped GaN.

A. Structural properties of the configurations studied

One of the objectives of this study was to explore the likelihood of the formation of complex Ti_Ga–V_N in the GaN matrix where V_N was introduced at the nearest-neighbor position of Ti_Ga, at a distance of 1.948 Å. In the case of Mn:GaN, the complex Mn_Ga–V_N has been experimentally pointed out as one of the stable defects in GaN, as Mn prefers to sit on the Ga lattice site at a place close to a vacant N site.^48,49 Moreover, this configuration keeps the separation between V_N and Ti_Ga below 3.00 Å which is appropriate to allow the conspicuous interaction of the wave functions of a defect to cause defect induced ferromagnetism.⁵⁰ The results indicated that the geometrically optimized configurations of the GaN supercell with V_N or Ti_Ga are considerably different from those of the starting lattice parameters. The optimized lattice parameters for all the studied supercells are listed in Table 1 whereas Fig. 1 shows the supercell with the positions of the dopants and vacancies highlighted.

Table 1 Lattice parameters and positions of defects in the GaN supercell for all configurations

	Configuration	a (Å)	c (Å)	Location in Fig. 1
				VN	TiGa
1	Ga16N16	3.186	5.186	—	—
2	Ga16N15 (Ga16N16 + VN)	3.067	5.100	1	—
3	Ti1Ga15N16 (Ga15N16 + TiGa)	3.176	5.176	—	3
4	Ti2Ga14N16 (Ga14N16 + 2TiGa)	3.149	5.163	—	3, 4
5	Ti1Ga15N15 (Ga15N15 + VN + TiGa)	3.002	5.040	1	3
6	Ti2Ga14N14 (Ga15N15 + 2VN + 2TiGa)	2.840	4.891	1, 2	3, 4

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Fig. 1 Wurtzite bulk GaN supercell showing the locations of the substituted dopants (Ti_Ga) and nitrogen vacancies (V_N) labeled as the 3, 4 and 1, 2 sites, respectively.

The observed gradual decrease in d_Ga–Ga along the a- and c-axes upon the introduction and increment of the defects may be attributed to a variant fashion of bonding between the fused atoms as the radius of Ga is 2.050 Å, that of Ti is 1.992 Å and that of N is 1.608 Å. The distance between two interacting atoms should be roughly equal to the sum of their surface radii, therefore, taking the radii into account, it can be said that in the current situation of Ti:GaN the bond length decreases as expected when compared with GaN. The reduction of the lattice dimensions is caused by the inter-ionic coulomb interactions due to the partly ionic nature of the material.⁵¹ The relaxed geometries of the supercells of the configurations used for this study are given in Fig. 2.

Fig. 2 Geometrically optimized supercell structures of the configurations studied.

In order to study the role of the nitrogen vacancies and Ti_Ga–V_N complex in the ferromagnetic ordering of Ti:GaN, we carried out detailed calculations on different configurations such as pure GaN, single V_N in GaN, single and double Ti_Ga in GaN and a pair of Ti_Ga–V_N complexes. The values of the Fermi energy, formation energy and magnetic moment calculated for different configurations are described in Table 2.

Table 2 The calculated values of energies and magnetic moments for different configurations

Configuration	Fermi energy: EF GGA (GGA + U) eV	Formation energy: Ef GGA (GGA + U) eV	Magnetic moment per supercell GGA (GGA + U) μB	Magnetic moment per dopant/vacancy GGA (GGA + U) μB
Ga16N16	−5.74	−24.85	0.00	0.00
Ga16VNN15	21.55	−186.47	0.00	0.00
Ga15Ti1N16	17.28 (18.45)	−200.45 (−194.09)	−1.00 (−1.00)	−0.98 (−1.14)
Ga15Ti1VNN15	19.00 (21.04)	−191.53 (−185.38)	0.882 (1.97)	0.58 (1.83)
ΔE = E(vac) − E(no vac)	1.72 (2.59)	8.92 (8.71)	—	—
Δμ = μ(vac) − μ(no vac)	—	—	1.88 (2.97)	1.56 (2.97)

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B. Properties of Ti:GaN

The configuration Ga₁₅Ti₁N₁₅ has the lowest formation energy pointing towards the solubility of Ti in the GaN matrix. Furthermore, the Fermi level shifted towards the conduction band upon vacancy introduction which predicts the n-type nature of GaN. Fig. 3 gives the DOS for Ga₁₅Ti₁N₁₆ representing a single Ti atom substitutionally placed on a cationic site of GaN without vacancies. The results (calculated with GGA) depict the dominance of a spin polarized Ti-3d impurity band system near the Fermi energy level at the CBM pointing towards possible ferromagnetic ordering. However, by switching on the Hubbard U correction, the self-interaction correction caused the delocalization of the d-states thereby providing an inverted hump near mid-band gap, in agreement with the Mn:GaN structure. The inversion corresponds to the conventional spin down state of singly occupying Ti-3d electrons.^57,58

Fig. 3 The DOS plots for (Ga₁₅Ti₁N₁₆) and (Ga₁₄Ti₂N₁₆) where the total density of states (TDOS) is represented by the solid line while the blue shaded area represents the partial density of states (PDOS) of N-2p and the red shading represents the Ti-3d states.

In order to investigate the magnetic coupling between the Ti atoms we performed calculations on two Ti_Ga atoms substitutionally doped in the form of supercell Ti₂Ga₁₄N₁₆ using GGA + U. The ferromagnetic (FM) state is found to be stable for being lower in energy than the antiferromagnetic (AFM) state, in agreement with Z. Xiong.³⁵ The induced ferromagnetism with a magnetic moment of 2.00 μ_B is calculated for the Ti₂Ga₁₄N₁₆ configuration having Ti-dopants introduced at the nearest neighboring positions. The spin polarized TDOS plots for Ti₁Ga₁₅N₁₆ and Ti₂Ga₁₄N₁₆ display the half metallic behavior (HMB) of the material representing these configurations. This ordering is due to p–d exchange interactions caused by hybridization between the Ti-3d majority spin components and N-2p band. It allows electron hopping between the Ti-3d states through intermediary N-2p orbitals thus aligning all 3d electrons in a ferromagnetic order via a double exchange mechanism in agreement with observations on several DMS materials like Ti:ZnO,^31,32 Ti:AlN,³³ V:GaN,⁶⁰ etc. The comparative view of TDOS for Ga₁₄Ti₂N₁₆ with Ti-3d and N-2p PDOS distinctly highlights the 3d–2p hybridization assisting the exchange interactions. In order to comprehend magnetic ordering in Ti₁Ga₁₅N₁₅, the TDOS and PDOS plots for TiGa₁₅N₁₅ are shown in Fig. 4.

Fig. 4 The TDOS plots for (Ga₁₅Ti₁N₁₅) with the N-2p, Ga-4p, Ga-4s and Ti-3d orbitals PDOS in comparison with pure GaN. The Ti-3d and Ga-4s states are shown shaded.

C. Properties of Ti:GaN with a single nitrogen vacancy

To shed light on the influence of nitrogen vacancies on the electronic structure of Ti-doped GaN, the plots of the calculated total and partial density of states were analyzed. The comparison of the DOS indicates that the introduction of a nitrogen vacancy into GaN (as Ga₁₆N₁₅ + V_N) causes a reduction in the bond length d_Ga–Ga along the c-axis from 5.186 Å to 5.100 Å and along the a-axis from 3.186 Å to 3.067 Å. Besides the structural changes, other material properties were also observed to change notably upon incorporation of nitrogen vacancies in the GaN matrix. V_N introduces fairly shallow donor levels in the bottom of the conduction band and causes a shift of the Fermi energy level towards the bottom edge of the conduction band which indicates the intrinsic n-type nature for GaN in line with the literature.^46,54 The observation of the partial DOS shows that the occupied CBM states come mainly from the Ga-4p and Ga-4s dangling bonds formed due to N vacancies. The reduction of d_Ga–Ga along the a- and c-axes causes an increased bonding trend between the first neighboring Ga atoms around the V_N lattice site due to far more extended donor states, in agreement with earlier studies.^55,56

In order to reveal the influence of the N vacancies on the magnetic properties of Ti:GaN, the calculated values of the magnetic moments for different configurations with and without vacancies are given in Table 2. The difference in the magnetic moments per dopant per supercell with and without N vacancies is notable. It has been observed that the value of the magnetic moments per supercell and per TM-ions is independent from the sites of the vacancies.³⁰ Similarly to the cases of Mn:GaN²⁷ and Cr:GaN,⁵⁰ the magnetic moment of Ti:GaN is observed to increase in the presence of V_N, but its value is comparatively smaller than those of the Cr and Mn doped GaN cases yet fairly larger than those of the Fe, Co, Ni, and Cu-doped GaN systems.³⁰

The results indicated that nitrogen vacancies enhance the magnetic moment in Ti:GaN. In order to model the situation, it should be useful to have a look into the electronic structure of the host and the way dopant is accommodated. In GaN, which is unintentionally n-doped, the nitrogen vacancy is a shallow donor and is an established cause of providing electrons to the conduction band.¹⁴ Like other TMs, Ti substitutes the cationic Ga-sites in the form of Ti³⁺ in GaN.^52,53 The available states in the Ti³⁺-3d¹ fivefold orbitals are occupied by additional electrons, in spin-up alignment, released from the dangling bonds in the vicinity of the nitrogen vacancies. It causes an increase in the magnetic moment from its expected value of 1 μ_B per dopant ion, in agreement with the cases of Mn- and Cr-doped GaN containing a single V_N.^27,50 The enhanced ferromagnetism also agrees with the magnetic behavior of Cr:AlN³⁰ and Ti:ZnO^31,32 in the presence of N and O vacancies, respectively. On the other hand, in the case of Fe³⁺, Co³⁺, Ni³⁺, or Cu³⁺, the extra available electrons occupy spin-down states after occupation of the spin-up states in the five d orbitals. This explains the probable cause of the reduction of the magnetic moment in TM ion (having filled spin-up bands) doped GaN with nitrogen vacancies when compared with the similar case of Ti:GaN.³⁰

From Table 2, it is interesting to note that the credit of the increased magnetic moment in Ga₁₅Ti₁N₁₅ configurations when compared with Ga₁₅Ti₁N₁₆ goes to V_N. As for the case without vacancies, for the Ga₁₅Ti₁N₁₆ configuration, Ti contributes to a major degree to the total magnetic moment of the supercell compared to the Ga₁₅Ti₁N_VN₁₅ configuration where Ti contributes to a relatively smaller degree and the rest of the contribution comes from the introduced nitrogen vacancy. Both the GGA and GGA + U calculations support it.

D. Properties of the Ti_Ga–V_N complex in Ti:GaN

After discussing the electronic structures for pure Ga₁₆N₁₆, Ga₁₆N₁₅ (Ga₁₆N₁₆ + V_N), TiGa₁₅N₁₆ (Ga₁₅N₁₆ + Ti_Ga) and Ti₂Ga₁₄N₁₆ (Ga₁₄N₁₆ + 2Ti_Ga), in the following section we describe the central configuration of our calculation, Ti₁Ga₁₅N₁₅ (Ga₁₅N₁₅ + Ti_Ga + V_N), forming the Ti_Ga–V_N complex. The DOS plots calculated on configuration Ga₁₅Ti₁N₁₅ + V_N using GGA and GGA + U for the Ti_Ga–V_N complex are shown in Fig. 5. The introduction of a nitrogen vacancy shows a significant change in the electronic structure of Ti:GaN as V_N induces states resonant with the impurity band formed near the Fermi level at the bottom of the conduction band. The bottom of the VB is mainly formed by N-2s, with a minor contribution from Ga-4s and Ga-4p. The top of the VB comprises a major contribution from N-2p and minor part from Ga-4p, Ga-4s and N-2s. The bottom edge of the CB consists of Ti-3d, N-2p, Ga-4p, and Ga-4s whereas the top of the CB consists mainly of Ga-4p. This spin polarized impurity band involves a major contribution from the Ti-3d orbitals and exhibits a half metallic behavior. This trend of half metallicity is more pronounced with the GGA + U calculation. Obviously, p–d exchange interactions resulting from hybridization between the Ti-3d and N-2p bands in the vicinity of the Fermi level should be responsible for the induced ferromagnetism like in previously discussed systems. The only difference now is the simultaneous presence of V_N which provides free carriers. The observed overlap of the Ti-3d states with V_N related donor states near the Fermi level predicts the transfer of carriers from the surroundings of the nitrogen vacancies to the Ti-3d band that enhances the local magnetic moment per Ti ion, causing carrier mediated exchange interactions.

Fig. 5 The TDOS plots for the Ti_Ga–V_N complex (Ga₁₅Ti₁N₁₅) in comparison with the N-2p, N-2s, Ga-4p, Ga-4s and Ti-3d orbital PDOS with GGA + U. The Ti-3d states are shown shaded.

Upon inclusion of a nitrogen vacancy near Ti_Ga, the formation of complex Ti_Ga–V_N is likely, which, when checked through DFT calculations in the form of the configuration Ga₁₅Ti₁V_NN₁₅, appeared as the most stable arrangement. This finding is consistent with the reported experimental finding that Mn prefers to substitute the Ga sites closest to the nitrogen vacancies.⁴⁹ Though the formation energy with a vacancy is comparatively higher, this configuration presents a higher magnetic moment pointing towards its significance for DMS materials. The role of vacancies in ferromagnetic ordering and exchange interactions in the case of TM doped compound semiconductors has been extensively studied and reported.^31,32 The difference in the values of the formation energies calculated with and without V_N for TM-doped GaN is 8.92 eV (8.71 eV) using GGA (GGA + U), which is interestingly smaller than the reported values for other TM doped GaN such as Cr (10.21 eV), Mn (9.78 eV), Fe (10.73 eV), Co (9.07 eV), Ni (9.60 eV) and Cu (9.19 eV).³⁰ It points towards the highest likelihood of the presence of Ti_Ga near an anionic vacancy, which favors the formation of the Ti_Ga–V_N complex in Ti:GaN.

The GGA + U calculated value of the magnetic moment per Ti for Ga₁₅Ti₁N₁₆ is 1 μ_B whereas its value for the Ti_Ga–V_N complex is 1.97 μ_B. The increase in the magnetic moment is due to vacancy introduction, thus predicting the role of nitrogen vacancies to enhance ferromagnetism in Ti doped GaN. This result is in accordance with a recent report by Zhenzhen Weng in which they considered that the anionic O vacancy increased the magnetic moments of Ti-doped ZnO.²⁵ For TiGa₁₅N₁₅, the formation of four humps near the Fermi level (Fig. 3) may help to describe this increased ferromagnetic behavior in Ti doped GaN in the case of the Ti_Ga–V_N complex. A hybridization between the impurity derived Ti-3d states (having a major contribution from the 3d xz, yz, zx antibonding orbitals) and V_N derived donor states (having a major contribution from the Ga-4p, Ga-4s and N-2p) is evident. It is observed that the first hump involves the hybridization between Ti-3d (xz) and N-2p, the second hump involves the hybridization between Ti-3d (xz), Ga-4p and Ga-4s, and the third hump involves the hybridization between Ti-3d (xy) and Ga-4p whereas the forth hump involves the hybridization between the Ti-3d (yz), Ga-4p and Ga-4s states. The charge transfer across these hybridized states by itinerant electrons supports a carrier mediated interaction establishing a long range magnetic order. The carrier mediated interaction is a function of the dopant-vacancy distance therefore, in order to strengthen the exchange interaction, it is suggested to introduce a nitrogen vacancy in the nearest neighbor site of the dopant in Ti:GaN. The hybridization of the Ti-3d and Ga-4p dangling states near the Fermi level and charge transfer from the donor derived band to the unoccupied Ti-3d states supports a long-range magnetic ordering between the Ti ions via the itinerant electrons.

E. Properties of Ti:GaN with a double nitrogen vacancy

To further study the prospects of Ti_Ga–V_N complexes in GaN, we studied another configuration of Ti₂Ga₁₄N₁₄ (Ga₁₄N₁₄ + 2Ti_Ga + 2V_N), thereby doubling the concentration of nitrogen vacancies, in order to further explore the characteristics of nitrogen vacancies in Ti doped GaN. It was very interesting to know that the magnetic moment becomes zero, which points to the information that upon doubling the concentration of V_N, the long range magnetic exchange interaction between substitutional Ti dopants extinguishes. This high Ti doping concentration in a Ga rich environment appears to arrange the dopants in antiferromagnetic ordering in the matrix.

It is suggested to material growers to optimize the ammonia flow rate in order to avoid the nitrogen deficient atmosphere that may cause harmful effects on the ferromagnetic spin coupling between the two nearest substitutional Ti atoms in the Ti:GaN diluted magnetic semiconductor. The suitably optimized growth conditions may provide the driving force to bring foreign Ti atoms to substitute cationically near nitrogen vacancies to set up Ti_Ga–V_N complexes in the material. It will be a stable complex and enhance ferromagnetism as V_N acts as the donor and Ti_Ga acts as the acceptor. It is therefore recommended to enhance the N partial pressure to reduce the appearance of N vacancies, avoiding the destruction of ferromagnetic coupling between the Ti atoms by the vacancies.

A long-range magnetic coupling is critical for achieving high temperature magnetism at low defect concentrations. The results suggest that a Ti_Ga–V_N complex supports long range magnetic interaction in favor of carrier mediated ferromagnetism.^59,60 Ultimately this intrinsic defect derived and impurity facilitated long-range magnetic coupling hosted at low defect concentrations opens a new route toward designing high T_c DMSs according to k_BT_c = 2ΔE/3c where c is the concentration of the dopant. The comparative view of the TDOS plots for all considered configuration is shown in Fig. 6.

Fig. 6 The TDOS plots for all considered configurations.

4. Conclusions

It is concluded that pure GaN demonstrates the absence of a magnetic moment but it becomes spin polarized with Ti doping. The un-doped GaN with a single V_N is again non-spin-polarized however it reveals an n-type conductivity which becomes more pronounced at a high vacancy concentration. However, upon simultaneous incorporation of Ti (on the cationic site) and V_N (on the neighboring sites) in the GaN matrix to realize Ti_Ga–V_N, a pronounced increase in the magnetic moment is observed. On the other hand, upon doubling the Ti and V_N concentrations in the same supercell, the magnetic moment became zero which is possibly due to antiferromagnetic coupling between dopants. It is therefore recommended that crystal growers should enhance the nitrogen partial pressure to limit the appearance of N vacancies which triggers AFM coupling in Ti:GaN. The hybridization of the dopant and host states causes a transfer of charge from the donor derived band to the unoccupied Ti-3d states, which supports a long-range magnetic ordering between the Ti ions via the itinerant electrons. The carrier mediated interaction is a function of the dopant-vacancy distance, therefore it is suggested that a nitrogen vacancy may be introduced in the nearest neighbor site of the dopant in order to strengthen the exchange interaction in Ti:GaN.

Acknowledgements

Salah Ud-Din Khan and U. A. Rana would like to sincerely appreciate the Deanship of Scientific Research at the King Saud University for funding of this research through the Research Group Project no. RGP-VPP-255.

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