The electronic and magnetic properties of transition metal (Mn and Cr) doped Aluminium Nitride (AlN) in the wurtzite structure is studied using the single impurity Anderson model (SIAM) extended to semiconductors by Haldane. An equation of motion method based on Green’s function was used to obtain the effective spin decomposed impurity levels. The calculated electronic density of states of Mn and Cr valence orbitals exhibit half metallic properties when the impurity is strongly coupled to the host. The effect of the Coulomb correlation and orbital hybridization on the formation of a localized moment in such systems is investigated. Magnetic impurities are often responsible for the inelastic scattering of conduction electrons. For a configuration averaged random ensemble of impurities, initially non-polarized host band develops a small moment presumably due to the potential scattering.

## I. INTRODUCTION

The ferromagnetism arising due to magnetic correlations of a transition metal (TM) impurity in a III-V semiconductor host, which in other words are called diluted magnetic semiconductors (DMSs), has received much attention in the field of spintronics.^{1,5} Among the III-V semiconductors, Aluminium Nitride (AlN) has the largest band gap experimentally determined to be 6.2 eV with many preferred properties such as high thermal stability, good thermal conductivity, low compressibility etc.^{4} AlN doped with TM atoms with a partially filled *d* orbital viz. Manganese, Chromium, Iron, Vanadium etc. belongs to this interesting class of semiconductors, that various studies has reported as good candidates for spintronic applications often owing to their half metallic properties.^{2,3}

Aluminium Nitride based DMSs has been extensively studied in recent years both theoretically and experimentally. Previously, Syrotyuk and Shved^{6} have shown by density functional theory (DFT) that Mn-doped AlN wurtzite is half metallic with a magnetic moment of 2.7*μ*_{B}. Li *et al.*^{7} have also synthesized Mn-doped AlN by solid-state reaction, and they have obtained a ferromagnetic material at room temperature. Ahmoum *et al.*^{8} has shown from a DFT calculation of Mn-doped AlN that Mn forms donor impurity levels in the wide direct band gap of AlN contributing a magnetic moment of 4.0*μ*_{B}. Kaczkowski and Jezierski,^{9} reported an increase in the magnetic moment on Cr ion in Cr doped AlN studied using DFT on including the Coulomb correction U at the impurity site. However, a DFT based mean field calculation is not enough to estimate the actual *T*_{c} or the effect of band structure and mixing energy on the magnetic properties of DMSs. In order to develop a realistic description of the problem it is necessary to treat the Coulomb interaction free from the mean field approximation as was done by Ohe *et al.* in their paper^{11} on Quantum Monte Carlo studies of DMSs with TM impurities described using Anderson impurity model.^{12} Our earlier work^{13} involved the use of Anderson impurity model for DMSs to describe the tunnelling barrier of a model magnetic tunnel junction (MTJ) containing TM atoms and study the effect of gap states on the electronic transport. It was found that the presence of impurity states in the gap of the semiconductor aided transport across the junction. However, a real simulation of the device requires the application of the model to real systems with a suitable DMS forming the barrier.

In the present study, a single impurity Anderson model (SIAM) extended to semiconductors by Haldane^{14} is solved within the Hartree-Fock approximation to describe the TM atom (Mn and Cr) for different concentrations in an Aluminium Nitride semiconductor host. The Bloch energies are obtained using a DFT calculation as implemented in Vienna *Ab initio* simulation package (VASP). This approach provides a mean field rationale for the existence of free local moments on transition metal impurities that result from the interplay of orbital hybridization and the Coulomb interaction of the electrons at the impurity. We have investigated the electronic structure of the impurity and their magnetic moments for various values of the parameters that describe the SIAM to obtain the conditions for local moment formation. Both (Al,Mn)N and (Al,Cr)N were observed to exhibit half metallic properties which could be exploited for spintronic applications. Moreover, with increasing concentration of Cr atoms replacing Aluminium in the semiconductor, a magnetic moment of 0.01*μ*_{B} was found to be induced in the host lattice. We also envisage the use of this model in tuning the parameters continuously to drive such systems from a magnetic to non-magnetic state or vice-versa.

## II. MODEL AND METHOD

The diluted magnetic semiconductors can be modelled by the single impurity Anderson Hamiltonian^{14}

where $cm\sigma \u2020(ck\sigma )$ is a creation (annihilation) operator of a host electron (AlN) with wave vector $k\u20d7$ and spin *σ* in the band *α*. $dm\sigma \u2020(dm\sigma )$ is a creation (annihilation) operator for a localized electron of the orbital m at the impurity (Mn or Cr) site. *ϵ*_{k} is the energy of host electrons, *E*_{d} is the bare energy of localized electrons and the number operators of host electron and localized electron are $nk\sigma =ck\sigma \u2020ck\sigma $ and $nm\sigma =dm\sigma \u2020dm\sigma $ respectively. U is the Coulomb energy of localized electrons at the impurity sites and *V*_{mk} is the mixing energy between the host electrons and the localized electrons. In order to construct a realistic model of (Al,Mn)N and (Al,Cr)N, we have obtained the electronic band structure of the host Aluminium Nitride (AlN) from the DFT calculations using Generalized gradient approximation (GGA) based on Perdew, Berke, and Ernzerhof (PBE)^{24} in VASP. The projector augmented wave (PAW) pseudopotential method^{25} was used to treat the valence electron configuration. The kinetic energy cut-off was 570 eV. The Brillouin zone integrations were performed on a 11x11x9 Γ-centered k-point grid using a tetrahedron method^{26} with a smearing width of 0.2 eV. Fig. 1 shows the ground state Bloch energy spectrum of AlN.

The Anderson Hamiltonian in Eq. (1) was solved in the unrestricted Hartree-Fock approximation (HFA), using a Green’s function method. To circumvent this many body problem involving complex magnetic and electronic correlations, it was important to decompose Eq. (1) to a single particle effective Hamiltonian,

which was solved in terms of the parameter $Em\sigma eff$. The parameters $Em\sigma eff$ were determined self consistently assuming initial values from

where ⟨*n*_{m′σ′}⟩ is the average occupation of the *m*th state of the impurity *d* electron and we know for a fact that the density of states is the imaginary part of the complex electron Green’s function. So ⟨*n*_{m′σ′}⟩ could be calculated using

and *G*_{mσ}(*ω*) is the impurity Green’s function renormalized by the thermodynamic self energy Σ(*ω*). It is obtained using the equation of motion method^{16} by evaluating,

The frequency variable *ω* is assumed to have an infinitesimal positive imaginary part, *ω* = *ɛ* + *iγ*. The thermodynamic self energy Σ(*ω*) accounts for the electron-electron and electron-host interaction and is calculated by integrating the quantity in Eq. (6). The self energy being a function of the complex energies has both a real part Σ_{R}(*ω*) and an imaginary part Σ_{I}(*ω*) given by

In the limit *γ* → 0, the real part takes the principal value of the sum in Eq. (7) and merely introduces a shift in the impurity *d*-levels. Whereas, the imaginary part in this limit becomes, Δ = *π*∑_{k}|*V*_{mk}|^{2}*δ*(*ɛ* −*ϵ*_{k}) = $\pi \u27e8|V\u0304km|2\u27e9\rho (\epsilon )$ and is the effective transition rate between the impurity and the conduction electrons. Now, it can be comprehended from Eq. (4) and Eq. (5) that the density of states of the impurity level reduces to the Lorentzian form,

The hybridization matrix or the transfer interaction *V*_{mk} in Eq. (8) broadens the impurity level in which case Δ appears as the half width of the *d*-level as expected. The localized orbitals of the impurity overlaps significantly only with the free states of the host bands that lie in the proximity of the Fermi energy, *E*_{F}. So five contributing bands (see Fig.1) were considered in the calculation of *ρ*(*ɛ*). Including interactions at the Hartree-Fock level shifts the impurity sharp peaks at *E*_{d} to $Em\sigma eff=Ed+U\u27e8nm,\u2212\sigma \u27e9$. Also the *n*_{mσ}’s in Eq. (4) for *σ* = +, − form a self consistent set of equations,

Hence the iterative calculation of these shifted energies $Em\sigma eff$ involves a criteria for the filling of *d* orbital states consistent with the convergence criteria. The interaction strength *U*/Δ appearing in Eq. (10) plays a key role in deciding the nature of the solutions. To study its effect on the magnetic properties the calculation was done for the strong (Δ > *U*/*π*) and weak coupling (Δ < *U*/*π*) limits of the interaction strength for values of *U*/Δ in the vicinity of the transition region, Δ ∼ *U*/*π*. The values of U for isolated atoms typically range from 0−20 eV but for transition metals in a semiconductor it reduces to few eV.^{18} The Hubbard U is the largest contributor to Coulomb interactions in a strongly correlated system and its value can be computed from constrained-density-functional calculations^{17} where U is defined as the second derivative of the total energy as a function of the localized states occupations of the Hubbard site. This definition holds true for the present Anderson model calculation as the only term that survives a double derivative in Eq. (1) is the term containing U renormalized by the different screening mechanisms in a solid. The values of Hubbard U evaluated self consistently for TM atoms in AlN by Kaczkowski and Jezierski^{9} using this method are used in our calculation. Typical values of *sp*^{3} − *d* hybridization of orbitals^{11} are assumed for the matrix elements *V*_{mk} as we are only interested in the physics of the material in the above mentioned coupling limits. The energy required to place an electron of either spin on the outermost orbital of the impurity, *E*_{d} is so chosen that it respects particle hole symmetry i.e., *E*_{d} = *E*_{F} − *U*/2 where *E*_{F}= 6.01 eV.

Magnetic impurities like Mn and Cr in AlN semiconductor host act as potential scattering centers to the conduction electrons in the lattice.^{23} Besides the local moment formation it is interesting to study the effect of scattering on the impurity and host bands. For a random distribution of impurities in the host lattice, the configuration-averaged Green’s functions over the impurity ensemble was calculated in the short range approximation using the Matsubara technique^{20} by Agafonov and Manykin.^{22} This method was numerically implemented to include these perturbations in the first order approximation of self energy. The impurity-induced modifications in the density of electronic states were thus obtained for an impurity concentration ranging from *N*_{im} = 0.0625 (1 Cr per 16 Al atoms) to *N*_{im} = 0.25 (4 Cr per 16 Al atoms).

## III. RESULTS AND DISCUSSION

The total density of states (TDOS) of pure AlN calculated using DFT (red line) and five bands around Fermi energy (blue line) considered in the calculation are plotted in Fig. 1(a). The band decomposed TDOS presents three regions: upper part of the valence band (VB) dominated by Al 3*p* and N 2*p* states (see Fig. 1(b)), the lower part of the conduction band (CB) of predominantly N 2*s*, 2*p* character and the middle part of the CB shows hybridization of Al 3*s* and N 2*p* states. The exact value of Fermi energy is presented in all figures in this paper unless mentioned otherwise. As it can be noted, the Fermi energy of pure AlN, *E*_{F}= 6.01 eV, is at the edge of the VB and AlN exhibits a direct band gap of *E*_{g}= 4.20 eV which concurs to the theoretical value (*E*_{g}= 4.26 eV) found by Jiao *et al.*^{19} Fig. 2 shows the Mn *d* orbital states in the coupling limits of interest. Mn has a stable half filled configuration with five electrons in the *d* orbital and as a consequence of this it hybridizes very little with the surrounding N 2*p* ligands resulting in rather localized peaks in both coupling limits. In the strong coupling regime (see Fig. 2(a)), the Fermi energy shifts to *E*_{F}= 11.5 eV crossing the conduction band (CB) of AlN in the spin down channel with minimal DOS $(\rho EF)$ in the spin up channel (see Table I). This is clearly an implication of half-metallic character displayed by some materials that are metallic in either of the two spin channels. Moreover, it contributes charge carriers that are predominantly spin down polarized to the system which can find applications in spintronics as spin injectors. The values of Fermi energy and DOS at *E*_{F} are in good agreement with a recent DFT calculation of Mn-doped AlN by Ahmoum *et al.*^{8} Also, the spin up peak at 6.5 eV lying near the VB edge confirms the origin of peaks in the optical absorption spectrum around 350 nm (corresponds to 3.5 eV) obtained by Tatemizo *et al.* in their study^{21} on sputter deposited Mn-doped AlN films. A similar peak is observed in the DFT+U study of AlMnN^{9} subsuming the corrections imparted by the self-consistently fixed Hubbard U in their calculation. In the weak coupling regime (Fig. 2(b)) when U dominates over Δ, there is practically no mixing of spin up and down channels of Mn *d* states. Mn prefers to retain its stable half filled configuration without being largely affected by the changes in the hybridization energy with respect to a fixed U. The Fermi level shifts to a higher value of 12.19 eV still crossing the CB with a lower DOS in the spin down channel resulting in a slightly higher magnetic moment of 0.45 *μ*_{B}/*cell* than the case of strong coupling (Table I).

(a) Strong coupling regime (Δ > U/π)
. | |||||
---|---|---|---|---|---|

Dopant atom | V_{mk} (eV) | E_{F} (eV); E-E_{F} (eV) | $\rho EF\u2191;\rho EF\u2193$ | n ↑; n ↓ | Mag. moment (μ_{B}/cell) |

Mn | 1.5 | 11.5 ; 5.49 | 0.012 ; 0.55 | 4.65 ; 0.325 | 0.43 |

Cr | 1.5 | 6.68 ; 0.67 | 2.35 ; 0.02 | 3.55 ; 0.40 | 0.31 |

(b) Weak coupling regime (Δ < U/π) | |||||

Mn | 0.9 | 12.19 ; 6.18 | 0.051 ; 0.144 | 4.795 ; 0.201 | 0.45 |

Cr | 0.7 | 5.50 ; 0.51 | 0.871 ; 0.029 | 3.90 ; 0.065 | 0.38 |

(a) Strong coupling regime (Δ > U/π)
. | |||||
---|---|---|---|---|---|

Dopant atom | V_{mk} (eV) | E_{F} (eV); E-E_{F} (eV) | $\rho EF\u2191;\rho EF\u2193$ | n ↑; n ↓ | Mag. moment (μ_{B}/cell) |

Mn | 1.5 | 11.5 ; 5.49 | 0.012 ; 0.55 | 4.65 ; 0.325 | 0.43 |

Cr | 1.5 | 6.68 ; 0.67 | 2.35 ; 0.02 | 3.55 ; 0.40 | 0.31 |

(b) Weak coupling regime (Δ < U/π) | |||||

Mn | 0.9 | 12.19 ; 6.18 | 0.051 ; 0.144 | 4.795 ; 0.201 | 0.45 |

Cr | 0.7 | 5.50 ; 0.51 | 0.871 ; 0.029 | 3.90 ; 0.065 | 0.38 |

Similar arguments hold for Cr doped AlN with the Fermi level crossing the VB in the spin up channel in the strong coupling limit (see Fig. 3(a)). Hence, (Al,Cr)N also exhibits a half-metallic behaviour with a spin up polarised DOS at *E*_{F}, $\rho EF\u2191=2.35$ (states/eV) (see Table I) which is in agreement with the DFT calculation of Cr-doped AlN in the zinc blende structure by Espitia *et al.*^{10} Moreover, the calculated position of the Cr-3*d* peak at 7.0 eV conforms to the observed absorption peak in the X-ray photo electron spectroscopy (XPS) studies on AlCrN thin films^{15} validated by DFT results. Unlike Mn, both Cr spin up and spin down *d* states distinctly hybridize with the VB of AlN which is dominated by the surrounding N 2*p* ligands. In the weak coupling limit, the Fermi energy crosses the VB at *E*_{F} = 5.5 *eV* with a majority of *d* electrons occupying the spin up channel in virtue of the weak orbital overlap or strong on site repulsion. This system may not be as good a metal as (Al,Mn)N for either values of the coupling ratio as Cr does not provide spin polarized free carriers to the system. The magnetic moments obtained in both systems with the value of U being 4.0 eV and 2.7 eV for Mn and Cr respectively are given in Table I and are in excellent agreement with the moments reported by previous calculations on the system.^{8,9} All the calculations in this study are carried out at T=0 K. Fig. 4 shows the effect of concentration of Cr impurities in AlN on the band DOS near the Fermi energy. The host band initially non-polarized undergoes an exchange splitting into spin up and spin down bands as seen in the figure. The splitting increases with the concentration and a small negative magnetic moment (∼ 0.01 *μ*_{B}) was found to be induced in the host lattice by the Cr atoms for a concentration of *N*_{im} = 0.25. Shi *et al.*^{2} had also reported from a DFT study of TM doped AlN nanosheets that unlike TMs like Mn, Fe, Co and Ni, Cr induced a negative moment of 0.05*μ*_{B} on the neighbouring N atoms due to the strong hybridization with the surrounding ligands. The three dimensional isosurfaces of charge density confirmed this observation showing spin down charge density near the N atoms attached to Cr. However the reason for the induced magnetism and the type of exchange interaction behind it is a subject of further study.

## IV. CONCLUSIONS

In summary, a single particle effective Anderson Hamiltonian was solved in the Hartree Fock approximation to obtain the electronic density of states of AlN doped with transition metal impurities, Mn and Cr for different doping concentrations. Both (Al,Mn)N and (Al,Cr)N exhibit a half-metallic band structure in the strong coupling limit with a completely spin polarized density of states in either of the two spin channels. The fact that Mn doped AlN is a good metal could be put to use in many applications that require spin polarized charge carriers. Doping of Cr on the other hand induces impurity states at the VB and CB edges lowering the band gap to almost 2.9 eV. It was also found that with increasing concentration of Cr atoms the exchange split bands of AlN developed a magnetic moment of the order of 0.01*μ*_{B}. Our study also implies the role of Coulomb correlation and the hybridization energy in deciding the chances of a local moment formation. This approach of solving for the electronic and magnetic properties of such DMSs is unerringly reliable to study strongly correlated systems if we could also try and include the long range electron-electron interaction and the exchange correlation that are omitted in the present formulation. We also intend to apply this model to the simulation of real MTJs to study the transport characteristics of the device containing Anderson like impurities in the barrier. The present formulation can be generalised to include the effect of multiple bands near the Fermi energy and also the higher order approximations for self energy to consider more impurities in the system. As most spintronic devices work at near room temperature it is necessary to study the effect of temperature on the magnetisation of an increased concentration of impurities as well which would be the focus of our future work.

## DATA AVAILABILITY

The data that support the findings of this study are available within the article.