Journal of Physics and Chemistry of Solids 72 (2011) 945–953

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Journal of Physics and Chemistry of Solids

0022-36

doi:10.1

n Corr

E-m

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

Pressure dependent elastic properties of diluted magnetic semiconductors:Hg1�xMnxS (x¼0.02 and 0.07)

Dinesh Varshney a,n, R. Sapkale a, G.J. Dagaonkar a, Meenu Varshney b

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

a r t i c l e i n f o

Article history:

Received 11 December 2010

Received in revised form

11 April 2011

Accepted 6 May 2011Available online 19 May 2011

Keywords:

A. Semiconductors

D. Phase transitions

D. Elastic properties

D. Mechanical properties

97/$ - see front matter & 2011 Elsevier Ltd. A

016/j.jpcs.2011.05.004

esponding author. Tel./fax: þ91 7312467028

ail address: vdinesh33@rediffmail.com (D. Va

a b s t r a c t

A theoretical study of the elastic properties in diluted magnetic semiconductors Hg1�xMnxS (x¼0.02

and 0.07) using an effective interionic interaction potential (EIoIP) in which long-range Coulomb

interactions, charge transfer mechanism (three body interaction) and the Hafemeister and Flygare type

short-range overlap repulsion extending up to the second neighbor ions and the van der Waals (vdW)

interaction is considered. Particular attention is devoted to evaluate Poisson’s ratio n, the ratio RS/B of S

(Voigt averaged shear modulus) over B (bulk modulus), elastic anisotropy parameter, elastic wave

velocity, average wave velocity and thermodynamic property as Debye temperature is calculated. By

analyzing Poisson’s ratio n and the ratio RS/B we conclude that Hg1�xMnxS is brittle in zinc blende (B3).

To our knowledge this is the first quantitative theoretical prediction of the pressure dependence of

ductile (brittle) nature of Hg1�xMnxS compounds and still awaits experimental confirmations.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Present information technology demands materials with bothmagnetism (spin), and semiconductivity (charge) combined inone device that exploits both charge and spin to process, andstores the information. Search for new materials and alloys are inprogress and AII

1�xMnxBVI as Hg1�xMnxS compounds belong to afamily of diluted magnetic semiconductors (DMS) might be thepromising materials for device applications. DMS materials can bemade by introducing magnetic ions like Mn, Cr, Co and Fe intonon-magnetic semiconductors and are extensively studied in therecent past [1].

As concerned with the transition metal doping, the Mn dopingis favorable because of its highest possible magnetic moment[2,3]. Apart from this the first half of the d-band is full, creating astable fully polarized state. We must mention that the strongexchange interactions between sp band electron and d electronassociated with Mnþþ results in interesting optical and electricalproperties like giant Faraday rotation and magnetic and fieldinduced metal–insulator transition. Henceforth, the materials ofthe type AII

1�xMnxBVI (Cd1�xMnxTe, Cd1�xMnxSe, Hg1�xMnxS) areof scientific and technological importance because their ternarynature gives the possibility of tuning the lattice constant andband parameters by varying the composition of the material.

ll rights reserved.

.

rshney).

The elastic moduli is important in assessing the competitionbetween the ductile and brittle failures, have been investigated inrelation to various microscopic characteristics of different sorts ofmaterials, such as metals and covalently bonded crystals. Theelastic modulus of simple and complex materials is usuallyevaluated following the ab-initio calculation techniques. Oneconstraint is the rationalization of the first principles calculationsoften requires profound understanding of the nature of thechemical bonding and its attributes in various solid systems.Henceforth, the lattice model calculations need to be elaboratedin concert with computational approaches and estimations.

A great deal of progress has recently been made in makingpressure dependent structural properties from ab-initio theories,however the anharmonic properties of solids and alloys bymeans of different forms of cohesion has also been useful dueto simple physical motivation, and inexpensive computationalmethods. The dominating component of cohesion in phenomen-ological models with two body interactions is the interactionpotential that incorporates long-range Coulomb interactions,which are counterbalanced by the short-range overlap repulsionowing its origin to the Pauli exclusion principle [4–8]. Despitetheir successes the basic nature of these interatomic potentials isinadequate to reveal a consistent picture of the interactionmechanism, static and dynamical properties of ionic solids. Theinadequacy of two-body interaction is clearly indicated by itsfailure to explain the Cauchy violations [9]. An acceptableexplanation of the Cauchy violations [10] was given by Lowdin[11] and Lundqvist [12] in terms of three-body interactions (TBI),

D. Varshney et al. / Journal of Physics and Chemistry of Solids 72 (2011) 945–953946

which have their origin in the non-orthogonality of electron wavefunctions.

The effects of three-body interaction following Hafemeisterand Flygare type overlap repulsion extended up to secondneighbor ions besides short range interactions are introduced[13] to discuss the mechanical properties of several solids andalloys. Motivated from these remarks [9–12] and the versatility ofmany-body interactions approach [13] for the successful descrip-tion of the high pressure elastic properties in binary semiconduc-tors, we thought it pertinent to make a comprehensive study ofcharge transfer effects on cohesive, thermo-physical, harmonicand anharmonic elastic properties of Mn doped HgS semiconduc-tor. The importance of three-body interaction arose from thecharge-transfer mechanism caused by the deformation of theelectron shells of the overlapping ions, to explain the variousthermodynamic properties and still needs detailed investigations.Besides charge transfer effects, we have incorporated the vdWinteraction effects to increase the accuracy.

The present investigations are organized as follows. In Section2, we introduce the assumptions for effective interionic potentialbetween a pair of ions and motivate them by physical arguments.We begin with the estimation of van der Waals coefficients (vdW)from the Slater–Kirkwood variational method with an idea thatboth the ions are polarizable. Later on, the elastic constants arededuced within the framework of Shell model, that incorporatesthe long-range Coulomb, charge transfer mechanism, van derWaals (vdW) interaction and the short-range overlap repulsiveinteraction up to second-neighbor ions within the Hafemeisterand Flygare approach. The main focus of the present investigationis to discuss the pressure dependent elastic properties as ductility(brittleness) and sound velocity of Hg1�xMnxS compound as nosystematic efforts have been made so far. The computed resultsand the numerical analysis of Hg1�xMnxS compounds are discussedin Section 3. Finally, conclusions are presented in Section 4.

2. The method of computation

Pressure dependent mechanical properties as ductility(brittleness), elastic anisotropy, longitudinal and transverse velo-city and thermodynamical property Debye temperature ofHg1�xMnxS are important and can be known from second orderelastic constants. To our knowledge this is the first attempt toreveal the pressure dependence of ductile (brittle) nature ofHg1�xMnxS compounds and still awaits experimental confirma-tions. In doing so, we first need to evaluate pressure dependentstructural properties. Many-body interaction approach is success-ful in description of the high-pressure phase. Usually the appliedpressures cause an increase in the overlap of adjacent ions in acrystal and hence, charge transfer takes place between the over-lapping electron shells. The transferred charges interact with allothers of the lattice via Coulomb’s law and give rise to many-bodyinteractions, of which the most significant is three-body interac-tion (TBI). The increased effect due to covalent nature and ofcharge transfer, thus obtained, lead to an obvious necessity oftheir inclusion in the high-pressure study of materials.

Usually, the thermodynamical properties are described by theappropriate thermodynamical potential relevant to the givenensemble. Either variable pressure or temperature acting on thesystem is altered, the free energy changes smoothly and con-tinuously. The stability of a particular structure is decided by theminima of Gibbs’s free energy. The relevant potential, e.g. Gibbsfree energy (G) can be written as G¼UþPV�TS, U being theinternal energy, which at 0 K corresponds to the cohesive energy,S is the vibrational entropy at absolute temperature T, pressure P

and volume V.

We must mention that the calculations presented here assumezero temperature i.e., the frozen ionic degrees of freedom.Although, the experimental results are obtained at ambienttemperature inferring a certain small temperature dependenceof the transition pressures in the range of low temperatures.Nevertheless, it is safe to consider the lattice model calculationresults as representative of the results that would be obtainedunder the actual experimental conditions. At zero temperature,the thermodynamically stable phase at a given pressure P is theone with the lowest enthalpy, and the thermodynamical potentialis the Helmholtz free energy (H).

The Gibbs free energy for zinc blende (B3) is given by Bornequation [14]

GB3ðrÞ ¼UB3ðrÞþPVB3, ð1Þ

here the abbreviation UB3(r) stands for the zinc blende (ZB) phase;VB3 (¼3.08r3) as the unit cell volume and r being the nearest-neighbor distance for B3 (ZB) phase. The lattice energy consideredconsisting of the long-range Coulomb, charging transfer mechan-ism, the short range Van der Waals and overlap repulsioneffective up to the second neighbor ions. Its relevant expressionis given by Born equation [14]

UB3 ¼ ð�aMZe2=rÞ½Zþ2nfT ðrÞ��Cr�6ij �Dr�8

ij

þnbbij exp½ðriþrj�rijÞ=r�þ n0b=2� �

½bii expðð2ri�krijÞ=rÞ

þbjj expðð2rj�krijÞ=rÞ�, ð2Þ

in terms of parameters b(r) as the hardness (range); ri(rj) as theionic radii; bij are the Pauling coefficient; n (¼4) and n0(¼6) arethe numbers of the nearest unlike (n) and like (n0) neighbors,respectively, for B3 (ZnS) structure. Here, the first term is thepotential energies due to long-range Coulomb, and the secondterm is due to the charge transfer caused by the deformation ofthe electron shells of the overlapping ions and the covalencyeffects. The diluted magnetic semiconducting compounds containcovalent bonds so that some electrons are distributed over theregion between neighboring atoms; in such a situation, the ioniccharge for each atom cannot be determined uniquely and hencethe calculation of the Madelung energy is modified by incorpor-ating the covalency effects [15]. We thus need to incorporate theeffective charge arose due to the polarization of a sphericalshaped dielectric in displacing the constituent positive ions. Thethird and fourth terms are the short-range (SR) vdW attractionenergies due to dipole–dipole and dipole–quadrupole interaction.The C and D are the overall vdW coefficients, which are evaluatedfrom the variational approach [16].

The vdW coefficients due to dipole–dipole and dipole–quadruple interactions are calculated from the Slater andKirkwood variational approach and are [16]

cij ¼3

2

e_ffiffiffiffiffiffiffimep aiaj

ai

Ni

� �1=2

þaj

Nj

� �1=2" #�1

, ð3Þ

dij ¼27

8

_2

meaiaj

ai

Ni

� �1=2

þaj

Nj

� �1=2" #2

ai

Ni

� �þ

20

3

aiaj

NiNj

� �1=2

þaj

Nj

� �" #�1

,

ð4Þ

cii ¼3

2

e_ffiffiffiffiffiffiffimep aiai

ai

Ni

� �1=2

þai

Ni

� �1=2" #�1

, ð5Þ

dii ¼27

8

_2

meaiai

ai

Ni

� �1=2

þai

Ni

� �1=2" #2

ai

Ni

� �þ

20

3

aiai

NiNi

� �1=2

þai

Ni

� �" #�1

,

ð6Þ

Table 1Van der Waals coefficients of Hg1�xMnxS semiconductors (cij in units of

10�60 erg cm6 and dij in unit of 10�76 erg cm8).

vdW coefficients Solids

HgS MnS Hg0.98Mn0.02S Hg0.93Mn0.07S

cii 48.23 3.54 47.04 44.11

cij 42.35 10.66 41.84 40.53

cjj 37.27 37.27 37.27 37.27

C 217.0 107.2 214.3 207.5

dii 61.31 1.88 59.30 54.44

dij 51.68 9.79 50.82 48.67

djj 43.48 43.48 43.48 43.48

D 224.1 78.33 220.3 210.9

D. Varshney et al. / Journal of Physics and Chemistry of Solids 72 (2011) 945–953 947

cjj ¼3

2

e_ffiffiffiffiffiffiffimep ajaj

aj

Nj

� �1=2

þaj

Nj

� �1=2" #�1

, ð7Þ

djj ¼27

8

_2

meajaj

aj

Nj

� �1=2

þaj

Nj

� �1=2" #2

aj

Nj

� �þ

20

3

ajaj

NjNj

� �1=2

þaj

Nj

� �" #�1

,

ð8Þ

where m, e and Z are the electron mass, charge and valence of theconstituent metallic element, respectively, (ai, aj) are the polar-izabilities of ith and jth ion, respectively; Ni and Nj are theeffective number of electrons responsible for polarization. Theoverall vdW coefficients C and D are then expressed in terms of cij

and dij, by an appropriate lattice sums (Sij, and Tij) the values ofwhich are taken from [5] such that and the expression is given by

C ¼ cijSijþciiSiiþcjjSjj, ð9Þ

D¼ dijTijþdiiTiiþdjjTjj: ð10Þ

The last two terms are SR repulsive energy due to the overlaprepulsion between ij, ii and jj ions. am is the Madelung constantfor B3 phase. bij are the Pauling coefficient defined as bij¼1þ(Zi/ni)þ(Zj/nj) with Zi(Zj) and ni(nj) as the valence and the numberof electrons in the outermost orbit, respectively. Ze is the staticcharge of the ion, k is being the structure factor for B3 structureand b(r) are the hardness (range) parameters. r is the nearestneighbor ion separations for B3 structure. For Z¼2, one findsbij¼1.0; bii¼1.5; bjj¼0.5, respectively.

The second term in Eq. (2) is an algebraic sum of three-bodyforce parameters f(r) and the force parameter arises due tocovalent nature i.e., fT(r)¼ f(r)þ fcov(r). The three-body force para-meter f(r) is expressed as [11,12]

f ðrÞ ¼ f0 expð�r=rÞ, ð11Þ

ri(rj) are the ionic radii of ions i(j). Keeping in mind that thediluted magnetic semiconducting compounds are partially ionicand partially covalent in bonding, and the attractive forces due tocovalency are important that modifies the effective charge. Thepolarization effects originates from changes in covalency due toelectric fields and the covalency term is expressed as [15]

fcovðrÞ ¼4e2V2

sps

r0E3g

, ð12Þ

Vsps is being the transfer matrix element between the outermost p

orbital of anion and the lowest excited of s state of cation, Eg is thetransfer energy of electron from anion to cation. The effectivecharge en

s of the host crystal is related with the number ofelectrons transferred to the unoccupied orbitals of a cation fromits surrounding anions at the nearest neighbor and is nc ¼ 1�en

s =e:

Furthermore, nc/12ffiV2sps/Eg

2 and the transfer matrix element Vspsand the transfer energy Eg is related to effective charge en

s

following [15]:

V2sps

E2g

¼1�en

s

12: ð13Þ

The transfer energy Eg is further expressed as

Eg ¼ E�Iþð2a�1Þe2

r, ð14Þ

here E is being the electron affinity for s i.e., for the non-metalatoms and I is the ionization potential of the constituentmetal atom.

The optical static dielectric constant es and the high frequencydielectric constant eN are intimately related to Szigeti effective

charge ens (¼Ze)n [13] as follows:

en2s ¼

9mo2TOðes�e1Þ

4pNkðe1þ2Þ2, ð15Þ

and

en2s

e2¼

9Vmo2TOðes�e1Þ

4pe2ðe1þ2Þ2, ð16Þ

here, m is the reduced mass, Nk is the number of atoms present perunit cell volume i.e., Nk¼1/V, oTO is the long wavelength trans-verse optical phonon frequency. For partially ionic as wellpartially covalent crystals as DMS, en

s deviates from Ze due tocovalency effects. The longitudinal optical (LO) phonon, trans-verse optical (TO) phonon frequency along with high frequencydielectric constant value leads to the optical static dielectricconstant using o2

LO/o2TO¼es/eN [17].

In order to demonstrate the effect of charge transfer mechan-ism, we have investigated the pressure variations of the second,third order elastic constants and their derivatives. The relevantexpressions for the elastic constants are illustrated in Appendix 1.The effective interionic potential described above for B3 phasecontains three material dependent parameters namely the hard-ness, range and three-body force parameters (b, r, f(r)). We canthen obtain these values from the equilibrium conditions [18].Having discussed the effective interionic potential and its appli-cations to various thermodynamical and elastic properties forchosen material, we shall now estimate and compute numericallythe high-pressure elastic properties for B3 phase.

3. Results and discussions

The effective interionic potential described in the earliersection for zinc blende (B3) phase contains three free parameters,namely the hardness, range and three body force parameters(b, r and fT(r)). While estimating the free parameters, we firstdeduce the vdW coefficients from the SKV method [16] as shownin Table 1. The values of electronic polarizabilities have beendirectly taken from a least-squares fit of experimental refractiondata [19,20] using additive rule and a Lorentz factor of 4p/3. Herewe have estimated the result from three-body interaction. Theinput data needed for the calculation can be directly measuredfrom Raman scattering or found from literatures for zinc blendeHg1�xMnxS. The LO and TO phonon frequency are related withhigh frequency dielectric constant value. Using o2

LO/o2TO¼es/eN

one obtain the optical static dielectric constant. For calculationswe have used eN¼8 for zincblende structure [17].

It is instructive to point that the mixed crystals, according tothe virtual crystal approximation [21], are regarded as an array ofaverage ions whose masses, force constants and effective chargesare considered to scale linearly with concentration. In addition,

0.12

0.18

0.24

0.30

0.36

0.42

0.48

0.54Hg0.98Mn0.02S

Cij

(1011

Nm

-2)

C11 C12 C44

0.0 0.5 1.0 1.5 2.0 2.5 3.00.00

0.13

0.26

0.39

0.52

Hg0.97Mn0.03S C11 C12 C44

Cij

(1011

Nm

-2)

P (GPa)

Fig. 1. Variation of second order elastic constants with pressure for Hg1�xMnxS

(x¼0.02 and 0.07) compounds.

D. Varshney et al. / Journal of Physics and Chemistry of Solids 72 (2011) 945–953948

pure HgS crystallize in B3 phase while to that MnS crystallize inB1 phase [22]. These facts allow us to first estimate the freeparameters for both binary compounds.

We believe that these parameters vary linearly with thedoping concentration (x), and hence they follow Vegard’slaw [23]:

bðHg1�xMnxSÞ ¼ ð1�xÞbðHgSÞþxbðMnSÞ, ð17Þ

rðHg1�xMnxSÞ ¼ ð1�xÞrðHgSÞþxrðMnSÞ, ð18Þ

and

fT ðrÞðHg1�xMnxSÞ ¼ ð1�xÞfT ðrÞðHgSÞþxfT ðrÞðMnSÞ: ð19Þ

We find the values of free parameters: hardness, b¼4.35(2.669)�10�12 erg and range, r¼5.68(4.63)�10�9 cm and threebody force parameter, f(r)¼7.556(1.337)�10�3 from the knowl-edge of the equilibrium distance and the bulk modulus followingthe equilibrium conditions for binary HgS (MnS).

dUðrÞ

dr

��������r ¼ r0

¼ 0, ð20Þ

and the bulk modulus (BT)

d2UðrÞ

dr2

��������r ¼ r0

¼ ð9kr0Þ�1BT : ð21Þ

While estimating these values we have used the inputparameters as C12¼0.62�1011 N m2, C44¼0.26�1011 N m2,r0¼2.53 A [17] and BT¼19.4 GPa [24]. Using these values as astarting point, we then estimate the free parameters asb¼4.317[4.233]�10�12 erg, r¼5.659[5.606]�10�9 cm andf(r)¼7.41[7.05]�10�3 for x¼0.02[0.07], respectively. We appeal

that the consistency of the results can be tested as a measure ofthe validity of these assumptions.

Usually, in alloy formation, the microscopic crystal structure ofalloys depends on the atomic properties and relative dopingconcentrations of the dopants. The relative atomic sizes of theconstituent elements in an alloy, the relative volume per valenceelectron in crystals of pure elements, the in-phase and out ofphase motion of constituent atoms at Brillouin zone, and theelectrochemical differences between the dopants influences thecrystal structure of an alloy. Vegard’s law is an approximateempirical rule, which holds that a linear relation exists, atconstant temperature (pressure), between the crystal latticeparameter of an alloy and the doping concentrations of theconstituent elements. This empirical rule holds good for most ofthe ionic solids, while to that the majority of the metallic alloysviolate it. Henceforth, the linear relationship in lattice constant-doping concentration is appropriate in determining the mechan-ical properties as functions of concentration of the constituentelements.

On the other hand in the virtual crystal approximation (VCA)one studies a crystal with the primitive periodicity, but composedof fictitious virtual atoms that interpolate between the behaviorof the atoms in the parent compounds. The VCA technique issimpler and takes care of the pseudopotential approximation.As pseudopotentials are used for determining the band structures,the VCA for substitutional effects in alloys is less accurate and alsoinadequate due to difference in between true and VCA pseudopotential.

Considering pressure and temperature as the external vari-ables (hydrostatic conditions), the free energy of a particularcrystal structure at a particular temperature and pressure isobtained by minimizing G (P, V, T) with respect to V. In an attemptto reveal the pressure dependent stable structure of the testmaterial, we minimize the Gibbs free energy GB3(r) for theequilibrium interatomic spacing (r). The value of r have beenobtained as 3.77 and 3.67 A for Mn doping concentration x¼0.02and 0.07, respectively.

In order to study the high-pressure elastic behavior of DMS asHg1�xMnxS, we have computed the second-order elastic con-stants (SOECs) and their variation with pressure is shown inFig. 1(a) and (b). We must mention that the elastic constant C11 isa measure of resistance to deformation by a stress applied on(1 0 0) plane with polarization in the direction /1 0 0S, and theC44 refers to the measurement of resistance to deformation withrespect to a shearing stress applied across the (1 0 0) plane withpolarization in the /0 1 0S direction. Henceforth, the elasticconstant C11 represents elasticity in length and a longitudinalstrain produces a change in C11. No doubt, the elastic constantsC12 and C44 are intimately related to the elasticity in shape, whichis a shear constant. However, a transverse strain causes a changein shape without a change in volume. Thus, the second orderelastic constants as C12 and C44 are less sensitive of pressure ascompared to C11 [18].

We note that C44 decrease linearly with the increase ofpressure away from zero at the phase-transition pressures. Onthe contrary, the value of C11 and C12 increases linearly withpressure. Furthermore, it is inferred from Fig. 1(a) and (b) that theincrement of elastic constants reveals that increased Mn dopingconcentration increases elastic constant and hence the enhancedstructural stability of HgS. In particular, the local environment ofthe tetrahedron changes with the concentration.

We argue that increased Mn doping concentration inHg–Mn–S increases elastic constant. In principle the reducedelastic constants allows a decrease in structural stability due tohybridization of Mn with S orbitals for tetrahedral bond forma-tion. To an end we comment that the larger the elastic constants

Table 2Second order elastic constants (C11, C12 and C44), bulk modulus (BT) and tetragonal

moduli (CS) (all are in 1011 N m�2) for Hg1�xMnxS compounds in B3 phase.

Solids C11 C12 C44 BT CS

Hg0.98Mn0.02S Present 0.38 0.16 0.12 0.23 0.11

Hg0.93Mn0.07S Present 0.41 0.17 0.11 0.25 0.12

Others 0.39 [3]

Table 3The values of pressure derivatives of SOECs (dBT/dP, dC44/dP and dCS/dP) and third

order elastic constants (C111, C112, C123, C144, C166 and C456) (in units of

1011 N m�2). (Others values are given in parenthesis).

Quantities Hg0.98Mn0.02S Hg0.93Mn0.07S

dBT/dP 4.20 4.14 (2.3 [3])

dC44/dP 8.91 9.64

dCS/dP �0.607 �0.58

C111 �0.82 �0.88

C112 �1.40 �1.50

C123 0.025 0.06

C144 �1.52 �1.82

C166 �2.95 �3.33

C456 �0.687 �0.90

-2.04

-1.53

-1.02

-0.51

0.00

0.51

1.02 C111 C112 C123 C144 C166 C456

Hg0.98Mn0.02S

Cijk

(1011

Nm

-2)

0.0 0.5 1.0 1.5 2.0 2.5 3.0-2.4

-2.0

-1.6

-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2 Hg0.93Mn0.07S C111 C112 C123 C144 C166 C456

Cijk

(1011

Nm

-2)

P (GPa)

Fig. 2. Variation of third order elastic constants with pressure for Hg1�xMnxS

(x¼0.02 and 0.07) compounds.

D. Varshney et al. / Journal of Physics and Chemistry of Solids 72 (2011) 945–953 949

reduction the larger the anion-mediated superexchange betweenMn ions in HgMnS semiconducting compounds.

It is informative to mention that the Born criterion for a latticeto be mechanically stable states is that the elastic energy densitymust be a positive definite quadratic function of strain. Thisrequires that the principal minors (alternatively the eigen values)of the elastic constant matrix should all be positive. The stabilityof a cubic crystal is expressed in terms of elastic constants asfollows [14]:

BT ¼ ðC11þ2C12Þ=340, ð22Þ

C4440, ð23Þ

and

s¼ ðC11�C12Þ=240, ð24Þ

Cij are the conventional elastic constants and BT is the bulkmodulus. The quantities C44 and s are the shear and tetragonalmoduli of a cubic crystal. Estimated values of bulk modulus, shearmoduli and tetragonal moduli for x¼0.02 and 0.07 are listed inTable 2, well satisfied the above elastic stability criteria.In addition, Vukcevich [25] proposed a high-pressure stabilitycriterion for ionic crystals, combining mechanical stability withminimum energy conditions. In accordance, the stable phase ofthe crystal is one in which the shear elastic constant C44 isnonzero (for mechanical stability) and which has the lowestpotential energy among the mechanically stable lattices.

We also intend to analyze the anharmonic properties ofHg1�xMnxS by computing the third order elastic constants(TOECs) and pressure derivatives of SOECs at zero pressure. Wehave, therefore, deduced the values of pressure derivatives ofSOEC (ds/dP, dBT/dP and dC44/dP) for x¼0.02 and 0.07 and arelisted in Table 3 and compared with available experimentalresults [3]. It is true that the agreement between the theoreticaland the experimental value of dBT/dP is not of the desired degreebut this may be because we have derived our expressionsneglecting thermal effects and assuming the overlap repulsionsignificant only up to nearest neighbors. We must also mentionthat the effective coupling of Mn between the spins s1 and s2 ofthe two electrons has not been incorporated in the effectivepotential and shall be taken care for future problems. The

variation of TOECs with pressure is shown in Figs. 2(a) and (b).It can be seen that the variation of third order elastic constantswith pressure points to the fact that the values of C111, C112, C123,C166, C456 are negative while that of C144 is positive after aparticular pressure as obtained from the effective interionicpotential at zero pressure (please see Table 3). Thus, we can saythat in II–VI semiconductors, the developed EIoIP consistentlyexplains the high-pressure elastic behavior.

Furthermore, the mechanical properties as ductility and brit-tleness of Hg1�xMnxS are of substantial importance and can beknown from second order elastic constants. A simple relationship,empirically linking the plastic properties of materials with theirelastic moduli has been mentioned [26]. The thermodynamicalproperty as the Voigt averaged shear modulus S representing theresistance to plastic deformation, while the bulk modulus B

represents the resistance to fracture. Accordingly, the ratioRS/Bo0.5, the material behaves in a ductile manner, otherwisethe material behaves in a brittle manner. From Fig. 3(a) and(b) the ratio RS/B shows that at low pressures, the compoundsshow brittle nature and with the further in increase in pressurethe compounds show ductile.

We may also refer to Frantsevich et al. [27] who distinguishthe ductility and brittleness of materials in terms of Poisson’sratio. According to Frantsevich rule the critical value of a materialis 0.33. For brittle materials, Poisson’s ratio is less than 0.33,otherwise the material behaves in a ductile manner. It is identi-fied from Fig. 3(a) and (b) that at low pressures, Poisson’s ratioshows the brittle nature and with the further increase in pressurethe compounds show ductile behavior.

Also, Poisson’s ratio n in terms of the bulk modulus B and theVoigt averaged shear modulus S as [28]

n¼ 1

23

B

S�2

� 3

B

Sþ1

� �1

: ð25Þ

0.24

0.26

0.28

0.30

0.32

0.34

R

νPois

sion

's ra

tio (ν

)

0.35

0.40

0.45

0.50

0.55

0.60

0.65Hg0.98Mn0.02S

Inve

rse

ratio

(R)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.24

0.26

0.28

0.30

0.32

0.34

0.36

R

ν

P (GPa)

Pois

sion

's ra

tio (ν

)

0.30

0.35

0.40

0.45

0.50

0.55

0.60Hg0.93Mn0.07S

Inve

rse

ratio

(R)

Fig. 3. Variation of Poisson’s ratio n and ratio RS/B with pressure.

0.0 0.5 1.0 1.5 2.0 2.5 3.00.0

0.2

0.4

0.6

0.8

1.0

1.2

Elas

tic A

niso

trop

y γ

P (GPa)

Hg0.98Mn0.02S Hg0.93Mn0.07S

Fig. 4. Variation of elastic anisotropy with pressure.

Table 4

Calculated elastic anisotropy g, Kleinman parameter x, longitudinal, transverse

and average elastic wave velocity of Hg1�xMnxS in m/s.

Compounds g x vl vt vm

Hg0.98Mn0.02S 0.053 0.557 4100 2240 535.94

Hg0.93Mn0.07S �0.049 0.551 4070 2170 519.75

D. Varshney et al. / Journal of Physics and Chemistry of Solids 72 (2011) 945–953950

It follows that the empirical Pugh’s critical value correspondsto n¼0.26, so that both the Pugh and Frantsevich empirical rulesonly differ on the exact border between the two types of behavior.Therefore the Hg1�xMnxS compounds that are semiconductingcompounds are brittle in nature in ZnS phase. The brittle nature isalso observed in face centered cubic intermetallic compounds asIr and Ir3X (X¼Ti, Ta, Nb, Zr, Hf, V) [29] and in III–V Ga1�xInxAssemiconducting compounds [30].

We may add that the value of the ratio RS/Bo0.57 is used forthe ductile behavior of intermetallics as MgCNi3, otherwise thematerial behaves in a brittle manner [28]. Henceforth, the criticalvalue of RS/B acts as a performance indicator to distinguish ductileand/or brittle transition of materials. It is known that anisotropicparameter g is zero for isotropic elasticity but still the cubiccrystal has elastic anisotropy. The elastic anisotropic parameter ofa cubic crystal is defined as [31].

g¼ 2C44þC12

C11�1: ð26Þ

Through the calculated elastic constants, we can obtain theelastic anisotropic parameter g at various pressures forHg1�xMnxS compounds. Fig. 4 shows the pressure dependenceof the elastic anisotropic parameter g up to 3.0 GPa. It is clearfrom the plot that the anisotropy increases with increase inpressure from o to 1.6 GPa in Hg0.98Mn0.02S and 1.8 GPa forHg0.93Mn0.07S and then g decreases, which indicates that theanisotropy is more obvious under low pressure. The anisotropyfactor drops slowly with pressure and then decrease more rapidlyat higher pressures. The values of anisotropic parameter g forthese compounds are given in Table 4 at zero temperature andpressure.

Usually, the elastic constants relate the properties of materialthat undergo stress, deform and then recover after returns to itsoriginal shape after stress ceases. The elastic constants are

emphasized in solids because they are closely intimated tovarious fundamental solid-state phenomena such as interatomicbonding, equations of state, and phonon spectra [32]. It isworth to mention that the elastic properties are also linkedthermodynamically with specific heat, thermal expansion, Debyetemperature, the relative positions of the cation and anionsublattices under volume-conserving strain distortions, andGruneisen parameter in semiconducting chalcogens, pnictidesand diluted magnetic semiconductors [15]. Most importantly,knowledge of elastic constants is essential for many practicalapplications related to the mechanical properties of a solid: loaddeflection, thermoelastic stress, internal strain, sound velocitiesand fracture toughness.

As suggested by Bouhemadou et al. [32], the above is applic-able for diluted magnetic semiconducting compounds also, wehave attempted to understand the Kleinman parameter, x, whichdescribes the relative positions of the cation and anion sublatticesunder volume-conserving strain distortions for which positionsare not fixed by symmetry. A low value of x implies a largeresistance against bond bending or bond-angle distortion and viceversa [33,34]. We used the following relation [35]:

x¼C11þ8C12

7C11þ2C12: ð27Þ

The Kleinman parameter, x, is calculated for Hg1�xMnxS andare illustrated in Table 4 at zero temperature and pressure forzinc-blende phase. A decreasing trend in x values is noticed onincreasing the concentration of Mn. Because of unavailability ofdata we could not compare them and can be considered as aprediction of elastic properties.

The average wave velocity vm has been approximately calcu-lated from [36]

vm ¼1

3

2

v3tþ

1

v3l

� �� �ð1=3Þ

, ð28Þ

where vl and vt are the longitudinal and the transverse elasticwave velocity, respectively, which are obtained from Navier’s

0.0 0.5 1.0 1.5 2.0 2.5 3.085

90

95

100

105

110

115

120

ΘD

(K)

P (GPa)

Hg0.98Mn0.02SHg0.93Mn0.07S

Fig. 6. Variation of Debye temperature (yD) with pressure.

D. Varshney et al. / Journal of Physics and Chemistry of Solids 72 (2011) 945–953 951

equation in the following forms:

vl ¼3Bþ4S

3r

� 1=2

, ð29Þ

vt ¼S

r

� 1=2

, ð30Þ

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

The pressure dependence of the sound velocity is shown inFig. 5(a) and (b), respectively. Fig. 5 shows that for Hg0.98Mn0.02S,vl increases from 0 to 2.2 GPa pressure range and then becomesaturated at higher pressure range while vt increases from 0 to1.2 GPa pressures and then starts decreasing in high pressurerange and for Hg0.93Mn0.07S, vl increases from 0 to 1.4 GPapressure range and then slowly tends towards saturation athigher pressure range while vt also increases from 0 to 1.4 GPapressures and then starts decreasing in high pressure range. Thevalues of longitudinal, transverse and average sound velocities aregiven in Table 4 at zero temperature and pressure.

The elastic constants determine the velocity of elastic wavesthrough the lattice and hence one can relate the Debye tempera-ture (yD) with the elastic constants. A semi-empirical relation hasbeen proposed by Blackman [37] to relate yD with the elasticconstants for a cubic as

y3D ¼

3:15

8ph

kB

� �3 r

M

�3=2 C11�C12

2

� �1=2 C11þC12þ2C44

2

� �1=2

C1=244 ,

ð31Þ

here, M is the acoustic mass of the compound; h and kB arethe Planck and Boltzmann constants, respectively. The Debyetemperature as functions of pressure is plotted in Fig. 6(a) and(b). It is inferred from the figure that yD increase with increasingin pressure for both the compounds, on the other hand thepressure dependence of yD at higher pressure suggests thatdecreasing of Debye temperature drives the system effectively

2400

3200

4000

4800

Vt

VlHg0.98Mn0.02S

Ela

stic

wav

e ve

loci

ty (m

sec

-1)

0.0 0.5 1.0 1.5 2.0 2.5 3.02000

3000

4000

5000

Vt

Vl

P (GPa)

Hg0.93Mn0.07S

Ela

stic

wav

e ve

loci

ty (m

sec

-1)

Fig. 5. Pressure dependence of elastic wave velocity vl and vt.

towards the softening of lattice with increasing pressure. However,we do not claim the process to be rigorous, but a consistentagreement following EIoIP is obtained on Debye temperature asthose revealed from experiments. Usually, the Debye temperature isalso a function of temperature and varies from technique totechnique as well depends on the sample quality with a standarddeviation of about 15 K. One thus define average elastic constant interms of Debye temperature as [38]

C ¼8p

3:15

� �2=3 kB

h

� �2 M

r

� �y2

D, ð32Þ

for Hg1�xMnxS (x¼0.02 and 0.07) compounds 0.17 and 0.68(1011 Nm�2), respectively. To correlate Cauchy discrepancy inelastic constant following [14]:

Cn ¼C12�C44

C12þC44, ð33Þ

at zero pressure. We have observed positive Cauchy discrepancy0.143 and 0.214 for Hg1�xMnxS compounds. It is worth to mentionthat some semiconductors with zinc blende and rock salt structurein binary alloys and mixed semiconductors [30,39,40] show apositive Cauchy deviation Cn. To an end we comment that, inDMS, the present effective interionic potential consistently explainsthe high pressure and elastic of Hg1�xMnxS (x¼0.02 and 0.07)semiconducting compounds.

4. Conclusions

The present study addresses for the first time, the pressuredependent ductile and brittle nature of Hg1�xMnxS (x¼0.02 and0.07) by formulating an effective interionic interaction potentialincorporating the long-range Coulomb, the covalency effects, thecharge transfer caused by the deformation of the electron shells ofthe overlapping ions and the Hafemeister and Flygare type short-range overlap repulsion extended up to the second neighbor ionsand the van der Waals (vdW) interaction. As a first step, weexercise for identifying the pressure dependent structure as wellas elastic properties. It is thus obvious from the overall achieve-ments that present charge transfer approach and covalencyeffects are essential for the description of mechanical properties.We stress that the charge transfer mechanism and covalencyeffects gives a realistic representation of interionic interactioncapable of explaining the elastic behavior. However, the devia-tions appearing may be ascribed to the extension of covalent andzero point motion effects.

From the computed values of pressure dependent Poisson’sratio n and the ratio RS/B we conclude that Hg1�xMnxS (x¼0.02

D. Varshney et al. / Journal of Physics and Chemistry of Solids 72 (2011) 945–953952

and 0.07) semiconducting compounds show brittle nature in B3phase. We stress that to the best of our knowledge the pressuredependence of ductile (brittle) nature of mixed valent compoundshave not been calculated and measured yet, hence the presentcalculations will inspire further experimental research on thesecompounds.

Acknowledgments

Financial support from DRDO New Delhi is gratefullyacknowledged.

Appendix 1

The expressions for the SOECs and TOECs for ZnS as

C11 ¼ L 0:2477ZðZþ8f ðr0ÞÞþ13 ðA1þ2B1Þþ

12ðA2þB2Þ

�þ5:8243Zaf 0ðr0Þ

, ðA:1Þ

C12 ¼ L �2:6458ZðZþ8f ðr0ÞÞþ13ðA1�4B1Þ

�þ1

4ðA2�5B2Þþ5:8243Zaf 0ðr0Þ , ðA:2Þ

C44 ¼ L �0:123ZðZþ8f ðr0ÞÞþ13 ðA1þ2B1Þþ

14ðA2þ3B2Þ

��1

3rð�7:53912ZðZþ8f ðr0ÞÞþA1�B1Þ , ðA:3Þ

C111 ¼ L 0:5184ZðZþ8f ðr0ÞÞþ19 ðC1�6B1�3A1Þþ

14ðC2�B2�3A2Þ

h�2ðB1þB1Þ�9:9326Zaf 0ðr0Þþ2:522Za2f 00ðr0Þ

, ðA:4Þ

C112 ¼ 0:3828ZðZþ8f ðr0ÞÞþ19 ðC1þ3B1�3A1Þþ

18ðC2þ3B2�3A2Þ

h�11:6482Zaf 0ðr0Þþ2:522Za2f 00ðr0Þ

, ðA:5Þ

C123 ¼ L 6:1585ZðZþ8f ðr0ÞÞþ19ðC1þ3B1�3A1Þ

��12:5060Zaf ’ðr0Þþ2:5220Za2f ðr0Þ

, ðA:6Þ

C144 ¼ L 6:1585ZðZþ8f ðr0ÞÞþ19ðC1þ3B1�3A1Þ�4:1681Zaf ’ðr0Þ

hþ0:8407Za2f ðr0Þþr �3:3507ZðZþ8f ðr0ÞÞ�

29C1

�þ13:5486Zaf ’ðr0Þ�1:681Za2f ðr0Þ

�þr

2�1:5637ZðZþ8f ðr0ÞÞð

þ23 A1�B1ð Þþ1

9C1�5:3138Zaf ’ðr0Þþ2:9350Za2f ðr0Þ��, ðA:7Þ

C166 ¼ L �2:1392ZðZþ8f ðr0ÞÞþ19 ðC1�6B1�3A1Þþ

18ðC2�5B2�3A2Þ

h�ðB1þB2Þ�4:1681Zaf 0ðr0Þþ0:8407Za2f 00ðr0Þ

þr �8:3768ZðZþ8f ðr0ÞÞþ23 ðA1�B1Þ�

29C1

�þ13:5486Zaf 0ðr0Þ�1:681Za2f 00ðr0Þ

�þr

2 2:3527ZðZþ8f ðr0ÞÞð

þ19C1�5:3138Zaf 0ðr0Þþ2:9350Za2f 00ðr0Þ

��, ðA:8Þ

C456 ¼ L 4:897ZðZþ8f ðr0ÞÞþ19ðC1�6B1�3A1Þ�B2

hþr �5:0261ZðZþ8f ðr0ÞÞ�

19C1

� �þr2 7:0580ZðZþ8f ðr0ÞÞð

þ13C1

�þr3

�4:8008ZðZþ8f ðr0ÞÞþ13 ðA1�B1Þ�

19C1

� ��: ðA:9Þ

Using the equilibrium condition

B1þB2 ¼�1:261Z½Zþ8f ðrÞ�: ðA:10Þ

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 /1 0 0S, and the C44 refers to themeasurement of resistance to deformation with respect to ashearing stress applied across the (1 0 0) plane with polarizationin the /0 1 0S direction. Henceforth, the elastic constant C11

represents elasticity in length and a longitudinal strain produces a

change in C11. No doubt, the elastic constants C12 and C44 areintimately related to the elasticity in shape, which is a shearconstant. However, a transverse strain causes a change in shapewithout a change in volume. Thus, the second order elasticconstants as C12 and C44 are less sensitive of pressure as comparedto C11.

Various symbols appear in the above expressions are asso-ciated with the crystal energy and have the following form:

A1 ¼ Aij ¼ L0d2

dr2VSR

ij ðrÞ

� �r ¼ r0

, ðA:11Þ

A2 ¼ Aii ¼ Ajj ¼ L0d2

dr2VSR

ii ðrÞþd2

dr2VSR

jj ðrÞ

� �r ¼ r0

, ðA:12Þ

B1 ¼ Bij ¼L0

a

d

drVSR

ij ðrÞ

� �r ¼ r0

, ðA:13Þ

B2 ¼ Bii ¼ Bjj ¼L0

a

d

drVSR

ii ðrÞþd

drVSR

jj ðrÞ

� �r ¼ r0

, ðA:14Þ

C1 ¼ Cij ¼ L0ad3

dr3VSR

ij ðrÞ

� �r ¼ r0

, ðA:15Þ

C2 ¼ Cii ¼ Cjj ¼ L0ad3

dr3VSR

ii ðrÞþd3

dr3VSR

jj ðrÞ

� �r ¼ r0

: ðA:16Þ

In addition, the pressure derivatives of second order elasticconstants are expressed as

3OdBT

dp¼�

20:1788ZðZþ8f ðr0ÞÞ�3ðA1þA2Þþ4ðB1þB2Þþ3ðC1þC2Þ

�104:8433Zaf 0ðr0Þþ22:7008Za2f 00ðr0Þ

" #,

ðA:17Þ

2Odsdp¼�

�11:5756ZðZþ8f ðr0ÞÞþ2ðA1�2B1Þ

þ23 A2�

72 B2þ

14C2þ37:5220Zaf 0ðr0Þ

" #, ðA:18Þ

and

OdC44

dp¼� 0:4952ZðZþ8f ðr0ÞÞþ

13 ðA1�4B1þC1Þþ

14 2A2�6B2

��þC2Þþ4:9667Zaf 0ðr0Þþ2:522Za2f 00ðr0Þ�

þr �17:5913ZðZþ8f ðr0ÞÞþA1�B2�23C1

hþ40:6461Zaf 0ðr0Þ�5:044Za2f 00ðr0Þ

þr

2 3:1416ZðZþ8f ðr0Þþ23 ðA1�B1Þþ

C13

h�15:9412Zaf 0ðr0Þþ8:8052Za2f 00ðr0Þ

, ðA:19Þ

r ¼�7:5391ZðZþ8f ðr0ÞÞþðA1�B1Þ

�3:141ZðZþ8f ðr0ÞÞþðA1þ2B1Þþ21:765Zaf 0ðr0Þ

� , ðA:20Þ

BT ¼13ðC11þ2C12Þ, ðA:21Þ

and for NaCl phase

C11 ¼ L �5:112ZðZþ12f ðr0ÞÞþA1þ12ðA2þB2Þþ9:3204Zaf 0ðr0Þ

� ,

ðA:22Þ

C12 ¼ L 0:226ZðZþ12f ðr0ÞÞ�B1þ14ðA2�5B2Þþ9:3204Zaf 0ðr0Þ

� ,

ðA:23Þ

C44 ¼ L 2:556ZðZþ12f ðr0ÞÞþB1þ14ðA2�5B2Þ

� , ðA:24Þ

Various symbols appear in Eqs. (A.22)–(A.24) are associatedwith the crystal energy and have the following form:

A1 ¼ Aij ¼ 2L0d2

dr2VSR

ij ðrÞ

� �r ¼ r0

, ðA:25Þ

D. Varshney et al. / Journal of Physics and Chemistry of Solids 72 (2011) 945–953 953

A2 ¼ Aii ¼ Ajj ¼ 4L0d2

dr2VSR

ii ðrÞþd2

dr2VSR

jj ðrÞ

� �r ¼ r0

, ðA:26Þ

B1 ¼ Bij ¼ 2L0d

drVSR

ij ðrÞ

� �r ¼ r0

, ðA:27Þ

B2 ¼ Bii ¼ Bjj ¼ 4L0d

drVSR

ii ðrÞþd

drVSR

jj ðrÞ

� �r ¼ r0

, ðA:28Þ

C1 ¼ 2L0 rd3

dr3VijðrÞ

� �r ¼ r0

, ðA:29Þ

C2 ¼ 4L0 rd3

dr3ViiðrÞþ

d3

dr3VjjðrÞ

� r ¼ r0

: ðA:30Þ

For both ZnS phase and NaCl phase L¼(e2/4a4) and L0 ¼(4a3/e2),and in terms of the short-range energy.

VSRij ðrÞ ¼ bbij exp

riþrj�rij

r

� ��

cij

r6ij

�dij

r8ij

: ðA:31Þ

The short-range interaction (SR) energy is expressed in termsof the overlap repulsion (first term) and the vdW d�d and d�qattractions (second and third terms), respectively.

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