inherent properties of ternary

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associated with the character of the chemical bond and thus provide means for ex- plaining and ... According to the Pauling8 the following relation gives the ...
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S0217979212500798

International Journal of Modern Physics B Vol. 26, No. 15 (2012) 1250079 (11 pages) c World Scientific Publishing Company

DOI: 10.1142/S0217979212500798

INHERENT PROPERTIES OF TERNARY (AN B 2+N C27−N ) TETRAHEDRAL SEMICONDUCTORS

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A. S. VERMA∗ , SHEETAL SHARMA and V. K. JINDAL Department of Physics, Panjab University, Chandigarh 160014, India ∗ ajay [email protected]

Received 29 June 2009 Revised 1 April 2010 Published 1 June 2012 In this paper we have evaluated inherent properties (i.e., heat of formation and cohesive energy) for chalcopyrite structured solids. We have presented two expressions relating the heat of formation and cohesive energy for AII B IV C2V and AI B III C2V I semiconductors with the product of ionic charge (ZA ZB ZC ) and nearest neighbor distance (d in ˚ A). The heat of formation and cohesive energy of these solids exhibit a linear relationship when plotted on a log–log scale against the nearest neighbor distance, but fall on different straight lines according to the product of ionic charge of the compounds. We have applied the modified relations on these solids and found a better agreement with experimental data as compared to the values evaluated by previous researchers so far. The results for heat of formation differ from experimental values by the following amounts: ZnSiP2 — 4.8%, ZnSnP2 — 0.4%, ZnSiAs2 — 0.7%, ZnGeAs2 — 2.6%, ZnSnAs2 — 1.2%, CdGeP2 — 3.8%, CuGaSe2 — 0.3% and AgInSe2 — 5% and the results for cohesive energy differ from experimental values by the following amounts: ZnSiP2 — 0.3%, ZnSnAs2 — 1.5%, CuGaSe2 — 3.7%, CuGaTe2 — 2%, CuInTe2 — 2.7%, AgGaTe2 — 0.7%, AgInSe2 — 3%, AgInTe2 — 3%. Keywords: Heat of formation; cohesive energy and chalcopyrites. PACS numbers: 65.40.-b, 65.40.De

1. Introduction Ternary (AN B 2+N C27−N ) tetrahedral semiconductors have attracted a lot of attention recently,1–6 since they are promising materials for modern microelectronic industries. The ternary compounds are direct gap semiconductors with tetragonal chalcopyrite crystal structure. The body-centered tetragonal chalcopyrite structure can be derived from the cubic zinc-blende structure by populating one of the face centered cubic sublattices with group B atoms and other one with equal ∗ Corresponding

author. 1250079-1

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A. S. Verma, S. Sharma & V. K. Jindal

amounts of group A and C atoms in a regular fashion. Since, generally, A–C and B–C bond lengths, denoted by dA−C and dB−C , respectively, are not equal, the mentioned substitution results in two different structural deformations: the first one is the relocation of anions in the x–y plane, which is characterized by the parameter u = 0.2 + (d2A−C − d2B−C )/a2 . Here, a is the lattice constant in the x- or y-direction. The second consequence of differing anion–cation bond lengths is a deformation of the unit cell to a length c which is generally different from 2a. This tetragonal distortion is characterized by the quantity η = c/a.7 The importance of these crystals has been well established,1–5 because of their useful design parameters like nonlinear coefficient, appropriate energy band-gap and birefringence. It is the last property which makes the chalcopyrites especially attractive for the various nonlinear laser devices,2,3 i.e., second harmonic generation, sum mixing, difference frequency generation and parametric oscillation covering a broad part of the electromagnetic spectrum from ultraviolet to the infrared through the visible region. Compared to their binary analogues these compounds have higher energy gaps and lower melting points, because of which they are considered to be important in crystal growth studies and device applications. Apart from it, the other important technological applications of these materials are in light emitting diodes, infrared detectors, infrared oscillations etc.3–6 A considerable amount of experimental and theoretical work has been done during the last few years on the structural, mechanical and optical properties of chalcopyrite semiconductors.2,4–6 The empirical relations have become widely recognized as the method of choice for computational solid-state studies. Modern computational methods have made it possible to study the structural, mechanical and optical properties of a wide variety of molecules and solids in great detail. There are however, instances where this level of detail either cannot be easily attained because of the complexity of the system or is not needed, as when studying broad trends in the behavior of a large set of systems. Empirical concepts such as valence, empirical radii, electronegativity and ionicity are then useful.8–10 These concepts are directly associated with the character of the chemical bond and thus provide means for explaining and classifying many basic properties of molecules and solids. In modern high-speed computer techniques, it allows researchers to investigate many structural and physical properties of materials only by computation or simulation instead of traditional experiments. Recently, Verma and co-authors11–14 have calculated the electronic, optical and mechanical properties of rock-salt, zinc blende, chalcopyrite and perovskite structured solids with the help of ionic charge theory of solids. It is now well established that ionic charge of a metal changes, when it undergoes a chemical combination and forms a compound. Therefore we thought it would be of interest to give an alternative explanation for inherent properties such as heat of formation and cohesive energy of ternary chalcopyrite semiconductors.

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Inherent Properties of Ternary (AN B 2+N C27−N ) Tetrahedral

2. Previous Models for Inherent Properties

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Inherent properties of the materials are crucial in many industrial applications. Inherent properties such as heat of formation and cohesive energy of most of the chalcopyrite family of semiconductors have not been determined experimentally because of various difficulties in growing single crystals of these compounds. Attempts have been made to fill this gap in the knowledge of the inherent properties of the chalcopyrites by theoretical calculations using different approaches, but mostly the results obtained differ considerably, and in many cases no satisfactory agreement has been achieved with existing experimental data.15–31 2.1. Heat of formation Heat of formation is an important thermal property of the materials and several researchers8,15,16 have discussed about it in terms of electro-negativity difference of the atoms constituting the system. According to the Pauling8 the following relation gives the energy of bond formation between atoms A and B, DAB = 1/2[D(A − A) + D(B − B) + 23Σ(XA − XB )] ,

(1)

where D(A − B), D(A − A) and D(B − B) are the energies of heteropolar bond A − B and homopolar bond A − A and B − B, respectively. XA and XB are electro negativities of atoms A and B. The second term on the right hand side of the equation (1) gives the value of standard heat of formation as given below: − ∆Hf = 23Σ(XA − XB )2 ,

(2) 15,16

where Σ is taken over all the bonds in the compound. Phillips investigated the heat of formation of few compound semi-conductors and proposed modified relation which is given as follows − ∆Hf = 2/3 × 23Σ(XA − XB )2

(3)

Phillips and Van Vechten17 have shown that heat of formation of binary compounds is also correlated to the bond length as follows − ∆Hf (AB) = ∆Ho (dGe /dAB )s D(AB)fiAB

(4)

where dGe and dAB are the bond length of germanium and the binary compound AB, respectively, fi,AB is the bond ionicity,18 and the D(AB) is given by the relation,18,19 D(AB) = 1 − b{2E2 (AB)/[Eo (AB) + E1 (AB)]}2

(5)

where Eo (AB) is the lowest direct energy gap, E1 (AB) and E2 (AB) are higher critical energies of the compound (AB), and b = 0.0467. The values of Eo (AB), E1 (AB) and E2 (AB) can be either taken from the experimental reflectivity data or calculated theoretically using equations given by Van Vechten20 and Neumann.19 In Eq. (4), the values of scaling factor ∆Ho and the exponent “s” have been found 1250079-3

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by Phillips and Van Vechten17 for two different sets of ∆Ho and s. For s = 4, ∆Ho = −300 kJ/mol and s = 3, ∆Ho = −287 kJ/mol. Later on Neumann19 has determined new empirical values of ∆Ho = −304 kJ/mol and s = 3.575 using the experimental values of −∆Hf for binary (AII B VI and AIII B V ) compounds reported in various references,21–23 and taking the ionicities, critical point energies and bond lengths from Van Vechten.18,20 Mulokozi24 has studied the heat of formation of rare earth compounds and proposed a relation for heats of formation in terms of nearest neighbor distance dR−X . According to him heat of formation may be expressed by the following relation, − ∆Hf = Ae2 (∆X)2 /dR−X + C .

(6)

In a series of rare earth compounds (where X does not vary) the partial charge ∆X will be constant (depending upon the electro negativity difference between the atoms R and X), e and A elementary charge, which are constants and C is covalent contribution. According to Mulokozi the value of Ae2 (∆X)2 and C depends upon cation. Recently, Kumar et al.,2 have developed a relation based on the plasma oscillations theory of solids for the calculation of heat of formation of AII B IV C2V and AI B III C2V I semiconductors. According to them heat of formation (−∆Hf ) of these semiconductors may be expressed as, − ∆Hf = A(hωp )B ,

(7)

where “A”’ and “B”’ are constants and the numerical values of the constants “A” and “B” are 0.3170 and 2.5310, respectively, for AI B III C2V I semiconductors and 11.4136 and 1.1624, respectively, for AII B IV C2V semiconductors. 2.2. Cohesive energy According to Sohn et al.,25 the bond-stretching potential acting on the nearest neighbors, the cohesive energy (Ecoh ) per unit cell, can be described as a sum of the Madelung energy and the Morse potential: Ecoh = −(AZ ∗2 e2 /r) + N D exp[−2β(r − rc )] − 2N D exp[−β(r − rc )] ,

(8)

where A is the Madelung constant, N is the number of bonds per unit cell and D is a bond energy. It is noted that Eq. (8) would approach the Born–Mayer-type cohesive energy without the last term, i.e., the Morse attraction potential. The equilibrium position rc in the Morse potential would be moved to the new equilibrium d (nearest neighbor distance) due to the Madelung energy. The constant β can be determined from the cohesive energy of group IV crystals that consists of only the Morse potential in Eq. (8): Ecoh (IV ) = N D exp[−2β(r − rc )] − 2N D exp[−β(r − rc )] 1250079-4

(9)

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Inherent Properties of Ternary (AN B 2+N C27−N ) Tetrahedral

where N = 4. The bulk modulus B is related to the cohesive energy per unit cell volume V : B = V (dr/dV )2 (d2 Ecoh /dr2 )r=rc .

(10)

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From Eqs. (9) and (10), we obtain the constant β with the covalent bond length rc : √ β = (2 3 rc B/D]1/2 (11) Aresti et al.26 have studied the cohesive energy of zinc-blende solids and proposed an empirical relation for cohesive energy in terms of nearest-neighbor distance d. According to them the cohesive energy may be expressed by the following relation: Ecoh = Ecoh (IV ) − B(d, R){1 − ΣEcoh (i)/Ecoh (IV )}

(12)

where Ecoh (IV ) is the cohesive energy of purely covalent crystals, ΣEcoh (i) is the sum of the standard heats of atomization of the cation and anion, and B(d, R) = Ecoh (IV ) − k(R) · d(BX)/d is now a parameter depending on d and R (row of element): k(R) = C exp(−Z 1/2 /4)

(13)

where C is a constant, which depends on the rows and Z = Z(A) + Z(B), the atomic number of atom A and atom B. Recently,25–29 many theoretical approaches have been reported to determine the value of cohesive energy of solid-state compounds in terms of nearest neighbor distance. Schlosser30,31 has studied the cohesive energy trends in rocksalt structure (alkali halides) in terms of nearest neighbor distance using the following relation, Ecoh = constant/d .

(14)

3. Curves Between the Inherent Properties and Nearest-Neighbor Distance We have plotted log −∆Hf (heat of formation) versus log d3 (nearest-neighbor distance in ˚ A) and log Ecoh (cohesive energy) versus log d2.5 , curves for AI B III C2V I and AII B IV C2V semiconductors, these are presented in the following Figs. 1 and 2. From the Fig. 1 it is quite obvious that the heat of formation (−∆Hf ) of chalcopyrite semiconductors exhibits a linear relationship when plotted against nearest-neighbor distance d (˚ A), but fall on different straight lines according to the ionic charge product of the compounds. The cohesive energy of AI B III C2V I and AII B IV C2V semiconductors also exhibits a linear relationship when plotted against nearest-neighbor distance and melting temperature, but fall on different straight lines according to the product of ionic charge of the compounds, this is presented in Fig. 2. 1250079-5

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I

2.55

log (∆H f )



CuGaSe2

ZnSiP2

III

A B C2

VI

ZnGeP2

2.45

CuInSe2



2.35 II

IV

ZnGeAs 2

AgInSe2

V

2.25

2.15

1.05

1.10

1.15

1.20

log d

1.30

1.25

3

Fig. 1. Plot of log ∆HF (heat of formation in kJ/mol) against log d3 (d = nearest neighbor distance in ˚ A) for AI B III C2V I and AII B IV C2V chalcopyrite semiconductors. Plots of AI B III C2V I II IV and A B C2V chalcopyrites are nearly parallel. All experimental data are taken from Refs. 2, 19, 33.

2.56 AgGaSe2

2.54

CuInSe2

2.52

ZnSiP2 CdSiP2

2.50

log Ecohesive

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A B C2

2.48 AgInSe2

2.46 2.44

CuGaTe2

AgGaTe2

2.42

ZnSnAs2 CuInTe2

2.40 0.90

0.92

0.94

0.96

0.98

1.00

log d

1.02

1.04

1.06

1.08

2.5

Fig. 2. Plot of log Ecohesive (cohesive energy in kcal/mol) against log d2.5 (d = nearest neighbor distance in ˚ A) for AI B III C2V I and AII B IV C2V chalcopyrite semiconductors. Plots of AI B III C2V I II IV and A B C2V chalcopyrites are nearly parallel. All experimental data are taken from Refs. 26. 1250079-6

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4. Concept of Ionic Charge Theory

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Any change in crystallographic environment of an atom is related to core electrons via the valence electrons. The change in wavefunction that occurs for the outer electrons usually means a displacement of electric charge in the valence shell so that the interaction between valence shell and core electrons is changed. This leads to a change in binding energy of the inner electron and to a shift in the position of the absorption edge. The valence electrons refer to the electrons that take part in chemical bonding. These electrons reside in the outer most electron shell of the atom. The participation of valence shell electrons in chemical bonding may be explained on the basis of following grounds. (i) The outermost-shell electrons are farthest away from the nucleus and therefore, are not very firmly bound to the nucleus. As such these are easier to remove due to low ionization energy. (ii) The outermost-shell electrons of an atom are also close to any foreign atom that may approach them and are therefore the first to be attracted by the approaching atom. Ionic charge also depends on valence electrons. Thus there must be a correlation between ionic charge and the inherent properties of solids. From the Fig. 1, the heat of formation (−∆Hf ) trends in these compounds decreases with increase in nearest neighbor distance and fall on straight lines according to the ionic charge product of the compounds. Similarly, from the Fig. 2, the cohesive energy trends in these compounds decreases with increase in nearest neighbor distance and fall on different straight lines according to the ionic charge product of the compounds. For getting better agreement with experimental and theoretical data for chalcopyrite type crystal structure compounds, relations (6), (7), (12) and (14) may be extended as, − ∆Hf = M/(ZA ZB ZC )N d3

(15)

Ecoh = R/(ZA ZB ZC )P d2.5 .

(16)

Here ZA , ZB and ZC are the ionic charge on the A, B and C2 , respectively. M , N , R and P are constants and the values are 5500, 0.08, 4000 and 0.1, respectively. It is well-known that in chalcopyrites each cation has four equal anion bonds but each anion has four (two + two) different cation bonds, this fact gives anion–cation distances dAC and dBC . In this relation d is average nearest neighbour distance and for AI B III C2V I and AII B IV C2V chalcopyrites can be calculated by (dAC + dBC )/2. A detailed study for ionic charge of chalcopyrite has been presented in previous works32 and the valence structures of the compounds can be written as A+ , B 3+ , C22− (A = Cu, Ag; B = Al, Ga, In; C = S, Se, Te) and A2+ , B 4+ C23− (A = Zn, Cd; B = Si, Ge, Sn; C = P, As). Therefore the product of ionic charge is 12 for AI B III C2V I and 48 for AII B IV C2V . The physical concept behind equation (15) is that the heat of formation (−∆Hf ) is related to the crystal ionicity, energy 1250079-7

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gaps and plasmon energy of the crystals.8,15–23 The crystal ionicity, energy gaps also depends on the product of ionic charges.32 Thus, there must be a correlation between −∆Hf and product of ionic charges. The main advantage of equations (15) is the simplicity of the formula, which does not require any experimental data except the nearest neighbor distance of AI B III C2V I and AII B IV C2V chalcopyrites. However, the previous models require the experimental values of Eo , E1 , E2 and fi in addition to bond length of these semiconductors. A detailed discussion of heat of formation and cohesive energy for these materials have been given elsewhere8,15–31 and will not be presented here. 5. Comparision Between Calculated and Experimental Values Although the properties of the AI B III C2V I and AII B IV C2V chalcopyrites have been extensively investigated and some of these compounds have attracted attention for Table 1. Values of heat of formation (−∆Hf in kJ/mol) for AI B III C2V I and AII B IV C2V semiconductors. Solids CuAlS2 CuAlSe2 CuAlTe2 CuGaS2 CuGaSe2 CuGaTe2 CuInS2 CuInSe2 CuInTe2 AgAlS2 AgAlSe2 AgAlTe2 AgGaS2 AgGaSe2 AgGaTe2 AgInS2 AgInSe2 AgInTe2 CuFeS2 ZnSiP2 ZnGeP2 ZnSnP2 ZnSiAs2 ZnGeAs2 ZnSnAs2 CdSiP2 CdGeP2 CdSnP2 CdSiAs2 CdGeAs2 CdSnAs2

d (˚ A) 2.29 2.40 2.58 2.30 2.42 2.60 2.40 2.51 2.68 2.40 2.51 2.68 2.42 2.53 2.69 2.49 2.61 2.78 2.30 2.31 2.35 2.45 2.41 2.44 2.53 2.40 2.44 2.54 2.49 2.53 2.62

∆Hf exp .2,19,33

317

267

446

242

312 293 275 290 271 252 289 270 290 266 247

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∆Hf 2

∆Hf 19

∆Hf

427.9 345.9 268.5 418.6 349.3 266.2 360.5 305 237.1 365.6 308.6 275.7 359.4 288.4 235.7 311.2 262.9 210.8 424.2 307.8 299.8 277.1 287.5 279.4 262 290.4 276.5 262 273 263.7 247.7

463.8 389.9 285.4 420.4 329.5 260.8 327.7 263.5 228.5 417.9 361.3 279.9 394.8 318.2 252.7 330.6 268 217.9

375 326 263 371 318 257 326 285 234 326 285 234 318 278 232 292 254 210 371 327 311 274 288 278 249 292 278 246 261 249 224

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Inherent Properties of Ternary (AN B 2+N C27−N ) Tetrahedral Table 2. Values of cohesive energy (Ecoh in kcal/mol) for AI B III C2V I and AII B IV C2V semiconductors.

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Solids CuAlS2 CuAlSe2 CuAlTe2 CuGaS2 CuGaSe2 CuGaTe2 CuInS2 CuInSe2 CuInTe2 AgAlS2 AgAlSe2 AgAlTe2 AgGaS2 AgGaSe2 AgGaTe2 AgInS2 AgInSe2 AgInTe2 ZnSiP2 ZnGeP2 ZnSnP2 ZnSiAs2 ZnGeAs2 ZnSnAs2 CdSiP2 CdGeP2 CdSnP2 CdSiAs2 CdGeAs2 CdSnAs2

d (˚ A) 2.29 2.40 2.58 2.30 2.42 2.60 2.40 2.51 2.68 2.40 2.51 2.68 2.42 2.53 2.69 2.49 2.61 2.78 2.31 2.35 2.45 2.41 2.44 2.53 2.40 2.44 2.54 2.49 2.53 2.62

Experimental26

330.4 ± 13.8 280.3 ± 4.8 341.2 ± 9.3 258.3 ± 11.3

348.7 ± 20.7 261.0 ± 3.7 292.5 ± 9.2 249.5 ± 10.2 336.0

262.9 ± 7.6 323.4 ± 11.8

This work 393.2 349.6 291.8 388.9 342.5 286.2 349.6 312.6 265.3 349.6 312.6 265.3 342.5 306.5 262.9 318.9 283.5 242.1 334.9 320.8 289.1 301.2 292.05 266.8 304.4 292.05 264.2 277.6 266.8 244.4

practical applications,7 the knowledge of their inherent properties such as heat of formation (−∆Hf ) and cohesive energy (Ecoh ) are rather incomplete. Experimental data are available for few compounds for chalcopyrite series AI B III C2V I and AII B IV C2V , so there are many properties of the solid solution, which have not been investigated. In Ref. 7 an analysis of the dependence of their chemical composition is given. In the present work it is shown that analogous relations exists for the ternary chalcopyrite semiconductors, which can be successfully employed to estimate the heat of formation (−∆Hf ) and cohesive energy (Ecoh ) from their ionic charges. In Tables 1 and 2 we have presented experimental heat of formation (−∆Hf ) and cohesive energy (Ecoh ) values evaluated by Kumar et al.,2 Neumman,19 Lide33 and other experimental values25,26 for the sake of comparison. We note that the evaluated values of heat of formation and cohesive energy by the proposed relations are in close agreement with the experimental data as compared to the values reported by previous researchers so far. For example, the results 1250079-9

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for heat of formation differ from experimental values by the following amounts: ZnSiP2 — 4.8%, ZnGeP2 — 6.1%, ZnSnP2 — 0.4%, ZnSiAs2 — 0.7%, ZnGeAs2 — 2.6%, ZnSnAs2 — 1.2%, CdGeP2 — 3.8%, CdSnP2 — 8.9%, CdGeAs2 — 6.4%, CuGaSe2 — 0.3%, CuInSe2 — 6.7% and AgInSe2 — 5% and the results for cohesive energy differ from experimental values by the following amounts: ZnSiP2 — 0.3%, ZnSnAs2 — 1.5%, CdSiP2 — 6%, CuGaSe2 — 3.7%, CuGaTe2 — 2%, CuInSe2 — 8.4%, CuInTe2 — 2.7%, AgGaTe2 — 0.7%, AgInSe2 — 3%, AgInTe2 — 3% in the current study. These results show that our current method is quite reasonable and can give us a useful guide in calculating and predicting the inherent properties of the more complex class of ternary chalcopyrite semiconductors. 6. Summary and Conclusions There are several methods in determining heat of formation and cohesive energy in semiconductors, but due to the small changes of the unit cell dimensions, the accuracy of determining these parameters always have been unpredictable. In previous models2,8,15–31 needs a powerful computer and efficient algorithms. In the proposed model, calculations are simple, fast and more accurate; in regard of the applications point of view it can be highly dependent. The only information needed for calculating inherent properties by proposed relations is the nearest neighbor distance and the product of ionic charge. We come to the conclusion that product of ionic charge of any compound is key parameter for calculating physical properties. It is also noteworthy that the proposed empirical relations are simpler widely applicable and values are in better agreement with experimental data as compared to empirical relation proposed by previous researchers. Since, we have been reasonably successful in calculating these parameters using the product of ionic charges and nearest neighbor distance of the materials for chalcopyrite crystals. It is natural to say that this model can easily be extended to rock-salt and zinc-blende crystals for which the work is in progress and will be appearing in forthcoming paper. Acknowledgments One of the authors (Dr. Ajay Singh Verma, PH/08/0049) is thankful to the University Grant Commission New Delhi, India for supporting this research under the scheme of U.G.C. Dr. D. S. Kothari Post Doctoral Fellowship. References 1. 2. 3. 4. 5.

J. Laz¨ yewski et al., J. Appl. Phys. 93, 3789 (2003). V. Kumar and B. S. R. Sastry, J. Phys. Chem. Solids 66, 99 (2005). T. Gurel and R. Eryigit, J. Phys. Condens. Matter 18, 1413 (2006). M. S. Omar, Mat. Res. Bull. 42, 319 (2007). A. V. Kosobutsky, Y. M. Basalaev and A. S. Poplavnoi, Phys. Status Solidi B 246, 364 (2009). 6. V. Kumar et al., Solid State Commun. 149, 1008 (2009). 1250079-10

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Inherent Properties of Ternary (AN B 2+N C27−N ) Tetrahedral

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