Physica B 406 (2011) 2158–2162

Contents lists available at ScienceDirect

Physica B

0921-45

doi:10.1

n Corr

Technol

422498

E-m

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

High pressure effect on structural and mechanical properties of someLnO (Ln¼Sm, Eu, Yb) compounds

Vipul Srivastava a,b,n, Sanjay Bhajanker a,b, Sankar P. Sanyal a

a Department of Physics, NRI Institute of Research & Technology, Raisen Road, Bhopal 462021, Indiab Condensed Matter Physics Laboratory, Department of Physics, Barkatullah University, Bhopal 462026, India

a r t i c l e i n f o

Article history:

Received 17 January 2011

Received in revised form

10 March 2011

Accepted 10 March 2011Available online 15 March 2011

Keywords:

Phase transition

Mechanical properties

Elastic constants

Interionic potential theory\

26/$ - see front matter & 2011 Elsevier B.V. A

016/j.physb.2011.03.022

esponding author at: Department of Physics

ogy Raisen Road, Bhopal 462021 Madhya P

9; fax: þ91 755 2677723.

ail address: vips73@yahoo.com (V. Srivastava

a b s t r a c t

The structural and mechanical properties of LnO (Ln¼Sm, Eu, Yb) compounds have been investigated

using a modified interionic potential theory, which includes the effect of Coulomb screening. We

predicted a structural phase transition from NaCl (B1)- to CsCl (B2)-type structure and elastic properties

in LnO compounds at very high pressure. The anomalous properties of these compounds have been

correlated in terms of the hybridisation of f-electrons of the rare earth ion with conduction band and

strong mixing of f-states of lanthanides with the p-orbital of neighbouring chalcogen ion. For EuO, the

calculated transition pressure, bulk modulus and lattice parameter are close to the experimental data.

The nature of bonds between the ions is predicted by simulating the ion–ion (Ln–Ln and Ln–O)

distances at high pressure. The second order elastic constants along with shear modulus and Young’s

modulus, elastic anisotropy and Poisson’s ratio are also presented for these oxides.

& 2011 Elsevier B.V. All rights reserved.

1. Introduction

In recent years there has been a renewed interest in theremarkable area of high-pressure behaviour of the rare-earth (RE)monochalcogenides both experimentally and theoretically. Out ofthese numerous RE compounds, monochalcogenides of samarium(Sm), europium (Eu) and ytterbium (Yb) have attracted physicistsand material scientists due to their mixed valence behaviour,increased magnetic ordering and possible applications in magneto-optic memories and modulators [1–4]. The interplay between thedivalent and trivalent state of RE-ion has also been observed in thisclass of materials experimentally. Most of the chalcogenides of Sm,Eu and Yb show such behaviour. These compounds exhibit semi-conducting character if the RE-ion is in divalent state (2þ), but showmetallic character if RE-ion is in trivalent state (3þ). Earlier studies[4] on Sm-, Eu- and Yb-chalcogenides reveal that some chalcogen-ides of Sm and Eu exhibit a valence transition followed by NaCl toCsCl-type transition at high pressure. Earlier studies also show [5]that all the Yb-chalcogenides (Yb-O, -S, -Se, -Te) undergo onlyvalence transition continuously with pressure. So far as the recentstudies on these oxide materials are concerned, EuO shows only B1

to B2 structural phase transition [6], while for SmO and YbO no suchstructural transition has been reported. However, the studies onelectronic structure [7,8] have shown that the divalent Yb config-uration is favoured as compared to the trivalent one by more than

ll rights reserved.

, NRI Institute of Research &

radesh, India. Tel.: þ91 755

).

20 mRy and is described in terms of pressure induced delocalisationof 4f-electrons in Yb-compounds [9]. Hence, the effective valency isassumed to be equal to 2 for these compounds. Most of the rareearth compounds have been found to exhibit small or negative valueof elastic constant, C12 [10,11], which, in general, can be attributedto the instability of f-electrons even at ambient pressure.

In a few earlier papers from our group we have successfullydeveloped an ionic model theory applicable to a large number ofrare earth compounds [12–16]. Motivated by the above men-tioned peculiarities in the present paper, we have used the sametheoretical approach with necessary modification in ionic chargedue to Coulomb screening by the delocalised f-electrons topredict the structural phase transition at high pressure and theelastic properties of Sm-, Eu- and Yb-oxides.

The paper is organised as follows: the theoretical model isdescribed briefly in Section 2; potentially interesting results anddiscussion are presented in Section 3.

2. Method of calculation

2.1. Cohesive energy and phase transition pressure

The interionic potential for Ln-oxide compounds in the frame-work of the rigid ion model [17] is expressed as

UðrÞ ¼X0

ij

Z2me2

rijþX

ij

bbij expriþrj�rij

rij

" #þX

ij

Cijr�6ij þ

Xij

Dijr�8ij ; i,j¼ 1,2 ð1Þ

which includes long-range Coulomb (first term), Hafemeisterand Flygare [18] form of short-range repulsive (second term)

V. Srivastava et al. / Physica B 406 (2011) 2158–2162 2159

and van der Waals multipole interactions [19] (third and fourthterms). Zme is the modified ionic charge, which parametricallyincludes the effect of Coulomb screening by the delocalised f-electrons [14]. This parameter has been determined following thetrend of variation of bulk modulus and considering that itdecreases (increases) with cation size [12–16]. In the case ofchalcogenides (pnictides), using such approximation has beenfound consistent with the number of electrons taking part in thecrystal bonding. This effective charge is not fitted to reproduceany crystal property. Two other parameters b and r are the short-range parameters, which can be determined from the bulkmodulus and equilibrium condition:

dUijðrÞ

dr

��������r ¼ r0

¼ 0 ð2Þ

The input constants and model parameters for LnO compoundsare presented in Table 1. The LnO compounds transform fromtheir initial B1 (NaCl) to B2 (CsCl) structure under pressure. Thestability of a particular structure is decided by the minima of theGibbs energy, given by

G¼UþPV�TS ð3Þ

where U is the internal energy, which at 0 K corresponds to thecohesive energy, S is the vibrational entropy at absolute T, pressure P

and volume V. The Gibbs free energies, GB1ðrÞ ¼UB1

ðrÞþ2Pr3, for NaCl

(B1) phase and GB2ðr0Þ ¼UB2ðr

0Þþ½8=ð3O3Þ�Pr03, for CsCl (B2) phase

become equal at the phase transition pressure P and temperature 0 K,

i.e. DGð ¼ GB1�GB2

Þ becomes zero. Here the abbreviations UB1and UB1

represent cohesive energies for the phases B1 and B2, respectively,and are written as

UB1ðrÞ ¼ �1:7475e2Z2

m

rþ6FijðrÞþ6FiiðrÞþ6FjjðrÞ ð4Þ

and

UB2ðr0 Þ ¼ �1:7627e2Z2

m

r0þ8Fijðr

0Þþ3Fiiðr0Þþ3Fjjðr

0Þ ð5Þ

Here r and r0 are the nearest neighbour (nn) separations correspond-ing to NaCl and CsCl phases, respectively. The short-range potentialsfor both the phases between the ions are written as

FijðrÞ ¼ bije�rij=rij�Cijr

�6ij �Dijr

�8ij , i,j¼ 1,2 ð6Þ

where bij and rij are short-range parameters as defined earlier. Topredict the transition pressure, we have minimised the Gibbs freeenergies with respect to interatomic separations and calculated

DGð ¼ GB1�GB2

Þ at various pressures.

2.2. Mechanical properties

To understand the interatomic forces in cubic solids (NaCl-type) the second order elastic constants (SOEC), i.e. C11, C12 and

Table 1Input and output data for LnO compounds.

Solids Crystal properties and model parameters

LnO r0 (A) B0 (GPa) Zm2 b (�10�19 J) r (A)

SmO 2.518a 106b 2.15 7.099 0.237

EuO 2.5725c 110c, 114d 118d 2.17 6.853 0.222

YbO 2.44e 130710e 2.30 9.336 0.219

a Ref. [24].b Ref. [22].c Ref. [4].d Ref. [6].e Ref. [23].

C44 are derived from crystal potential energy by the method of thehomogeneous deformations as follows:

C11 ¼e2

4r40

!�5:112Z2

mþA12þ1

2ðA11þA22þB11þB22Þ

� �ð7aÞ

C12 ¼e2

4r40

!�0:225Z2

mþB12þ1

4ðA11þA12�

5

4ðB11þB12ÞÞ

� �ð7bÞ

C44 ¼e2

4r40

!2:556Z2

mþB12þ1

4ðA11þA12þ

3

4ðB11þB12ÞÞ

� �ð7cÞ

where Aij and Bij (i,j¼1,2) are the short-range force constants andare uniquely determined from the following equations

Aij ¼4r3

0

e2

� �@2FSR

ij ðrÞ

@r2

" #r ¼ r0

; Bij ¼4r3

0

e2

� �@FSR

ij ðrÞ

r@r

" #r ¼ r0

ð8aÞ

Aij ¼4r3

0

e2

� �@2FSR

ii ðrÞ

@r2

" #r ¼

ffiffi2p

r0

; Bij ¼4r3

0

e2

� �@FSR

ii ðrÞ

r@r

" #r ¼

ffiffi2p

r0

ð8bÞ

where FSRij (r) is the short-range potential defined by the second

term of Eq. (1) and the bulk modulus is derived from elasticconstants as

B0 ¼1

3ðC11þ2C12Þ ð9Þ

Under ambient conditions the elastic stability criteria for a cubiccrystal [20] are (C11þ2C12)40, C4440 and (C11�C12)40. Inaddition another physical quantity, namely the anisotropic ratio,defined as: A¼2C44/(C11�C12) has also been calculated. For anideal isotropic system, A is unity and deviation from unitymeasures the amount of elastic anisotropy. The Poission’s ratio, n,is calculated using the relation

n¼ 3B�2G

2ð3BþGÞð10Þ

where B is the bulk modulus and G is the average shear modulus. Asper Hill [21] average shear modulus, G is defined as the arithmeticmean of Voigt, GV, and Reuss, GR, values, which can be expressed interms of elastic constants as

GV ¼1

5ðC11�C12þ3C44Þ ð11aÞ

GR ¼5ðC11�C12ÞC44

3ðC11�C12Þþ4C44ð11bÞ

Young’s modulus, E is calculated using the expression

E¼9BG

3BþGð12Þ

3. Results and discussion

The pressure–volume relationships and mechanical propertiesof LnO (Ln¼Sm, Eu, Yb) compounds have been obtained for thefirst time from interionic potential theory upto 100 GPa. The inputparameters and calculated crystal parameters of these com-pounds are listed in Table 1. In Table 2 we present interatomicseparations ðrB1

and rB2Þ, transition pressure and cohesive energies

for both the B1 and B2 phases and percentage volume change forall the three oxides. The calculated values of interatomic separa-tions in the B1 phase are in good agreement with the experi-mental values [4,6,23,24], while the same for high pressure phaseare not available and hence could not be compared. The inspec-tion of Table 2 further reveals that EuO could only be comparedwith the experimental values [4,6] of transition pressure and

Table 2Phase transition pressure and cohesive energy of LnO compounds.

Solids Equilibrium interionic distance (A) Cohesive energy (kJ/mol) Transition

pressure (GPa)

%Vol. change

LnO rB1rB2

UB1UB2

PT DV/V0%

SmO

Present 2.53 2.66 �1880.2 �1814.8 43 8.0

EuO

Present 2.58 2.72 �1873.4 �1804.7 45 7.9

Expt. 47a 8.0

YbO

Present 2.45 2.58 �2085.7 �2008.5 61 6.8

Expt 440 b

a Ref. [6].b Ref. [23].

Fig. 1. Variation of ionic charge parameter Zm2 as a function of atomic number of

rare earth ion.

V. Srivastava et al. / Physica B 406 (2011) 2158–21622160

volume collapse. Nevertheless, more discussion on this with otherfamily members can be found in this section later.

The variation of ionic charge parameter, Zm2 , with the atomic

number of the rare earth ions (Sm, Eu, Yb) is shown in Fig. 1. It canbe seen from this figure that the value of Zm

2 increases linearly fromSm to Yb, which is according to the increasing trend of bulk modulusin this series. The large values of bulk modulus in LnO show theirstrong ionic character, whereas small values of modified ioniccharge parameter for these oxides indicate strong delocalisation off-electrons. In all the LnO compounds, divalency of the Ln-ion ismost favourable. This fact is consistent with our calculated values ofscreened charge parameter as reported in Table 1.

Earlier studies on the high pressure behaviour of Ln-mono-chalcogenides (LnX) reveal [4] that some of the Sm–X compoundsare semiconductors (except SmO) with a large band gap [7], anddo not show any structural phase transition but SmTe showsstructural phase transformation followed by valence change [4].High pressure phase for SmO has not been determined so far. Onthe other hand, we have predicted structural phase transition inSmO. The equation of state for SmO is presented in Fig. 2. We havepredicted a B1 to B2 phase transition at 43 GPa with 8% volumediscontinuity and bulk modulus is calculated to be 102 GPa. Ourcalculated bulk modulus is in good agreement with the othertheoretically calculated value of 106 GPa reported by Savane et al.[22]. EuO is the only compound amongst the chalcogenides ofEu, which shows a valence transition at �30 GPa followed by

NaCl-to-CsCl-type transition around 40 GPa [6–9]. Heathmanet al. [6] further extended the range of investigation upto63 GPa using energy dispersive X-ray diffraction (EDXD) andsynchrotron radiation and showed that no transformation ofEu2þ to Eu3þ occurs at the pressure region of 28 to 40 GPa andobtained a smooth compression curve until 47 GPa. When CsClphase appears, the bulk modulus is calculated to be 105 GPa,which is in good agreement with the experimental and othertheoretical results [4,6]. The Yb monochalcogenides (Yb-S, -Se, -Te)undergo only valence transformation continuously with pressure,because of a change in the valence state from 2þ to 3þ asobserved by Jayaraman et al. [4]; YbO synthesised and studiedupto little bit high pressure of 35 GPa by Werner et al. [23], whoobtained similar results as that of the other Yb-chalcogenides. Wehave, therefore, extended the pressure range up to 70 GPa for YbOand observed structural phase transition at very high pressure of61 GPa with 6.8% volume collapse (Fig. 4). It can also be notedfrom Fig. 4 that our results are in excellent agreement with theexperimental data (using pressure medium as Si-grease andmethanol, ethanol mixture) at lower pressure but differ after8 GPa. Our results after 8 GPa in B1 phase do not follow theexperimental path since we do not observe any 2þ to 3þ valencechange in YbO. However, one can see that it is in agreement withthe Murnaghan fit. The bulk modulus is calculated to be 131 GPa,which agrees with the experimental value of 130 GPa [23]. TheP–V relationships obtained in earlier study [4] were not fineenough to conclude whether the transitions were abrupt orcontinuous. Due to non availability of the experimental P–V dataof SmO, the comparison is only possible for YbO and EuO as inFigs. 2–4.

In order to understand the bonding properties of LnO com-pounds, we have calculated the interionic Ln–Ln and Ln–O distancesfor Sm-, Eu- and Yb-O. In case of SmO, the Sm–O distance in NaCl-type structure is 2.53 A at ambient pressure, which is in goodagreement with the Wyckoff value of 2.518 A [24]. The bondbetween Sm and O is found to be partially covalent in nature. TheSm–O distance is larger than the sum of covalent radii of Sm(1.62 A) and O (0.73 A). Also the Sm–O distance is much shorterthan the sum of ionic radii of Sm2þ (0.96 A) and O2� (2.32 A) andslightly less than the sum of atomic radius of Sm (2.59 A) andcovalent radius of O (0.73 A). Under the application of pressure Sm–O distance decreases from 2.53 to 2.32 A at ambient to transitionpressure of 43 GPa in B1 phase. At B1–B2 phase transition pressurethis distance suddenly increases and becomes 2.45 A in B2 phase.Similarly, the Sm–Sm distance in SmO abruptly decreases from 3.28to 2.83 A at B1–B2 phase transition pressure. We have also carriedout similar analysis of Ln–O and Ln–Ln distances for other two LnOcompounds and summarised them in Table 3.

Fig. 2. Pressure–volume relationship for SmO. Solid line represents calculated

B1 phase while dash line represents B2 Phase.

Fig. 3. Pressure–volume relationship for EuO. Solid circles are experimental data

taken from Ref. [6].

Fig. 4. Pressure–volume relationship for YbO. Experimental data points are taken

from Ref. [23].

Table 3Ln–O and Ln–Ln distances of LnO compounds.

Properties SmO (A) EuO (A) YbO (A)

Ln–O

P¼0 (B1 phase) 2.53 2.58 2.45

P¼Pt (B1 phase) 2.32 2.37 2.21

P¼Pt (B2 phase) 2.45 2.49 2.33

Ln–Ln

P¼0 (B1 phase) 3.58 3.65 3.46

P¼Pt (B1 phase) 3.28 3.35 3.12

P¼Pt (B2 phase) 2.83 2.82 2.69

Table 4Elastic constants and bulk moduli (in GPa) under ambient conditions.

Solids C11 C12 (¼C44) C111 C112 C123

SmO 216 45 �17 �1.5 0.84

EuO 232 41 �22 �1.4 0.77

YbO 284 54 �25 �1.8 1.01

Table 5Mechanical properties of LnO compounds. Shear (G) and Young’s modulus (E) in

GPa unit.

Solids GV GR G E n A

LnO

SmO 61 55 58 146 0.260 0.519

EuO 63 53 58 147 0.266 0.434

YbO 78 69 74 186 0.263 0.472

V. Srivastava et al. / Physica B 406 (2011) 2158–2162 2161

It is revealed from the literature that the role of f-electrons onthe electronic properties in divalent to trivalent transition in RE-ion has been extensively studied. However, their role in otherbulk properties such as elastic constants, phonon spectra, etc. isyet to be explored in detail. Elastic constants play significant rolein determination of the mechanical properties. Therefore, effort ismade to calculate shear modulus, (G), Young’s modulus (E), elasticanisotropy (A) and Poisson’s ratio (n) using elastic constants andstandard relations as mentioned in earlier section. In Table 4 wehave presented second order elastic constants (SOEC) and thirdorder elastic constants (TOEC) for three LnO compounds. Since weused two-body interaction potential between the ions, the calcu-lated values in SOEC of C12 and C44 are equal; while in TOEC thecalculated value of C112¼C116 and C123¼C144¼C456. An inspectionof Table 4 reveals the elastic stability criteria for a cubic crystal[20] as mentioned in previous section is followed by the calcu-lated SOECs. The predicted values of modulus (shear and Young’s)along with anisotropic factor and Poisson’s ratio are presented inTable 5. One can understand the stiffness of the materials bydefining Young’s modulus, E. It is seen from Table 5 that E

increases from Sm to Yb. If E is large, material is stiff and if thevalue of E increases, the covalent nature of the materials alsoincreases, which is also supported by the calculated values ofscreened charge parameter, Zm

2 (Table 2). Poisson’s ratio, n, definesthe stability of solid against shear. The value of Poisson’s ratio liesin between �1 and 0.5. Materials do not change shape at lowerbound and at the upper bound and the volume remainsunchanged. Interestingly, the calculated values of n (Table 5) in

V. Srivastava et al. / Physica B 406 (2011) 2158–21622162

LnO compounds do not fall in this range and reveals a possibilityof instability of structure. It is therefore clear from the presentinter-ionic potential theory that the divalent nature of Ln-ion ismost favourable and is supported in terms of screened chargeparameter. Among the three oxides YbO shows much stiffnessand covalent nature. The calculated value of Poisson’s ratiodefines instability of the structure.

Acknowledgements

The present work is financially supported by Madhya PradeshCouncil of Science and Technology (MPCST), Bhopal, India. One ofthe authors SB, acknowledges MPCST, Bhopal for the award ofresearch project fellowship (Project no. 1904/CST/R&D/2009) andthankful to Shri D. Subodh Singh and Dr. B.B. Saxena, NRI Instituteof Research & Technology, Bhopal, India for their constant supportin research. SPS gratefully acknowledges Council of Scientific andIndustrial Research (CSIR), New Delhi, India.

References

[1] Ryouichi Akimoto, Michihiro Kobayashi, Takashi Suzuki, J. Phys. Soc. Jpn. 62(1993) 1490.

[2] C.J.M. Rooymans, Solid State Commun. 3 (1965) 421.[3] A. Chatterjee, A.K. Singh, A. Jayaraman, Phys. Rev. B 6 (1972) 2285.

[4] A. Jayaraman, A.K. Singh, A. Chatterjee, S.U. Devi, Phys. Rev. B 9 (1974) 2513;A. Jayaraman, Phys. Rev. Lett. 29 (1972) 1674.

[5] B.M. Angelov, J. Phys. C: Solid State Phys. 13 (1980) L433.[6] S. Heathman, T.Le. Bihan, S. Darracq, C. Abraham, D.J.A. De Ridder,

U. Benedict, K. Mattenberger, O. Vogt, J. Alloys Compd. 230 (1995) 89.[7] A. Svane, G. Santi, Z. Szotek, W.M. Temmerman, P. Strange, M. Horne,

G. Vaitheeswaran, V. Kanchana, L. Petit, H. Winter, Phys. Status Solidi (b)241 (2004) 3185.

[8] A. Svane, P. Strange, W.M. Temmerman, Z. Szotek, H. Winter, L. Petit, Phys.Status Solidi (b) 223 (2001) 105.

[9] W.M. Temmerman, Z. Szotek, A. Svane, P. Strange, H. Winter, A. Delin,B. Johansson, O. Eriksson, L. Fast, J.M. Wills, Phys. Rev. Lett. 83 (1999) 3900.

[10] I. Vedel, K. Oki, A.M. Redon, J.M. Leger, Physica B 139 and 140 (1986) 361.[11] P.K. Jha, R.K. Singh, S.P. Sanyal, Proc. INSA 68B (2002) 57.[12] G. Pagare, D. Sen, V. Srivastava, S.P. Sanyal, J. Phys. Conf. Ser. 215 (2010)

012114.[13] G. Pagare, V. Srivastava, S.P. Sanyal, R.K. Singh, Phys. Status Solidi B 241

(2004) 3193.[14] V. Srivastava, S.P. Sanyal, J. Alloys Compd. 366 (2004) 15.[15] V. Srivastava, A.K. Bandyopadhyay, P.K. Jha, S.P. Sanyal, J. Phys. Chem. Solids

64 (2003) 907.[16] A. Singh, V. Srivastava, M. Aynyas, S.P. Sanyal, Physica B 404 (2009) 1852.[17] R.K. Singh, Phys. Rep. 82 (1982) 259.[18] W. Hafemeister, W.H. Flygare, J. Chem. Phys. 43 (1965) 795.[19] W. Cochran, Crit. Rev. Solid State 2 (1971) 1.[20] J. Wang, S. Yip, Phys. Rev. Lett. 71 (1993) 4182.[21] R. Hill, Proc. Phys. Soc. London A 65 (1952) 349.[22] A. Svane, V. Kanchana, G. Vaitheeswaran, G. Santi, W.M. Temmerman,

Z. Szotek, P. Strange, L. Petit, Phys. Rev. B 71 (2005) 045119.[23] A. Werner, H.D. Hochheimer, A. Jayaraman, Solid State Commun. 38 (1981)

325.[24] R.W.G. Wyckoff, Crystal Structures, 2nd edn., vols. 1–2, Inter-science,

1963–1965.