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Solid State Sciences 12 (2010) 864–872

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Solid State Sciences

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

Pressure induced mechanical properties of boron based pnictides

Dinesh Varshney a,b,*, G. Joshi a, Meenu Varshney c, Swarna Shriya a

a School of Physics, Vigyan Bhawan, Devi Ahilya University, Khandwa Road Campus, Indore 452001, Indiab School of Instrumentation, USIC Bhawan, Devi Ahilya University, Khandwa Road Campus, Indore 452001, Indiac Department of Physics, M.B. Khalsa College, Indore 452002, India

a r t i c l e i n f o

Article history:Received 14 October 2009Received in revised form27 January 2010Accepted 2 February 2010Available online 8 February 2010

Keywords:PnictidesComputational techniquesDuctilityMechanical propertiesPhase transitions

* Corresponding author at: School of Physics, VUniversity, Khandwa Road Campus, Indore 452001, In

E-mail addresses: vdinesh33@rediffmail.com,Varshney).

1293-2558/$ – see front matter � 2010 Elsevier Massdoi:10.1016/j.solidstatesciences.2010.02.003

a b s t r a c t

The elastic and thermodynamical properties of the III–V semiconductors as BY (Y¼N, P, As) are calcu-lated in zincblende and NaCl phases by formulating an effective interionic interaction potential. Thispotential consists of the long-range Coulomb, the Hafemeister and Flygare type short-range overlaprepulsion extended up to the second neighbour ions and the van der Waals (vdW) interaction. Thevariations of elastic constants with pressure follow a systematic trend identical to that observed in othercompounds of ZnS type structure family and the Born relative stability criteria is valid in boronmonopnictides. From the elastic constants the Poisson’s ratio n, the ratio RS/B of S (Voigt averaged shearmodulus) over B (bulk modulus), elastic wave velocity, average wave velocity and thermodynamicalproperty Debye temperature are calculated. By analyzing the Poisson’s ratio n and the ratio RS/B weconclude that at low pressures the boron monopnictides are brittle in nature in ZnS phase and ductilenature at high pressures in both ZnS and NaCl phases. To our knowledge this is the first quantitativetheoretical prediction of the pressure dependence of ductile (brittle) nature of BY compounds.

� 2010 Elsevier Masson SAS. All rights reserved.

1. Introduction

BN material is extremely hard and has very high meting point. Itdoes not dissolve in iron and steel, so it could be used for coating ofhigh duty tools. BP is one of the promising III–V semiconductors inthe zincblende structure. Technological interest in it has beenstimulated in recent years by its potential use in optoelectronic andmicroelectronic devices working under extreme conditions. BAs isof particular interest as it has the Phillips lowest ionicity fi¼ 0.002and shares the lowest Pauling ionicity fi¼ 0.00 with BP [1]. Theseunique properties of the III–V boron compounds are due to theirsmall core size and the absence of p electrons in the cores of theatoms, which belong to the second row of the periodic table.

Using generalized gradient approximation (GGA) Hassan et al. [2]calculated the transition pressures for BN (BP, BAs) as 624 (128,95) GPa respectively. Wentzcovitch et al. showed the phase transi-tion of these compounds from zincblende to rocksalt at 1110 (160,110) GPa by employing the total-energy pseudopotential techniquewithin the local-density approximation [3]. Such high-pressuretransitions cannot be achieved experimentally. The modelling oflattice models in III–V compounds is a complicated task and, in many

igyan Bhawan, Devi Ahilyadia.vdinesh33@gmail.com (D.

on SAS. All rights reserved.

instances, must be guided by experimental evidence of the lowdegree of freedom in order to obtain a correct minimal model, whichwill capture the observed effect and will make useful predictions. Incomparison to the other theoretical models phenomenologicallattice models have proven very successful in obtaining qualitativeand quantitative understanding with proper parameterisation of theinput parameters.

To discuss the mechanical properties of several solids and alloyswe have used charge transfer approach [4], following Hafemeisterand Flygare [5] type overlap repulsion extended upto second-neighbour ions besides short-range interactions. We refer to thepioneering work of Fumi and Tosi [6], who properly incorporate vander Waals interaction along with d–d and d–q interactions to revealthe cohesion in several semiconductor solids. In trying to under-stand the structural aspects, we admit that the vdW attractions arethe corner stone of lattice phenomenological models that is ignoredin the first principle microscopic calculations.

We are motivated from the remarks on phases of BY (Y¼N, P,AS) compounds [1–3] and the charge transfer effect approach [4]for the successful description of the phase transition and high-pressure behaviour of binary compounds, we thought it is pertinentto employ an effective interionic interaction potential that includesvdW attraction, which is not explicitly accounted in band structurecalculation in boron compounds. It is worth noting that vdWinteraction appears to be effective in revealing the elastic andstructural properties of III–V compounds. While computing the

D. Varshney et al. / Solid State Sciences 12 (2010) 864–872 865

pressure and structure dependent mechanical properties asductility (brittleness) and wave velocities of B pnictides, we need tofirst understand the pressure dependent structures and thus thepressure dependent structural phase transition is of importance.

The main focus of the present investigation is to discuss thepressure dependent elastic properties as ductility (brittleness),elastic anisotropy and sound velocity of B pnictides as no systematicefforts have been made so far. To our knowledge this is the firstattempt to reveal the pressure dependence of ductile (brittle) natureof BY compounds and still awaits experimental confirmations. Thepresent paper is organized as follows. The theory and technicaldetails to estimate the pressure dependent structure and elasticproperties are given in Section 2. In a next step, the phase-transitionpressures, the elastic constants, wave velocities and Debye temper-ature are deduced within the framework of Shell model that incor-porates the long-range Coulomb, van der Waals interaction, and theshort-range overlap repulsive interaction up to second-neighbourions within the Hafemeister and Flygare approach. In Section 3, wepresent our results and the comparison with the experimental andother theoretical studies. Finally, conclusions are presented inSection 4.

2. Theory and method of computation

Pressure dependent mechanical properties as ductility and brit-tleness, longitudinal and transverse velocity and thermodynamicalproperty as Debye temperature of BY compounds are important andcan be known from second-order elastic constants. In doing so, wefirst need to evaluate pressure dependent structural properties. Theidea we have in mind 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 harmonicelastic forces without any internal strains within the crystal.

An isolated phase is stable only when its free energy is mini-mized for the specified thermodynamic conditions. As thetemperature or pressure or any other variable acting on the systemsis altered, the free energy changes smoothly and continuously. Aphase transition is said to occur when the changes in structuraldetails of the phase are caused by such variations of free energy. Thetest materials transform from their initial B3 to B1 structure underpressure. The stability of a particular structure is decided by theminima of Gibbs’ free energy, G¼Uþ PV� TS, U is internal energy,which at 0 K corresponds to the cohesive energy, S is the vibrationalentropy at absolute temperature T, pressure P and volume V.

The Gibbs’ free energies GB3(r)¼UB3(r)þ 3.08Pr3 for ZnS (B3)phase and GB1(r0)¼UB1(r0)þ 2Pr03 for NaCl (B1) phase become equalat the phase-transition pressure P and at zero temperature i.e.,GB1¼GB3. Here, UB3 and UB1 represent cohesive energies for B3 andB1 phases, and are

UB3ðrÞ ¼ �aMZ2e2

r�X

ij

Cij

r6ij

�X

ij

Dij

r8ij

þbX

ij

bijexp�

riþ rj� rij

r

�

þbX

ii

biiexp�

2ri�krij

r

�þb

Xjj

bjjexp�

2rj�krij

r

�ð1Þ

UB1ðr0Þ ¼ �a0MZ2e2

r0�X

ij

C0ijr06ij�X

ij

D0ijr08ijþb

Xij

bijexp

riþ rj� r0ij

r

!

þbX

ii

biiexp

2ri�k0r0ij

r

!þb

Xjj

bjjexp

2rj�k0r0ij

r

!ð2Þ

Here the first two term is the potential energy due to long-rangeCoulomb interaction. The second and third terms are the

short-range (SR) vdW attraction energies due to dipole–dipole anddipole–quadrupole interaction. The C and D are the overall vdWcoefficients which are evaluated from the variational approach [7].

The vdW coefficients due to dipole–dipole and dipole–quadruple interactions are calculated from the Slater and Kirkwoodvariational approach and are

cij ¼32

eZffiffiffiffiffiffiffimep aiaj

"�ai

Ni

�1=2

þ

aj

Nj

!1=2#�1

; (3)

dij ¼278

Z2

meaiaj

"�ai

Ni

�1=2

þ

aj

Nj

!1=2#2

"�ai

Ni

�þ 20

3

aiaj

NiNj

!1=2

þ

aj

Nj

!#�1

: ð4Þ

cii ¼32

eZffiffiffiffiffiffiffimep aiai

��ai

Ni

�1=2

þ�

ai

Ni

�1=2��1

; (5)

dii ¼278

Z2

meaiai

��ai

Ni

�1=2

þ�

ai

Ni

�1=2�2

��ai

Ni

�þ 20

3

�aiai

NiNi

�1=2

þ

ai

Ni

!��1

ð6Þ

cjj ¼32

eZffiffiffiffiffiffiffimep ajaj

" aj

Nj

!1=2

þ

aj

Nj

!1=2#�1

; (7)

djj ¼278

Z2

meajaj

" aj

Nj

!1=2

þ

aj

Nj

!1=2#2

" aj

Nj

!þ 20

3

ajaj

NjNj

!1=2

þ

aj

Nj

!#�1

: ð8Þ

where m and e are the electron mass and charge respectively, (ai, aj)are the polarizabilities of ith and jth ion respectively; Ni and Nj arethe effective number of electrons responsible for polarization. Theoverall vdW coefficients C and D are then expressed in terms of cij

and dij, by a appropriate lattice sums (Sij, and Tij) the values of whichare taken from [6] such that and the expression is given by,

C ¼ cijSij þ ciiSii þ cjjSjj (9)

D ¼ dijTij þ diiTii þ djjTjj (10)

The last term is SR repulsive energy due to the overlap repulsionbetween ij, ii and jj ions. am (am

0) are the Madelung constants for B3(B1) phases. The bij is the Pauling coefficient defined as bij¼ 1þ (Zi/ni)þ (Zj/nj) with Zi(Zj) and ni(nj) as the valence and the number ofelectrons in the outermost orbit. Ze is the ionic charge, k(k0) is beingthe structure factor for B3 (B1) structures, and b(r) are the hardness(range) parameters. r(r0) are the nearest neighbour ion separationsfor B3 (B1) structures.

In the absence of any barrier, a transition occurs when the ther-modynamical potential relevant to the given ensemble of the lower-pressure phase equals that for some other structure, which becomesthe stable phase above this coexistence pressure. As a next step, wehave investigated the pressure variations of the second-order elasticconstants (SOECs). The relevant expressions for the SOECs and thepressure derivatives of SOECs are given in Appendix. We shallnow compute numerically the high-pressure phase transition, the

Table 2Input crystal data and model parameters for boron compounds.

Compound Input parameters Model parameters

ri (Å) rj (Å) a0 (Å) BT (GPa) Zm2 b (10�12 erg) r (10�1 Å)

BN 1.06 0.3 3.615 [1] 369 [10] 1.96 2.31 2.23BP 1.06 0.35 4.53 [1] 173 [11] 2.79 8.432 3.0BAs 1.06 0.26 4.78 [1] 145 [3] 3.26 18.77 2.85

-2700

-2400

-2100

-1800

-1500

-1200

-900

-600

-300

Pt

a

b

BP B3 B1

Gib

b's

Free

Ene

rgy

(KJ/

mol

e)

P (GPa)0 50 100 150 200 250

-2205

-1890

-1575

-1260

-945

-630

-315

Pt

BAs B3 B1

ree

Ener

gy (K

J/m

ole)

D. Varshney et al. / Solid State Sciences 12 (2010) 864–872866

associated volume collapses, elastic properties and thermodynam-ical properties of BY (Y¼N, P and As) compounds for B3 and B1phases in the next section.

3. Results and discussion

Knowledge of the force constants is important parameter forfinding out the stability of the different structures at differentvolumes. Two different factors determine the response of anycrystal structure to pressure. First, changes in nearest-neighbourdistances, which affect the overlaps and bandwidths of the bands.Second, changes in symmetry, which affects the hybridization andbond-repulsion. The formalism described in Appendix is applied toboron compounds belonging to the cubic crystal system. We haveundertaken such structural and elastic properties in an orderedway.

The thermodynamical potential G or H have been computedusing the values of the three material dependent parametersnamely, modified ionic charge (Zm), range (r) and hardness (b)which have been evaluated from the equilibrium condition�����dUðrÞ

dr

�����r¼r0

¼ 0 (11)

and the bulk modulus (BT):�����d2UðrÞdr2

�����r¼r0

¼ ð9kr0Þ�1BT (12)

The values of vdW coefficients C and D involved in Eqs. (1) and(2) have been evaluated from the Slater–Kirkwood variationalmethod [7] using electronic polarizabilities of B, N, P, and As aretaken from references [8,9] and are listed in Table 1. The input dataequilibrium distance (r0) [1], the bulk modulus (BT) [3,10,11] and thededuced model parameters are given in Table 2.

In an attempt to reveal the structural phase transition of thetest materials, we minimise the Gibbs’ free energies GB3(r) andGB1(r0) for the equilibrium interatomic spacing (r) and (r0). Upto250 GPa there is no phase transition occurred in BN compound,whereas BP and BAs show the first-order phase transition.Fig. 1(a–b) shows Gibbs’ free energies GB3(r) and GB1(r0) as func-tions of pressure (P) using the interionic potential discussed abovefor BP and BAs compounds. At zero pressure, the Gibbs’ freeenergy for B3 crystal phase is more negative therefore it is ther-modynamically and mechanically stable, while the B1 is not. Aspressure increases, beyond the phase-transition pressure (Pt), theGibbs’ free energy for B1 system becomes more negative than B3phase, so B1 will be more stable. In boron compounds a crystal-lographic transition from B3 to B1 occurs in certain pressure range.The phase-transition pressure (Pt) thus obtained for BP and for BAsis 160 and 110 GPa respectively. The transition pressures for BPand BAs compounds are consistent with the known theoreticalpressures. We must mention that for BN the transition pressure is1110 GPa [2,3], which is difficult to obtain theoretically usingcharge transfer approach.

Let us now estimate the values of relative volumes associatedwith various compressions following Murnaghan equation of state[12].

Table 1van der Waals coefficients of BY (Y¼N, P and As) (cij in units of 10�60 erg cm6 and dij

in unit of 10�76 erg cm8). C and D are the overall van der Waals coefficients.

Compound cii cij cjj C dii dij djj D

BN 0.396 3.064 40.889 29.075 0.047 1.342 22.835 8.37BP 0.396 3.256 47.759 32.528 0.047 1.509 28.089 9.715BAs 0.396 3.971 79.748 47.911 0.047 2.225 55.831 16.148

VV0¼�

1þ B0

B0P��1=B0

; (13)

with V0(B0) is being the cell volume (bulk modulus) at ambientconditions and B0 is the pressure derivative of the bulk modulus.The estimated value of pressure dependent radius for both struc-tures, the curve of volume collapse with pressure to depict thephase diagram are illustrated in Fig. 2(a–b) for BP and BAs. Themagnitudes of the discontinuity in volume at the transition pres-sure are 2.7% and 4.2% respectively.

The elastic properties define the properties of material thatundergoes stress, deforms and then recovers and returns to itsoriginal shape after stress ceases. These properties play an importantpart in providing valuable information about the binding charac-teristic between adjacent atomic planes, anisotropic character ofbinding and structural stability. Hence, to study the stability of thesecompounds in ZnS (B3) and NaCl (B1) structures, we have calculatedthe elastic constants at normal and under hydrostatic pressure byusing developed effective interionic potential. Fig. 3(a–c) shows thevariation of second-order elastic constants (SOECs) with pressure inB3 phase for BN and in B3 and B1 phase for BP and BAs, respectively.

0 50 100 150 200 250

-2835

-2520

Gib

b's

F

P (GPa)

Fig. 1. (a–b): Variation of Gibbs’ free energy for B3 and B1 phases with pressure.

0 5 0 1 00 150 200 250

0. 6

0. 7

0. 8

0. 9

1. 0

a

b

B1 B3

BA s

Volu

me

Col

laps

e

0 5 0 100 15 0 2 00 25 0

0. 6

0. 7

0. 8

0. 9

1. 0

B1 B3

BP

Volu

m e

Col

laps

e

P( GP a)

P(GPa)

Fig. 2. (a–b): Equation of state of BY (Y¼N, P and As) compound.

0

4

8

1 2

1 6

2 0a

b

c

B 3 P h a s e

B N C

1 1

C1 2

C4 4

Cji

01(11

mN

2-)

0

4

8

1 2

1 6

2 0

2 4

2 8

B 1B 3

B P C

1 1

C1 2

C4 4

Cji

01(11

mN

2-)

0 50 1 00 1 50 2 00 2 500

3

6

9

1 2

B 1B 3

B A s C1 1

C1 2

C4 4

Cji

01(11

mN

2-)

P (G P a )

Fig. 3. (a–c): Variation of second-order elastic constants with pressure for BY (Y¼N, Pand As) compound.

D. Varshney et al. / Solid State Sciences 12 (2010) 864–872 867

We must mention that the elastic constant C11 is a measure ofresistance to deformation by a stress applied on (1,0,0) plane withpolarization in the direction C100D, and the C44 refers to themeasurement of resistance to deformation with respect toa shearing stress applied across the (100) plane with polarization inthe C010D direction. Henceforth, the elastic constant C11 representselasticity in length and a longitudinal strain produces a change inC11. No doubt, the elastic constants C12 and C44 are intimately relatedto the elasticity in shape, which is a shear constant. However,a transverse strain causes a change in shape without a change involume. Thus, the second-order elastic constants as C12 and C44 areless sensitive of pressure as compared to C11. We notice that thepressure dependent behaviour of second-order elastic constants ofBAs in B1 phase differs from the BP behaviour. The above is attrib-uted to the less stability of BAs in NaCl structure due to largedifference of cohesive energy for BAs as compared to BP as evidentfrom Fig. 1. The same is reflected in the pressure dependentbehaviour of Bulk modulus (please see Fig. 4).

In B3 and B1 phases, C11 and C12 increases linearly with pressureand C44 decreases with increasing pressure, away from zero at thephase-transition pressures. We note that C44 shows a change in theslope as functions of near to transition pressure in B3 phase, while tothat an increase with increasing pressure, away from the phase-transition pressures in B1 phase is noticed. At transition pressures,BP and BAs have shown a discontinuity in SOECs, which is inaccordance with the first-order character of the phase transition. Itis useful to mention that the Born criterion for a lattice to be in themechanically stable states is that the elastic energy density must be

a positive definite quadratic function of strain. This requires that theprincipal minors (alternatively the eigenvalues) of the elasticconstant matrix should all be positive. Further the stability of a cubiccrystal is expressed in terms of elastic constants as follows [13]:

BT ¼ ðC11 þ 2C12Þ=3 > 0; (14)

C44 > 0; (15)

and

CS ¼ ðC11 � C12Þ=2 > 0: (16)

Estimated values of bulk modulus (BT), shear moduli (C44) andtetragonal moduli (CS), well satisfied the above elastic stabilitycriteria for boron compounds. The calculated values of second-order elastic constants, bulk modulus (BT), tetragonal moduli (CS),the pressure derivatives of second-order elastic constants (dBT/dP,dC44/dP and dCS/dP) for B3 phase are given in Table 3 and are alsocompared with available other experimental [10,11,14–16] andtheoretical results [2,17–31], the deviations might be ascribed tothe zero point motion effects.

The variation of bulk modulus with pressure is shownin Fig. 4(a–c).The figures show linear increase in bulk modulus with increase inpressure even after transition pressure, reveal that the structurebecomes more harder and more strengthens. Furthermore, Vukcevich[32] proposed a high-pressure stability criterion, combiningmechanical stability with minimum energy conditions. In accordance,

4

6

8

1 0

1 2

1 4

1 6

1 8

B 3 P h a s e

B Na

01( TB11

mN

2-)

4

8

1 2

1 6

2 0 B Pb

B 1B 3

01( TB11

mN

2-)

0 50 1 0 0 1 5 0 2 0 0 2 5 0

3

6

9 B A sc

B 1B 3

01( TB11

mN

2-)

P ( G P a )

Fig. 4. (a–c): Variation of bulk modulus with pressure.

Table 3Second-order elastic constants (C11, C12 and C44), bulk modulus (BT), tetragonalmoduli (CS) (all are in 1011 Nm�2) and pressure derivatives of SOECs (dBT/dP, dC44/dPand dCS/dP) for Boron compounds in B3 phase.

BN BP BAs

C11

Present 9.478 4.07 4.1Theoretical 8.37 [17] 8.30 [18] 3.59 [20] 3.60 [19]

3.56 [21]2.79 [19] 2.914 [22]2.95

9.90 [19] 8.44 [20] 3.589 [22] 3.57 [23]3.37 [23]

[23] 2.75 [23]2.51 [2]

8.178 [22] 7.51 [2] 3.29 [2]Experimental 3.15 [11]

7.12 [14] 8.20 [15]

C12

Present 2.121 1.66Theoretical 1.51 0.81 [20] 1.55 [19]

0.68 [21]1.20 [19] 0.728 [22]0.78

1.82 [17] 4.20 [18] 0.806 [22] 0.87 [23]0.78 [23]

[23] 0.63 [23]0.798 [2]

4.41 [19] 1.90 [20] 0.975 [2]Experimental 1.796 [22] 1.765 [2] 1.00 [11]

1.90 [15]

C44

Present 24.89 0.55 0.51Theoretical 4.93 [17] 4.50 [18] 2.02 [20] 1.46 [19]

2.05 [21]1.13 [19] 1.579 [22]1.77

3.94 [19] 4.83 [20] 1.967 [22] 1.50 [23]2.00 [23]

[23] 1.50 [23]1.27 [2]

4.699 [22] 4.36 [2] 1.54 [2]Experimental 4.80 [15] 1.60 [11]

BT

Present 4.16 2.77 2.48Theoretical 3.86 [20] 3.67 [24] 1.72 [20] 1.64 [21]

1.77 [23]1.52 [23] 1.50 [23]1.33 [23]

3.95 [28] 3.97 [29] 1.76 [23] 1.64 [23]1.60 [23]

1.34 [23] 1.45 [24]1.44 [26]

3.70 [30] 3.92 [31] 1.65 [24] 1.66 [24]1.60 [23]

1.37 [2]

3.68 [2] 1.70 [2]Experimental 3.69 [11] 1.73 [11], 2.67 [16]

CS

Present 3.98 0.97Theoretical 1.39 [20] 1.44 [21]

1.16 [2]Experimental 1.08 [11]

dBT/dPPresent 4.489 4.28 5.02Theoretical 3.65 [28] 3.59 [29] 3.68 [23] 4.02 [23]

3.07 [25]3.65 [23] 3.75 [23]4.00 [26]

3.60 [24] 3.80 [30] 3.76 [26] 3.70 [27] 3.49 [25]3.31 [31]

Experimental 4.00 [10] 3.15 [16]

dC44/dPPresent 1.936 11.75 15.34

dCS/dPPresent 0.019 �0.00004 0.107

D. Varshney et al. / Solid State Sciences 12 (2010) 864–872868

the stable phase of the crystal is one inwhich the shearelastic constantC44 is nonzero (for mechanical stability) and which has the lowestpotential energy among the mechanically stable lattices.

On the other hand, C44 is a small quantity, the calculated value of[(4r0/e2)C44� 0.556Zm

2 ] is found to be a negative quantity so that(A2� B2) is negative. This suggests that these terms belong to anattractive interaction and possibly arise due to the van der Waalsenergy. The van der Waals energy converges quickly, but the overlaprepulsion converges much more quickly. This means that the second-neighbour forces are entirely due to the van der Waals interactionand the first neighbour forces are the results of the overlap repulsionand the van der Waals attraction between the nearest neighbors.However, at high pressure the short-range forces for thesecompounds increase significantly, which, in turn, is responsible forchange in the coordination number and phase transformation. Otherthan deriving the equation of states correctly from a model approachand then to analyze the variation of short-range forces, at present wehave no direct means to understand the interatomic forces at highpressure.

Furthermore, the mechanical properties as ductility and brit-tleness of BY (Y¼N, P, AS) pnictides are of substantial importanceand can be known from second-order elastic constants. A simplerelationship, empirically linking the plastic properties of materialswith their elastic moduli has been mentioned [33]. The thermo-dynamical property as the Voigt averaged shear modulus S repre-senting the resistance to plastic deformation, while the bulkmodulus B represents the resistance to fracture. Accordingly, theratio RS/B< 0.5, the material behaves in a ductile manner, otherwisethe material behaves in a brittle manner. From Fig. 5(a–c) the ratio

RS/B shows the ductile nature of BY (Y¼N, P, As) in ZnS phase andBP, BAs show the ductile nature in NaCl phase also.

We may also refer to Frantsevich [34] who distinguish theductility and brittleness of materials in terms of Poisson’s ratio.According to Frantsevich rule the critical value of a material to bebrittle (ductile) is 0.33. For brittle materials, the Poisson’s ratio is lessthan 0.33; otherwise the material behaves in a ductile manner. Thepressure dependency of Poisson’s ratio according to Frantsevich ruleis illustrated in Fig. 5(a–c). Accordingly, for BN compound, Fig. 5(a)shows that it is of brittle in nature from 0 to 160 GPa and is ductilewith an increase in pressure up to 250 GPa. For BP (BAs), theFig. 5(b, c) illustrates that in the pressure range [0� P� 40(60) GPa],

0 .2 8

0 .2 9

0 .3 0

0 .3 1

0 .3 2

0 .3 3

0 .3 4

0 .3 5

R

ν

ν

ν

Poi

ssio

n's

ratio

(ν)

Poi

ssio

n's

ratio

(ν)

Poi

ssio

n's

ratio

(ν)

0 . 3 5

0 .4 0

0 .4 5

0 .5 0

0 .5 5B N

B 3 P h a s e

Inverse ratio (R)

Inverse ratio (R)

Inverse ratio (R)

0 . 2 8

0 .3 2

0 .3 6

0 .4 0

0 .4 4

R

0 .2

0 .3

0 .4

0 .5B P

B 1B 3

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0

0 .2 7

0 .3 0

0 .3 3

0 .3 6

0 .3 9

R

P ( G P a )

0 .3

0 .4

0 .5

0 .6B A s

B 1B 3

Fig. 5. (a–c): Variation of Poisson’s ratio n and ratio RS/B with pressure.

5 0 0 0

1 0 0 0 0

1 5 0 0 0

2 0 0 0 0

2 5 0 0 0

Vt

Vl

B 1B 3

B P

cesm(

yticoleveva

wcitsalE

1-)

1 0 0 0 0

1 5 0 0 0

2 0 0 0 0

2 5 0 0 0a

b

c

Vt

Vl

B 3 P h a s e

B N

cesm(

yticoleveva

wcitsalE

1-)

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0

3 0 0 0

6 0 0 0

9 0 0 0

1 2 0 0 0

Vt

Vl

P ( G P a )

B 1B 3

B A s

cesm(

yticoleveva

wcitsalE

1-)

Fig. 6. (a–c): Pressure dependence of elastic wave velocity vl and vt.

D. Varshney et al. / Solid State Sciences 12 (2010) 864–872 869

the compounds are brittle in ZnS phase and further increase inpressure [>40(60) GPa], the ductile behaviour in both B1 and B3phase exists. We can observe that there is a contradiction inbetween Pugh [33] and Frantsevich [34] empirical rules.

The Poisson’s ratio n in terms of the bulk modulus B and theVoigt averaged shear modulus S as [35]

n ¼ 12

�3

BS� 2��

3BSþ 1

��1

(17)

It follows that the empirical Pugh’s critical value corresponds ton¼ 0.26, so that both the Pugh and Frantsevich empirical rules onlydiffer on the exact border between the two types of behaviour.Therefore, as per the empirical Pugh’s rule, the boron monopnic-tides that are semiconducting compounds are ductile in nature at0 GPa in ZnS phase and for high pressures in NaCl phase.

We may add that the value of the ratio RS/B> 0.5 is used for theductile behaviour of intermetallics as MgCNi3, otherwise thematerial behaves in a brittle manner [35]. The pressure dependenceof the ratio RG/B is further illustrated in Fig. 5(a–c). It is evident thatat low pressures, i.e., pressure less than 35 GPa for BN, pressure lessthan 20 GPa for BP, and pressure less than 40 GPa for BAs the ratioof RS/B is large than 0.5, which means materials behaviour is ina brittle manner. The critical value of RS/B¼ 0.5 which separatesductile and brittle nature of materials is one criterion not thesufficient condition.

The average wave velocity vm has been approximately calculatedfrom

vm ¼"

13

2v3

t

þ 1v3

l

!#�13

(18)

where vl and vt are the longitudinal and the transverse elastic wavevelocity respectively, which are obtained from Navier’s equation inthe following forms:

vl ¼�

3Bþ 4S3r

�12

(19)

vt ¼�

S�1

2

(20)

r

where S is the Voigt averaged shear modulus, B is the bulk modulus,and r is the density.

The pressure dependence of the sound velocity is documentedin Fig. 6(a–c) respectively. It is noticed that in B3 phase an increasedbehaviour of vl is clearly seen. The transverse elastic wave velocity,vt in B3 phase for BN shows saturating nature, while pressureslarger than 50 GPa for BP, and pressure larger than 75 GPa for BAs,a change in slope is noticed. Both vl and vt show a increasing trendaway from transition pressure in B3 phase. Saturating trend of vt isseen more in BAs as compared to BP in B1 phase. The values oflongitudinal, transverse and average sound velocities are given inTable 4 at zero temperature and pressure.

Apart from elastic constants, we have investigated variousimportant physical properties like force constant (f), Gruneisenparameter (g), Reststrahlen frequency (n0), compressibility (b) andthe Debye Temperature (qD). The relevant expressions used [36] inour calculations are given below. The molecular force in theabsence of the Lorentz effective field is given by

Table 4Calculated values of wave velocities and thermodynamical properties of Boroncompounds with zincblende structure.

Properties BN BP BAs

Longitudinal wave velocity vl (m/s) 14,100 11,300 8350Transverse wave velocity vt (m/s) 6950 4990 4500Average wave velocity vm (m/s) 1660 1200 948.57Debye temperature qD (K) 750 454.5 329.19Force constant (f) (105 dyne/cm) 7.03 4.63 4.96Reststrahlen frequency (n0) (1012 Hz) 50.16 35.5 33.85Gruneisen parameter (g) 2.03 1.78 1.921Compressibility (b) (10�11 Pa�1) 0.33 0.66 0.62

D. Varshney et al. / Solid State Sciences 12 (2010) 864–872870

f ¼ 13

"d2

dr2USRðrÞ �2r0

ddr

USRðrÞ#

r¼r0

; (21)

which consists of the SR overlap repulsion and the van der Waalsinteraction potentials between the unlike ions. The force constantin turn gives the Reststrahlen frequency as

n0 ¼1

2p

�fm

�12

; (22)

with m is being the reduced mass.In order to describe the anharmonic properties of a crystal, we

have calculated g from the relation

g ¼ �r0

6

�U000ðr0ÞU00ðr0Þ

�(23)

and compressibility

7 0 0

8 0 0

9 0 0

10 0 0

11 0 0

12 0 0a

b

c

B 3 P h a s e

B N

D(K)

4 2 0

5 6 0

7 0 0

8 4 0

9 8 0

11 2 0

12 6 0

B 1B 3

B P

D(K)

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0

3 0 0

3 5 0

4 0 0

4 5 0

5 00

B 1B 3

BAs

D(K)

P (G Pa )

Fig. 7. (a–c): Variation of Debye Temperature (qD) with pressure.

b ¼"

r20 fU00ðr Þg

#�1

(24)

9V 0

The above thermodynamic parameters are listed in Table 4. Wehave used the effective interionic potential to predict successfullythe elastic and anharmonic properties of semiconductingcompounds under consideration.

Apart from phase transition and pressure dependence of SOECs,we also estimate Debye temperature (qD) from the presentapproach. We define

q3D ¼

3:158p

�hkB

�3� rM

32ðC11�C12Þ

12ðC11þC12þ2C44Þ

12C

1244; (25)

where M is the acoustic mass of the compound. To explain thevariation of qD with the pressure, we attempt to analyze ourresults in the framework of dynamics of lattice with pressure. Thechange in the force constants induced by pressure increases qD butin mid-pressure range qD starts decreasing, again after transitionpressure for BP and for BAs, it starts increasing which drives thesystem effectively toward the hardening of lattice with increasingpressure. However, we do not claim the process to be rigorous, buta consistent agreement following EIoIP is obtained on Debyetemperature as those revealed from experiments. The availablecalculated values of Debye temperature at 0 K for BY (Y¼N, P, As)compounds are given in Table 4 and is plotted in Fig. 7(a–c). Thisresult motivates us for the definition of an ‘average’ elasticconstant as

C ¼�

8p

3:15

�23�

kB

h

�2�Mr

�q2

D; (26)

which in turn is calculated as 2.49, 1.25, 1.28 (1011 Nm�2) from theDebye temperature allow us to correlate Cauchy discrepancy inelastic constant following

C* ¼ C12 � C44

C12 þ C44; (27)

at zero pressure. We have observed positive Cauchy discrepancy0.37, 0.59, 0.53 for BY compounds. It is worth to mention that theIII–V semiconductors with zincblende structure (B3 to B1 structuralphase transition) [37–40] and most of the body centered cubictransition metals shows a positive Cauchy deviation C*.

4. Conclusion

The present study addresses for the first time, the pressuredependent ductile and brittle nature of III–V compound BY (Y¼N, Pand As) by formulating an effective interionic interaction potential.The obtained values of free parameters allow us to predict phase-transition pressure and associated volume collapse. As a first step,we exercise for identifying the pressure dependent structures. Wehave found the volume discontinuity in pressure volume phasediagram identifies the structural phase transition from ZnS (B3) toNaCl (B1) structure. It can be emphasized that the present approachreproduced the structural properties at high pressure consistently,in terms of the screening of the effective Coulomb potentialthrough modified ionic charge (Zm

2 ).An immediate consequence of our lattice model calculations is

the validity of Born criterion. The second-order elastic constants C11

and C12 increase with increase in pressure up to phase-transitionpressure that supports high-pressure structural stability of boroncompounds. The computational methods for the determination ofcohesive, structural and mechanical properties of semiconductors

D. Varshney et al. / Solid State Sciences 12 (2010) 864–872 871

under pressure are successfully performed by means of ab initiocalculations as the local-density approximation and the moleculardynamics methods. Despite the rapid development of computa-tional techniques, the nature of interatomic forces is not wellunderstood about these materials and lattice dynamical models areimportant in interpreting and covered the chemical trends in thestructural stabilities.

From the computed values of pressure dependent Poisson’s ration and the ratio RS/B we conclude that BY (Y¼N, P, As) are brittle innature in ZnS phase at low pressures. However, at high pressuresthe boron monopnictides are ductile in both the zincblende (B3)[Sodium Chloride (B1)] phases. We note that the variation of qD

with pressure infers that the system moves toward the hardeningof lattice with increasing pressure. We stress that to the best of ourknowledge the pressure dependence of ductile (brittle) nature ofmixed valent compounds have not been calculated and measuredyet, hence the present calculations will inspire further experi-mental research on these compounds. It has been found that thissimple model as compared to complicated and time-expensiveband structure calculations may account for a considerable partof the available results for the high-pressure studies on boroncompounds. The present study of BY thus presents a quantitativedescription of the physical thermodynamical parameters and teststhe appropriateness of the effective interionic potential.

Acknowledgements

Financial support from DRDO, New Delhi is gratefullyacknowledged.

Appendix

The expressions for the SOECs for AIIIBV with ZnS structure are

C11 ¼e2

4a4

�0:2477Z2 þ 1

3ðA1 þ 2B1Þ þ

12ðA2 þ B2Þ

�; (A.1)

C12 ¼e2

4a4

�� 2:6458Z2 þ 1

3ðA1 � 4B1Þ þ

14ðA2 � 5B2Þ

�; (A.2)

C44 ¼e2

4a4

�� 0:123Z2 þ 1

3ðA1 þ 2B1Þ þ

14ðA2 þ 3B2Þ

� 13

V�� 7:53912Z2 þ A1 � B1

�: ðA:3Þ

The expression for pressure derivatives of second-order elasticconstants follows

3UdBT

dp¼ �

h20:1788Z2 � 3ðA1 þ A2Þ þ 4ðB1 þ B2Þ

þ 3ðC1 þ C2Þi; ðA:4Þ

2UdCS

dp¼ �

�� 11:5756Z2 þ 2ðA1 � 2B1Þ þ

23

A2 �72

B2 þ14

C2

�;

��

0:4952Z2 þ 13ðA1 � 4B1 þ C1Þ

1� (A.5)

þ4ð2A2 � 6B2 � C2Þ

UdC44

dp¼þVh�17:5913Z2þA1�B2� 2

3C1

iþV2

h3:1416Z22

3ðA1�B2Þþ

C1

3

�): (A.6)

Various symbols appear in the above expressions are associatedwith the crystal energy and have the following form

A1 ¼ Aij ¼4a3

e2

d2

dr2VijðrÞ!

r¼a

; (A.7)

A2 ¼ Aii ¼ Ajj ¼4a3

e2

d2

dr2ViiðrÞ þd2

dr2VjjðrÞ!

r¼a

; (A.8)

B1 ¼ Bij ¼4a2

e2

�ddr

VijðrÞ�

r¼a; (A.9)

B2 ¼ Bii ¼ Bjj ¼4a2

2

�d

ViiðrÞ þd

VjjðrÞ�

; (A.10)

e dr dr r¼a

V ¼"�7:5391Z2 þ ðA1 � B1Þ�3:141Z2 þ ðA1 þ 2B1Þ

#; (A.11)

U ¼ �5:0440Z2 þ ðA1 þ A2Þ � 2ðB1 þ B2Þ; (A.12)

The expressions for the elastic constants can be obtained ina similar fashion for rocksalt structure

C11 ¼e2

4a4

�� 5:112Z2 þ A1 þ

ðA2 þ B2Þ2

�; (A.13)

C12 ¼e2

4a4

�0:226Z2 � B1 þ

ðA2 � 5B2Þ4

�; (A.14)

C44 ¼e2

4a4

�2:556Z2 þ B1 þ

ðA2 þ 3B2Þ4

�: (A.15)

The expressions for the pressure derivatives of the second-orderelastic constants of NaCl structure solids under hydrostatic pressurep are obtained in the form

3UdBT

dp¼ �

n13:980Z2 þ C1 � 3A1 þ C2 � 3A2

o(A.16)

2UdCS

dp¼ �

23:682Z2 þ C1 þ

C2 þ 6A2 � 6B2

4

�(A.17)

UdC44

dp¼�

�11:389Z2þA1�3B1þ

C2þ2A2�10B2

4

�(A.18)

where

U ¼ �2:330Z2 þ A1 þ A2 (A.19)

A1 ¼8a3

e2

"d2

dr2VijðrÞ#

r¼a

; (A.20)

A2 ¼16a3

e2

"d2

dr2ViiðrÞ þd2

dr2VjjðrÞ#

r¼ffiffiffi2p

a

; (A.21)

B1 ¼8a3

e2

�1r

ddr

VijðrÞ�

r¼a; (A.22)

D. Varshney et al. / Solid State Sciences 12 (2010) 864–872872

B2 ¼16a3

e2

�1r

ddr

ViiðrÞ þ1r

ddr

VjjðrÞ�

r¼ffiffiffi2p

a; (A.23)

whereas

BT ¼13ðC11 þ 2C12Þ (A.24)

and

CS ¼12ðC11 � C12Þ; (A.25)

Various symbols appear in the above expressions are associatedwith the crystal energy and have the following form in terms of theshort-range energy

VijðrÞ ¼X

ij

bbijexp�

ri þ rj � rij

r

��X

ij

cij

r6ij

�X

ij

dij

r8ij

(A.26)

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