Influence of crystal structure and formation energies of impurities (Mg，Zn and Ca) in zinc blende GaN

XIONG Zhi-hua(熊志华), JIANG Feng-yi(江风益), WAN Qi-xin(万齐欣), RAO Jian-ping(饶建平)

Research Center for Luminescence Materials and Devices of Education Ministry, Nanchang University, Nanchang 330047, China

Received 10 April 2006; accepted 25 April 2006

Abstract: First-principles calculations on neutral metal impurities (Mg, Zn and Ca) in zinc blende GaN were studied. Formation energies were calculated for substitution on the gallium site, the nitrogen site and incorporation in the octahedral interstitial site and the tetrahedral interstitial sites. The calculated results show that the major defects studied have a high formation energy in excess of 5 eV, and the gallium substitutional site is favorable for incorporation. Mg_Ga has particularly low formation energy 1.19 eV and can be expected to incorporate readily into GaN. The local crystal structural changes around the impurity in the lattice were studied after metal atoms occupying the gallium substitutional site. It shows that the lattice constant becomes bigger and the tetrahedral angle between impurities and its nearest N atom becomes smaller mainly due to the extended M—N bond length and big size of impurities atoms, which results in a local lattice distortion. The Zn-N (2.04 ?) bond strength is the smallest among the three impurities which raises the formation energy. Ca_Ga is unfavorable due to a large size mismatch in spite of a large bond strength (2.25 ?). The calculated results identify the two key factors determining impurities incorporation in zinc blende GaN: the atomic size of impurities comparing to that of host atoms and the bond strength between the impurities and its neighbors. The results are in well agreement with other calculated and experimental results.

Key words: formation energy; crystal structure; zinc blende GaN

1 Introduction

The zinc blende GaN is believed to be better suited for n and p-type doping than its wurtzite counterpart due to the isotropy of the cubic GaN lattice, with may exhibit higher electron drift velocities and reduced phonon scattering giving rise to potentially higher mobility materials[1]. More and more experiments showed that the epitaxy of thin film were demonstrated to crystallize in the cubic zinc blende structure[2,3], and zinc blende GaN-based light emitting diodes grown on GaAs substrates were also reported[4]. As the capabilities of growing zinc blende nitrides were demonstrated in recent years[5, 6], the information on the structural, impurity formation energy, formation enthalpy etc of this new class of materials was strongly desirable. Compared to the wurtzite GaN, people knew little physical information about these. To our knowledge, some of these results were briefly reported earlier[7, 8]. In the present paper, we performed a systematic investigation of the atomic structure and defect formation energies, selected impurities (Mg, Zn and Ca) using first- principles pseudopotential calculations.

First-principles calculations of formation energies in GaN were performed using a plane-wave pseudo- potential method as implemented in the VASP[9] code. For the exchange correlation potential, the generalized gradient approximation(GGA) was employed, and the GGA functional given by Perdew and WANG[10] was used. Based on the optimized structure parameters for the perfect crystal, GaN supercell is constructed for defect calculations, which contains 64 atoms in the zinc blende lattice. For the  case of GaN∶Mg, one Ga atom in the center of the lattice with fractional coordinate (0.5, 0.5, 0.5) was replaced by one Mg atom. The convergence tests of the total energy with respect to the planewave energy cutoff and k-point sampling have been carefully examined. The final set of energies were computed with an energy cutoff of 500 eV, and integration using 5×5×5 k-points sampling over the irreducible Brillouin zone, generated by the Monkhorst-Pack scheme[11]. Forces on atoms were calculated, and atoms were allowed to relax using a conjugate gradient technique until their residual forces had converged to less than 5 meV/?. The calculations were found to be converged to better than 2 meV per atom with these parameters. The electronic ground state was determined through conjugating gradient minimization of the total energy with respect to the plane wave coefficients.

Both formation energy and formation enthalpy are calculated, for the former determines the equilibrium concentration of impurities in zinc blende GaN and the latter reflects the bond strength between the impurities and N atoms around the impurities.

Formation energy E_f is substituted on the gallium site, the nitrogen site and for incorporation on the octahedral interstitial site. We calculate the formation energy by assuming the M(Mg, Zn and Ca) atom, and Ga atom reservoirs are bulk M and Ga, thus the formation energies of defect are obtained from the following expression[12].

where  E_GaN:M, E_GaN, E_M, E_Ga andare the total energy of M doped in GaN, pure GaN, bulk M, orthorhombic Ga and N₂, respectively. The integers n, m, and l are the number of doped impurities atoms and substituted Ga and N atoms, respectively.

Formation enthalpiesper atom pair of the M₃N₂ compound are calculated as[13]

where  is the total energy per atom pair of the compound M₃N₂; E_M is the energy per atom of bulk metal impurities (Mg, Zn and Ca); E_N is the energy per N atom in the N₂ dimer or the condensed N₂ phase.

3.1    Formation energy

Four high symmetry sites are consider: two substitutional(M_Ga and M_N) and two interstitial M_i (tetrahedral and octahedral) sites. Table 1 shows the calculated formation energies of defect for the four cases.

It is first noted that the formation energies of Mg, Zn and Ca interstitials is high (majority of them more than 5 eV) in zinc blende GaN. The results are similar to the results obtained from the gallium, and the formation

Table 1 Calculated formation energies of defect for impurities M on different sites

energy of nitrogen self-interstitials is also greater than 5 eV in the neutral charge state[14,15]. Therefore it is concluded that Mg, Zn and Ca interstitials may not be able to form significant concentrations in zinc blende GaN.

However, for the cases of Mg, Zn and Ca on gallium site, the situation is different. The most notable result from Table 1 is the low formation energy for Mg on gallium site (1.19 eV). The energies are lower than both the V_N and V_Ga vacancy formation energies reported by Neugebauer and Walle[14], which suggests that Mg_Ga could be readily formed in high concentrations in GaN. Also, for the cases of Zn and Ca on gallium site, the energies are also lower comparing to the other doped cases .Therefore, we can identify that Mg, Zn and Ca impurities occupy the gallium site in zinc blende GaN.

Furthermore, we observe that the formation energy increases with the size of the impurities ion, the energy cost involved in relaxation. Therefore, we definitively conclude that Zn and Ca atoms heavier than Mg would exhibit even poorer solubility.

3.2    Local structural distortion around impurity of crystal

As the metal impurities are implanted into the compound semiconductors and replaced another cation in the lattice, the most direct influence is to the local structural distortion of the crystal. We will study detaily the metal impurities on the gallium site due to the low formation energies. The cubic structure of the zinc blende GaN structure with space group  can be described with a=b=c and α=β=γ=90?. In the 2×2×2 supercell, after one Ga atom was replaced by one M atom, the shape of the cubic structure is unchanged (α=β=γ=90?), and only the lattice constant and the local atomic positions are changed.

Table 2 shows the optimized crystal parameters of GaN and GaN∶M in the selected 2×2×2 supercell used in the calculations. From the table, we can see: 1) comparing the lattice constant of pure GaN with the experimental value[16], the calculated values are higher about 0.26%, which is within the scope of the allowed errors and in good agreements with other calculated results[8,16]; 2) the lattice constants become longer (1.15%-1.63%) when the metal impurities are doped in

Table 2 Relaxed crystal structures of GaN and GaN∶M, lattice constant, tetrahedron angle, bond length of N-M and Ga-N of formation enthalpies

zinc blende GaN; 3) For the GaN∶M, the bond length of M-N become longer and the neighbor bond length of N-Ga changes slightly. However, the extend bond lengths of Ga-N become longer; 4) Mg₃N₂ has the big formation enthalpy which leads to the strong bond strength, while the situation of Zn₃N₂ is different though Zn atom has the optimize size compared with host Ga atom. It is also found that our formation enthalpies agree well with the results reported by DEAN[18].

In order to obtain the clear picture about the doped cases, Fig.1 shows the distorted local structure of GaN doped Mg atoms on the central substitutional Ga site in the lattice. From checking the relative atomic positions around the Mg atom, we can easily find out that the increased lattice constant is mainly due to the increased bond length of Mg-N and the Ga-N around the impurity. It can be seen from Fig.1 and Table 2, the Mg-N bond length is 2.050 ?, which is bigger than that of the average Ga-N bond length. This weakens Mg-N interactions in the lattice. The tetrahedral structures around the N atoms connected with the centered Mg atom are distorted. The Ga-N bond length for the N atoms connected with the Mg atom is about 1.961 ?, which is little shorter than the average Ga-N bond length. However, the N-Ga bond length increases to 1.992 ? when it extends far away from the central Mg atom. Furthermore, the distorted tetrahedral structure can also be characterized by the angles, as shown in Fig.1, which is deviated from the standard tetrahedron’s 109.47?. Big size of the impurities ion compared with the Ga ion induce the longer bond length of the M-N bonds, which also in turn breaks the crystal symmetry in a local area, as shown in Fig.1.

It is found, for zinc blende GaN, the bond length change is isotropic when the metal impurities occupy the substitutional gallium site. Zn actually exhibits the most optimal size match with the Ga host atom, however, the Zn—N bond strength is low for the small formation enthalpy, which raises the formation energy. Ca also is unfavorable due to a large size mismatch in spite of a large bond strength with high formation enthalpy. Mg, finally, is favorable to incorporating for its suitable size and large bond strength of Mg-N. Therefore, we can conclude that the two key factors determined the doping effects are suitable size and strong bond strength.

Fig.1 Distorted local structure of GaN:Mg around central Mg atom replacing Ga atom in lattice

Low formation energies of Mg substitution gallium site suggests that Mg can readily form significant concentrations in GaN. The Zn—N bond strength is the lowest among the three impurities which raises the formation energy. Ca_Ga is unfavorable due to a large size mismatch in spite of a large bond strength. The local crystal structural changes obviously, the lattice constant become bigger and the tetrahedral angle become smaller mainly due to the longer M-N bond length and the big size radius of impurities. The calculated results identify the two main factors determining acceptor incorporation are the atomic size of impurities comparing to that of host atoms and the bond strength between the acceptor and its neighbors. The experiment results are in good agreement with other calculated and experimental results.

References

[1] Alves J L A, Leite A H W, de Oliverira C. Zine-blende GaN: ab initio calculations [J]. Materials Science and Engineering, 1997, B50: 57.

[2] Okumura H, Hamaguchi H, Koizumi T, BALAKRISHNAN K, ISHIDA Y, ARITA M, CHICHIBU S, NAKANISHI H, NAGATOMO T, YOSHIDA S. Growth of cubic III-nitrides by gas source MBE using atomic nitrogen plasma: GaN, AlGaN and AlN [J].Yoshida, J Cryst Growth, 1998, 189-190: 390.

[3] Brandt O, Yang H, Jenichen B, SUZUKI Y, PLOOG K H. Surface reconstructions of zinc-blende GaN/GaAs(001) in plasma-assisted molecular-beam epitaxy [J]. Phys Rev B, 1995, 52: R2253-R2256.

[4] Yang H, Zheng L X. Cubic-phase GaN light-emitting diodes [J]. Appl Phys Lett, 1999, 74: 2498-2500.

[5] Harima H, Inoue T, Nakashima S. Raman scattering characterization of group III-nitride epitaxial layers including cubic phase [J]. J Cryst Growth, 1998, 189-190: 435-438.

[6] Lima A P, Tabata A, Leite J R. Growth of cubic InN on InAs(001) by plasma-assisted molecular beam epitaxy [J]. J Cryst Growth, 1999, 201-202: 396-398.

[7] Karch K, Bechstedt F, Pletl T. Lattice dynamics of GaN: Effects of 3d electrons [J]. Phys Rev B, 1997, 56: 3560-3563.

[8] Li S T, Ouyang C Y. First principles study of wurtzite and zinc blende GaN: a comparison of the electronic and optical properties [J]. Physics Letters A, 2005, 336(2-3): 145-151.

[9] Kresse G, Hafner J. Ab initio molecular-dynamics simulation of the liquid-metal–amorphous-semiconductor transition in germanium [J]. Phys Rev B, 1994, 49: 14251-14269.

[10] Perdew J P, Chevary J A, Vosko S H. Atoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation [J]. Phys Rev B, 1992, 46: 6671-6687.

[11] Hendrik J. Monkhorst, James D P. Special points for Brillouin-zone integrations [J]. Phys Rev B, 1976, 13: 5188-5192.

[12] Cui X Y, Medvedeva J E, Delley B. Role of embedded clustering in dilute magnetic semiconductors: Cr doped GaN [J]. Phys Rev Lett, 2005, 95: 256404-256407.

[13] Agostino Z, Fabio B, Paolo R. First-principles prediction of structure, energetics, formation enthalpy, elastic constants, polarization, and piezoelectric constants of AlN, GaN, and InN: Comparison of local and gradient-corrected density-functional theory [J]. Phys Rev B, 2001, 64: 452081(1-6).

[14] Neugebauer J, Chris G, Van de Walle. Atomic geometry and electronic structure of native defects in GaN [J]. Phys Rev B, 1994, 50: 8067-8070.

[15] Bogusl/awski P, Briggs E L, Bernholc J. Native defects in gallium nitride [J]. Phys Rev B, 1995, 51: 17255-17258.

[16] Stampfl C, Van de Walle g. Density-functional calculations for III-V nitrides using the local-density approximation and the generalized gradient approximation [J]. Phys Rev B, 1999, 59: 5521-5535.

[17] Vincenzo F, Michael M, Matthias S. Electronic and structural properties of GaN by the full-potential linear muffin-tin orbitals method: The role of the d electrons [J]. Phys Rev B, 1993, 47: 13353-13362.

[18] DEAN J A. Lange’s Handbook of Chemistry [M]. New York: McGraw-Hill, 1985.

(Edited by LI Yan-hong)

Foundation item: Porject supported by National High-Tech Research and Development Program of China

Corresponding author: JIANG Feng-yi; Tel: +86-791-8304441; Fax: +86-791- 8304441; E-mail: jiangfy@ncu.edu.cn