ARTICLE

First principles study on structural and electronic propertiesand defect formation enthalpies of cubic Hf3N4 under highpressureJin-Ping Zhang, Xiao-Ling Zhu, Yang-Yang Zhang, and Jing-Xia Gao

Abstract: Using the self-consistent density functional method, we investigate the structural and electronic properties of cubicHf3N4 with Th3P4 structure at ambient and high pressures. The lattice parameters, cell volume, bulk modulus, and pressurederivative at ambient pressure are obtained, which are in excellent agreement with the available measurements. The change ofbond lengths for two different types of Hf–N bond with pressure suggests that the tetrahedral Hf–N bond is slightly lesscompressible than the octahedral ones, which agree well with the Zr3N4 and Zn3N2 results. The band gap pressure coefficient forc-Hf3N4 are fitted, which are 8.5 × 10−2 eV/GPa and –7.0 × 10−5 eV/(GPa)2, respectively. Based on the density of states analysis, bandstructure suggests that the investigated material can be used as a semiconductor optical material. Mulliken population analysisshows that the charge density of the Hf atom is less sensitive to pressure variation than that of the N atom. At last, the defectformation enthalpies of the cubic Hf3N4 are calculated.

PACS Nos.: 62.50.−p, 71.15.Mb, 73.20.At.

Résumé : L’utilisation de la méthode de la fonctionnelle de densité auto-cohérente nous permet d’étudier les propriétésstructurelles et électroniques du Hf3N4 cubique avec la structure Th3P4 a pression ambiante et haute. Nous obtenons lesparamètres du réseau, le volume de la cellule, le module de compression volumique et la dérivée de la pression, le tout a pressionambiante et ces résultats sont en excellent accord avec les mesures disponibles. La variation avec la pression de la longueur delien entre deux liens Hf–N différents suggère que le lien Hf–N tétraédrique est légèrement moins compressible que l’octaédrique,ce qui agrée avec les résultats pour Zr3N4 et Zr3N2. Les coefficients de pression de la bande interdite de c-Hf3N4 sont alors ajustésnumériquement : 8.5 × 10–2 eV/GPa et –7.0 × 10–5 eV/GPa respectivement. Sur la base de l’analyse de la densité d’états, la structurede bande suggère que ce matériau peut être utilisé comme semi-conducteur optique. L’analyse des charges de Mulliken montreque la densité de charge de l’atome Hf est moins sensible aux variations de pression que celle de l’atome N. Finalement, nouscalculons l’enthalpie de formation de défauts dans Hf3N4 cubique. [Traduit par la Rédaction]

1. IntroductionNitrides of transition metals that are used as hard materials

have attracted great interest because of their high elastic moduli,ultrahardness, high melting points, and superconductivity [1–5].Novel cubic hafnium nitride (c-Hf3N4) with Th3P4 structure wasfirst obtained by Zerr et al. [6] via chemical reaction of hafniumwith molecular nitrogen at high pressure (18 GPa) and high tem-perature (2800 K) in a laser-heated diamond anvil cell. Preliminarycompressibility measurements have indicated high bulk moduli,B0, of about 260 GPa (with B ′ = 4) for c-Hf3N4, only 10% below the B0

of �-Si3N4, which suggests that c-Hf3N4 may be nearly as hard as�-Si3N4. c-Hf3N4 can potentially be used in industry for coatingcutting tools used for machining ferrous alloys. Hence, subse-quently a number of studies on c-Hf3N4 have been carried outexperimentally [7–9]. Additionally, the mechanical, electronic,and optical properties of c-Hf3N4 have been investigated theoret-ically [10–17]. But, as far as we know, little research has beenreported on the electronic properties of c-Hf3N4 under conditionsof high pressure. Therefore, this work is focused on investigatingthe physical properties under high pressure from first principles.Recently, first principles calculations have been successfully usedto reveal and predict the structural, mechanical, and physicalproperties of different matters [18, 19]. As is well known, first

principles offer one of the most powerful tools for carrying outtheoretical studies of these properties even at higher pressure[20–22].

In this paper, using the standard Kohn–Sham self-consistentdensity functional theory based on SIESTA (Spanish Initiative forElectronic Simulations with Thousands of Atoms) code [23, 24], wecalculate the electronic structure and density of states (DOS) ofc-Hf3N4 at both ambient and high pressures. In addition, thechanges of the bond angles and the bond lengths under differentpressures are listed to study the changes of structural propertieswith pressure. We have fitted the calculated direct band gap to aquadratic function to determine the band gap pressure coeffi-cient. What is more, the ionic configurations of cubic Hf3N4 atdifferent pressures are estimated. In ionic crystal, there are twotypes of disorder: Schottky-type and Frenkel-type disorders. It isknown that there are different causes of different types of disor-der in crystal; these two types of disorder have different defectformation enthalpies. The Schottky and Frenkel disorders inc-Hf3N4 crystals are reported at the end of our work.

2. Computational methodIn this paper, the physical properties of the cubic hafnium ni-

tride with Th3P4 structure at both ambient and high pressures are

Received 19 January 2014. Accepted 26 May 2014.

J.-P. Zhang, Y.-Y. Zhang, and J.-X. Gao. College of Information Engineering, Huanghe Science and Technology College, Zhengzhou 450006, China.X.-L. Zhu. School of Electronic and Information Engineering, Chengdu University, Chengdu 610106, China.Corresponding author: Jin-Ping Zhang (e-mail: jinping213@163.com).

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Can. J. Phys. 92: 1581–1586 (2014) dx.doi.org/10.1139/cjp-2014-0032 Published at www.nrcresearchpress.com/cjp on 27 May 2014.

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investigated by using the SIESTA simulation program [23, 24],which is based on the standard Kohn–Sham self-consistent den-sity functional method. The generalized gradient approximation(GGA) of Perdew et al. [25] is employed to deal with the electron–electron exchange correlation energies. Norm-conserving pseudo-potentials are adopted to represent the electron–ion interaction.The split-valence double-zeta plus polarization orbitals (DZP) basisset with an energy shift of 0.02 Ry is treated as our atomic orbitalbasis set in all the computations. The numerical integrals areperformed on a real-space grid with equivalent cut-off energy of250 Ry. The Brillouin zone sampling is carried out using the 5 × 5 ×5 set of Monkhorst–Pack mesh [26]. The structural optimizationsfor the positions of atoms and lattice vectors of c-Hf3N4 are ob-tained using the conjugate gradient algorithm, until the maxi-mum atomic force is less than 0.20 eV/Å. In our structuraloptimization course, the experimental structural parameters (a =b = c = 5.803 Å) are used as initial structural of calculations, and theunit cell of c-Hf3N4 crystal comprises 14 atoms, with space groupI-43d.

3. Results and discussion

3.1. Structure properties at ambient and high pressuresIn this work, using the conjugate gradient minimization

method in GGA forms of exchange and correlation, we have per-formed a structural investigation of c-Hf3N4 at different high pres-sures. To confirm the precision of our calculations, the latticeparameters, cell volume, bulk modulus, and pressure derivativeunder zero pressure are determined. These results as well as theavailable experimental results [6–8] and other theoretical values[10–16] are listed in Table 1. From Table 1, we can find that anumber of theoretical investigations have been made for c-Hf3N4.Of these investigations, Feng et al. [10] and Chihi et al. [11] obtainedthe lattice constants using the pseudopotential plane-wavemethod with the GGA. Their results are inferior to ours whencompared with the measurements [6–8]. Obviously, other param-eters, such as V0, dtet(Hf–N), and doct(Hf–N) were not derived in theirwork. Several theoretical investigations [12–14, 16] obtained onlypartial parameters. By comparison, one can find that the mostcomplete and the best results were determined by Mattesini et al.[15] using Blöchl’s projector-augmented wave method with GGA.As shown in Table 1, it can be clearly seen that our calculatedlattice constants and bulk modulus are in good agreement withthe experimental [7, 8], the cell volume and Hf–N bond lengthsagree well with the ref. 6, and the percent errors are within 2%. Asa result, we have reason to believe that this method is especiallyaccurate.

To understand the structure properties of c-Hf3N4 with theTh3P4 structure under high pressures, we investigate the bondlengths and bond angle stability of c-Hf3N4 in our studied pressurerange (0–50 GPa). For the cubic hafnium nitride structure, the Hfatom is in octahedral coordination with six N atoms and simulta-neously in tetrahedral coordination with four N atoms. Figure 1depicts how the octahedral Hf–N bond lengths and the tetra-hedral Hf–N bond lengths change with pressure. It shows that thebond lengths decrease with increasing pressure, but the octa-hedral Hf–N bond lengths decrease faster than the tetrahedral Hf–Nbond lengths. The octahedral Hf–N bond lengths show a steeperchange, especially in the region of 0–30 GPa. Correspondingly, thetetrahedral Hf–N bond length decreases gently under compres-sion and is shortened by about 2.80% up to 50 GPa, so that thedifference of the octahedral and tetrahedral Hf–N bond lengthsbetween 5 GPa becomes smaller and smaller when pressure in-creases from 0 to 50 GPa. This indicates that tetrahedral Hf–N isslightly less compressible than the octahedral. This conclusionagrees well with the discussion in refs. 6 and 7 about the bondlengths for the octahedral and tetrahedral sites in Zr3N4 and Zn3N2.At the same time, we find that the octahedral Hf–N bond angleand the tetrahedral Hf–N bond angle almost do not change in thewhole pressure range (0–50 GPa) because of symmetry. And wealso find that the structure of c-Hf3N4 is stable and there is nophase transformation in our studied pressure range (0–50 GPa).

To understand the hardness of cubic Hf3N4, we calculate itsbulk modulus, which is a material property indicating the degreeof resistance of a material to compression. Based on the pressure–volume data, which are shown in Fig. 2, we obtained the bulkmodulus, B0, and its pressure derivative, B ′, by fitting the third-order Birch–Murnaghan equation [27] of state

P �3

2B0(v

�7/3 � v�5/3)�1 �3

4(B ′ � 4)(v�2/3 � 1)� (1)

where v = V/V0, with V0 fixed at the value determined from thezero pressure data. The fitting results (bulk modulus B0, and itspressure derivative B ′) are collected in Table 1 along with theexperimental data [6–8] and other theoretical values [11, 12, 14, 15].It can be clearly seen that the bulk modulus, B0, is in good agree-ment with the experimental [7, 8]; the deviation from the experi-ments [7, 8] is only 1.34% for the B0. As for the pressure derivative,B ′, different experiments [6–8] give different results, and the dif-ferences among these experiments [6–8] are very large, as can beseen in Table 1. This makes it difficult for us to accurately compare

Table 1. Lattice constants, geometries, and bulk modulus and its pressure derivative compared withavailable experimental data and other theories for c-Hf3N4.

Source a0 (Å) V0 (Å3) dtet(Hf–N) (Å) doct(Hf–N) (Å) B0 (GPa) B ′ (GPa)

This work 6.817 158.46 2.25 2.48 230.05 4.72Experimental[8] 6.702 — — — 227(7) 5.3(6)[7] 6.707(9) — — — 227(7) 5.3(6)[6] 6.701 150.45 2.17 2.47 260 4-fixedTheory[10](GGA) 6.837 — — — — —[11](GGA) 6.578 — — — 232 5.95[12](LDA) 6.583 — 2.09 — 268 6.56[13](GGA) 6.843 — — — 251.64 —[14](LDA) 6.583 122.59 — — 268 6.56[15](LDA) 6.586 142.84 2.19 2.37 275.6 5.27[15](GGA) 6.719 151.67 2.21 2.45 228.5 5.22[16](LDA) 6.589 — — — 283 —[16](GGA) 6.7159 — — — 215 —

Note: GGA, generalized gradient approximation; LDA, local-density approximation.

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the present values with them. We find that the pressure deriva-tive, B ′, is somewhat small when compared with some experi-ments [7, 8] and lager when compared with others [6]. It is easy tofind that the present investigations are accurate. It is evident thatthe high bulk modulus of cubic Hf3N4 indicates very high hard-ness.

3.2. Electronic properties under high pressureFor hafnium nitrides a small band gap of 0.88 eV is calculated at

zero pressure, which is in good agreement with the previous the-oretical result (0.84 eV) [15]. The pressure-induced energy shifts ofthe optical transition related to the indirect energy gap are plot-ted in Fig. 3. This shows that the band gap increases with increas-ing pressure and that this dependence is slightly nonlinear. Todetermine the band gap pressure coefficient, we have fitted thecalculated indirect band gap, Eg(�), with a quadratic function

Eg(P) � Eg(0) � aP � bP2 (2)

and obtained a = 8.50 × 10−2 eV/GPa and b = –7.0 × 10−5 eV/(GPa)2.

Previous investigations [28–30] have demonstrated that the DOScan come to a better understanding of the electronic proper-ties. So the DOS for c-Hf3N4 at different high pressures is calculated.Figures 4 and 5 show the total and projected density of states (PDOS),respectively, of c-Hf3N4 at different high pressures. The electronicproperties of c-Hf3N4 are basically characterized by the DOS.

To understand the variations of the DOS of c-Hf3N4 with highpressure, first of all, we should analyze the DOS under zero pres-sure. As shown in Fig. 4, the zero-point energy (vertical dashedline) is aligned at the Fermi level. It can be seen that in the upperpart of the valence band (VB), just below the Fermi energy (Ef), weidentify three distinct peaks (labeled V1 to V3 in Fig. 4) at –1.30,–2.32, and –4.73 eV at zero pressure. From Fig. 5, we can see thatthe peak is contributed to mainly by the N 2p orbital electrons,and Hf 5d orbital electrons contribute very little. From Fig. 4, wealso distinguish four important conduction band (CB) regions,which we call for simplicity C1, C2, C3, and C4 from the bottom ofthe CB. The CB is mainly attributed to the unoccupied Hf 5d and N2p states. This is similar to the DOS of c-Zr3N4 found in Guo et al.’s[4] investigation.

Fig. 1. The bonds length changes of Hf–N with pressure.

Fig. 2. Birch–Murnaghan equation of state for cubic Hf3N4.

Fig. 3. Change of the band gap energy of c-Hf3N4 with pressure.

Fig. 4. DOS for c-Hf3N4 at (a) 0, (b) 25, and (c) 50 GPa; the Fermi levelis indicated by the vertical dashed line at 0 eV.

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Comparing the DOS of c-Hf3N4 under different pressures, thefollowing two results can be obtained in Fig. 4: the CB is shifted tohigher energies and the valence band is shifted to lower energieswith increasing pressure. So as the difference of energy betweenthe bottom of the CB and the top of the valence band increases,such change indicates that the band gap increases a little withpressure. At the same time, as can be seen from Fig. 4, the numberof peaks of the CB increases, however, their intensity is graduallyweakened. The analysis of the CB and valence band DOS underhigh pressure suggests that cubic Hf3N4 can be used as the semi-conductor optical material. These results also have been discussedin related systems like c-Zr3N4 [4] and Zn3N2 [5].

To provide further insight into the electronic structure andchanges of two different bond lengths with pressure, we considerthe PDOS of c-Hf3N4 at high pressure. As shown in Fig. 5, thezero-point energy (vertical dashed line) is aligned at the Fermilevel. From Fig. 5, under zero pressure, it can be seen that thePDOS of Hf at the tetrahedral and octahedral sites in c-Hf3N4 aresomewhat different in their relative intensities for several peaks,reflecting the difference in the local environment. With increas-ing pressure, it also can be seen that the difference between thetwo curves of the PDOS of tetrahedral Hf 5d and the octahedral Hf5d become smaller and smaller. In particular, at 50 GPa, the twocurves of the PDOS of tetrahedral Hf 5d and the octahedral Hf 5dalmost coincide with each other in the region from –6 to 0 eV. Inother words, as the pressure increases, the difference between theoctahedral and tetrahedral Hf–N bond lengths becomes smallerand smaller. It can be noticed that the PDOS of N 2p orbital and Hf5d has prominent overlaps, such as those near –6, 0, and 7 eV.However, the degree of overlaps on the tetrahedral Hf 5d and N 2pis greater than that of the octahedral Hf 5d and N 2p, which meansthat the bonding of the former is stronger than the latter.

The Mulliken population analysis is often used for discussingthe relative covalency and ionicity of materials [31–34]. The chargepopulations as well as orbital occupancies of the valence orbitalsare calculated by the Mulliken population analysis. Because of thelimitations of this type of analysis [35], the charges can only be

considered in qualitative terms. Here, we estimated the ionic con-figuration for c-Hf3N4 in the fundamental state to be Hf+0.783N–.587

by using the Mulliken population. A negative value representscharge gaining, whereas a positive value corresponds to chargeloss. We find that there is a large charge transfer from the Hfatoms to the N atoms. To further understand the behaviour ofelectron transfer with high pressure, charge population as a func-tion of pressure is plotted in Fig. 6. We have fitted the calculatedcharge population with a quadratic function

Nx(P) � Nx(0) � axP � bxP2 (3)

where x is either N or Hf. The fitting results are NN(0) = 5.59, aN =–3.0 × 10−4, bN = –1.06 × 10−5 and NHf(0) = 3.22, aHf = 5.24 × 10−4, bN =1.03 × 10−5. It can be seen that the electronic charge of Hf transfersto N atoms, and the charge transfers decrease from the cation tothe anion with increasing pressure. In addition, the charge popu-lations of the Hf atoms become larger and larger with increasingpressure. However, the charge populations of N atoms get smallerand smaller with increasing pressure. The charge change of Hfatoms is a little smaller than that of N atoms. In other words, thecharge density of Hf atoms is less sensitive to pressure variationthan that of N atoms. These results indicate that c-Hf3N4 has in-creasingly covalent character. As we know, the charge popula-tions are largely dependent on the selected basis set, but thetrends observed in the charge transfer process are reliable.

3.3. Schottky and Frenkel disorder in c-Hf3N4 crystalsDifferent causes lead to different disorders in crystals, so

Schottky-type and Frenkel-type disorders have different disorderformation enthalpies. The Frenkel-type disorder results from anion being moved from its natural place to the interspaces of theionic crystal, and the reaction formula is as follows:

HfHfx ↔ Hfi

• � VHf′ (4)

Fig. 5. PDOS of some orbitals of Hf and N atoms at (a) 0, (b) 10, (c) 30, and (d) 50 GPa; the Fermi level is indicated by the vertical dashed line at0 eV.

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NNx ↔ Ni

′ � VN• (5)

where the superscript “•” represents disorder with positive charge,while the superscript “′” represents disorder with negative charge.In addition, the superscript “x” represents that the ion is electroneu-tral, while the superscript “i” represents the ion-occupied inter-space of the ionic crystal. For Frenkel-type disorder of c-Hf3N4crystals, we only investigate the case of an anion being movedfrom its natural place to the interspace. The Schottky-type disor-der results from an ion being moved from its natural place to thesurface of the ionic crystal and the reaction formula is as follows:

Null ↔ 4VN• � 3VHf

′ (6)

The formation enthalpy of the defect is the total (internal) en-ergy difference between the ideal system and the system withvacancy. The calculations of the vacancy formation enthalpyshould be performed in a large enough supercell lattice to ensurethat the interaction between the defects can be neglected. Theformation energies of the Schottky-type and Frenkel-type defectsin c-Hf3N4 lattice are calculated by

�Hf � Edisorder(Hf3N4) � Enormal(Hf3N4) (7)

where �Hf is the formation enthalpy of the defect; Enormal(Hf3N4)is the normal crystal lattice energy; and Edisorder(Hf3N4) is theenergy of the c-Hf3N4 lattice with a defect. According to (7), wecalculated that the Frenkel-type and Schottky-type defect forma-tion enthalpies in the Hf3N4 lattice are 4.8 and 78.8 eV, respec-tively. The huge formation enthalpy of Schottky-type defect(78.8 eV) practically rules out this defect formation mechanism.This indicates that Frenkel-type disorder is the dominant type ofatomic disorder.

4. ConclusionIn this paper, the structural and electronic structure changes of

c-Hf3N4 crystal under high pressure were first studied by densityfunctional theory calculations, and our study shows that bondlengths of two different types of Hf–N bond decrease with increas-ing pressure and the difference between them decreases. Thetetrahedral Hf–N bond is slightly less compressible than the octa-hedral ones. Meanwhile, the octahedral and the tetrahedral Hf–N

bond angles almost do not change in the whole pressure range. Inaddition, a bulk modulus of c-Hf3N4 of B0 = 230 GPa, is obtained;such a high bulk modulus indicates very high hardness, similar tothat of �-Si3N4 and c-Zr3N4. The DOS analysis shows that c-Hf3N4

can be used as a semiconductor optical material. To determine theband gap pressure coefficient, we have fitted the calculated indi-rect band gap to a quadratic function. Based on the Mullikenpopulation analysis, it is found that the covalent character ofc-Hf3N4 will increase with increasing pressure, and the charge ofthe N atom is more sensitive to pressure variation than that of theHf atom. At the end of our work, the formation energies of theFrenkel-type and Schottky-type defects in a c-Hf3N4 lattice arecalculated. We find that Frenkel-type disorder is the major type ofatomic disorder.

AcknowledgementsThis work is financially supported by the Program of Henan

Educational Committee (Nos. 13B140986 and 13B430985) andthe Program of Zhengzhou Science and Technology Bureau(Nos. 121PPTGG359–3 and 121PYFZX178). The authors thank R. Martinand P. Ordejón for help using SIESTA.

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Fig. 6. Charge population versus pressure (star: first-principlescalculated results; solid line: fitted by quadratic function): (a) N; and(b) Hf.

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