Structural phase stability and elastic properties of lanthanum monochalcogenides at high pressure

Journal of Molecular Structure: THEOCHEM 777 (2006) 5–10 www.elsevier.com/locate/theochem

Structural phase stability and elastic properties of lanthanum monochalcogenides at high pressure A. Bouhemadou b

a,*

, R. Khenata b, M. Maamache

a

a Department of Physics, Faculty of Science, University of Setif, 19000 Setif, Algeria Laboratoire de Physique Quantique et de Mode´lisation Mathe´matique (LPQ3M), De´partement de Technologie, Centre Universitaire de Mascara, 29000 Mascara, Algeria

Received 9 July 2006; accepted 1 August 2006 Available online 17 August 2006

Abstract Ab initio calculations have been performed on the structural, elastic and high pressure properties of cubic inter-metallic LaS, LaSe and LaTe compounds. The Kohn–Sham equations were solved by applying the full-potential (linear) augmented plane wave method plus local orbitals (FP-(L)APW+lo) as implanted in the WIEN2K package. In this approach, the generalized gradient approximation (GGA) and the local density approximations (LDA) are used for the exchange-correlation (XC) potential. Results are given for lattice constant, bulk modulus and its pressure derivative. The pressure transition at which these compounds undergo structural phase transition from NaCl-type (B1) to CsCl-type (B2) structure are calculated and compared with previous calculations and available experimental data. The elastic constants at room conditions and under pressure in both B1 and B2 phases are also calculated.  2006 Elsevier B.V. All rights reserved. Pacs: 62.20. Dc; 64.70 Kb; S1.5 Keywords: Lanthanum monochalcogenides; FP-(L)APW+lo; High pressure; Phase transition; Elastic constants

1. Introduction High pressure research on structural phase transformations and behavior of materials under compression based on their calculations or measurements have become quite interesting in the recent few years as it provides insight into the nature of the solid state theories, and determine the values of fundamental parameters [1]. The lanthanum monochalcogenides LaX (X = S, Se, Te) are some of the structurally simplest materials which have been extensively studied, both experimentally as well as theoretically [2–9]. However, the high-pressure structural properties of lanthanum monochalcogenides have not been studied much. The lanthanum monochalcogenides like the most of the rare-earth monochalcogenides crystallize in NaCl-type structure (B1) at ambient conditions with space *

Corresponding author. Fax: +213 36 92 52 01. E-mail address: [email protected] (A. Bouhemadou).

0166-1280/$ - see front matter  2006 Elsevier B.V. All rights reserved. doi:10.1016/j.theochem.2006.08.031

group Fm3m (225) [10]. The rare-earth atom is positioned at (0, 0, 0) and the chalcogen at (1/2, 1/2, 1/2). Under pressure, the most of the rare-earth monochalcogenides compounds have been found to undergo a first-order phase transition from the sixfold-coordinated NaCl structure to the eightfold-coordinated CsCl-type structure (B2) with the space group symmetry Pm3m (221) in which the chalcogen atom is positioned at (1/2, 1/2, 1/2). X-ray diffraction measurements under pressure performed on LaS have shown that this compound undergoes a phase transformation from its ambient B1 phase to B2 phase around 25 GPa in a silicon oil pressure medium [6]. However, no high-pressure experimental studies of LaSe and LaTe are known at present. The aim of this paper is to give a comparative and complementary study of the structural, elastic and phase transition to both experimental and theoretical works for lanthanum monochalcogenides by using the full-potential (linear) augmented plane wave plus local orbitals method

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A. Bouhemadou et al. / Journal of Molecular Structure: THEOCHEM 777 (2006) 5–10

(FP-(L)APW+lo). This paper is organized as follows: In Section 2, we briefly described the computational parameters used in the calculations. In Section 3, we discussed the calculated results of structural, phase transition and elastic properties. Finally, we give a brief conclusion. 2. Calculation method The first-principle calculations are performed by employing a full-potential (linear) augmented plane wave plus local orbital (FP-(L)APW+lo) [11–13] method, based on density functional theory [14,15] and implemented in the most recent version of the WIEN2K package [16]. The exchange-correlation potential for structural properties was calculated by the local-density approximation (LDA) [17] with and without generalized gradient approximation (GGA) based on Perdew et al. [18]. In the FP-(L)APW+lo method, the unit cell is divided into non-overlapping spheres centered at atomic sites (muffintin spheres) of radius RMT and an interstitial region. In the muffin-tin spheres, the Kohn–Sham wave functions are expanded in a linear combination of radial functions time spherical harmonics, and in the remaining space of the unit cell a plane wave basis set is chosen. The basis set inside each muffin-tin sphere is split into core and valence subsets. The core states are treated within the spherical part of the potential only and are assumed to have a spherically symmetric charge density totally confined inside the muffin-tin spheres. The valence part is treated within a potential expanded into spherical harmonics. The valence wave functions inside the spheres are expanded up to lmax = 10. The La (5s25p65d16s2), S (3s23p4), Se (3d104s24p4) and Te (4d105s25p4) states are treated as valence electrons. The RMT are taken to be 2.8, 2.4, 2.6 and 2.8 atomic units (a.u.) for La, S, Se and Te, respectively, in the B1 phase. In the B2 phase, we have adopted the values of 3.0, 2.6, 2.8 and 3.0 Bohr for La, S, Se and Te, respectively. A plane wave cut-off Kmax = 3.5 a.u.1 is chosen for the expansion of the wave functions in the interstitial region. The k integrations over the Brillouin zone is performed up to 8 · 8 · 8 MonkorstPack [19] mesh for both B1 and B2 phases. The self-consistent calculations are considered to be converged when the total energy of the system is stable within 105 Ry. 3. Results and discussion 3.1. Structural and phase transition properties In order to calculate the ground states properties of LaS, LaSe and LaTe, the total energies are calculated in both phases B1 and B2 for different volumes around the equilibrium cell volume V0. The plots of calculated total energies versus reduced volume for these compounds in both phases are given in Fig. 1a. It is seen from these E–V curves, that the B1 structure is the most stable at room conditions, which is consistent with exper-

Fig. 1. (a) Variation of total energies as a function of volume of unit cell in LaS, LaSe and LaTe compounds in both structures, B1 and B2. The solid curve is a least-squares fit to Murnaghan’s equation of state. (b) The variation of relative volume V/V0 of LaS, LaSe and LaTe with pressure in B1 and B2 structures. V0 is the zero pressure equilibrium volume of the B1 phase. The arrow marks the calculated transition pressure PT and the corresponding relative volume V/V0.

imental results and other theoretical works [4–6]. The calculated total energies are fitted to the Murnaghan’s equation of state [20] to determine the ground state properties such as the equilibrium lattice constant a0, the bulk modulus B0 and its pressure derivative B 0 . The calculated equilibrium parameters (a0, B0 and B 0 ) in both structures are given in Table 1 which also contains results of previous calculations as well as the experimental data. From Table 1, we can observe that the error in the calculated value of lattice parameters, bulk moduli is around 0.3% and 1.5%, respectively, compared to experimental results. The pressure derivatives of the bulk moduli come around the value 5 for all three compounds. Under compression, the calculation shows that LaS, LaSe, and LaTe will undergo a structural phase transition from NaCl-type to CsCl-type structure. In order to determine the transition pressure at T = 0 K, the enthalpy, H = E + PV should be calculated [21]. The stable structure at a given pressure is the structure for which the enthalpy has its lowest value. The transition pressure corresponding to the phase transition from B1 to B2 phases is obtained from the relation HB1 (P) = HB2 (P), where HB1 and HB2 are the enthalpies of the B1 and B2 phases, respectively. Hence the transition pressure (PT) is determined by the common tangent of the two E–V curves, B1 and B2 structures. The results of the calculated phase transition parameters are listed in Table 2 together with theoretical and available experimental ones. In lanthanum monosulphide

A. Bouhemadou et al. / Journal of Molecular Structure: THEOCHEM 777 (2006) 5–10 Table 1 Calculated lattice constant (a0), bulk modulus (B0) and its pressure derivative (B 0 ) compared to experiment and other theoretical works of LaS, LaSe and LaTe for B1 and B2 structures Compounds

Present work LDA

GGA

LaS (B1) ˚) a0 (A B0 (GPa) B0

5.843 83.51 5.00

5.895 81.53 4.67

LaS (B2) ˚) a0 (A B0 (GPa) B0

3.511 86.59 4.69

3.586 79.17 4.52

LaSe (B1) ˚) a0 (A B0 (GPa) B0

6.062 71.93 4.93

6.126 68.40 4.28

LaSe (B2) ˚) a0 (A B0 (GPa) B0

3.647 75.33 4.73

3.722 69.94 4.82

6.455 55.00 4.87

6.512 55.34 4.96

3.881 61.30 4.17

3.947 58.30 4.63

LaTe (B1) ˚) a0 (A B0 (GPa) B0 LaTe (B2) ˚) a0 (A B0 (GPa) B0 a b c d e

Ref. Ref. Ref. Ref. Ref.

Experiment

5.842a, 5.852 86.00a

Other works

b

5.727a, 5.851c 107.0a, 69.0d

3.301a 228a

6.059a, 6.066b

5.957a, 6.063c 97.74a, 70.0d

3.471a 151.08a

7

and V0 is the zero pressure equilibrium volume). Experimentally, LaS has been found to undergo first-order phase transition from B1 structure to the B2 structure at pressure equal to 25 GPa [6]. Theoretically, the modified inter-ionic potential theory [8] finds the transition pressure PT equal to 25.5 GPa and the tight binding linearized-muffin-tin orbital (TB-LMTO) calculation [7] finds it equal to 24.9 GPa. The calculated pressures transition from B1 phase to B2 phase are found to be equal to 14.0 and 6.0 GPa for LaSe and LaTe, respectively. These pressures are accompanied by a volume collapse equal to 8.3% for LaSe and 9.90% for LaTe. For these compounds, there is no experimental data available to us to check our results. From the Fig. 2, it is interesting to note that the calculated transition pressure decreases linearly with the increase in the size of the chalcogen atoms. It is reported in Ref. [7] that the transition pressure increases from LaSe to LaTe which is in disagreement with our result. Hence an experimental verification is needed. 3.2. Elastic properties

6.408a, 6.435b 55.0e

6.255a, 5.851c 74.02b, 35.0d

3.621a 150.43a

[7]. [9]. [4]. [8]. [5].

a crystallographic transition from B1 to B2 is found to be around 23 GPa with a volume collapse DV/V0 of about 8.9% (DV is the change in volume at transition pressure

The elastic properties define the properties of material that undergoes stress, deforms and then recovers and returns to its original shape after stress ceases. These properties play an important part in providing valuable information about the binding characteristic between adjacent atomic planes, anisotropic character of binding and structural stability and are usually defined by the elastic constants Cij. Hence, to study the stability of these compounds in B1 and B2 structures, we have calculated the elastic constants at normal and under hydrostatic pressure by using the method developed recently by Thomas Charpin and integrated it in the WIEN2K package [16]. The elastic moduli require knowledge of the derivative of the energy as a function of lattice strain. In the case of cubic system, there are only three independent elastic constants,

Table 2 The calculated transition pressures (PT), relative volume of the B2 to B1 phase at zero pressure, relative volume change (%) in the LaX (X = S, Se, Te) compared to available experimental and theoretical data {V(B1)0, V(B2)0, V(B1)T, and V(B2)T are the volume of the B1 and B2 at zero pressure and at the transition pressure, respectively} Compounds

PT (GPa)

V(B1)T/ V(B1)0

V(B2)T/ V(B2)0

V(B2)0/ V(B1)0

LaS Present Expt. [6] Ref. [8] Ref. [7]

23.0 25.0 25.5 24.9

0.83

0.82

0.899

LaSe Present Ref. [8] Ref. [7]

14.0 12.4 12.7

0.85

0.86

0.897

8.3 10.4 11.0

LaTe Present Ref. [8] Ref. [7]

6.0 16.8 16.5

0.91

0.91

0.891

9.9 7.2 8.2

DV/ V(B1)0 8.9 8.1 8.4

Fig. 2. Variation of phase transition pressure from B1 to B2 structures with lattice constants for lanthanum monochalcogenides LaS, LaSe and LaTe.

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namely, C11, C12 and C44. So a set of three equations is needed to determine all the constants, which means that three types of strain must be applied to the starting crystals. The first equation involves calculating the bulk modulus B0, which is related to the elastic constants by the following formula:

and the Voigt modulus is given by:

B0 ¼ ðC 11 þ 2C 12 Þ=3

As A approaches unity the crystal becomes isotropic, and the gap between the bounds vanishes. We also calculate the Young modulus E and the Poisson ratio m, which are frequently measured for polycrystalline materials when investigating their hardness. These quantities are related to the bulk modulus B0 and the shear modulus G by the following equations [25]

ð1Þ

The second type involves performing volume-conservative tetragonal strains given by the following tensor: 2 3 d 0 0 60 d 7 0 ð2Þ 4 5 0

0

1 ð1þdÞ2

1

ð3Þ

where V0 is the volume of the unit cell. Finally, for the last type of deformation, we use the volume-conserving rhombohedral strain tensor given by: 2 3 1 1 1 d6 7 ð4Þ 41 1 15 3 1 1 1 which transforms the total energy to EðdÞ ¼ Eð0Þ þ 16ðC 11 þ 2C 12 þ 4C 44 ÞV 0 d2 þ Oðd3 Þ

ð5Þ

For cubic crystal, the shear constant CS is given by: C S ¼ 12ðC 11  C 12 Þ

ð6Þ

Another important parameter is the Kleinman parameter which describes the relative positions of the cation and anion sublattices under volume conserving strain distortions for which positions are not fixed by symmetry. We use the following relation [22]: f¼

C 11 þ 8C 12 7C 11 þ 2C 12

ð7Þ

Once the elastic constants are determined, we would like to compare our results with experiments, or predict what an experiment would yield for the elastic constants. A problem arises when single crystal samples cannot be obtained, for then it is not possible to measure the individual elastic constants Cij. Instead, the isotropic bulk modulus B0 and shear modulus G are determined. These quantities cannot in general be calculated directly from the Cij, but we can use our values to place bounds on the isotropic moduli. Reuss found lower bounds for all lattices, while Voigt discovered upper bounds [23]. For the specific case of cubic lattices, Hashin and Shtrikman [24] found stricter bounds. The isotropic bulk modulus for cubic system is given exactly by 1. The Reuss modulus GR is given by: GR ¼ 5ðC 11  C 12 ÞC 44 =½4C 44 þ 3ðC 11  C 12 Þ

ð9Þ

The width of the bounds on the shear modulus is related to the anisotropy constant A ¼ 2C 44 =ðC 11  C 12 Þ

ð10Þ

E ¼ 9B0 G=ð3B0 þ GÞ and m ¼ ð3B0  EÞ=ð6B0 Þ

Application of this strain has an effect on the total energy from its unstrained value as follows: EðdÞ ¼ Eð0Þ þ 3ðC 11  C 12 ÞV 0 d2 þ Oðd3 Þ

GV ¼ ðC 11  C 12 þ 3C 44 Þ=5

ð8Þ

ð11Þ

In our calculations, we consider only small lattice distortions in order to remain within the elastic domain of the crystal. The calculated values of elastic constants (C11, C12, and C44), the shear constant and the Kleinman parameter in both B1 and B2 phases as well as the available theoretical ones are listed in Table 3. It is seen that the calculated C44 values are slightly smaller than those obtained by Varshney et al. [26] by using the effective interionic interaction potential (EIOP) approach. On the contrary our calculated C12 values are twice larger than those obtained by the same author. This is consistent with the fact that the methods for calculations of elastic constants are different. On the other hand, we can state that this difference is a consequence found in the bulk modulus and lattice constant values. We are not aware of any experimental data on the elastic properties. Future experimental measurements will, however, testify all calculated results. We consider the present elastic constants as a prediction study

Table 3 The calculated elastic constants (C11, C12, and C44), the shear constant (CS) and the Kleinman parameter (f) of lanthanum monochalcogenides LaS, LaSe and LaTe in B1 and B2 structures Compounds NaCl (B1) LaS Present Ref. [26] LaSe Present Ref. [26] LaTe Present Ref. [26] CsCl (B2) LaS Present LaSe Present LaTe Present

C11 (GPa)

C12 (GPa)

C44 (GPa)

Cs (GPa)

f

234 140

23 36

25 46

105.5 53

0.248

203 110

21 10

22 32

71.0 50

0.254

171 50

12 10

8 27

79.5 20

0.219

167

56

47

55.5

0.480

155

47

34

54

0.45

128

40

32

44

0.459

A. Bouhemadou et al. / Journal of Molecular Structure: THEOCHEM 777 (2006) 5–10

for these compounds, hoping that our present work will stimulate some other works on these compounds. To study the stability of these compounds in the B1 and B2 structures, we have calculated the elastic constants at the equilibrium lattice and compared our results to the stability criteria [27,28] using the following relations: ðC 11  C 12 Þ > 0; C 44 > 0; B0 > 0

ð12Þ

We have found that in both structures, these criteria are satisfied, indicating that both phases are elastically stable. From Table 3, we can remark that the elastic constants increase in magnitude as a function of the chalcogen chemical identity as one moves upwards within period VI, i.e., from Te to S. For lanthanum monochalcogenides, with the increase of atomic number, the metallic characteristic increases from La–S bond to La–Te bond, accompanied by a decrease of the magnitude of elastic constants. We note a linear increase of the elastic constants with the decrease of the nearest-neighbor distance from LaTe to LaS as shown in Fig. 3. The predicted isotropic moduli and anisotropy constants are shown in Table 4. Now we are interested to study the pressure dependence of the elastic properties. In Figs. 4 and 5, we present the variation of the elastic constants (C11, C12, and C44) and the bulk modulus B0 of LaS, LaSe and LaTe with respect to the variation of pressure in B1 and B2 phases, respectively. We clearly observe a linear dependence in all curves of these compounds in the considered range of pressure, confirming

9

Table 4 Predicted isotropic moduli (G, shear modulus; E, Young’s modulus; m, Poisson’s ratio) and anisotropy constant (A) of lanthanum monochalcogenides LaS, LaSe and LaTe in B1 and B2 structures Compounds

G (GPa)

E (GPa)

m

A

NaCl (B1) LaS LaSe LaTe

46.6 40.6 24.56

117.8 102.5 64.1

0.26 0.26 0.30

0.24 0.244 0.104

CsCl (B2) LaS LaSe LaTe

50.26 40.96 36.4

126.3 104.0 91.1

0.26 0.27 0.25

0.85 0.63 0.73

Fig. 4. Calculated pressure dependence of elastic constants (C11, C12, and C44) and bulk modulus (B0) for LaS, LaSe and LaTe compounds in B1 structure.

Fig. 3. Variation of elastic moduli B0, C11, C12 and C44 with lattice constants for lanthanum monochalcogenides LaS, LaSe and LaTe in B1 and B2 structures.

the idea of Polian et al. [29], Harrera-Cabrira et al. [30] and Kanoun et al. [31] of the no responsibility of the soft acoustic mode on the phase transition. In Table 5, we listed our results for the pressure derivatives oC11/oP, oC12/oP, oC44/oP and oB0/oP, for LaS, LaSe and LaTe in B1 and B2 phases. It is easy to observe that the elastic constants C11, C12 and bulk modulus B0 increase when the pressure is enhanced. Moreover, the shown shear mode modulus C44 decreases linearly with the increasing of pressure. This behavior is also observed in the work of Varshney et al. [26]. To our knowledge no experimental or theoretical data for the pressure derivative of elastic constants of these compounds are given in the literature. Then, our results can serve as a prediction for future investigations.

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Acknowledgement We would like to thank Doctor Claude Demangeat (Institut de Physique et Chimie des Mate´riaux de Strasbourg) for the critical reading of the manuscript. References

Fig. 5. Calculated pressure dependence of elastic constants (C11, C12, and C44) and bulk modulus (B0) for LaS, LaSe and LaTe compounds in B2 structure. Table 5 Calculated pressure dependence of the elastic constants (C11, C12, and C44) and bulk modulus (B0) for LaS, LaSe and LaTe compounds in B1 and B2 structures Compounds

oB0 oP

oC 11 oP

oC 12 oP

oC 44 oP

NaCl (B1) LaS LaSe LaTe

3.32 4.45 4.38

10.657 11.317 11.004

1.179 1.022 1.067

0.682 0.837 0.877

CsCl (B2) LaS LaSe LaTe

4.13 4.26 4.23

6.074 4.183 5.612

3.167 4.274 3.568

2.703 3.151 4.097

4. Conclusion In this paper, we have reported calculations of structural, phase transition and elastic properties of LaS, LaSe and LaTe under hydrostatic pressure using the full-potential (linear) augmented plane wave plus local orbitals (FP(L)APW+lo) method within LDA and GGA approximations. The calculated structural and the high pressure structural phase transition properties are in good agreement with the available experimental data. We are not aware of any experimental data on the elastic properties of these compounds neither at ambient conditions nor under pressure effect, so our calculations can be used to cover this lack of data for these compounds.

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