Computational Materials Science 67 (2013) 341–345

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Computational Materials Science

journal homepage: www.elsevier .com/locate /commatsci

First-principles study on the elastic and electronic properties of hexagonal e-Fe3N

Yao-Jun Shi a,b, Yu-Lei Du a,b,⇑, Guang Chen a,b

a Engineering Research Center of Materials Behavior and Design, Ministry of Education, Nanjing University of Science and Technology, Nanjing 210094, Chinab Jiangsu Institute of Advanced Materials, Danyang 212300, Jiangsu, China

a r t i c l e i n f o

Article history:Received 17 August 2012Received in revised form 13 September 2012Accepted 14 September 2012Available online 26 October 2012

Keywords:Elastic propertiesElectronic propertiese-Fe3NFirst-principles

0927-0256/$ - see front matter � 2012 Elsevier B.V. Ahttp://dx.doi.org/10.1016/j.commatsci.2012.09.012

⇑ Corresponding author at: Engineering Researchand Design, Ministry of Education, Nanjing UniversitNanjing 210094, China. Tel./fax: +86 25 84315159.

E-mail address: yldu_njust@njust.edu.cn (Y.-L. Du)

a b s t r a c t

Using first-principles GGA+U calculations, we systematically studied the electronic structure, mechanicalproperties and Debye temperature (hD) of the hexagonal e-Fe3N. The structural and elastic properties ofe-Fe3N were well described by considering the Fe 3d on-site Coulomb energy. The bulk modulus, shearmodulus, Young’s modulus, and Poisson’s ratio determined by GGA+U method are 191.5 GPa, 79.3 GPa,209.0 GPa and 0.32, respectively. They are in better agreement with the experimental values comparedwith GGA. e-Fe3N exhibits virtually isotropic properties in compression and slightly anisotropic in shear.The modest in-plane elastic anisotropy in a–c plane of e-Fe3N was also revealed. The Debye temperatureof e-Fe3N was calculated from the elastic moduli and sound velocities. It is indicated that the hD valueobtained by GGA+U method is in better agreement with experimental data.

� 2012 Elsevier B.V. All rights reserved.

1. Introduction

Iron nitrides exhibit interesting physical, chemical, mechanicaland magnetic properties and therefore deserve scientific interest.In particular, nitriding of iron and steel can improve the surfacehardness, tribological properties, fatigue endurance, and corrosionresistance [1–3]. Besides these technological aspects, the iron ni-trides have also attracted interest due to their excellent magneticproperties, which make them suitable for applications in magneticdevices [4–6]. Within this system, e-iron nitride is a hexagonalphase extending over a wide, temperature dependent domain ofexistence. This phase is generally represented by the Fe3N stoichi-ometric composition where nitrogen occupies alternate octahedralinterstices in an ordered manner [7]. The e-Fe3N phase often ap-peared in other iron nitrides and influenced the purity and mag-netic properties of the final product [8]. Experimental andtheoretical studies have been made on the crystal structures, theelectronic and magnetic properties of the e-phase [9–12]. Addi-tionally, the mechanical properties of e-Fe3N, such as bulk modulusand Young’s modulus, have been predicted by first-principles cal-culations [13,14]. However, the experimental value is much lowerthan expected [15]. The reason for these large discrepancies is stillnot clear up to now.

It is noted that the previous calculations on e-Fe3N phase areperformed within the local density approximation (LDA) or thegeneralized gradient approximation (GGA) to electron exchange

ll rights reserved.

Center of Materials Behaviory of Science and Technology,

.

and correlation [11–14]. However, the Fe atom in e-Fe3N containsmany electrons in partially filled d shells. The d electrons are inher-ently localized and the associated intra-atomic exchange and Cou-lomb energy terms are both numerous and large. To better accountfor the on-site d-electron correlations, we have adopted as a suit-able model the density functional theory (DFT)+U approach. TheDFT+U method has proven to be an efficient and reliable tool forcalculating the electronic structure of systems where the Coulombinteraction is strong. It was successfully applied to investigate theelectronic structure and elastic properties of other iron nitrides andthe calculated results are in good agreement with the experimentalvalues [16–18]. In the present work, a first-principles GGA+Uinvestigation was performed in order to reveal the electron corre-lation effects on the structural, electronic, elastic properties andDebye temperature of e-Fe3N. The bulk modulus, shear modulus,Young’s modulus, and Poisson’s ratio were determined. The elasticproperties of e-Fe3N were analyzed and discussed.

2. Method of calculations

The first-principles calculations are conducted by the WIEN2kcode using the full-potential linearized augmented plane-wavemethod (FLAPW) plus local orbital based on the density functionaltheory (DFT) [19]. The muffin-tin sphere radii (Ri) of Fe and N havebeen chosen as 1.82 and 1.61 a.u., respectively. The convergenceparameter RmtKmax, which controls the size of the basis set forthe wave function, was set to 8.0, where Rmt is the minimumspheres radii and Kmax is the maximal value of the reciprocal latticevector used in the plane wave expansion. In the DFT calculations, itis important to check the convergence of the crystal energy as a

Fig. 1. The total energy per unit cell with respect to the number of k-points.

342 Y.-J. Shi et al. / Computational Materials Science 67 (2013) 341–345

function of k-points sampling in the whole Brillouin zone (BZ). Inthe present work, we firstly make a convergence test to determinethe minimum number of k-points needed. Fig. 1 shows the varia-tion of ground state energies per e-Fe3N unit cell as a function ofk-points. At and above 1200 k-points, the changes in the groundstate energies were less than 30 lRy. Therefore, 1200 k-pointswere used in our calculations. The self-consistent cycle was con-verged to within 0.1 mRy/unit cell. In order to take into accountthe strong on-site Coulomb interaction (U) presented in the local-ized 3d electrons of Fe atom, the GGA+U method was performed.The effective Coulomb exchange interaction Ueff = U–J was usedhere for the calculation, where the U is the Coulomb-energetic costto place two electrons at the same site, and J is an approximation ofthe Stoner exchange parameter. The key to perform an accurateDFT+U calculation lies in selecting suitable value for Ueff. Two

Fig. 2. (a) The unit cell volume of e-Fe3N as a function of Ueff. (b) The bulk modulusof e-Fe3N as a function of Ueff.

methods are commonly used to determine this value [20]. In onecase, Ueff is treated as a tunable parameter, which is chosen to givethe best agreement with known properties of the material. In theother case, Ueff is determined by the constrained DFT calculations.However, the obtained Ueff values based on the above two methodsare of questionable accuracy due to the limitations of the methoditself. In the present work, the empirical method was used to deter-mine the suitable value of Ueff [21]. Fig. 2a and b illustrates the var-iation of equilibrium volume and bulk modulus of e-Fe3N as afunction of Ueff, respectively. The equilibrium volume increaseswith increasing strength of the on-site correlations. The calculatedvolumes are 82.99 Å3 for Ueff = 1.25 eV and 83.56 Å3 for Ueff = 1.5 -eV, respectively. Both of them are very close to the experimentalvalue of 83.26 Å3 [22]. The bulk modulus decreases with increasingstrength of the on-site correlations. The bulk modulus was calcu-lated to be about 178 GPa for Ueff = 1.5 eV, closest to the experi-mental value of 172 GPa [15]. Thus, these two facts takentogether, the equilibrium structure is best described withUeff = 1.5 eV, and the Ueff was determined to be 1.5 eV in ourcalculations.

The strain-energy method based on Hooke’s Law was employedto calculate the elastic constant matrix. An appropriate set ofstrains was applied to the undeformed unit cell lattice with the re-laxed structure. Then, the elastic constants were determined fromthe resulting change in total energy on the deformation. For thehexagonal crystals, there are five independent elastic constants,which are usually referred to as c11, c12, c13, c33, and c44. Five distor-tions involving (2c11 + 2c12 + 4c13 + c33), c11 + c12, c11 � c12, c33, andc44 were employed to determine all the elastic constants. The de-tailed procedures can be found in the literature [23].

3. Results and discussion

e-Fe3N crystallises in a hexagonal crystal structure (space groupP6322, No. 182) [22] illustrated in Fig. 3. It has six Fe atoms and twonitrogen atoms in one unit cell. Nitrogen atoms occupy only cor-ner-sharing octahedra. The Fe atoms are positioned at (x,0,0),and the N atoms at (1/3, 2/3, 1/4). Thus, the structure is definedby the lattice constants a and c, and the internal atomic positionx. The structure parameters for e-Fe3N were first optimized inWIEN2k to ensure that the calculations were based on equilibriumlattice constants. The experimental lattice parameters were takenas the basis for doing the optimization by energy minimizationfor the hexagonal symmetry. The optimized structural parametersby GGA and GGA+U methods are listed in Table 1, respectively, to-gether with the experimental data. It can be seen that the com-puted equilibrium lattice constants a and c by GGA+U are 4.7172and 4.3358 Å, respectively. Moreover, the values by GGA are

Fig. 3. Crystal structure of e-Fe3N.

Table 1Calculated lattice parameters and optimized atomic positions of e-Fe3N.

Method Unit cell (Å) Atom Site x y z

GGA a = 4.6578 Fe 6g 0.3267 0 0c = 4.3040 N 2c 1/3 2/3 1/4

GGA+U a = 4.7172 Fe 6g 0.3287 0 0c = 4.3358 N 2c 1/3 2/3 1/4

Experiment a = 4.6919a Fe 6g 1/3b 0 0c = 4.3670a N 2c 1/3 2/3 1/4

a Ref. [22].b Ref. [9].

Y.-J. Shi et al. / Computational Materials Science 67 (2013) 341–345 343

4.6578 and 4.3040 Å, respectively. The equilibrium lattice con-stants calculated by GGA+U are in better agreement with theexperimental values (a = 4.6919 Å, c = 4.3670 Å). The variationtrend of optimized internal position x values by GGA and GGA+Uare the same in contrast to experimental data. Note that the impor-tant thing is the deviation of x-coordinate from the ideal value of 1/3.Thus, the value obtained by GGA+U is closer to experimental data.

Fig. 4 shows the spin-resolved total density of states of e-Fe3Nand partial DOS of Fe 3d states using the GGA approximation.The lowest-lying states from about �18 to �16 eV correspond toN 2s orbitals. For the energy range from �9 to �6 eV, the N 2pstates hybridize with the Fe 3d states. The states just below theFermi level are dominated by Fe 3d bands. The Fermi level locatedexactly at a peak shoulder between two sharp peaks is mostlyoccupied by Fe 3d electrons. The calculated DOS profiles are consis-tent with the previous GGA calculation [14]. Fig. 5 gives the totalDOS of e-Fe3N calculated by using GGA+U method. The partialDOS of Fe 3d states was also shown in the figure. As can be seen,the profile of DOS by GGA+U is similar to that by GGA. The sharppeaks in the total DOS are derived from the localized Fe 3d states.The most striking difference is that the Fermi level shifts upwards

Fig. 4. DOS of e-Fe3N using GGA.

Fig. 5. DOS of e-Fe3N using GGA+U.

and lies at valley bottom with a minimum of the DOS between twosharp peaks. The value of density of states at Fermi level N(EF) fore-Fe3N calculated in this study by GGA is 5.50 (states/eV unit cell),while the value of N(EF) for e-Fe3N is 4.59 (states/eV unit cell) fromthe present GGA+U calculations. It is known that the relativestability correlates with the N(EF). The lower the N(EF) is, the morestable the corresponding phase is [24]. The lower value of N(EF) byGGA+U method indicates the stability of the system.

In order to reveal the electron correlation effects on the elasticproperties, the elastic constants for e-Fe3N were calculated by thestrain-energy method using GGA+U approximation. Table 2 liststhe calculated elastic constants by GGA+U and GGA, respectively.For comparison, we also listed the calculated elastic constants inprevious reports and the available experimental values of the elas-tic moduli. The mechanical parameters including bulk modulus,shear modulus, Young’s modulus, and Poisson’s ratio were calcu-lated. Bulk modulus and shear modulus were calculated usingthe Voigt–Reuss–Hill (VRH) approximation [25], while the Young’smodulus and the Poisson’s ratio were calculated by the followingformulae [26]:

Table 2Calculated and experimental elastic constants (cij in GPa), bulk modulus (B), shear modulus (G), Young’s modulus (E) (in GPa), and Poisson’s ratio (t) of e-Fe3N.

c11 c12 c13 c33 c44 B G E m

Calc. GGA (this work) 341.6 157.5 171.6 374.8 110.4 228.4 99.3 260.2 0.31GGA+U (this work) 277.7 162.0 134.6 305.5 104.6 191.5 79.3 209.0 0.32GGAa 313.72 141.82 131.50 329.17 105.65 196.2 95.8 247.2 0.29GGAb 276.0 94.5 120.6 325.3 109.1 170.9 97.3 245.3 0.26

Expt.c 172 78 203 0.32

a Ref. [13].b Ref. [14].c Ref. [15].

150 100 50 50 100 150a, GPa

200

100

100

200

c, GPa

Fig. 6. Direction dependent Young’s modulus of e-Fe3N.

344 Y.-J. Shi et al. / Computational Materials Science 67 (2013) 341–345

E ¼ 9BG3Bþ G

and m ¼ 3B� 2G2ð3Bþ GÞ ð1Þ

The elastic stability is a necessary condition for a crystal toexist. Assuming that both c12 and c33 are positive, the requirementsof elastic stability for a hexagonal crystal lead to the followingrestrictions on the elastic constants [27]:

c11 þ c12 �2c2

13

c33> 0 ð2Þ

c44 > 0 ð3Þ

c66 ¼c11 � c12

2> 0 ð4Þ

In the present work, it is obvious that the calculated elastic con-stants by GGA or GGA+U all fulfil the elastic stability criteria, whichmeans that e-Fe3N is mechanically stable. This conclusion is con-sistent with the previous study on the mechanical stability [14].

When the interactions of 3d electrons in Fe atom were consid-ered, the bulk modulus, shear modulus, and Young’s modulus werecalculated to be 191.5, 79.3 and 209.0 GPa, respectively. These val-ues are all smaller than that calculated by GGA, which are 228.4,99.3 and 260.2 GPa, respectively. It is found that the calculated val-ues of elastic moduli by GGA and GGA+U are all larger than exper-imental values. Obviously, compared to the previously calculatedelastic moduli and obtained values by GGA in this work, the pres-ent GGA+U values are in better agreement with the experimentalvalues when the effective Coulomb exchange interaction is takeninto account. Additionally, it is found that calculated bulk modulusis about 11% larger than that of experimental value even usingGGA+U approximation. The discrepancy between experimentaland calculated bulk modulus of e-Fe3N may be due to the temper-ature effect. The calculated bulk modulus is for T = 0 K in the usualfirst-principles calculations. The bulk modulus at room tempera-ture will be lower than that of 0 K, which attributes to the thermalexpansion.

The elastic anisotropy (A) of crystal is also important for evalu-ating mechanical properties of materials. The compression andshear anisotropic factors provide measures of the degrees ofanisotropy in atomic bonding in different crystallographic planes.For a transversely isotropic crystal (hexagonal), the elastic aniso-tropies in compressibility (Acomp) and in shear (Ashear) are givenby [23]:

Acomp ¼S33 þ 2S13

S11 þ S12 þ S13ð5Þ

Ashear ¼2c44

c11 � c12ð6Þ

where Sij denotes as elastic compliance constants and cij denotes aselastic constants, respectively. The computed values of Acomp andAshear by GGA+U are 0.998 and 1.808, respectively. It is indicatedthat there is very little compressibility anisotropy and slightly

anisotropy in shear. The analysis of feature about volume and c/adata in Figs. 1 and 2 from Ref. [15] indicates that c/a increases withincreasing pressure. In other word, the compressive strain in c axisis smaller than that in a axis when under the same pressure in bothdirections. Therefore, the compressibility observed experimentallyis not totally isotropic, and we can estimate that c33 > c11. Thus,our calculated results are in good agreement with the experimentaldata in Ref. [15]. Due to the hexagonal symmetry of e-Fe3N, thein-plane anisotropy in a–b plane is nonexistent, while a significantin-plane elastic anisotropy in a–c plane is usually revealed [28]. Anillustrative way of describing this in-plane elastic anisotropy is thedirectional dependence of Young’s modulus. The directionaldependence of Young’s modulus for a hexagonal crystal can bedetermined by the following equation [27]:

1E¼ 1� l2

3

� �S11 þ l43S33 þ l2

3 1� l23

� �2S13 þ S44ð Þ ð7Þ

where Sij are the elastic compliance constants and l3 is the directioncosine with respect to the c axis. Fig. 6 illustrates the directionaldependence of the e-Fe3N in the elastic modulus calculated usingthe compliance constants computed from the present GGA+U calcu-lation on the a–c plane. The variation in magnitude of the elasticconstant with respect to direction would indicate the overall anisot-ropy exhibited by e-Fe3N. The Young’s moduli are 170.1 and223.1 GPa in a and c axes, respectively. Thus, a modest in-planeelastic anisotropy in a–c plane is revealed.

Debye temperature (hD) is one of the most useful parameter tounderstand thermodynamic properties of solids such as specificheat, stability of lattices, melting points. There are various methodsto obtain the values of Debye temperature. A usual method for esti-mating the hD value can be obtained from the values of the elastic

Table 3Calculated and experimental longitudinal, transverse, average sound velocity (vl, vt,vm in m/s) and the Debye temperature (hD in K) from polycrystalline elastic modulusof e-Fe3N.

vt vl vm hD

Calc. GGA (this work) 3644 6946 4076 561GGA+U (this work) 3310 6409 3706 505GGAa 3656 6426 4064 555

Expt.b 3232 6081 3612 497

a Ref. [14].b The experimental values are obtained from elastic moduli in Ref. [15].

Y.-J. Shi et al. / Computational Materials Science 67 (2013) 341–345 345

constants and the wave velocities such as average wave velocity,traverse and longitudinal elastic wave velocities. The hD of e-Fe3Ncould be calculated by using the average sound velocity (vm)according the following equation [29]:

hD ¼hkB

3n4p

NAqM

� �� �1=3

mm ð8Þ

where h is Planck’s constant, kB is Boltzmann’s constant, n is thenumber of atoms in cell, NA is the Avogadro’s number, q is the den-sity and M is the molecular weight of the solid. The average soundvelocity in the polycrystalline material is given by [29]:

mm ¼13

2m3

tþ 1

m3l

� �� ��1=3

ð9Þ

where ml and mt are the longitudinal and transverse sound velocityobtained using the shear modulus G and the bulk modulus B fromNavier’s equation [29]:

mt ¼

ffiffiffiffiGq

sand ml ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiBþ 4

3G

� �=q

sð10Þ

The calculated and experimental sound velocity and Debyetemperature of e-Fe3N are given in Table 3. The calculated hD ofe-Fe3N are 561 K by GGA and 505 K by GGA+U, respectively. Theexperimental value of hD is obtained from elastic moduli in Ref.[15]. Therefore, the experimental result is reliable. The discrepancybetween theory and experiment is 12.88% for GGA and 1.61% forGGA+U, respectively. Obviously, the calculated result of Debyetemperature by GGA+U method may be more accurate than GGAfor e-Fe3N as compared with the experimental value.

4. Conclusions

In summary, the electronic structure, elastic properties and De-bye temperature of e-Fe3N were studied by first-principles calcula-tions using GGA+U approximation. The five independent elasticconstants were determined and the bulk modulus B, Young’s mod-ulus E, shear modulus G, and Poisson’s ratio t were computed. Thecrystal structure is demonstrated to be elastically stable. With the

help of first-principles GGA+U calculations, the present values ofelastic moduli are in better agreement with the experimental val-ues comparing with GGA calculations and previous reports. Inaddition, a modest in-plane elastic anisotropy in a–c plane is re-vealed. The calculated results of Debye temperature are indicatedthat the hD value obtained by GGA+U method is in better agree-ment with experimental data. Based on the present work, we pro-pose that the electron correlation effects in e-Fe3N cannot beignored.

Acknowledgments

This work has been supported by the NUST Research Fundingthrough Grant No. 2010ZDJH10, and the NUST Excellent Ph.D. Stu-dents Foundation.

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