Journal of Nuclear Materials 410 (2011) 46–51

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Journal of Nuclear Materials

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Structural, electronic, mechanical, and thermodynamic properties of UN2:Systematic density functional calculations

Yong Lu a,b, Bao-Tian Wang b,c, Rong-Wu Li a, Hong-Liang Shi b, Ping Zhang b,d,⇑a Department of Physics, Beijing Normal University, 100875, People’s Republic of Chinab LCP, Institute of Applied Physics and Computational Mathematics, Beijing 100088, People’s Republic of Chinac Institute of Theoretical Physics and Department of Physics, Shanxi University, Taiyuan 030006, People’s Republic of Chinad Center for Applied Physics and Technology, Peking University, Beijing 100871, People’s Republic of China

a r t i c l e i n f o

Article history:Received 30 August 2010Accepted 30 December 2010Available online 6 January 2011

0022-3115/$ - see front matter � 2011 Elsevier B.V. Adoi:10.1016/j.jnucmat.2010.12.308

⇑ Corresponding author. Address: LCP, Institute oftational Mathematics, P.O. Box 8009, Beijing 100088Tel.: +86 10 62014411x2208.

E-mail address: zhang_ping@iapcm.ac.cn (P. Zhang

a b s t r a c t

We have performed a comparative study of UN2 and b-U2N3using the generalized gradient approximation(GGA) and the GGA + U approaches based on the density functional theory (DFT). The lattice parametersobtained from the GGA + U calculations can be ameliorated appreciably, however, the density of states(DOS) is insensitive to Hubbard U for both compounds. Our Bader analysis shows that the effectivecharges increase with decreasing U:N ratio (from UN to UN2). The f electrons in UN2 and b-U2N3 seemless localized than that in UN phase. The effects of Hubbard U on mechanical properties is evident, whilethe phonon dispersion depends weakly on the changes of Hubbard U. Based on our phonon dispersiondata, the lattice vibration energy, thermal expansion, and specific heat are obtained by utilizing the quasi-harmonic approximation (QHA).

� 2011 Elsevier B.V. All rights reserved.

1. Introduction

Actinide elements and compounds possess particular interest-ing physical behaviors due to the 5f states and have always at-tracted extensive attention because of their importance innuclear fuel cycle [1–3]. In the case of uranium nitrides, uraniummononitride (UN), uranium dinitride (UN2), and diuranium trinit-ride (b-U2N3) have been found to be three stable U–N phases. Theyhave some superior thermal physical properties compared to oxi-des fuels, such as high melting point, high thermal conductivity,and high metal density [4,5] as well as good compatibility withthe sodium coolant. For that reason they are considered alternativematerials for potential applications in Generation-IV reactors [6],and a lot of experimental and theoretical studies have beenperformed.

For actinide compounds, part of the 5f electrons are neitherfully localized nor completely itinerant [7]. There is strong Cou-lomb correlation among the partially filled 5f electrons. This factmakes the conventional DFT schemes fail to capture the electroniclocalization effects, and will lead to produce unreasonable groundstate properties, such as the insulator nature and the magneticconfigurations [8–11]. In order to overcome these shortcomings,various approaches have been proposed and applied to actinium

ll rights reserved.

Applied Physics and Compu-, People’s Republic of China.

).

nitrides and oxides, such as calculations involving self-interactioncorrection (SIC) [12,13], hybrid exchange–correlation functionals[14,15], the combination of dynamical mean-field theory (DMFT)with LDA [16,17], as well as the DFT + U approach [18–20]. Espe-cially, the DFT + U formalism has been extensively used for a widepanel of correlated materials. However, the introduction of param-eter U induces an increase in the number of metastable stateswhich makes the convergence to the ground state difficult [21].Consequently, different starting points of the calculation will leadto the discrepancy of final state reached by the self-consistentalgorithm and its associated total energy. While this problem canbe solved by performing a procedure that is based on the monitor-ing of the occupation matrices of the correlated orbitals. Such aprocedure can unequivocally determine the ground state by com-paring the energies of all energy minima, as presented by Jomardet al. [22] and Dorado et al. [23]. In our previous work, we havesuccessfully investigated the structural, electronic and thermody-namic properties of UN by using the LDA/GGA + U formalism withU = 2 eV. In this work, we will report on a comparative study of theconventional GGA and the GGA + U calculations for UN2 and b-U2N3.

We have performed lattice parameter, electronic structure,mechanical and thermodynamic properties calculations of UN2

and b-U2N3 using GGA and GGA + U approaches. The lattice param-eters obtained from the GGA + U calculations are amelioratedappreciably by choosing Hubbard U parameter around 2 eV. How-ever, the density of states (DOS) is insensitive to Hubbard U forboth compounds. Our Bader analysis shows that the effective

Y. Lu et al. / Journal of Nuclear Materials 410 (2011) 46–51 47

charges increase with decreasing U:N ratio (from UN to UN2). Sincethat the f electrons in UN2 and b-U2N3 seem less localized than thatin UN phase. Our calculated elastic constants confirm that UN2 ismechanically stable. The variety of C11 and C12 with Hubbard U issmall, while the C44 decreases clearly with increasing U. The calcu-lated phonon dispersions of UN2 with and without turning on Hub-bard U are also displayed, and the discrepancy between them isvery small. Furthermore, we predicted the lattice vibration energy,thermal expansion, and specific heat by using the QHA based onthe phonon DOS. The calculated specific heat capacity CP withinGGA + U is in good agreement with the all-electron results, andthe discrepancy between them can be ignored, indicating thatthe effects of Hubbard U on thermodynamic properties isinsignificant.

The paper is organized as follows: The computational details arebriefly introduced in Section 2. The calculated results are presentedand discussed in Section 3. Finally, a summary of this work is givenin Section 4.

2. Computational methods

Our present first-principles DFT total energy calculations wereperformed by using the Vienna ab initio simulations package(VASP) [24,25] with the projected-augmented-wave (PAW) pseud-opotentials [26] and plane waves. The exchange and correlation ef-fects were described within GGA [27], in which the uranium6s26p66d25f27s2 and nitrogen 2s22p3 electrons were treated as va-lence electrons. The strong on-site Coulomb repulsion among thelocalized U 5f electrons were described by using the formalismdeveloped by Dudarev et al. [18–20]. The electron wave functionwas expanded in plane waves up to a cutoff energy of 500 eV. Fur-ther increase of the value of cutoff energy does not change the re-sults. We adopt the single unit cell to perform the structural,electronic and mechanical properties calculations. For UN2, weused a Monkhorst–Pack [28] 9 � 9 � 9 mesh in Brillouin zone(BZ) integration and for b-U2N3, a 9 � 9 � 6 mesh was adopted.Since we have performed numerous convergence studies on deter-mining the influence on the total energy of the k-point mesh, andthese k-point meshes were sufficient to get results converged toless than 0.001 eV per atom. The corresponding electronic DOSswere obtained with 15 � 15 � 15 k-point mesh and 11 � 11 � 9k-point mesh, respectively. In order to calculate the elastic con-stants for UN2, we enforced small strains on the equilibrium cellaccording to the following law [29]:

EðV ; eiÞ ¼ EðV0Þ þ VX6

i¼1

riei þV2

X6

i¼1

X6

j¼1

Cijeiej þ . . . ð1Þ

where E(V0) and V0 are the total energy and volume of the equilib-rium cell without strains, respectively, E(V,ei) is the total minimumenergy of the distorted cell under the strain vector

Table 1Calculated lattice constants a0 or c0 (Å), volume V0 (Å3) and bulk modules B (GPa) for UNexperimental and other theoretical values are also listed.

Compound Property GGA GGA + U

UN2 a0 5.284 5.308B 252.4 253.5

b-U2N3 a0 3.695 3.725c0 5.779 5.758V0 68.33 69.19B 203.2 209.2

a Ref. [32].b Ref. [36].c Ref. [31].d Ref. [33].

� = (e1,e2,e3,e4,e5,e6), and Cij are the elastic constants. For the fluoritestructure, there are only three independent elastic constants, C11,C12, and C44. Therefore, we used three different strains listed inthe followings to calculate them: �1 = (d,0,0,0,0,0), �2 = (d,d,0,0,0,0),�3 = (0,0,0,d,d,0). The strain amplitude d is varied in steps of 0.02from �0.06 to 0.06.

3. Results and discussions

3.1. Atomic and electronic structures of UN2 and b-U2N3

At ambient conditions, UN2 crystallizes in a CaF2-type ionicstructure with space group Fm�3m (No. 225) and b-U2N3 in a hexag-onal b-type structure with space group P�3m1 (No. 164). Throughcomparing the total energies of the antiferromagnetic (AFM), ferro-magnetic (FM), and nonmagnetic (NM) phases within the GGA andGGA + U approaches, we find that the ground state of UN2 is in aNM configuration and b-U2N3 in a FM. The theoretical equilibriumlattice constants and bulk modulus B are obtained by fitting thethird-order Brich–Murnaghan equation of state (EOS) [30]. The re-sults are presented in Table 1. For comparison, the correspondingexperimental values and other theoretical data are also listed. Ascan be seen from Table 1, the GGA approach predicts the latticeparameter of UN2 to be 5.284 Å, which is somewhat smaller thanthe experimental value of 5.31 Å [31]. When we turned on theHubbard U parameter to around 2 eV, the lattice parameter a0 isenhanced to 5.308 Å, which is in good agreement with the experi-mental result. Within GGA approach, our calculated B equals to252.4 GPa, and the GGA + U with U = 2 eV gives the similar result,which is 253.5 GPa. The both results lie within the range of valuesobtained by Kotomin et al. [32] for a suite of different functionals(from 235.8 to 264.6 GPa). As for b-U2N3, the GGA approach resultsof a0 (c0) and volume V0 are equal to 3.695 Å (5.779 Å) and68.33 Å3, respectively. Within GGA + U formalism, the correspond-ing values are a0 = 3.725 Å (c0 = 5.758 Å) and V0 = 69.19 Å3, respec-tively. All these values are consist with the correspondingexperimental data, i.e., a0 = 3.700 Å (c0 = 5.825 Å) and V0 =69.06 Å3 [33]. The melioration of Hubbard U is puny. Similarly,the bulk modulus B are predicted to be 203.2 GPa within GGA ap-proach and 209.2 GPa within GGA + U with U = 2 eV, and the vari-ation is small.

In order to carry out a further analysis of the electronic struc-tures, we present the DOS and orbital-resolved site-projectedDOS (PDOS) of UN2 and b-U2N3 in Fig. 1a and b, respectively. Wenote that the DOSs are insensitive to the choice of Hubbard U. Ascan be seen from Fig. 1a, the occupied DOS near the Fermi levelis featured by the five well-resolved peaks for UN2. The two near�0.5 and �1.8 eV are principally U 5f and N 2p in character, whichcorrespond to the hybridization between U 5f and N 2p orbits,while the other three peaks respectively near �2.8, �3.4 and�4.2 eV are mainly the mixture of N 2p- and U 6d-orbital. The

2 and b-U2N3 within the GGA and GGA + U with Hubbard U = 2 eV. For comparison,

PAW-GGA US-GGA LCAO Expt.

5.259a 5.256a 5.26b 5.31c

264.6a 235.8a 270b

3.647a 3.741a 3.66b 3.700d

5.802a 5.722a 5.6b 5.825d

66.83a 69.35a 64.96b 69.06d

208.9a 153.6a 232b

(a) (b)

Fig. 1. Total and projected DOSs for UN2 and b-U2N3 by the GGA + U method.

Fig. 2. Valence charge densities of UN2 in (1 �10) plane and b-U2N3 in (110) plane within GGA + U with U = 2eV.

Table 2Bader effective atomic charges of UN2 and b-U2N3. Our previous results of UN (Ref.[11]) and the results calculated within LCAO (Ref. [36]) and PW91 (Ref. [32]) are alsolisted for comparison.

Methods Bader charge UN b-U2N3 UN2

GGA QU +1.65 +2.19 +2.44QN -1.65 �1.43,�1.52 �1.22

GGA + U QU +1.71 +2.24 +2.55QN �1.71 �1.465,�1.55 �1.275

LCAO QU +1.58 +2.29 +2.78QN �1.58 �1.46,�1.55 �1.39

PW91 (PAW) QU +1.66 +2.08 +2.48QN �1.66 �1.34,�1.50 �1.24

48 Y. Lu et al. / Journal of Nuclear Materials 410 (2011) 46–51

amplification of band gap with Hubbard U is very small, from0.35 eV with U = 0 eV to 0.75 eV with U = 4 eV. This makes UN2

crystal exhibit semiconducting nature, in agreement with the re-sults of all-electron calculations [34]. For conduction band, the to-tal DOS is mainly featured by U 5f-orbital in a wide band range.With the transformation of phase state, in the b-U2N3 phase, theU 5f-orbital still occupies the most conduction band near the Fermilevel, while the valence band is mainly marked by the N 2p-orbitalwith somewhat mixture of U 6d- and 5f-orbital. Similar to UN, theb-U2N3 shows a clear metallic conductivity too.

Also, we plot in Fig. 2 the valence charge density to analyze thechemical bonding nature of UN2 and b-U2N3. In Fig. 2, the bondingbridges between U and N atoms can be clearly seen and the bond-ing symmetry of UN2 is more evident than that of b-U2N3. This kindof symmetry can lead to high bulk modulus, as shown in the abovediscussion. Obvious covalent nature of chemical bonds can befound in both UN2 and b-U2N3. Besides, we also calculated effective(Bader) charges [35] for these two uranium nitrides within theGGA and GGA + U approaches. The calculated valance charges arelisted in Table 2, together with the LCAO [36] and PW91 [32] re-sults for comparison. Within GGA approach, the U ions in UN2

and b-U2N3 can be represented as U2.44+ and U2.19+, respectively.And the corresponding results for GGA + U are U2.55+ and U2.24+,respectively. Clearly, the effective charges of U ions increase as afunction of decreasing U:N ratio, indicating that the ionicity inUN2 and b-U2N3 is somewhat stronger than that in UN. This con-

clusion is in qualitative agreement with previous LCAO and PAWresults.

3.2. Mechanical properties

Our calculated elastic constants C11, C12, and C44 for UN2 withinGGA and GGA + U approaches are listed in Table 3. As can be seenfrom Table 3, the calculated elastic constants C11 and C44 decreasewith increasing the Hubbard U, while the C12 improves by enlarg-ing the U parameter. Since there are no corresponding experimen-

Table 3Calculated elastic constants, bulk modulus B, shear modulus G, Young’s modulus E,Poisson’s ratio t, and anisotropic factor A for Fm�3m UN2.

GGA C11 C12 C44 B G E t A

U = 0 495.4 137.3 65.6 256.7 99.4 264.2 0.328 0.366U = 2 488.2 140.5 55.3 256.4 89.4 240.2 0.344 0.318U = 4 483.8 146.2 41.3 258.7 75.7 207.0 0.367 0.245

Y. Lu et al. / Journal of Nuclear Materials 410 (2011) 46–51 49

tal results so far, we can only give the dependence of elastic con-stants on Hubbard U and the reference value. At U = 2 eV, the valueof C11, C12, and C44 are 488.2 eV, 140.5 eV and 55.3 eV, respectively.All the calculated elastic constants satisfy the following mechani-cal stability criteria [29] of cubic structure:

C11 > 0;C12 > 0;C11 > C12; ðC11 þ 2C12Þ > 0: ð2Þ

Under the Voigt approximation [37], the effective bulk modulus BV

and shear modulus GV for cubic phase can be expressed asBV ¼ 1

3 ðC11 þ 2C12Þ and GV ¼ 15 ðC11 � C12 þ 3C44Þ, respectively. Under

Reuss approximation [38], the Reuss bulk modulus BR and Reussshear modulus GR are expressed as BR ¼ 1

3 ðC11 þ 2C12Þ andGR ¼ 5ðC11�C12ÞC44

4C44þ3ðC11�C12Þ, respectively. Hill [39] proved that the Voigt and

Reuss equations represent upper and lower limits of the true poly-crystalline materials, and recommended that the bulk modulus Band shear modulus G were arithmetic averages of Voigt and Reusselastic modulus, i.e., B ¼ 1

2 ðBR þ BV Þ and G ¼ 12 ðGR þ GV Þ. From that,

the Young’s modulus E and Poisson’s ratio m can be given by,E ¼ 9BG

3BþG and m ¼ 3B�2G2ð3BþGÞ. Using these functions, the bulk modulus B,

shear modulus G, Young’s modulus E, and Poisson’s ratio m for UN2-

have been calculated and tabulated in Table 3. The variety of bulkmodulus B is small with amplifying the Hubbard U. At U = 2 eV,the bulk modulus B deduced by elastic constants is 256.4 GPa,which is consistent well with that obtained by EOS fitting. Mean-while, the relatively larger value of bulk modulus B for UN2 com-pared with UO2 (207 GPa) [10] and UC2 (216 GPa) [40] indicatesthat the resistance to fracture of UN2 is better than UO2 and UC2.The calculated Poisson’s ratio m = 0.344 with U = 2 eV is larger than

Fig. 3. Calculated phonon dispersion curves (left panel) and corresponding phonon DOS (curves are also presented.

0.18 of UN [41]. The larger value of Poisson’s ratio indicates that thestability of UN2 is relatively weaker than UN. Since microcracks inmaterials can be easily induced by significant elastic anisotropy, itis important to evaluate anisotropic factors to understand theirmechanical durability. With a cubic system, the anisotropic factorA can also be calculated from elastic constants, i.e., A ¼ 2C44

C11�C12. Our

calculated value of A decrease with enhancing the Hubbard U. AtU = 2 eV, the corresponding value is equal to 0.318, indicating thatUN2 is elastically anisotropic. The factor A is equal to 1.0 for isotro-pic crystals while a value different from 1.0 indicates elasticanisotropy.

3.3. Thermodynamic properties of UN2

We employ the Hellmann–Feynman theorem and the directmethod [42] to calculate the phonon dispersion curves of UN2. Inthe BZ integration, the 3 � 3 � 3 Monkhorst–Pack k-point meshis used for the 2 � 2 � 2 UN2 supercell containing 96 atoms. Wedisplace two atoms (one U and one N atoms) from their equilib-rium positions to calculate the Hellmann–Feynman forces, andthe amplitude of all the displacements is 0.03 Å. The calculatedphonon dispersion curves along C�X � K � C � L � X �W � Ldirections for UN2 are displayed in Fig. 3 within GGA and GGA + Uapproaches. For UN2, nine phonon modes exist in the dispersionrelations. As can be seen from Fig. 3, there are significant LO–TOsplitting after including Born effective charge of UN2, the value ofwhich are Z�N = � 4.904 and Z�U = 9.048 with U = 0 eV, andZ�N = �4.524 and Z�U = 9.807 with U = 2 eV. The both phonon disper-sions with U = 0 eV and U = 2 eV are similar to each other, and thediscrepancy between them is insignificant. The total and projectedphonon DOS for UN2 are also plotted in Fig. 3. Since the phononDOSs for U = 0 and 2 eV are nearly the same, we plot here onlythe one with U = 2 eV. Because of the fact that the uranium atomis heavier than nitride atom, the phonon DOS is split into twoparts: one part in the range of 0–7 THz where the vibrations of ura-nium atoms are dominant; the other part is in the domain of 7–22 THz where the vibrations mainly come from nitride atoms.

right panel) within GGA + U scheme at U = 2eV for UN2. The GGA phonon dispersion

Fig. 6. Heat capacity of UN2 with U = 2 eV within GGA + U. Experimental data fromCousell et al. [44] and Tagawa [45], and theoretical results from Weck et al. [34] andChevalier et al. [46] are also displayed for comparison.

50 Y. Lu et al. / Journal of Nuclear Materials 410 (2011) 46–51

The phonon dispersion illustrates the stability of UN2, and furtherindicates that our following thermodynamic calculations arereliable.

The thermodynamical quantities such as the lattice vibrationenergy, thermal expansion, and specific heat are calculatedthrough QHA [43] within GGA + U approach. In our previous studyof UN [11], we have systematically given the calculation method.For present study of UN2, the energy of electron excitation is ig-nored in calculation of the Helmholtz free energy due to the insu-lator nature for UN2. The calculated free energy versus volumecurves for UN2 is plotted in Fig. 4, from which the volume expan-sion upon the temperature increase can be derived. Fig. 5 showsthe temperature dependence of the bulk modulus B and its valuedecreases with increasing the temperature T. The calculated spe-cific heat capacity CP of UN2 is displayed in Fig. 6, together withthe all-electron calculation results by Weck et al. [34]. The exper-imental results [44,45] and theoretical data [46] for UN phase arealso listed for comparison. One can see that the CP of UN2 is higherthan that of UN up to around 2000 K. This can be attributed to thefact that the percentage of N element in UN2 is bigger than that inUN. However, as the T increases, in the high-temperature domain,the U 5f electron begin to show its great effect on CP in metallic UN(corresponding to an electron heat coefficient c = 49.2 mJ mol�1

K�1 [47]), which largely enhances the value of CP. Our calculatedresult is in agreement with the all-electron methods conclusion,and the effect of Hubbard U on CP is insignificant. As we have ob-

Fig. 4. Dependence of the Helmholtz free energy F(T,V) on crystal volume at varioustemperatures and the locus of the minimum of the free energy for UN2.

Fig. 5. Temperature dependence of the bulk modulus for UN2.

served above that the effects of Hubbard U on phonon dispersionand phonon DOS are very small.

4. Conclusion

In summary, we have performed systematic first-principles cal-culations on the structural, electronic, mechanical, and thermody-namic properties of UN2 using the GGA and GGA + U approaches.With the Hubbard U correction, the atomic structures, includinglattice parameters and bulk modulus, of UN2 and b-U2N3 can bereasonably reproduced with respect to experimental results. Calcu-lated DOSs show the important role that the 5f electrons play in theconduction band of UN2 and b-U2N3. The 5f electrons in UN2 and b-U2N3 seem less localized than that in UN phase, since our Baderanalysis shows that the effective charges increase with decreasingU:N ratio (from UN to UN2). The effects of Hubbard U on elasticconstants are evident, especially on C44. Our phonon dispersion re-sults explicitly indicate that UN2 is thermodynamically stable, andthe Hubbard U parameter contributes little to the phonon disper-sion and phonon DOS. Finally, we calculated the thermodynamicalquantities such as the lattice vibration energy, thermal expansion,and specific heat using QHA. The effect of Hubbard U on CP is insig-nificant. By comparison the CP curves of UN2 and UN, we find thatthe U 5f electrons do a great contribution to the thermal properties.We expect that these calculated results will be useful for the appli-cation of uranium nitrides in the Generation-IV reactor and nuclearindustry.

Acknowledgment

This work was supported by NSFC under Grant Nos. 90921003,60776063 and 51071032.

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