1. Introduction
Since the discovery of giant magnetoresistance (GMR), spintronics has became a landmark of solid-state physics and information science [1, 2]. In spintronic applications, charge carrier electrons and/or holes are controlled by their spin degree of freedom, and where the spin polarization is controlled either by using ferromagnetic layers or by exploiting the coupling of spins to orbital moments [1, 2]. In these applications, materials characterized by half-metallic behavior are much in demand and worthy of attention [3, 4] due to their particular property of playing the role of both conductor and insulator of charge carriers at the same time. More specifically, the substance acts as a conductor in one spin orientation, and as an insulator or a semiconductor in the opposite spin orientation [3, 4].
Many Heusler alloys [5, 6], zinc-blende pnictides and chalcogenides [7, 8], manganites [9], perovskites [10, 11], diluted magnetic semiconductors (DMS) [12–14] and diluted magnetic oxides (DMO) [15, 16], show half-metal characteristics with peculiar intrinsic properties. In DMS and DMO, the central vision is to conceive magnetic compounds by arbitrarily injecting a fraction of non-magnetic host semiconductors or oxides by magnetic impurities, namely transition metal or rare earth elements [17, 18]. This injection gives rise to the apparition of localized magnetic moments in the system due to a change in charge carrier concentration [19].
III-V semiconductors including AlP, GaP, AlAs, InP, and many more other composites, offer numerous advantages as well as some peculiar properties. GaAs has a remarkable electron mobility (even larger than diamonds), and poor thermal conductivity, which makes it effective in radio frequency and microwave devices [20, 21]. Ga-based compounds like GaAs, GaP and GaN are very appealing in high-performance optoelectronic devices because of their direct band gap and low exciton binding energy [22–24]. GaN is a promising material in power electronics, where it can be used as an ultra charger [25].
In order to generate high-temperature ferromagnetic phases in III-V hosts, several studies have been conducted by injecting a small portion of transition metal impurities such as (Ga,Fe)Sb [26, 27], (Ga,Cr)N [28], (Ga,Mn)N [29], (Ga,Mn)As [29–31], and (Ga,Mn)P [29]. Furthermore, introducing isoelectronic Fe and Be donors into InAs results in an s-d exchange interaction around 2.8 eV, which exceeds regular values found in conventional DMS materials [31]. Implanting Mn impurities into the InP epilayer brings a ferromagnetic phase with a Curie temperature of 40 ± 52 K [32]. A high Curie temperature of 585 K and 953 K was generated in ferromagnetic half-metal AlP doped with 10% of V and Cr, respectively [33]. The Curie temperature reached a value of 670 K for Al0.9Cr0.1S [33]. The co-doping with (V, Ti) significantly increases the Curie temperature in ferromagnetic AlAs with respect to a single substituent [34].
Aluminum nitride (AlN ) is another element of the III-V family, and has many outstanding properties that make it very useful in optoelectronics [35] and hard protective coatings [36]. Several experimental and theoretical studies have been conducted in order to explore the properties of AlN. Experimentally, magnetization is found to be sensitive to the external electric field applied in parallel to the surface of graphene nanoribbons posed on an aluminum nitride nanosheet [37]. Using x-ray diffraction, infra-red absorption and wafer curvature techniques, the dependence of the evolution of residual stress on thickness has been studied in sputter-deposited AlN thin films on Si [38]. Doping in III-nitride materials including (Al,Ga)N is thoroughly investigated in order to understand how to achieve a satisfactory doping level [39].
AlN is a wide band gap semiconductor [40] adequate for exploitation in high-temperature and high-power electronic devices [41], as well as solar energy [42], high-frequency devices [43], long-wavelength optoelectronic devices [44], nonlinear optics [45] and multi-functional sensors [46]. It is also characterized by a high temperature stability, good thermal conductivity, and a high elasticity [47, 48]. AlN can be crystallized in several forms, but the more stable phases are zinc-blende and wurtzite structures that can be obtained at ambient pressures and temperature [49].
Theoretically, per-atom-pair binding energy and melting temperature are shown to be quadratic functions of the inverse of the size of AlN nano-particles [50]. On the other hand, the full potential linearized augmented plane wave (FP-LAPW) method in the framework of density functional theory (DFT) reveals that the band gap, the static dielectric constant and static refractive index vary quadratically along with the variation of impurities concentration in the zinc-blende AlN doped by In [51]. Doping AlN, wurtzite and zinc-blende structures with rare-earth element Er induces magnetism and increases the static dielectric constant and the absorption in the visible spectrum [52].
Inspired by the appealing results and properties of AlN, we study in this work for the first time the structural, magnetic and electronic properties of V- and Cr-doped AlN with the intention of exploring and generating half-metallic ferromagnets that can be exploited in the spintronics domain. Based on the Korringa-Kohn-Rostoker method (KKR), we show that V and Cr impurities bring a half-metallic ferromagnetic behavior to AlN, and that the double exchange mechanism is the interaction responsible for magnetism in this system. The formation energy values are found to be negative in all doping cases, meaning that the compounds are stable and acquire energy in all situations. The total moments increase linearly while increasing doping impurities, and the Curie temperature is found to be above room temperature in most cases.
This manuscript is divided as follows: the next section will discuss the methods of calculation details and the structural properties of zinc-blende aluminum nitride. Section 3 discusses the results attained such as band structure, density of states, formation energies and stability phase to have more understanding of the electronic and magnetic properties of AlN, (Al1−xVx)N and (Al1−xCrx)N, then we present the total magnetic moment, and the Curie temperature of the system according to different concentrations.
2. Method of calculations and crystalline structure
Due to its high capacity, speed and accuracy in dealing with disorder in materials [53, 54], the self-consistent KKR Green's function method [55–57] is used to calculate the structural, electronic and magnetic properties of the host semiconductor zinc-blende aluminum nitride AlN, and V and Cr impurities randomly doped the cation sites of AlN. This disordered system is substantially described by the CPA method, where the average electronic properties are assessed rather than properties of the individual doping element [58]. Generalized gradient approximation (GGA) based on Perdew, Burke and Ernzerhof parameterization (PBE) is used for exchange-correlation energy functional in the calculation [59]. The form of the potentials are treated as muffin tin (MT), where in this approximation the potential is considered to be spherically symmetric inside the atomic sphere and constant in the interstitial regions. The electronic wave functions are computed considering the angular momentum quantum number defined at each atomic site up to ℓ = 2 for d electrons. 500 K points are collected in the irreducible first Brillouin zone of the reciprocal space, and the relativistic effects are considered using the scalar relativistic approximation (SRA) for the valence states. The PBE-KKR-CPA is implemented in the MACHIKANEYAMA2002 package produced by Akai of Osaka University, Japan [54].
As stated above, in this work, AlN is an aluminum pnictide, that has a zinc-blende structure and crystallizes in the cubic $F\bar{4}3m$ space group, as shown in figure 1. In zinc-blende structures, ions have a tetrahedral coordination, where each cation Al3+ is surrounded by four equivalent anions N3− and vice versa, Al-ions occupy (4a) Wyckoff positions: $\left(0,0,0\right)$ and N-ions occupy (4c): $\left(\displaystyle \frac{1}{4},\displaystyle \frac{1}{4},\displaystyle \frac{1}{4}\right)$. The nature of bonding between Al and N is covalent; this is due to the difference between the electo-negativity χ values of these two components, which is greater than 0.5 on the Pauling scale.
Figure 1. Unit cell of AlN. The symbol X denotes the empty spheres (defaults in crystal). |
In order to study the stability of the pure host AlN, the formation energy, which is defined by equation (1 ), is computed:
$\begin{eqnarray}{E}_{\mathrm{For}}={E}_{{AlN}}-\left({E}_{\mathrm{Bulk}}^{{Al}}+{E}_{\mathrm{Bulk}}^{N}\right),\end{eqnarray}$
Thereafter, we have calculated the energy of formation of the host doped by impurities: $\begin{eqnarray}\begin{array}{rcl}{E}_{\mathrm{For}}&=&{E}_{{AlN}}-\left[\left(1-x\right){E}_{\mathrm{Bulk}}^{{Al}}\right.\\ & & \left.+{{xE}}_{\ \mathrm{Bulk}}^{\mathrm{impurty}}+{E}_{\mathrm{Bulk}}^{N}\right],\end{array}\end{eqnarray}$
where x is the impurity concentration. During the calculations all components are taken to be body-centered cubic structures. In order to study the magnetic phase stabilization, we compute the total energy difference ΔE between the ferromagnetic and the disordered local moment (DLM) energies, known also as the spin glass phase [60, 61]: $\begin{eqnarray}{\rm{\Delta }}E={E}_{\mathrm{DLM}}-{E}_{\mathrm{Ferro}}.\end{eqnarray}$
The positive value of ΔE indicates that the system is stable in the ferromagnetic phase. Otherwise, the DLM phase is the stable one. In the DLM phase, the impurity spins are divided into two, where half of the spins point out to the up direction, and the other half to the down direction. The total magnetization in this phase is equal to zero. If the ferromagnetic phase is the stable phase, we can define and compute the Curie temperature (transition temperature from ferromagnetic phase to DLM phase). This quantity is computed by using the mean-field approximation (MFA): $\begin{eqnarray}{T}_{c}=\displaystyle \frac{2}{3{K}_{B}}\displaystyle \frac{{\rm{\Delta }}E}{x},\end{eqnarray}$
KB is the Boltzmann constant and x is the impurity concentration.3. Results and discussion
We start by studying the structural properties of AlN. This material is an aluminum pnictide where Al and N atoms are arranged in a zinc-blende structure. Using the energy optimization method, the parameter lattice that minimizes the total energy is found to be a0 = 4.41 Å as shown in figure 2. This value is in good agreement with the experimental and theoretical ones reported in [49, 51, 62]. The difference between the electro-negativity χ values indicate that the nature of the bonding between Al and N is covalent.
Figure 2. Optimization of the parameter lattice. |
Figure 3(a) displays the total density of states (TDOS) of AlN material and partial density of states (PDOS) of its Al and N p-orbitals. The TDOS is given per unit cell, while PDOS are computed per atomic orbitals. Near the Fermi level one can observe a wide gap region of Eg = 3.01 eV in good agreement with previous works based on the DFT method, where the observed values vary from 3.25 eV to 3.38 eV [52, 63, 64]. Furthermore, the orbitals above the Fermi level are unoccupied states, while those under the Fermi level are fully occupied. It follows that the AlN host is an n-type semiconductor. We notice in passing that Eg is found to be ≃5 eV in [51, 65]. This difference in band gap energy values is due to the fact that the values in DFT methods are always smaller than the experimental ones.
Figure 3. (a) TDOS and PDOS of AlN and (b) band structure. |
From figure 3(b), AlN has an indirect band gap located at Γ in the valence band and at X point in the conduction band. It follows that AlN cannot play a significant role as an optoelectronic device, in particular as a light emitter, due to the fact that the interband transition requires phonons, and thus the transition rate is too low. However, it can be used as a light absorber, where AlN must absorb a photon with an energy higher than Eg to let the electrons existing at the Γ quasi-moment at the valence band jump toward the valley existing at the Γ -point in the conduction band. Furthermore, as reported in [16], the wavelength (λ) of the photon is equivalent to Eg, and thus λ( AlN) is found to be equal to 411 nm. This value belongs to the visible range, indicating that AlN can be a good light absorber for the visible and the infrared spectrum. Furthermore, by examining both the conduction and valence band, we notice that N(2p) states are dominant, and the symmetry between the spin-up and the spin-down DOS reveals the non-magnetic character of the AlN compound. Using equation (1 ), the formation energy of AlN is found to be equal to −6 eV; this negative value confirms that the host semiconductor is a stable component.
After doping the host matrix with transition metals, we vary impurity concentrations in a range between 2% and 25%. The compound turns out to be a magnetic component. For the purpose of realizing the source of ferromagnetism in the studied system, we have computed the TDOS and PDOS of the 3d orbital of the impurity. In figures 4(a) and (b), we show the TDOS and PDOS of Cr- and V-doped AlN around the Fermi level for a concentration of 16%. In both components, at the Fermi level, we see the apparition of only one spin direction in the semiconducting gap region giving a half metallic behavior of the doped compound; we note that the same behavior appears in all concentration percentages from 2% to 25%. The narrow and sharp peaks of the 3d electronic states transcend the Fermi level with unequal DOS, giving rise to a net magnetization. Furthermore, the sharp form of the 3d orbitals of the impurities indicates the localized character of the d-electron DOS compared to the wide and flat p-electron DOS. The carriers can then occupy either localized impurity states or the delocalized states in the conduction or valence band. The doping does not affect the nature of the bonding in the whole system, because the difference in the electro-negativity is still greater than 0.5 for both components.
Figure 4. Total and partial DOS of V-doped (a) and Cr-doped (b) AlN (16%). |
However, substituting the cation sites in the host semiconductor with impurities leads to a splitting of the d-orbitals of these impurities and they become degenerate, and two new sub-orbitals are created: the t2 -orbital triply degenerate states and e-orbital doubly degenerate, separated due to a tetrahedral crystal field. The origin of this crystal field is the attraction between the positive charge of cation and negative charge electrons of the impurities, which are mostly anti-bonding states. When the TM impurities get close to the center of the cation, the degeneracy d-orbital is broken due to the static electric field produced by a surrounding charge distribution, and the newly produced e and t2 states take a low and high energy, respectively. In table 1, we present the strength of this crystal field splitting evaluated by the relation ${{\rm{\Delta }}}_{{CR}}=E({e}^{+})-E({t}_{2}^{+})$; furthermore, we evaluate the exchange splitting energy presenting the separation between the spin-up and spin-down of e-orbitals given by ΔEX = E(e+) − E(e−).
Table 1. Crystal field and exchange splitting for V- and Cr-doped AlN as a function of the concentrations. |
| Materials (Al1−xTMX)N | ΔECR(eV) | ΔEEX (eV) |
|---|---|---|
| (Al0.98V0.02)N | 0.86 | 1.19 |
| (Al0.92V0.08N | 1.19 | 1.19 |
| (Al0.90V0.10)N | 1.20 | 1.20 |
| (Al0.84V0.16)N | 1.53 | 1.19 |
| (Al0.80V0.20)N | 1.70 | 1.18 |
| (Al0.75V0.25)N | 1.89 | 1.20 |
| (Al0.98Cr0.02)N | 1.03 | 1.70 |
| (Al0.92Cr0.08)N | 1.19 | 1.70 |
| (Al0.90Cr0.10)N | 1.36 | 1.87 |
| (Al0.84Cr0.16)N | 1.53 | 2.04 |
| (Al0.80Cr0.20)N | 1.53 | 2.04 |
| (Al0.75Cr0.25)N | 1.70 | 2.04 |
Vanadium has five valence electrons. When it is introduced in the host semiconductor it loses three electrons to replace the electrons of Al and become V3+. The two remaining electrons fill the up spin direction of the e-orbital; this is shown in figure 4(a) by the location of the e-orbital in the valance band, and in figure 5. In turn, Cr loses three of its six valence electrons to Al; the remaining three electrons completely fill the e-orbital and partially the t2-orbital of Cr as shown in figure 5. In figure 4(b) we notice that the filled part of the t2-orbital is located in the valence band, while the empty part is located in the conduction band. In both V and Cr components, the down sub-orbitals are completely empty due to their existence in the conduction band. These empty and partially empty degenerate energy levels are the main source of half-metallicity in the whole system.
Figure 5. Electron configuration of transition metal impurities in AlN, where ΔCR is the crystal field splitting between t2 and e. |
In order to identify the exchange coupling mechanism responsible for magnetism in the doped system, we analyze the PDOS of the d-orbitals of impurities as function of the concentrations as plotted in figures 6(a) and (b). We note that the spikes are decreased and became wider as long as we augment the impurity concentrations, and we can then apparently recognize the type of the exchange coupling responsible for magnetism as a double exchange mechanism due to a risen hybridization between N(2p) and V(3d) or Cr(3d). This interaction is a short-range interaction that can only happen when the impurities are generated. We should also note that the p-d hybridization is much stronger in the Cr case than in the V case because of more interacting electrons in the (Al1−xCrx)N system.
Figure 6. PDOS comparison of d-orbitals of V (a) Cr (b) in terms of concentrations. |
The energy of formation of the doped system (see table 2) is computed by using equation (2 ). For both components, the formation energy is negative, indicating that the whole system is stable, and the values of formation energy start from −5.8 eV for 2% of impurities to reach a value of −3.7 eV for 25%.
Table 2. Local moments for V- and Cr-doped AlN, the total energy differences, and formation energy as a function of concentrations. |
| Materials (Al1−xTMXN | TM moment $\left({\mu }_{B}/\mathrm{atom}\right)$ | Al moment $\left({\mu }_{B}/\mathrm{atom}\right)$ | N moment $\left({\mu }_{B}/\mathrm{atom}\right)$ | ΔE (meV) | EFor (eV) |
|---|---|---|---|---|---|
| (Al0.98V0.02)N | 1.35 | 0.00069 | −0.00027 | 0.31 | −5.81 |
| (Al0.92V0.08)N | 1.33 | 0.00274 | −0.00229 | 5.09 | −5.27 |
| (Al0.90V0.10)N | 1.32 | 0.00337 | −0.00313 | 7.00 | −5.01 |
| (Al0.84V0.16N | 1.31 | 0.00503 | −0.00599 | 11.03 | −4.56 |
| (Al0.80V0.20)N | 1.28 | 0.00575 | −0.00787 | 11.99 | −4.20 |
| (Al0.75V0.25)N | 1.20 | 0.00610 | −0.01009 | 11.74 | −3.77 |
| (Al0.98Cr0.02)N | 2.16 | 0.00088 | 0.00072 | 1.30 | −5.82 |
| (Al0.92Cr0.08)N | 2.16 | 0.00332 | 0.00195 | 5.49 | −5.29 |
| (Al0.90Cr0.10N | 2.17 | 0.00408 | 0.00215 | 6.69 | −5.11 |
| (Al0.84Cr0.16N | 2.17 | 0.00620 | 0.00203 | 9.00 | −4.57 |
| (Al0.80Cr0.20N | 2.17 | 0.00748 | 0.00132 | 9.30 | −4.21 |
| (Al0.75Cr0.25N | 2.11 | 0.00854 | −0.00058 | 9.37 | −3.75 |
The value of the partial moments of the transition metals is much greater than the sum of the partial moments of Al and N with an order of magnitude of two; for this reason the primary contribution in magnetism in the whole system comes from the impurities. The impurity moment values of V and Cr are about 1.3 μB/atom and 2.1 μB/atom, respectively. The partial moment values and total moments of the system containing Cr are bigger than that of V because of the extra electron existing in the t2 − orbital of Cr. We notice also from figure 7(a) that the total moment raises linearly at the same time as the concentrations of the transition metals.
Figure 7. (a) Total moments for (Al1−xVx)N, (Al1−xCrx)N and (b) Curie temperature TC (in K) against V and Cr doping concentrations. |
In order to study the stabilization of the magnetic states following equation (3 ), we compute the total energy difference ΔE between the energy spin glass phase and the ferromagnetic energies. The results are shown in table 2; for both systems the total energy of the ferromagnetic state is lower than that of the DLM state, and ΔE > 0 in all cases, and increases alongside with the TM concentrations. Therefore, the ferromagnetic phase is the stable phase. Moreover, Shi et al [66] have also reported the ferromagnetism at room temperature in Cr-doped AlN. This behavior has been confirmed by Espitia R et al [67] for both compounds Al0.0625V0.9375N and Al0.0625Cr0.9375N, respectively. Because of the strong p-d hybridization in the Cr case, the energy difference as well as the Curie temperature are higher than the V doped case. Due to this result, we are in a position where we can compute the Curie temperature, which presents the temperature above which a transition from a ferromagnetic phase to paramagnetic phase happens; this magnitude is given by equation (4 ).
As shown in figure 7(b), in the V case, TC in the lower range concentration increases rapidly starting from 119.49 K at x = 2%, and reaches its maximal value TC = 559.78 K at x=12 %. After this obtained value, TC starts decreasing until it reaches a value of 363.19 K at x = 25%; this behavior means that Tc is saturated at a V concentration of x = 12%. Similarly to the case of V, (Al1−xCrx)N is saturated at 5% of Cr concentration with a Curie temperature of 533.03 K. It starts from 504.35 K for x = 2% and decreases to 290.07 K for x = 25%, but meanwhile most of the Curie temperature values are above room temperature (dashed lines in figure 7(b)), which make both components suitable for use in the spintronics domain. However, the Cr-doped system exhibits TC higher than room temperature for most concentration cases. It is therefore easier to use such alloys in spintronic applications at ambient conditions.
4. Conclusion
Based on the density functional theory (DFT), namely the KKR-CPA-PBE method, the parameter lattice of the host AlN compound is optimized as a function of the energy. The DOS and band structure are calculated for the pure case and compared to AlN doped with V and Cr. It turns out that impurities influence the electro-magnetic properties of our semiconductor. Indeed, the arisen impurity orbitals cross the Fermi level in the band gap region, meaning that they are responsible for ferromagnetism and half-metallicity in our DMS. Magnetic properties, electronic structures, formation energy, phase stabilization and Curie temperature of diluted aluminum nitride have been analyzed and discussed. It is found that TC increases up to a certain level by increasing doping concentrations and starts to decrease after they became saturated. Thus, V- and Cr-doped AlN are potential candidates for high Curie temperature ferromagnetic materials. The calculated properties of DMS may be investigated in the spintronics field, where spin control plays a major role.