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Physica B 404 (2009) 1852–1857

Contents lists available at ScienceDirect

Physica B

0921-45

doi:10.1

� Corr

E-m

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

High pressure structural phase transition and elastic propertiesof yttrium pnictides

Archana Singh a,�, Vipul Srivastava b, Mahendra Aynyas a, Sankar P. Sanyal b

a Department of Physics, Sadhu Vaswani College, Bairagarh, Bhopal 462030, Indiab Computational Laboratory, Department of Physics, Barkatullah University, Bhopal 462026, India

a r t i c l e i n f o

Article history:

Received 23 September 2008

Received in revised form

23 February 2009

Accepted 2 March 2009

Keywords:

Phase transition

Equation of state

Elastic constant

High pressure

Yttrium pnictides

26/$ - see front matter & 2009 Elsevier B.V. A

016/j.physb.2009.03.003

esponding author. Tel.: +91755 4244797; fax

ail address: archana.v.singh22@gmail.com (A.

a b s t r a c t

We have investigated, the cohesive energies, equilibrium lattice constants, pressure–volume relation-

ship, phase transition pressure (PT), elastic constants and variation of elastic constants with pressure for

yttrium pnictides (YX; X ¼ N, P, As and Sb), using an interionic potential theory with modified ionic

charge which, includes Coulomb screening effect due to d-electrons. These compounds undergo

structural phase transition from NaCl (B1) to CsCl (B2) structure at high pressure (ranges 20–130 GPa).

We have also calculated bulk (BT), Young (E), and shear moduli (G), Poisson ratio (u) and anisotropic ratio

(A) in NaCl (B1)-type structure for yttrium pnictides and compared them with other experimental and

theoretical results which show a good agreement.

& 2009 Elsevier B.V. All rights reserved.

1. Introduction

The structural phase transformation and mechanical proper-ties of binary rare earth and transition metal pnictides under highpressure, both theoretically and experimentally, have receivedconsiderable attention in recent years. At ambient condition, thepnictides of yttrium (YN, YP, YAs and YSb) crystallize in six-foldcoordinated NaCl-type crystal structure with space group sym-metry Fm3m (225) and undergo pressure-induced first orderphase transition to eight-fold coordinated CsCl-type structurewith space group symmetry Pm3m (221). Experimental study onYSb has been carried out by Hayashi et al. [1] using synchrotronX-ray diffraction techniques, who have observed a first orderstructural phase transformation from NaCl to CsCl structure at apressure of around 26–28 GPa, while the bulk modulus (BT) wasfound to be 58 GPa at ambient pressure. It was also found that theB2 phase appears at 26 GPa for YSb with coexistence of both lowand high pressure phases between 26 and 36 GPa. Theoretically,there have been a number of first principles-calculations reportedso far based on the density functional theories to reveal the crystalstructure, high pressure phase transition, electronic and elasticproperties of yttrium pnictides [2–8]. The structural properties ofYN have been studied by Takeuchi et al. [2], Stampfl et al. [3], DeLa Cruz et al. [4] and Mancera et al. [5] using full-potentiallinearized augmented plane wave (FP-LAPW) method. Rodriguez-Hernandez and Munoz [6] have studied the pressure-induced

ll rights reserved.

: +91755 2581277.

Singh).

structural phase transition of YSb using Trouiller–Martins non-local non-conserving pseudo-potential. The structural phasestability and elastic constants of YSb under pressure effect havebeen studied using scalar relativistic full-potential augmentedplane wave method by Bouhemadou and Khenata [7]. Thestructural and electronic properties of all yttrium compoundshave been also reported using FP-LAPW+lo method by Amrani andHaj Hassan [8]. Earlier Hasgawa [9] has performed one-electronenergy band structure calculations within local-spin-density-approximation exchange potential for YSb to predict its metalliccharacter. Very recently studies on elastic constants and structuralphase transition in yttrium pnictide compounds have beenreported by Bouhemadou [10] who has used pseudo-potentialplane wave approach within DFT.

Although the first principles-calculations can predict theelectronic properties of a wide class of rare earth compoundssatisfactorily [2–10], the interionic potential theories have beenfound more accurate to explain the structural phase transition andelastic properties of rare earth chalcogenides and pnictidecompounds [11]. It is well known that the electronic configurationin rare-earth atom hybridize between 5fn 6dm and 5fn�16dm+1

states, even at ambient condition. The interionic potential in thesetheories considers Coulomb screening effect on the atomic coredue to the hybridization of f-electrons of rare earth atomphenomenonlogically through a charge parameter Zm.

In the present paper, we use an interionic potential theory, asreferred above [11], to calculate self-consistently, the highpressure structural, elastic and thermal properties of yttriumpnictide compounds and compare them with available theoreticaland experimental data to judge the suitability of the potential and

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A. Singh et al. / Physica B 404 (2009) 1852–1857 1853

our predictions for these properties. The organization of the paperis as follows: the method of calculation of structural phasetransition pressure and model theory will be given in Section 2,while in Section 3 we present potentially interesting results onstructural, mechanical and thermal properties of yttrium pnic-tides

Table 1Input parameters and generated model parameters for yttrium pnictides.

Solids Input parameters Model parameters

r0 (A) BT (GPa) Zm2 b (�10�19J) r (A)

YN 2.4385a 162b 2.43 0.365 0.291

YP 2.8219b 87b 2.82 0.381 0.308

YAs 2.8832b 77b 2.88 0.374 0.314

YSb 3.0775c 62b 3.06 1.471 0.332

a Ref. [8].b Ref. [10].c Ref. [1].

2. Methodology

2.1. Calculation of structural phase transition and elastic properties

The interionic potential for the yttrium pnictides in theframework of the rigid ion model is expressed as

UðrÞ ¼X

ij

Z2me2

rijþX

ijbbij exp

½ðri þ rj � rijÞ�

rij

þX

ijCijr�6ij þ

XijDijr

�8ij (1)

which includes the long-range Coulomb interaction (first term),Hafemesiter and Flygare form of short range repulsive (secondterm) and van der Waals multipoles interactions (third and lastterm). Zme is the modified ionic charge and parametricallyincludes the effect of the Coulomb screening due the d-electronfor these compounds. b and p are short range parameters, whichcan be determined from the equilibrium condition.

dUðrÞ

dr

��������r¼r0

¼ 0 (2)

Thermodynamically, a phase transition is said to occur whenchange in the structural details of the phase are caused byvariation of the free energy. The yttrium pnictides transform fromtheir initial NaCl (B1) to CsCl (B2) type structure under pressure.The stability of a particular structure is decided by the minima ofthe Gibbs energy, given by

G ¼ U þ PV � TS (3)

where U is the internal energy at 0 K temperature correspondingto the cohesive energy, S is the vibrational entropy at absolute T,pressure P and volume V. The Gibbs free energies for NaCl (B1)structure is

GB1ðrÞ ¼ UB1ðrÞ þ 2r3P (4)

and

GB2ðr0Þ ¼ UB2ðr

0Þ þ8

3ffiffiffi3p

� �r03P (5)

for CsCl (B2) phase become equal at the phase transition pressureP and temperature 0 K, i.e. DG becomes zero.

DG ¼ GB1 � GB2 ¼ 0 (6)

Here the abbreviations UB1 and UB2 represent cohesive energiesfor the B1 and B2 phases respectively, and are written as

UB1ðrÞ ¼ �1:7475e2Z2

m

rþ 6FijðrÞ þ 6FiiðrÞ þ 6FjjðrÞ (7)

UB2ðr0Þ ¼ �1:7627

e2Z2m

r0þ 8Fijðr

0Þ þ 3Fiiðr0Þ þ 3Fjjðr

0Þ (8)

here r and r0 are the nearest neighbours (nn) separationcorresponding to NaCl- to CsCl-type structures, respectively. Theshort range potential for both phases between the ions are writtenas

FijðrÞ ¼ b exp�rij

r

� �� Cijr

�6ij � Dijr

�8ij ; i; j ¼ 1;2 (9)

For the calculation of transition pressure, we have minimized theGibbs free energies with respect to inter-atomic separation andcalculated GB1 and GB2 at various pressures [12].

The study of second order elastic constants (SOEC) C11, C12 andC44 and their pressure derivatives at 0 K is quite important forunderstanding of the inter-atomic force in solids. The secondorder elastic constants and their pressure derivatives can bederived from crystal potential energy by the method of thehomogeneous deformations as follows:

C11 ¼e2

4r40

!½�5:112Z2

m þ A12

þ 0:5ðA11 þ A22 þ B11 þ B22Þ� (10)

C12 ¼e2

4r40

!½�0:225Z2

m þ B12

þ 0:25ðA11 þ A12 � 1:25ðB11 þ B12ÞÞ� (11)

C44 ¼e2

4r40

!½2:556Z2

m þ B12

þ 0:25ðA11 þ A12 þ 0:75ðB11 þ B12ÞÞ� (12)

and bulk modulus is given by [13]

BT ¼13ðC11 þ 2C12Þ (13)

where Aij and Bij (ij ¼ 1, 2) are short-range force constant anduniquely determined from the relation

A11 ¼4r3

0

e2

� �d2FSR

ij ðrÞ

dr2

" #r¼r0

; B11 ¼4r3

0

e2

� �dFSR

ij ðrÞ

rdr

" #r¼r0

(14a)

A22 ¼4r3

0

e2

� �d2FSR

ij ðrÞ

dr2

" #r¼

ffiffiffiffiffiffi2r0

p; B22 ¼

4r30

e2

� �dFSR

ij ðrÞ

rdr

" #r¼ffiffi2p

r0

(14b)

where Fij is short range potential given by Eq. (9). The inputparameters and generated model parameters for yttrium pnictidecompounds are tabulated in Table 1.

2.2. Calculation of Debye temperature (yD)

The Debye temperature (yD) is an important physical para-meter of solids, which defines a demarkation line betweenquantum mechanical and classical behaviour of phonons. Wehave estimated Debye temperature of yttrium pnictides using thefollowing expression [14,15]:

yD ¼h

kB

3n

4pVa

� �1=3

sm (15)

where h is a Plank’s constant, kB is Boltzmann’s constant and Va isthe atomic volume and sm is average sound velocity. The average

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A. Singh et al. / Physica B 404 (2009) 1852–18571854

sound velocity of the polycrystalline material is given by [14,16]

sm ¼1

3

2

s3T

þ1

s3L

!" #�1=3

(16)

where sT and sL are the longitudinal and transverse soundvelocities expressed in terms of elastic constants as follows:

sL ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC11 þ

25ð2C44 þ C12 � C11Þ

� r

s(17)

sT ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC44 �

15ð2C44 þ C12 � C11Þ

� r

s(18)

here C11, C12 and C44 are second order elastic constants (see Eqs.(10)–(12)) and r is mass density per unit volume.

3. Results and discussion

3.1. Structural properties

The input crystal properties and calculated model parametersfor the interionic potential model of yttrium pnictides are given inTable 1. We present the calculated properties on lattice constants,cohesive energies, structural phase transition pressures andrelative volume change for yttrium pnictides for initial NaCl-and final CsCl-type structures in Table 2. We have also comparedour calculated results with the available theoretical [2–8] andexperimental [1] data of yttrium pnictide compounds.

It is seen from Table 1 that calculated values of modelparameters for interionic potential follow a systematic trend of

Table 2Cohesive energies and phase transition properties for yttrium pnictides.

Solids Equilibrium inter-ionic distance (A) Cohesive en

R1 (B1) R2 (B2) U1 (B1)

YN

Present 2.460 2.600 �3177.29

Expt. 2.438a

Other 2.465b 3.01b

2.385c 3.002e

2.425d

2.457e

YP

Present 2.830 3.00 �2655.31

Expt. –

Other 2.821f 3.455f

2.841e 3.437e

YAs

Present 2.89 3.070 �2450.22

Expt. –

Other 2.883f 3.543f

2.915e 3.582e

YSb

Present 3.090 3.240 �2147.55

Expt. 3.061g 3.53g

Other 3.07f 3.76f

2.425h 3.402e

a Ref. [8].b Ref. [5].c Ref. [3].d Ref. [3].e Ref. [8].f Ref. [10].g Ref. [1].h Ref. [7].

variation for yttrium pnictide compounds. The modified ioniccharge parameter Zm

2 is derived from the values of bulk modulus(BT) and the short-range hardness (b) and range parameters (r) donot differ much in the series. It is clear from Table 2 that thecalculated values of equilibrium lattice parameters in NaCl- andCsCl-type structures are in good agreement with availableexperimental [1] and theoretical [2–8] results. We have computedtotal energies in B1 and B2 phases upon compressions. It is foundthat the stable phase of yttrium pnictides at ambient condition isB1 phase, as the energy is minimum in B1 phase as compared to B2

phase for all yttrium pnictides. Under high pressure yttriumpnictides undergo a structural phase transformation from initialNaCl- to CsCl-type structure. Structural phase transition pressureof these yttrium compounds are listed in Table 2 and comparedwith the available theoretical and experimental values. In order toobtain reasonable agreement with other experimental andtheoretical values of transition pressure, we have assumed thationic charge parameter Zm

2, vary linearly with atomic number ofanion for the pnictide series. Such an approximation works wellfor these compounds and the slope of the curve is found to benegative, which is in accordance with the decreasing value of thebulk modulus and it is evident from Fig. 1. Since the parameter Zm

2

has been derived from calculated values of bulk modulus,therefore, it is easy to point out its variation in yttrium pnictidesseries is related to Kondo effect [17]. Equation of states for yttriumcompounds are presented in Figs. 2(a)–(d). In the case of YN, wehave predicted a transition pressure of 128 GPa with volumecollapse of 3.5%. There is no experimental evidence for compar-ison, but other theoretical studies reported by Mancera et al. [4],Amrani et al. [8] and Bouhemadou et al. [10] by usingPP–PW method have predicted PT, 138, 136.39 and 130.43 GPa,

ergy (KJ/mol) Phase transition

pressure (GPa), PT

Relative volume

collapse (%)

U2 (B2)

�3060.006 128 3.5

138b 1.7b

136.39e

�2552.69 66 4.9

63.8f 3.2f

55.94e

�2355.22 56 4.5

58.25f 2.57f

50.45e

�2079.857 24 6.9

26.3g

33.6f 3.4f

28.61e

28h

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respectively. For the similar phase transition in YP, YAs and YSb byour theory, the value of transition pressure are calculated to be 66,56 and 24 GPa with volume collapse of 4.9%, 4.5% and 6.9%,respectively. We could compare our results for YP, YAs and YSbwith others theoretical work [7,8,10], who have calculated (B1 toB2) transition pressure of 55.94 and 63.8 GPa for YP, 50.45 and58.25 for YAs and 28, 28.61 and 33.6 GPa for YSb. On the otherhand, experimental study for YSb only is reported by Hayashi et al.[1], in which they observed B1 to B2 phase transition at a pressure

Fig. 2. Equation of states for YN (a

Fig. 1. Variation of modified ionic charge parameter with atomic number of

yttrium pnictides.

of 26.3 GPa. Our calculated phase transition pressure for YSbdiffers by 7% from the experimental value.

3.2. Elastic properties

The elastic properties define the properties of materials, when itundergo stress, deform and then recover and return to its originalshape after stress ceases. These properties play an important role inproviding valuable information about binding characteristics be-tween the adjacent atomic plane, structural stability, specific heat,thermal expansion, Debye temperature and Gruneisen constant. Wehave calculated the elastic constants of yttrium pnictides at normalas well as at high pressure by using the methodology expressed inSection 2. In the case of cubic system, there are only threeindependent SOEC namely C11, C12 and C44. Since we use two-bodyinteraction potential between the species, the calculated values ofC12 and C44 are equal. The calculated values of SOEC are tabulated inTable 3 and compared with other theoretical values [10]. There is noexperimental observation available on elastic properties of thesepnictides. One can see, however, from Table 3 that the elasticconstants increase in magnitude as we go from -N to -Sb in the seriesof yttrium pnictides. The stability of these pnictides can also bedefined by the SOEC. We have compared our results to the stabilitycriteria [18,19] using the following relation:

C11 � C1240; C444B040

We have found that in NaCl-type structure, these criteria aresatisfied, indicating that this phase is elastically stable. We havecalculated the pressure dependence of the SOEC, namely C11 andC44 in Fig. 3 for all the yttrium pnictides. It is clear from the Fig. 3that C11 varies largely under the effect of the pressure as comparedwith C44. The elastic constant C11 represents the variation on thelength with pressure; a longitudinal strain is produced, which brings

), YP (b), YAs (c) and YSb (d).

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Fig. 3. Variation of elastic constants with pressure for yttrium pnictides.

Table 4Calculated values of density of mass density (r in kg/m3), the longitudinal,

transverse and average sound velocity (sL, sT and sm in m/s) and Debye temperature

(yD in K) for yttrium pnictides.

Solids r SL ST Sm yD

YN

Present 5.75 7029 4059 4458 856

Other 5.86a 7417a 4558a 5030a 973a

YP

Present 4.62 5815 3357 3697 617

Other 4.82a 5852a 3551a 3904a 673a

YAs

Present 5.52 5076 2930 3226 527

Other 5.67a 5072a 3034a 3357a 551a

YSb

Present 6.02 4152 2397 2639 403

Other 6.10a 4298a 2438a 2711a 419a

a Ref. [10].

Fig. 4. Variation of Debye temperature with pressure for yttrium pnictides.

Table 3Calculated values of elastic constants (C11, C12 and C44), bulk modulus (BT), Young

modulus (E), shear modulus (G) in (GPa), Poisson ratio (n) and anisotropic ratio (A)

for Yttrium pnictides.

Solids C11 C12 C44 BT E G n A

YN

Present 293 89 89 157 247 94 0.244 0.867

Other 314a 126a 83a 162b 291a 122a 0.196a 1.090a

YP

Present 170 47 47 88 147 53 0.227 0.770

Other 199a 47a 28a 87a 146a 60a 0.213a 0.543a

YAs

Present 156 41 41 79 137 47 0.218 0.711

Other 183a 39a 23a 77a 127a 52a 0.221a 0.487a

YSb

Present 116 29 29 57 104 35 0.200 0.67

Other 151a 24a 21a 62a 91a 34a 0.262a 0.369a

58b

a Ref. [10].b Ref. [1].

A. Singh et al. / Physica B 404 (2009) 1852–18571856

change in C11. On the other hand the elastic constant C44 is related tothe elasticity in shape of the compound, which is a shear constant; atransverse strain that causes a change in shape without change involume. Therefore, C44 is a less sensitive to pressure as compared toC11. As pressure increases in B1 phase, C11 also increases linearly butC44 decreases linearly. There is no experimental data available inliterature to compare our predictions. We have also calculated theYoung modulus (E), shear modulus (G), Poisson ratio (u) andanisotropic ratio (A) for yttrium pnictides as follows:

E ¼ðC11 � C12ÞðC11 þ 2C12Þ

C11 þ C12

G ¼C11 þ C12 þ C44

5

u ¼ C12

C11 þ C12

A ¼2C44

C11 � C12

and values are tabulated in Table 3.

3.3. Thermal property

Thermal property with respect to Debye temperature has beendiscussed for the yttrium pnictides and presented in Table 4. The

Debye temperature (yD) is obtained by incorporating the elasticconstants from present interionic potential theory. In Table 4, wehave also estimated the values of average sound velocity (sm),longitudinal sound velocity (sL) and transverse sound velocity (sT),density of mass (r) and compared our results with the otherstheoretical results and found a good agreement. One can under-stand the decreasing trend of yD in the four yttrium pnictides bylooking the trend of variation of elastic constants under ambientcondition. We have also calculated the variation of yD with respectto pressure for all the four yttrium pnictides and predicted themin Fig. 4. It is clear from the figure that yD increases as pressureincreases for all the yttrium pnictides, which show quite normalbehaviour in this kind of compounds.

4. Conclusion

A two-body inter-ionic potential is formulated to analyse thestructural as well as elastic properties of yttrium pnictides. Weidentify the structural phase transformation from NaCl- to CsCl-type structure through pressure–volume relationship. From ourcalculated results it can be emphasized that the present approachreproduces the structural properties at high pressure consistently,in terms of the Coulomb screening effect through the modified

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A. Singh et al. / Physica B 404 (2009) 1852–1857 1857

ionic-charge parameter. An immediate consequence of our latticemodel is the Born criterion for crystal stability depicted by theelastic constants. Our calculated high pressure properties are ingood agreement with the available experimental and theoreticalresults. We are not aware of any experimental data on the elasticand thermo dynamical properties of yttrium pnictides at highpressure.

Acknowledgements

The authors are thankful to Madhya Pradesh Council of Scienceand Technology (MPCST) for the financial support. V. Srivastava isthankful to DST, New Delhi, for the financial support under FastTrack scheme. S.P.S. and A.S. are thankful to CSIR, New Delhi, forpartial financial support.

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