Available at: http://publications.ictp.it IC/2008/022

United Nations Educational, Scientific and Cultural Organization and

International Atomic Energy Agency

THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

STRUCTURAL AND ELECTRONIC PROPERTIES OF LaN

Mohamed Ghezali Centre Universitaire de Béchar, Département de Sciences Exactes,

BP 417 Rue de Kanadissa, 08000 Bechar, Algeria,

Bouhalouane Amrani Laboratoire de Traitement de Surface et Sciences des Matériaux, Département de Physique,

Faculté des Sciences, Université des Sciences et de la Technologie d’Oran (U.S.T.O.), Oran 31000, Algeria,

Youcef Cherchab Centre Universitaire de Béchar, Département de Sciences Exactes,

BP 417 Rue de Kanadissa, 08000 Bechar, Algeria

and

Nadir Sekkal* Département de Physique-Chimie, Ecole Normale Supérieure de l’Enseignement Technique,

BP 1523, El M’Naouer, 31000 Oran, Algeria, Physia-Laboratory, BP 47 (RP), 22000 Sidi Bel Abbès, Algeria

and The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.

MIRAMARE – TRIESTE

May 2008

* Corresponding author: nsekkal@yahoo.fr and nsekkal@ictp.it On leave of absence from : Computational Materials Science Laboratory, Département de Physique, Institut de Sciences Exactes, Université de Sidi Bel Abbès, 22000 Sidi Bel Abbès, Algeria and Applied Materials Laboratory, Centre de Recherches (ex CFTE), Université de Sidi Bel Abbès, 22000, Sidi Bel Abbès, Algeria

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Abstract

Using two different first principles methods, the full potential linear augmented plane waves (FPLAPW) and a version of the full potential linear muffin-tin orbitals method (FPLMTO) which enables an accurate treatment of the interstitial regions, the structural properties of LaN are investigated. It is predicted the possibility of an additional local minimum in the wurtzite (B4) phase, approximately like for ScN and YN for which it was found a second minimum for the hexagonal A3 phase. A competition between the rocksalt (B1) and the wurtzite (B4) as the ground state phase is found depending on whether LDA (local density approximation) or GGA (generalized gradient approximation) is used. The electronic properties are also discussed.

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1 Introduction

Three important points make LaN an attractive topic for theoretical investigations: (i) it is a transition metal nitride, (ii) it can be a superconductor [1], and (iii) it presents many similarities with both ScN [2] and YN [3]. It is widely accepted that LaN crystallizes in the rocksalt (B1) phase with a possible phase transition to the CsCl (B2) under pressure [1] but has been found recently to crystallize also in other phases [4]. On the other hand, both ScN and YN have been predicted to have a second local minimum in the hexagonal A3 phase [3, 5] and it will be justified to check if this is not also the case with LaN since it appears from literature that this material has been synthetized just few times and that most of works which addressed the ground state phase problem have focused only on the rocksalt (B1) and the CsCl (B2) structures. One of the purposes of the present work is to investigate this problem and to calculate the pressure which induces the phase transition from B1 to B2. We have also investigated the electronic structure of LaN for both the ground state rocksalt (B1) phase and the zinc blende (B3) phase to check if it presents wide bandgaps as for ScN and YN [2, 3].

2 Methods

Two different full potential first principle methods have been employed in the present work. The FPLAPW [6] and the plane wave version of FPLMTO in which the non overlapping muffin tin spheres potential is Fourier transformed in the interstitial regions and hence treats the interstitial regions on the same footing with the core regions [7]. The exchange correlation energy of electrons is described in both the local density approximation (LDA) [8, 9] and the generalized gradient approximation (GGA96) using the parameterisation of Perdew et al [10, 11]. FLAPW and FPLMTO have been carried out using the WIEN2K [12] and the lmtART [7, 13] codes respectively. In FPLMTO, the non overlapping muffin tin spheres MTS potential is expanded in spherical harmonics inside the spheres of radius RMTS. In the interstitial region, the s, p and d basis functions are expanded in a number (NPLW) of plane waves determined automatically by the cut-off energies. In FPLAPW, the Kohn-Sham wave functions are expressed in spherical harmonics within spheres and in plane waves in the remaining space of the unit cell. The details of calculations for both methods are summarized in Table 1 and Table 2. Notice that the RMTS can be different for each atomic specie in different phases since the full potential ensures the no dependency of calculations on the RMTS.

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Table 1: Parameters used in the FPLMTO calculations. NPLW is the number of plane waves used in the interstitial regions, Ecut is the cut-off energy in Rydbergs, RMTS is in atomic units and K-Point represents the number of special K –points in the irreductibe BZ involved in the calculations.

NaCl (B1) CsCl (B2) Zinc blende(B3) Wurtzite (B4) Hexagonal (A3) β-Sn (A5) parameters

LDA GGA LDA GGA LDA GGA LDA GGA LDA GGA LDA GGA lmax 6 6 6 6 6 6 6 6 6 6 6 6 NPLW (s) 136 136 92 92 180 180 304 304 278 278 172 172 NPLW (p) 180 180 170 170 306 306 576 576 490 514 298 298 NPLW (d) 338 330 256 256 536 536 976 952 824 908 534 534 Total NPLW 2974 6566 3070 7152 5064 9984 11442 24780 9756 21700 5032 9864 RMTS (La) 2.884 2.989 2.914 2.933 2.657 2.705 2.665 2.741 2.74 2.773 2.672 2.709 RMTS (N) 2.004 1.993 2.133 2.147 1.846 1.879 1.852 1.905 1.904 1.927 1.857 1.882 Ecut (s) 9.91 9.54 9.30 9.18 11.7 11.3 10.6 10.2 9.76 9.46 11.7 11.3 Ecut (p) 13.2 12.7 12.8 12.6 15.7 15.1 15.3 14.7 13.9 13.8 15.7 15.1 Ecut (d) 19.8 17.5 18.6 18.4 23.3 22.5 22.0 21.9 20.3 20.2 23.3 22.5 K-Point (8,8,8) (8,8,8) (10,10,10) (10,10,10) (10,10,10) (10,10,10) (16,16,16) (16,16,16) (12,12,12) (12,12,12) (16,16,16) (16,16,16)

Table 2: Parameters used in the FPLAPW calculations. NPLW is the number of plane waves used in the interstitial regions. RMTS is in atomic units and K-Point represents the number of special K –points in the irreductibe BZ involved in the calculations.

NaCl (B1) CsCl (B2) Zinc blende (B3) Wurtzite (B4) Hexagonal (A3) β-Sn (A5) parameters

LDA GGA LDA GGA LDA GGA LDA GGA LDA GGA LDA GGA

lmax 10 10 10 10 10 10 10 10 10 10 10 10

RMT KMAX 8.5 8.5 8.5 8.5 8.5 8.5 8.5 8.5 8.5 8.5 8.5 8.5

Total NPLW 335 351 289 306 709 745 2718 2843 1338 1400 1790 1888

RMTS (La) 2.6 2.6 2.6 2.6 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4

RMTS (N) 2.2 2.2 2.2 2.2 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0

K –points (10,10,10) (10,10,10) (10,10,10) (10,10,10) (10,10,10) (10,10,10) (9,9,5) (9,9,5) (8,8,6) (8,8,6) (9,9,9) (9,9,9)

3 Results

The total energy was calculated for different values of the lattice constant, the lowest energy corresponding to the equilibrium. We have investigated the rocksalt (B1), the CsCl (B2), the zinc blende (B3), the β-Sn (A5), the wurtzite (B4) and also the hexagonal A3 structure that has recently

been found in MgO. The latter is nearly five-times coordinated [14] and has been theoretically confirmed to be stable in ScN and YN [3, 5]. A3 belongs to the hc class of hexagonal phases. Its direct Bravais primitive lattice vectors are ax= ( )0,2/3,2/1 ! a0, ay= ( )0,2/3,2/1 a0 and az=c0, a0 and c0 being the

two different lattice parameters, c0/a0 being the axial ratio. The primitive unit cell contains two La atoms at r1=0 and r2=(2/3,1/3, c0/2.a0) a0, and two N atoms at r3=(0, 0, u. c0/a0) a0 and r4=(2/3,1/3, c0(u+1/2)/a0) a0, u being the internal parameter (dimensionless). In Fig. 1, and for each method, we show the minimization curves for the different phases. Volume and energy are per single formula unit. Both LDA and GGA96 calculations show that the A3 phase is very improbable for LaN since its minimization curve lays far above those belonging to the other phases.

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However, our calculations show controversial results for the wurtzite (B4) phase. LDA combined with both FPLAPW and FPLMTO shows that the ground state configuration is the rockalt (B1) structure but with a minimum close to that of wurtzite (B4) phase, the difference between them is found to be small (0.030 eV/unit cell for LDA+FPLMTO and 0.096 eV/unit cell for LDA+FPLAPW). The same FPLAPW and FPLMTO methods combined with GGA96 gives wurtzite (B4) as the ground state phase. It is known that compared to experiment, GGA do not in general lead to more accurate structural or elastic parameters than LDA. We have checked our results by modifying our input parameters beyond what is necessary in such a way to improve our calculations but the results remained approximately the same. Thus, we are encouraged to predict that the wurtzite phase is at least a metastable phase for LaN and should present a second local minimum. This is also probable since a similar result has been obtained for YN and ScN [3, 5, 15]. It should be noted that similar controversial results have been found for different materials [16-18] and it was attributed to GGA which increases the correction to the total energy if the electron density inhomogeneity is great. For LaN, and according to experimental works, it is clear that the rock salt ground state phase given by LDA is the correct one but also that with both LDA and GGA, the equilibrium total energy for wurtzite is close to that of rock salt so that it is probably a metastable phase. The GGA calculations are here to strength this finding. Using these minimization curves, the equilibrium volume, the equilibrium lattice constant, the bulk modulus B and its derivative have been calculated by fitting to the Murnaghan equation of state [19]. The results are summarized in Table 3 where we notice a good agreement with literature (when available). To determine the most stable structure at finite pressure and temperature, we have used the enthalpy H=E+PV instead of the free energy G=E+PV-TS since we consider the temperature constant. Enthalpy was calculated for both B1 and B2 phases and from their curves crossing, the pressure giving the B1 to B2 phase transition is inferred (Table 4). Our FPLAPW+LDA value agrees well with literature while FPLMTO+GGA96 value is different.

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Table 3: The structural parameters of LaN calculated in the present work with the indicated methods. Data from other works are indicated when available. V0 is the equilibrium volume, a the lattice constant, B the bulk modulus and B’ is its pressure derivative). V0 is taken equal to a3.c/2 for β-Sn, a3/4 for both zinc blende and NaCl phases, a3 for the CsCl phase and 1/2.[a2.c.(3/4)½] for the two hexagonal phases. In all cases, the volume per unit formula is taken into account.

Approach a (Å) c / a u V0 B(GPa) B’ NaCl (B1)

FPLMTO 5.182 34.802 158.140 4.166 LDA FPLAPW 5.227 35.702 138.246 3.724 FPLMTO 5.282 36.851 124.453 4.448 GGA96 FPLAPW 5.322 37.685 121.099 4.012

Other works 5.17a, 5.311c, 5.307b

152a, 148d 3.657b

CsCl (B2) FPLMTO 3.089 29.473 139.605 4.239 LDA FPLAPW 3.112 30.138 151.558 4.355 FPLMTO 3.110 30.104 173.145 4.468 GGA96 FPLAPW 3.185 32.309 125.571 4.205

Other works 2.974a 282a, 114.18b

4.406b

ZnS (B3) FPLMTO 5.513 41.904 104.740 3.468 LDA FPLAPW 5.511 41.844 97.557 3.563 FPLMTO 5.611 44.184 98.709 3.439 GGA96 FPLAPW 5.623 44.447 90.354 3.062

Wurtzite (B4) FPLMTO 4.084 1.43 0.41 42.197 126.512 4.008 LDA FPLAPW 3.949 1.592 0.3883 42.461 102.365 3.622 FPLMTO 4.181 1.423 0.396 45.060 105.034 3.483 GGA96 FPLAPW 4.071 1.533 0.39679 44.782 92.448 3.731

Hexagonal MgO-h ( A3) FPLMTO 4.263 1.22 0.5 40.936 131.514 5.850 LDA FPLAPW 4.243 1.248 0.5 41.275 115.139 3.417 FPLMTO 4.332 1.22 0.5 42.947 120.039 4.014 GGA96 FPLAPW 4.298 1.259 0.5 43.329 99.544 6.037

β-Sn(A5) FPLMTO 3.900 1.414 41.952 104.740 3.502 LDA FPLAPW 3.895 1.4138 41.773 96.811 3.427 FPLMTO 3.970 1.414 44.246 98.856 3.303 GGA96 FPLAPW 3.964 1.429 44.506 87.026 2.936

a) Ref [1]. b) Ref [20]. c) Ref [21]. d) Ref [4].

Table 4: The B1 to B2 transition pressure PT calculated within the different methods. This work, LDA This work, GGA96 Approach

FPLMTO FPLAPW FPLMTO FPLAPW Other works

PT:B1 � B2 (GPa)

33.77 25.61 44.73 31.48 26.9a, 25.25b

a) Ref [1]. b) Ref [20].

It is more probable that LaN is a semiconductor with an indirect bangap than a semi-metal [22]. However, our calculations of the band structure of LaN in its equilibrium volume in the rocksalt (B1) phase give a semi-metallic behavior since the Fermi level Ef crosses the valence bands (VB). This is

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probably due to the underestimation of the gap by LDA because the obtained indirect negative bandgap is small, it is around 0.3 eV in FPLMTO+LDA (Fig. 2). The present result is similar to what has been found for YN [3] while it was more clear that ScN was a probably a semiconductor from FPLMTO+LDA calculations [2]. The incomplete cancellation of self interaction and also the discontinuity of the exchange correlation potential with respect to the number of electrons are both behind the inability of LDA to predict correctly the band gaps of semiconductors and insulators. In the wurtzite (B4) phase and in its equilibrium volume, LaN is found to be a semiconductor with a wide and indirect band gap of 0.99 eV, the top of VB being at K and the bottom of the conduction band (CB) at Γ (Fig. 3).

In Fig. 4, we show the band structure of LaN in its equilibrium volume in the zinc blende (B3) phase. We remark a large and indirect fundamental gap of about 1.07 eV with the top of the VB in X and the bottom of CB in Γ. Similar results have been found for ScN and YN, for which the top of VB is also at

X and the bottom of its CB is at W for ScN and W for YN with a larger indirect gap of 2.36 eV (ScN) and close to 2 eV for YN [2, 3].

4 Conclusion

In summary, we have benchmarked the performance of two of the most widely used first principles techniques (FPLAPW and FPLMTO) combined with both LDA and GGA96 for the calculation of the structural and phase stability properties of LaN. The most important result is the prediction of the possibility of a second local minimum in the wurtzite (B4) phase for LaN while the hexagonal A3 phase is found to be less stable. The electronic properties of bulk LaN show similarities with ScN and LaN.

Acknowledgments. One of the authors, N.S., thanks CTAPS of Irbid (Jordan) and the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy, for their hospitality in 2005 and 2007/2008. He also thanks M. Poropat and V. Kravtsov for help in ICTP and S.Y. Savrasov for his Mindlab software freely available. This work has been supported by the Algerian national research projects CNEPRU (J 3116/02/05/04, J 3116/03/51/05 and D05520060007).

References

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Phys. Stat. Sol. (b) 243 (2006) 2788.

[3] Youcef Cherchab, Bouhalouane Amrani, Nadir Sekkal, Mohamed Ghezali and Khadija Talbi,

Physica E 40 (2008) 606.

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[4] M. Hasegawa, K. Nivwa and T. Yagi, Solid State Commun. 141 (2007) 267. [5] N.Farrer and L.Bellaiche, Phys. Rev. B 66 (2002) 201203. [6] J.C.Slater, Adv. Quant. Chem. 1 (1994) 5564. [7] S.Y.Savrasov, Phys Rev B 54 (1996) 16470. [8] P.Hohenberg and W.Kohn, Phys. Rev. 136 (1964) B864. [9] W.Kohn and L.S.Sham, Phys. Rev. 140 (1965) A1133. [10] J.P.Perdew and Y.Wang, Phys. Rev. B 45 (1992) 13244. [11] J.P.Perdew, S.Burke and M.Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865. [12] P.Blaha, K.Schwarz, G.K.H. Madsen, D.Kvasnicka, J.Luitz, WIEN2k, An Augmented Plane

Wave Plus Local Orbitals Program for Calculating Crystal Properties, Vienna University of technology, Vienna, Austria, 2001.

[13] http://www.physics.ucdavis.edu/∼mindlab/

[14] S.Limpijumnong and W.R.L.Lambrecht, Phys. Rev. B 63 (2001) 104103.

[15] N.Takeuchi, Phys. Rev. B 65 (2002) 045204-1.

[16] L.A.Palomino-Rojas, M.Lo´pez-Fuentes, Gregorio H.Cocoletzi, Gabriel Murrieta, Romeo de

Coss and Noboru Takeuchi, Solid State Sci. (2007), doi:10.1016/j.solidstatesciences.2007.11.022.

[17] Joongoo Kang, E.-C Lee and K.J.Chang, Phys. Rev. B 68 (2003) 054106.

[18] N.Moll, M.Bockstedte, M.Fuchs, E.Pehlke, M.Sheffler, Phys. Rev. B 52 (1995) 2550. [19] F.D.Murnaghan, Proc. Natl. Acad. Sci. USA 30 (1944) 5390. [20] Y.O.Ciftci, K.C.Olakoglu, E.Deligoz, H.Ozisik, Materials Chemistry and Physics 108 (2008)

120. [21] R.W.G.Wyckoff, in Crystal Structures, (Wiley, New York, 1963), Vol. 1, 2nd Ed., p. 86. [22] C.Stampfl, W.Mannstadt, R.Asahi and A.J.Freeman, Phys. Rev. B 63 (2001) 155106.

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180 210 240 270 300 330-17091.50

-17091.48

-17091.46

-17091.44

-17091.42

-17091.40

FPLMTO+LDA

To

tal

En

erg

y (

Ry

d)

Volume (a.u)3

180 210 240 270 300 330 360

-17105.02

-17105.00

-17104.98

-17104.96

-17104.94

-17104.92

-17104.90

-17104.88

FPLMTO+GGA

To

tal

En

erg

y (

Ry

d)

Volume (a.u)3

180 210 240 270 300

-17091.52

-17091.48

-17091.44

FPALPW+LDA

To

tal

En

erg

y (

Ry

d)

Volume (a.u)3

180 210 240 270 300 330

-17105.04

-17105.02

-17105.00

-17104.98

-17104.96

-17104.94

-17104.92

Volume (a.u)

3

Tota

l Energ

y (

Ryd)

FPLAPW+GGA

-5

0

5

10

Rock salt

Ef

L K XWX !!

En

erg

y (

eV

)

Fig. 1: Calculated total energy versus relative volume for LaN within the different methods. The stars are for the B1 phase, the filled squares for B2, the open circles for B3, the filled diamonds are for B4, the open diamonds are for A3 and the filled triangles are for A5. The fit to the Murnaghan equation of state is presented with full lines.

Fig. 2: The FPLMTO+LDA band structure of LaN in the rocksalt (B1) phase for the equilibrium volume.

9

-5

0

5

10

15

20

L

Wurtzite

Ef

M A K!K "M

En

erg

y (

eV

)

-5

0

5

10

15

20Zinc blende

Ef

L K XWX !!

En

erg

y (

eV

)

Fig. 3: The FPLMTO+LDA band structure of LaN in the wurtzite (B4) phase for the equilibrium volume.

Fig. 4: The FPLMTO+LDA band structure of LaN in the zinc blende (B3) phase for the equilibrium volume.