Structural phase transition in lanthanum monochalcogenides induced by hydrostatic pressure

phys. stat. sol. (b) 241, No. 14, 3179–3184 (2004) / DOI 10.1002/pssb.200405241

© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Structural phase transition in lanthanum monochalcogenides induced by hydrostatic pressure

Dinesh Varshney*, 1, N. Kaurav1, P. Sharma1, S. Shah2, and R. K. Singh3 1 School of Physics, Vigyan Bhawan, Devi Ahilya University, Khandwa Road Campus, Indore 452017,

India 2 Department of Physics, P. M. B. Gujarati Science College, Indore-452001, India 3 M. P. Bhoj (Open) University, Shivaji Nagar, Bhopal-462016, India

Received 25 June 2004, revised 22 August 2004, accepted 7 September 2004 Published online 22 October 2004

PACS 61.50.Ks

An effective interionic interaction potential (EIOP) is developed to investigate the pressure induced phase transitions from NaCl-type (B1) to CsCl-type (B2) structure in lanthanum monochalcogenides LaX [X = S, Se, Te] compounds. The long range Coulomb, van der Waals (vdW) interaction and the short-range repulsive interaction up to second-neighbor ions within the Hafemeister and Flygare approach with modified ionic charge are properly incorporated in EIOP. The vdW coefficients are computed following the Slater–Kirkwood variational method, as both the ions are polarizable. The estimated value of the phase transition pressure (Pt) and the magnitude of the discontinuity in volume at the transition pressure are consistent with the reported data. A large volume discontinuity in the pressure-volume phase diagram identifies the structural phase transition from B1 to B2 structure. We also study the second order elastic constants for these La compounds. It appears that the vdW interaction is effective in determining the elas-tic and structural properties of these test compounds.


1 Introduction

Most of the rare earth mono-chalcogenides crystallise in NaCl type structure at ambient pressure and are semiconducting if the rare earth ion is in the divalent state [1, 2]. Furthermore, when the rare earth ele-ment is trivalent, the volumes of the corresponding monochalcogenides show the usual lanthanide con-traction and the compounds have metallic properties. For example, LaS, LaSe and LaTe have electrical resistivities of the order of 250 µΩ cm at room temperature [2, 3]. The pressure-volume relationships for divalent rare-earth monochalcogenides have been extensively investigated by high-pressure X-ray dif-fraction and thereby bulk moduli have been evaluated [4]. Ultrasonic measurements [5] have been used to determine the elastic constants of many of the divalent and some trivalent rare earth monochalco-genides and some trivalent pnictides. The high-pressure structural properties of lanthanum monochalco-genides have not been studied much. Quite generally, the chalcogenides, LaX (X = S, Se, Te) crystallise in NaCl type structure. LaS shows phase transformation from its ambient NaCl structure (B1 phase) to CsCl type structure (B2 phase) around 25 GPa in a silicone oil pressure medium (28 GPa in Argon pressure medium) [6]. However, no high-pressure experimental studies of LaSe and LaTe are known at present. It should be noted that den-sity functional theory within the tight binding linear muffin-tin orbital approach (TB-LMTO) [7] has widened the scope of theoretical and accurate experimental investigations of crystallographic phase tran-sitions from B1 to B2 in these compounds. Further, Lu et al. [8] reported the local-density approximation

* Corresponding author: e-mail: [email protected], Phone: +91 731 2467028, Fax: +91 731 2465689

3180 D. Varshney et al.: Structural phase transition in lanthanum monochalcogenides


(LDA) results for electronic structure of LaS and SmS. Although LDA successfully predicts the equilib-rium lattice constant of the non-f-electron system LaS, it is inadequate to describe the localization of the f-electrons in SmS. Hence, this failure of LDA precludes its use in an adequate description of the struc-tural properties. On the other hand, the many-body interactions [9] describe successfully the structural properties of several binary solids. The idea we use in this paper is to employ the two body interactions including vdW attraction, which is not explicitly accounted for in first-principles pseudopotential calculations for the estimation of struc-tural and elastic properties in rare earth compounds. This paper is organized as follows: In section 2, we introduce the model and supply technical details to estimate the pressure dependent elastic properties and support them by physical arguments. As the next step, the phase transition pressures and the elastic con-stants are deduced within the framework of Shell model, that incorporates the long-range Coulomb, van der Waals interaction, and the short-range overlap repulsive interaction up to second-neighbour ions within the Hafemeister and Flygare approach. In section 3, we return to the details of the numerical analysis and conclusions are presented in section 4.

2 The model

The understanding of the thermodynamical properties of La-chalcogenides compounds needs the formu-lation of an effective interionic potential. The idea we have in mind is as follows: the change in force constants is small, the short range interactions are effective up to the second-neighbour ions, and the atoms are held together with harmonic elastic forces without any internal strains within the crystal. The effective interionic potential is expressed as

βρ

− −

+ −⎛ ⎞= + + +⎜ ⎟

⎝ ⎠∑ ∑ ∑ ∑

26 8( ) exp .i j ijm

ij ij ij ij ijij ij ij ijij

r r rZ eU r b c r d r

r (1)

Here, long-range Coulomb is represented by first term, second term correspond to Hafemeister and Fly-gare form of short-range repulsive energies and van der Waals multipole are represented by third and fourth terms respectively. The symbols cij and dij are the van der Waals coefficients and βij are the Pauling coefficients. Zme is the modified ionic charge and parametrically includes the effect of Coulomb screening effect, b and ρ are short-range parameters. Thus, the effective interionic potential contains only three free parameters (Zm, b and ρ) which can be determined from the crystal properties. Details regard-ing the method of calculation of model parameters are reported elsewhere [10]. An isolated phase is stable only when its free energy is minimum for the specified thermodynamic conditions. As the temperature or pressure or any other variable acting on the systems is altered, the free energy changes smoothly and continuously. A phase transition is said to occur when such smooth and continuous variations of free energy causes a change in the structural details of the phase. The test mate-rials, under consideration in this paper, transform from their initial B1 to B2 structure under pressure. The stability of a particular structure is decided by the minima of Gibbs free energy, G = U + PV – TS, where U is internal energy, which at 0 K corresponds to the cohesive energy, S is the vibrational entropy at absolute temperature T, pressure P and volume V.

The Gibbs free energies GB1(r) = UB1(r) + 2Pr3 for NaCl (B1) phase and GB2(r′) = UB2(r′) + [8/3 3 ]Pr′3 for CsCl (B2) phase become equal at the phase-transition pressure, PT, and at zero temperature i.e., ∆G (= GB2 – GB1) = 0 at PT when T = 0. Here, UB2 and UB1 represent cohesive energies for B1 and B2 phases, and are

= − + + +

2 2

1( ) 1.7475 6 ( ) 6 ( ) 6 ( ) ,mB ij ii jj

e ZU r V r V r V r

r (2)

′ ′ ′ ′= − + + +′

2 2

2 ( ) 1.7627 8 ( ) 3 ( ) 3 ( ) .mB ij ii jj

e ZU r V r V r V r

r (3)

phys. stat. sol. (b) 241, No. 14 (2004) / www.pss-b.com 3181


In Eq. [2] and [3], r and r′ are nearest-neighbour (nn) separations corresponding to NaCl and CsCl phases, respectively. Thus, the short-range term of interionic potential reads

βρ

− −

+ −⎛ ⎞= + + =⎜ ⎟

⎝ ⎠

6 8( ) exp ; , 1,2 .i j ijij ij ij ij ij ij

r r rV r b c r d r i j (4)

The study of the second-order elastic constants (C11, C12 and C44) and their pressure derivatives at 0 K is quite important for understanding the nature of the interatomic forces. Since these elastic constants are functions of the first- and second-order derivatives of the short-range potentials, their calculations will provide a further check on the accuracy of short-range forces in these materials. Following Singh [9] and subjecting the dynamical matrix to the long-wavelength limit, we obtain

+⎡ ⎤

= − + +⎢ ⎥⎣ ⎦

22 2 2

11 140

( )5.112 ,

4 2m

e A BC Z A

r (5)

−⎡ ⎤

= − +⎢ ⎥⎣ ⎦

22 2 2

12 140

( 5 )0.226 ,

4 4m

e A BC Z B

r (6)

⎡ + ⎤

= + +⎢ ⎥⎦⎣

22 2 2

44 140

( 3 )2.556 .

4 4m

e A BC Z B

r (7)

First term in equation [5]–[7] is the long-range Coulomb interaction. The symbols (A1, B1) and (A2, B2) are the short-range parameters for the nearest (nn) and the next nearest neighbors (nnn) defined as

= =

⎡ ⎤ ⎡ ⎤= = +⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦0 0

3 2 3 2 20 0

1 22 2 2 2 2

2

4 d 4( 2) d d( ) , ( ) ( ) ,

d d dij ii jj

r r r r

r rA V r A V r V r

e r e r r (8)

= =

⎡ ⎤ ⎡ ⎤= = +⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦0 0

2 20 0

1 22 22

4 d 4( 2) d d( ) , ( ) ( ) ,

d d dij ii jj

r r r r

r rB V r B V r V r

e r e r r (9)

where Vij(r) and Vii(r) [Vjj(r)] are the overlap potentials between the nn and nnn, respectively.

3 Results and discussion

The effective interionic potential described in the preceding section for NaCl (B1) and CsCl (B2) phases contains three free parameters, namely modified ionic charge, range and hardness parameters (Zm, b and ρ). While estimating the free parameters, we first deduce the vdW coefficients from the Slater–Kirkwood variational method [9] and are listed in Table 1. While performing the numerical computations we consider the crystal as purely ionic, i. e., the electronic polarizabilities of the individual ions. In an attempt to reveal the structural phase transition of the test materials, we minimize the Gibbs free energies GB1(r) and GB2(r′) for the equilibrium interatomic spacing (r) and (r′). As the pressure is in-creased, ∆G decreases and approaches zero at the phase transition pressure. Beyond this pressure, where

Table 1 Van der Waals coefficients of La monochalcogenides (cij in units of 10–60 erg cm6 and dij in unit of 10–76 erg cm8).

compound cii cij cjj dii dij djj

LaS 35.2 98.6 305.1 20.4 86.8 332.9 LaSe 35.2 178.7 1205.8 20.4 237.6 2080.6 LaTe 35.2 213.6 1856.5 20.4 327.9 3689.9



0 5 10 15 20 25 30

-20

0

20

40

60

80

LaSe LaTe LaS

∆ G(k

J/m

ol.)

P (GPa) phase B2 is more stable, ∆G becomes negative. The Gibbs free energy difference ∆G [= GB2(r) – GB1(r′)], has been plotted as a function of pressure (P) in Fig. 1 by using the interionic potential discussed above. The pressure corresponding to ∆G approaching zero is the phase transition pressure (Pt) at which a crys-tallographic transition from B1 to B2 occurs in lanthanum monochalcogenides. A comparison of the present results with the available experimental data and other theoretical work on these compounds is presented in Table 3. We point out that no experimental measurements of phase transition pressure of LaSe and LaTe are known. Furthermore, the estimated values of Pt, from our calculations, are in agree-ment with those obtained from TB-LMTO [7]. It is interesting to note that the transition pressure in-creases from LaSe to LaTe. It is noteworthy to comment that there is a transfer of electrons from chalco-gen s and p like states to the La-d like states continuously under pressure, which shall be responsible for the observed structural transformation. We now estimate the relative volumes, V(P)/V(0), associated with various compressions following Murnaghan’s equation of state [9]. The relative volume is plotted as a function of pressure in Fig. 2. We notice that our calculation has predicted correctly the relative stability of competing crystal structures, as the values of ∆G are positive. The magnitude of the discontinuity in volume at the transition pressure Pt and the values of Pt, obtained from Fig. 2, are tabulated in Table 2. As can be seen, the values obtained in our calculations are in good agreement with those reported from TB-LMTO approach [7]. In order to study the high-pressure elastic behaviour of these compounds, we have computed the sec-ond-order elastic constants and their variation with pressure. We have plotted the variation of CL = (C11 + C12 + 2C44)/2 and CS = (C11 – C12)/2 in Fig. 3. We note that CL and CS increase linearly with increase in pressure and in accordance with the first-order character of the phase transition. This feature is similar to the previously reported pressure dependence of elastic stiffness for PbTe and SnTe possessing the NaCl structure at ambient pressure and undergoing B1 to B2 structural phase transition at Pt [12]. The Born criterion for the mechanical stability of a lattice is that the elastic energy density be a posi-tive definite quadratic function of strain. This requires that the principal minors (alternatively the eigen values) of the elastic constant matrix should all be positive. We note that the stability of a cubic crystal is expressed in terms of elastic constants as follows [9]:

( )= + >11 122 /3 0 ,TB C C >44 0 ,C and = − >11 12( )/2 0 .SC C C (10)

Table 2 Input crystal data and model parameters for lanthanum monochalcogenides.

input parameters model parameters compound

ri (Å) rj (Å) a0 (Å) BT (GPa) Zm2 b (10–12 erg) ρ (10–1 Å)

LaS 1.04 [11] 1.24 [11] 5.85 [7] 89 [6] 3.18 3.05 7.83 LaSe 1.04 [11] 1.4 [11] 6.06 [7] 97.7 [7] 3.11 5.82 2.74 LaTe 1.04 [11] 1.52 [11] 6.43 [7] 55 [7] 3.04 6.41 3.6

Fig. 1 Variations of Gibbs free energy differ-ence ∆G with pressure.

phys. stat. sol. (b) 241, No. 14 (2004) / www.pss-b.com 3183


0.6

0.7

0.8

0.9

1.0

LaS

V(P

)/V

(0)

0.7

0.8

0.9

1.0

LaSe

V(P

)/V

(0)

0 10 20 30

0.7

0.8

0.9

1.0

LaTe

V(P

)/V

(0)

P (GPa)

In Eq. [10], Cij are the conventional elastic constants, C44 and CS are the shear and tetragonal moduli of a cubic crystal and BT is the bulk modulus. Estimated values for LaS (LaSe, LaTe) are: the bulk modulus BT = 0.69 (0.7, 0.35) × 1011 Nm–2, shear moduli C44 = 0.46 (0.32, 0.27) × 1011 Nm–2, and the tetragonal moduli, CS = 0.5 (0.55, 0.2) × 1011 Nm–2; these values satisfy Born’s elastic stability criterion. Vukcevich [13] proposed a high-pressure stability criterion for ionic crystals, combining mechanical stability with minimum energy conditions. In accordance, the stable phase of the crystal is one in which the shear elastic constant C44 is nonzero (for mechanical stability) and has the lowest potential energy among the mechanically stable lattices. However, since C44 is a very small quantity, the calculated value of [(4r0/e

2) C44 – 0.556Zm2] is found to

be negative, so that (A2 – B2) is negative. This suggests that these terms belong to an attractive interaction and possibly arise due to the van der Waals energy. The van der Waals energy converges quickly, but the overlap repulsion converges much more quickly. This means that the second neighbour forces are en-tirely due to the van der Waals interaction; the first neighbour forces are the results of the overlap repul-sion and the van der Waals attraction between the nearest neighbours. However, at elevated pressure, the

Fig. 2 Variation in relative volume V(P)/V(0) with pressure.

0

1

2

3

CL

CS

LaS

CL

and

CS

(1011

Nm

-2)

0

1

2CL CS

LaSe

CL

and

CS

(101

1N

m-2)

0 10 20 300

1

2

CS

CL

LaTe

CL

and

CS

(1011

Nm

-2)

P (GPa)

Fig. 3 Variation of the combination of elastic constants with pressure.



Table 3 Calculated transition pressures and volume collapse of lanthanum chalcogenides (The quanti-ties given in brackets are aexpt. [7] and bothers works [8]).

compound transition pressure (GPa) volume collapse (%)

LaS B1 → B2 25.5 (25a, 24.9b) 8.1 (8.4b) LaSe B1 → B2 12.4 (12.7b) 10.4 (11.0b) LaTe B1 → B2 16.8 (16.5b) 7.2 (8.2b)

short-range forces increase significantly, which, in turn, is responsible for the change in the coordination number and the structural phase transformation in these compounds. Other than deriving the equation of states correctly from a model approach and then using it to analyse the variation of short-range forces, at present we have no direct means to understand the interatomic forces at high pressure.

4 Conclusion

In the present investigation, an effective interionic interaction potential is formulated for analyzing the structural as well as the elastic properties in lanthanum monochalcogenides. The obtained values of the free parameters in the interionic potential, allow us to predict the phase transition pressures and the asso-ciated volume collapse. It is believed that the vast volume discontinuity in the pressure-volume phase diagram identifies the structural phase transi-tion from NaCl to CsCl structure. The calculated values of the phase transition pressures and the magnitudes of the volume discontinuity are in agreement with those calculated previously from TB-LMTO calculations. We also check the validity of Born’s mechanical stability criterion by computing the second order elastic constants and find that the high-pressure structure of LaX compounds are stable. Furthermore, the combination elastic constants CL and CS increase linearly with the increase of pressure in accordance with the first-order character of the structural phase transition observed in compounds that have NaCl structure at ambient pressure. We should emphasize that our conclusions have been established only within the framework of shell model, which includes overlap repulsion up to second nearest neighbour interactions. Nevertheless, it has been found that this simple model, as compared to complicated band structure calculations, may account for a considerable part of the available experimental and theoretical results for the high-pressure studies.

Acknowledgements One of the authors (RKS) is thankful to Distance Education Council, New Delhi, for gener-ous financial support.

References

[1] J. Flahaut, in: Handbook on the Physics and Chemistry of Rare Earths, edited by K. A. Gschneider and L. Erying (North-Holland, Amsterdam, 1979), Vol. 4, pp. 3–81.

[2] A. Jayaraman, P. D. Dernier, and L. D. Longinotti, High Temp. High Press. 7, 1 (1975). [3] J. M. Leger, P. Aimonino, J. Loriers, P. Dordor, and B. Coqblin, Phys. Lett. A 80, 325 (1980). [4] A. Jayaraman, A. K. Singh, A. Chatterjee, and S. U. Devi, Phys. Rev. B 9, 2513 (1974). [5] H. R. Ott and B. Lüthi, Z. Phys. B 28, 141 (1977). [6] T. Le Bihan, A. Bombardi, M. Idiri, S. Heathman, and A. Lindbaum, J. Phys.: Condens. Matter 14, 10595 (2002). [7] G. Vaitheeswaran, V. Kanchana, and M. Rajagopalan, J. Phys. Chem. Solids 64, 15 (2003). [8] Z. W. Lu, D. J. Singh, and H. Krakauer, Phys. Rev. B 37, 10045 (1988). [9] R. K. Singh, Phys. Rep. 85, 259 (1982). [10] Dinesh Varshney, N. Kaurav, P. Sharma, S. Shah, and R. K. Singh, Phase Transit. 77 (2004). [11] D. R. Lide (ed.), CRC Handbook of Chemistry and Physics, 79th ed. (CRC Press, New York, 1999). [12] A. J. Miller, G. A. Saunders, and Y. K. Yogurtcu, J. Phys. C 14, 1569 (1981). [13] M. R. Vukcevich, phys. stat. sol. (b) 54, 435 (1972).

Structural phase transition in lanthanum monochalcogenides induced by hydrostatic pressure

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