Magnetic properties in a IIIA-nitride monolayer doped with Cu: a density functional theory investigation

Abstract

Using first-principles calculations, we have studied the electronic structure and magnetic properties in a IIIA-nitride monolayer doped with Cu. The substitutional Cu impurity induces a global magnetic moment of 2.00 μ_B. All the doped systems are half metallic at the GGA level, but are magnetic semiconductors at the HSE06 level. As the atomic number of the IIIA elements increases, the increasing covalency leads to a tendency towards delocalization of the local magnetic moment. The ground state magnetism is determined by the competition between the ferromagnetic p–p/p–d hybridization interaction and the anti-ferromagnetic super-exchange interaction. By using external strain, the magnetic ground state can be deliberately tuned.

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1. Introduction

Dilute magnetic semiconductors (DMSs), which utilize both the charge and the spin freedom of electrons to create new functionalities beyond conventional semiconductor devices, have attracted much attention recently because of their promising application in spintronic devices.^1–3 Theoretical predictions point out that DMSs with a high Curie temperature (T_C) can be fabricated by doping wide band-gap III–V and II–VI binary compounds with transition metals (TM).⁴ However, the concept of “d⁰” magnetism⁵ implies that ferromagnetism can be induced easily by a nonmagnetic impurity, such as Cu,^6–10 Li,^11,12 Mg¹³ and so on. All these works suggest the possibility of fabricating DMSs using nonmagnetic dopants. Although it seems less straightforward and more challenging, it has multi-facet great significance: from the theoretical aspects, it provides an approach to understand the origin of the magnetism in materials without magnetic elements;¹⁴ from a technical point of view, it offers a new intriguing strategy to design magnetic semiconductors, and to avoid forming TMs clusters or secondary phases in TM-doped DMSs.¹⁵

Since the discovery of graphene,¹⁶ other two-dimensional (2D) layered materials such as transition metal dichalcogenides,^17,18 and other group III–V binary compounds^9,19–25 have recently been extensively studied, owing to their exotic physical properties arising from their dimensional reduction systems, and their potential applications in future spintronic devices,^26,27 Regular spin ordering in 2D materials caters to the tendency of development in electronic memory related research toward miniaturization.²⁸ Recently, the IIIA-nitride honeycomb sheets were theoretically reported as stable single-layer materials, and their synthesis could be achieved on suitable metal substrates.^19,29 Soon after intensive efforts are being dedicated to designing the promising FM orders in such as-yet hypothetical materials. To achieve FM properties, the effective ways include substitutional doping^,21 chemical decoration,^20,23,30 applied external field or strain, and so on.^31,34 Even though intensive efforts are being dedicated to manipulating the spin ordering in such materials in experiment and theory, great breakthrough to date has been relatively rare. Furthermore, in most existing studies, the substitution Cu for IIIA elements that is applied to the regulation of FM properties has been limited to bulk IIIA-nitrides. Therefore, a systematic analysis and understanding for the substitution effect on the electronic structure and magnetic properties will be essential for manipulation of the magnetic properties in IIIA-nitride monolayer.

2. Models and computational details

As shown in Fig. 1, we construct 4 × 4 × 1 with 32 atoms and 6 × 6 × 1 with 72 atoms supercells to simulate the IIIA-nitride monolayer. The vacuum space between two sheets is set as 12 Å in the direction normal to the sheet, to avoid interactions between two adjacent images. Our calculations are performed using the projector augmented wave method with a plane-wave cutoff energy of 460 eV, as implemented in the VASP code.^32,33 The generalized gradient approximation (GGA) with the Perdew–Burke–Ernzerhof (PBE) functional³⁴ is selected to describe the electron–electron exchange and correlation energies in the DFT calculations. The conjugate gradient technique is applied to the structural optimization with 0.01 eV Å⁻¹ convergence threshold for the Hellmann–Feynman force on each atom. The gamma-centered k points are sampled for the 2D Brillouin zone according to the Monkhorst–Pack method.³⁵ The k-points are set to 5 × 5 × 1 and 9 × 9 × 1 meshes for the structural relaxations and physical properties, respectively. Energy-bands are obtained by using 75 k points along highly symmetric directions in Brillouin zone. For comparison, we also perform calculations based on the HSE hybrid-DFT functional³⁶ with parameters: α = 0.25 for the fraction of Fock exchange and μ = 0.2 Å⁻¹ for the range separation.

Fig. 1 (a) Schematic view of geometric structure for a 6 × 6 × 1 InN monolayer, calculated spin density for Cu-doped (b) AlN, (c) GaN and (d) InN monolayers, respectively. The big and small balls denote IIIA and N atoms, respectively. The Cu substitutes IIIA elements at positions denoted by 1–8, respectively. The isosurface is 0.002 e Å⁻³.

3. Results and discussions

3.1. Stabilities and geometry structures of Cu-doped monolayers

Firstly, we consider the doped IIIA-nitride monolayer with one substitutional Cu atom. To determine the energetic stabilities of these doped systems, the formation energy (E_f) is evaluated from E_f = E(doped) − E(pure) − E(Cu) + E(IIIA), where E(doped) and E(pure) are the total energies of the corresponding supercell with and without Cu dopant, respectively. The E(Cu) and E(IIIA) are the total energy of an isolated Cu and IIIA atom, respectively. In Table 1, we summarize the ralaxized Cu–N and IIIA–N bond lengths, formation energies, total energy difference between the spin-polarized state and non-spin polarized one (ΔE_spin), total magnetic moment (M_tot), magnetic moment located at the Cu atom (M_Cu) and its nearest neighboring N atoms (M_N). The Cu atoms were initially placed at the top of the IIIA atom sites with a distance of 1.5 Å. After structural optimization, it was found that the Cu atoms almost move back to initial IIIA atom sites, especially for the case of InN matrix. The formation energies follow the order: E_f (AlN) > E_f (GaN) > E_f(InN). Of course, the formation energy strongly depend on bond strength of IIIA–N thus the sequence of the formation energy could be is Al > Ga > In. Due to the largest lattice constant, the Cu-doped InN monolayer has the lowest formation energy of 1.56 eV per Cu atom, which indicates that Cu-dopant can be easily implanted into InN matrix in experiment. From Table 1, we found that the optimized Cu–N bond length is 1.858, 1.865 and 1.882 Å, respectively. Apart from InN, for AlN and GaN, the Cu–N bond lengths are slightly larger than the pristine Al–N and Ga–N bond lengths. This means a slight expansion of local structure around the Cu dopant with the nearest neighboring N atoms moving away from the Cu impurity in AlN or GaN, but a contraction for InN case. Although the local structure around the Cu dopant experiences a breathing relaxation, it still remains the D_3h symmetry without Jahn–Teller distortion.

Table 1 The optimized Cu–N (d_Cu–N) and IIIA–N (d_III–N) bond lengths, formation energy, total energy difference between the spin-polarized and non-spin polarized states (ΔE_spin), total magnetic moment (M_tot), magnetic moment located at IIIA element (M_IIIA) and its nearest neighboring N atom (M_N) at PEB and HES levels

III–N	d Cu–N (Å)	d IIIA–N (Å)	E f (eV)	ΔEspin (meV)	M tot (μB)	M IIIA (μB)	M N (μB)
						PBE/HES	PBE/HES
AlN	1.858	1.806	6.17	−268	1.9998	0.743/0.825	0.232/0.245
GaN	1.865	1.848	3.47	−150	2.0003	0.687/0.803	0.208/0.232
InN	1.882	2.057	1.56	−111	1.9991	0.554/0.773	0.234/0.207

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3.2. Electronic structures and magnetic properties of Cu-doped monolayers

As expected, the Cu-doping induces magnetism.³⁷ For all the dope cases, the spin-polarized states strongly favor over the non-spin-polarized states in energy, which is reflected by the total energy difference (ΔE_spin) between this two states (see Table 1). We note that as the row numbers of IIIA elements increases, ΔE_spin decreases continuously. Calculations on effective charges show that the covalent character of IIIA-nitride monolayer becomes stronger when going from AlN to InN,¹⁹ which is in full agreement with our analyses based on charge density difference. The more intense covalence means the stronger hybridization. As a result, the bandwidth of the p orbital is relatively widened, which impedes the spontaneous polarization due to the higher electronic kinetic energy. The calculated total magnetic moment is 2.0 μ_B for all the doped systems with one Cu dopant at both GGA-PBE and HSE06 levels. The Cu dopant is a main contributor to the total magnetic moment. The neighboring N atoms also hold a little contribution to the total magnetic moment. The rest contributions originate from the interstitial spaces around these atoms. The spin-resolved band structure and density of states (DOS) of the supercell doped with a single Cu atom are presented in Fig. 2–4. These doped systems show a half-metallic behavior with the spin-down channel being metallic and the spin-up channel being semiconducting with sufficient unfilled states above the Fermi level, at GGA-PBE limit. The 100% spin polarization of conduction carriers suggests that Cu-doped materials can be used for spin injection where highly polarized spin current is desired. At the HSE06 level, the doped systems revert to semiconductors. Our further studies show the half-metallic behavior still retain for Cu-doped AlN and GaN bulk at the HSE06 limit. Although calculations at the GGA and HSE levels predicted the different electrical conductivity, they presented qualitatively consistency in magnetic properties prediction.

Fig. 2 Electronic band structures and total density of states for Cu-doped AlN monolayer obtained from GGA-PBE and HSE06 calculations. The Fermi level is indicated by the dashed line. The black ↑ and red ↓ represent spin-up and spin-down channels, respectively.

Fig. 3 Electronic band structures and total density of states for Cu-doped GaN monolayer obtained from GGA-PBE and HSE06 calculations. The Fermi level is indicated by the dashed line. The black ↑ and red ↓ represent spin-up and spin-down channels, respectively.

Fig. 4 Electronic band structures and total density of states for Cu-doped InN monolayer obtained from GGA-PBE and HSE06 calculations. The Fermi level is indicated by the dashed line. The black ↑ and red ↓ represent spin-up and spin-down channels, respectively.

To obtain the intuitive physical picture, we present the spin-density distributions for Cu-doped systems in Fig. 1. The magnetic moments in the supercell have the same direction, indicating the dopants couple ferromagnetically to their neighboring N atoms; at the xy plane the p_x + p_y character of the spin-polarized orbitals of N atoms is clearly visible. In contrast, no significant spin polarization was found in the p_z orbital. Quiet interestingly, we find that the ratio of (M_Cu : M_tot) tends to increase (see Table 1), and the spin-density distribution extends from the central Cu to farther N atoms, when passing from AlN to InN. As motioned above, the increasing covalency can account for this trend. The higher covalency between the IIIA elements and N atoms means that both the IIIA element and N atoms share the hole induced by Cu-doping, and thus delocalize hole on N-2p orbitals. Also, the increasing screening effect implies that much more outer electrons of IIIA elements will be captured by localized N-2p orbitals, and hence leads more delocalized spin density distributions when IIIA elements change from Al to In. It's important to note that this trend facilitates a long-range magnetic interaction between local magnetic moments, although it is not beneficial for spontaneous polarization.

The total magnetic moment is not equal to the number of the holes, which is in agreement with the Cu-doped IIIA-nitride bulk.^6–8 The origin of the local magnetic moment in Cu-doped IIIA-nitride monolayer can be understood by analyzing the crystal-field splitting, according to the molecular orbital theory. The IIIA-nitride bulk with zincblende structure has a T_d symmetry. Under the T_d symmetry the 2p orbitals form the triply degenerate t_2p states, while the Cu-3d orbitals are split into a doubly e_g (d_z², d_x²−y²) state and a triply degenerate t_2g (d_xy, d_xz, and d_yz) states. The doubly degenerated e levels are lower in energy than the singlet t₂ levels. Due to the effective hybridization between Cu 3d-states and host p-valence states,^6–8 the electronic configuration of divalent Cu can be identified as (e^2↑e^2↓t^3↑₂t^2↓₂), which corresponds a magnetic moment of 1.0 μ_B on Cu site. This also indicates that one hole accumulated near nitrogen atoms, due to the substitution of Cu²⁺ for trivalent Al or Ga. The electron configuration (t^↑↑↑_2pt^↓↓↓_2p) are responsible for the magnetic moment of 1.0 μ_B around N atoms. As mentioned above, the local structure of the Cu-doped monolayers maintain D_3h symmetry. Under this symmetry of crystal field, the atomic Cu-3d levels are split into a single a₁ (d_z²) state and two twofold degenerate e₁ (d_xy, d_x²−y²) and e₂ (d_xz, d_yz) states. Taking Cu-doped InN case as a prototype, they can be seen in Fig. 5, where the calculated density of states of Cu-3d and N-2p are shown. The d_z² orbital, which is situated furthest away from the Fermi level, is perpendicular the xy plane and therefore has lower overlap with neighboring p orbitals than the d_xz, and d_yz orbitals. As a result, a₁ states are filled in both spin-up and spin-down channels and located at the highest energy. The Simultaneously, the triply degenerate t_2p state can be classified to one singlet a₂ (p_z) state and one doubly degenerate e_x,y (p_x, p_y) states. The doubly degenerated e_x,y level is higher in energy than the singlet a₂ level. The p_x and p_y mixing with s orbitals are strongly hybridized with d_xy and d_x²−y² orbitals to make three planar spd-like orbitals which are directed from the central Cu atom at the corners of the hexagons to its three nearest neighbors. The relevant electronic states of a₂ (p_z) from 2.0 eV to the Fermi level make π-bonding network, which maintains the stability of the planar geometry.¹⁹ To conclude, we can conclude that the Cu²⁺ electronic configuration can be written in the following way: a^↑₁a^↓₁e^2↑₂e^2↓₂e^2↑₁e^↓₁. The additional states related to N-2p states can be expressed as: a^↑₁a^↓₁e^2↑₁e^↓₁. Therefore, similar to bulk AlN and GaN, the ligand field theory account for the total magnetic moment of 2.0 μ_B.

Fig. 5 Density of states of the a₁, e, e₁, and e₂ states of Cu and its nearest neighboring N calculated using (a) GGA-PBE and (b) HSE06, respectively for Cu-doped InN monolayer.

In practice, a long-ranged FM order with Curie temperature above room temperature is desirable. In order to explore the preferred magnetic coupling between separated Cu atoms in IIIA-nitride monolayer, we calculated the energies of ferromagnetic (FM) and antiferromagnetic (AFM) states of the three IIIA-nitride monolayers. To study the magnetic interaction, two Cu atoms are implanted in the 6 × 6 × 1 supercell as motioned above. Five configurations are considered and marked as C(i, j) according to the order of Cu–Cu separation as shown in Fig. 1(a). To estimate the strength of exchange coupling, we also calculated the energy differences between the FM state and the AFM state for each configuration. In principle, the energy difference contains all exchange interactions of the infinite lattice together with the periodic arrangement of the impurity spins. The energy difference ΔE (=E_FM − E_AFM) are listed in Table 2. It can be seen from Table 2 that the C(1, 2) always has the lowest total energy and the largest ΔE, indicating that the FM state is most stable for C(1, 2) with a clustering trend. In doped InN system, the FM state is energetically more favorable than the AFM state, hence, the state prefers prefer to an AFM ordering in Cu-doped AlN. There seems to be a magnetic transition from the AFM state to the FM state, when the row number or atomic radii of cation increase. On the whole, the AFM interaction is relatively weak with respect to the FM coupling. Method based on the mean-field theory and Heisenberg model using the relation γk_BT_C/2 = (E_AFM − E_FM) allows us to estimate the Curie temperature (T_C) of Cu-doped systems.^20,38 Here γ is the dimension, k_B is Boltzmann constant. To obtain room temperature ferromagnetism, the ΔE should be greater than 105 meV, when the systems under study are reckoned as 2D materials. It is reasonable to believe that the Cu-doped InN sheet will possess the characteristics of room temperature ferromagnetism.

Table 2 The calculated IIIA–IIIA distance (d_IIIA–IIIA), relative stability energy (E_RS eV), the difference between FM and AFM (ΔE = E_FM − E_AFM), total magnetic moment (M_tot), and magnetic state for each C(i, j) structure of the two-Cu-doped IIIA–N binary compounds sheets. The relative stability energy refers to the energy of C(1, 2) structure in ground state

Type	C(i, j)	d IIIA–IIIA	E RS	ΔE	M tot	Magnetic state
AlN	C(1, 2)	2.780	−0.48950166	−237	2.0	FM
	C(3, 4)	5.480	−0.48846061	32	4.0	AFM
	C(5, 6)	6.250	−0.48854348	32	4.0	AFM
	C(7, 8)	9.383	−0.48849037	26	4.0	AFM
GaN	C(1, 2)	2.780	−0.41297865	−200	4.0	FM
	C(3, 4)	5.630	−0.41206831	−62	4.0	FM
	C(5, 6)	6.419	−0.41206752	−58	4.0	FM
	C(7, 8)	9.608	−0.41193688	30	4.0	AFM
InN	C(1, 2)	3.324	−0.35928923	−272	4.0	FM
	C(3, 4)	6.221	−0.35877998	−97	4.0	FM
	C(5, 6)	7.096	−0.35873044	−198	4.0	FM
	C(7, 8)	10.70	−0.35862785	−150	4.0	FM

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The origin of the magnetism in DMS is still under active debate, although several theoretical models have been proposed during past years. In the GGA limit, our calculated results show that the magnetic ground state in Cu-doped IIIA-nitrides is stabilized by the competition between the p–p/p–d hybridization mechanism^39,40 and the anti-ferromagnetic super exchange interaction. As shown in Fig. 1(d) that Cu dopant couples to its nearest neighboring N ferromagnetically via p–d hybridization,^39,40 and the intermediate localized spins on N-2p couple with each other via p–p interaction,^41,42 similar to p–d hybridization in transition-metal (TM)-doped semiconductors or oxides. Therefore, the intermediate p–p interaction mediates an indirect ferromagnetic coupling between the p–d hybridized CuN₃ clusters. On the other hand, CuN₃–N–CuN₃ mediates the AFM coupling. We now take AlN-C(7, 8) as a paradigm, and visualize its spin charge density in inset of Fig. 6(a). The carrier on intermediate N between two CuN₃ clusters delocalized over the CuN₃–N–CuN₃ unit, thus lowering the kinetic energy and resulting in an indirect AFM coupling between CuN₃ clusters according to Anderson–Goodenough–Kanamori rule,⁴³ similar to TM–anion–TM superexchange interaction. In Cu-doped AlN, the AFM coupling between CuN₃ clusters plays a dominant role in magnetic interaction. Thus, apart from C(1, 2), other configurations are AFM.

Fig. 6 Strain dependence of the total energy and energy difference between FM and AFM order AlN-C(7, 8) and GaN-C(7, 8). The insets in left panel shows the spatial spin density distributions of AlN-C(7, 8) without strain. The yellow and blue isosurfaces correspond to the spin-up and spin-down components, respectively. The insets in right panel illustrates schematic of the exchange mechanism for C(1, 2), virtual hopping results in an FM ground state.

As mentioned above, when the system goes from AlN to InN, the covalency and lattice parameter enhance accordingly. Thus the hole in N will become more delocalized when the covalency increases. This tendency is against forming localized magnetic moment, which is reflected by the ΔE_spin in Table 1, but it facilitates the long-range magnetic coupling. On the other hand, the increasing lattice parameter and decreasing ionicity will weaken superexchange interaction, but it has little effect on the p–p hybridizations since the p–p interaction is relatively long ranged compared to p–d hybridization. Consequently, FM coupling will gradually play a dominant role when doped systems go along the line: AlN → InN. Therefore, it is difficult to understand that the C(7, 8) shows AFM and FM properties for GaN and InN, respectively. From Table 2, it is easy to see that the AFM interaction is predominant over the FM one in AlN matrix, whereas the situation is contrary in InN matrix. However, for all the doped systems, the C(1, 2) is an exception. The separation between two Cu atoms in C(1, 2) is close to the value (2.56 Å) of bulk Cu in face centered cubic structure. Thus, the d_xy/d_x²−y² of Cu impurity pair will overlap each other, giving rise to direct FM interaction between Cu atoms. As shown in the inset of Fig. 6(b), the Cu e₁ state is exactly half filled; virtual hopping is allowed in the FM arrangement but not allowed in the AFM configuration, resulting in a lower energy FM state.

If our assumption is reliable, the magnetic order can be tuned by applying biaxial tensile strain. The biaxial tensile strain here is defined as ε = Δa/a₀, where lattice constants of the unstrained and strained supercell are equal to a₀ and a = Δa + a₀, respectively.^23,26 The C(7, 8) of AlN and GaN (AlN-C(7, 8) and GaN-C(7, 8)) are taken as examples. Fig. 6(a) exhibits a monotonically increased tendency of total energies with tensile strain. Fig. 6(b) shows the variation in energy difference ΔE (=E_FM − E_AFM) of AlN-C(7, 8) and GaN-C(7, 8) as a function of strain which is in the range from 0% to 10% of its equilibrium lattice constant a₀. According to our GGA-PBE results, it is found that the magnetic transition between FM and AFM orders in AlN-C(7, 8) and GaN-C(7, 8) can be induced by applying strain. The FM coupling is favored over the AFM coupling (i.e., ΔE < 0) under the tensile strain form 3% to 5% for GaN-C(7, 8), and the energy difference can reach ΔE can reach −81 meV when the tensile strain is at 5%. Interestingly, only when the strain is increased to 6%, the AFM state → FM state occurs for AlN-C(7, 8), and the energy difference ΔE can reach a minimum of −131 meV at the tensile strain ε = 7%. After carefully examining the lattice constant a, we find it is 3.31 Å and 3.30 Å for GaN and AlN unit cells at the strain 3% and 6%, respectively, which indicates that the magnetic transition is intimately associated with the lattice constants. Under the increasing tensile strain, the covalency is reduced while the ionicity is increased. Therefore, AFM coupling eventually plays a leading role at ε ≥ 8%. This mechanism also well holds for the AlN-C(7, 8) under strain in the range from 1% to 3%. As described above, the lattice constants are increased and the superexchange coupling is thus efficiently weakened. Under a relatively moderate tensile strain, such as 4% for GaN-C(7, 8), the FM coupling is predominant over the AFM coupling. The above competition mechanism can provide a convincing explanation for the magnetic ground state of IIIA-nitride monolayer doped with Cu.

4. Conclusion

Based on first-principles calculations, we have conducted a theoretical study of the structural, electronic, and magnetic properties of IIIA-nitride monolayer doped with Cu atoms. The substitution Cu for IIIA atom induces a total magnetic moment of 2.00 μ_B. All the doped systems show half metallic behavior and magnetic semiconductor at GGA and HSE06 levels, respectively. When the system goes from AlN to GaN, cooperation between the increasing covalency and the elongated lattice results in various magnetic ground states. The ground state magnetism is eventually determined by the competition between the FM p–p/p–d hybridization interaction and the AFM super-exchange interaction. The doped AlN system favors an AFM ordering, while the FM state is stabilized by the p–p/p–d hybridization mechanism in doped InN system. The predicted strong room-temperature ferromagnetism suggests that Cu-doped InN monolayer might have important potential application in spintronics. By using external strain, the magnetic ground state can be deliberately tuned, which would be propitious to their advanced applications.

Acknowledgements

This work was supported by The Joint Fund of Guizhou Department of Science and Technology (Grant No. LH [2014] 7227), the Specialized Research Fund for the Doctoral Program of Kaili University (Grant No. BS [2013] 30), the National Natural Science Foundation of China (Grant No. 11074069, and 61176116), the Natural science research project in Guizhou province department of education (Grant KY [2012] 061) and the Natural Science Foundation of Technology Department of GuiZhou Province(No. J [2014] 2147).

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