© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Phys. Status Solidi B 246, No. 2, 345–353 (2009) / DOI 10.1002/pssb.200844337 p s sbasic solid state physics

b

statu

s

soli

di

www.pss-b.comph

ysi

ca

An empirical model for bulk modulus and cohesive energy of rocksalt-, zincblende- and chalcopyrite-structured solids

A. S. Verma*, 1, 2

1 Department of Physics, B.S.A. College, Mathura 281004, India 2 Department of Physics, Sanjay Institute of Engineering and Management, Mathura 281406, India

Received 22 August 2008, revised 5 October 2008, accepted 3 November 2008 Published online 12 December 2008

PACS 62.20.de, 71.15.Nc * e-mail ajay_phy@rediffmail.com, Phone: +91 565 2423417, Mob.: +91 9412884655

© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction Alkali halides, alkaline-earth chalco-genides and transition-metal nitrides (TMN) have attracted the attention of physicists due to many practical uses and they have rocksalt crystal structures. Compounds of alkali metals with halogen elements are crystalline solids, which are of great interest from both the theoretical and experi-mental standpoints and have a special place in chemistry since they are simple ionic molecules. Alkali halides are relatively easy to subject to theoretical treatment since they have simple structure and are bound by the well-under-stood Coulomb forces between the ions [1–3]. Alkaline-earth compounds are important in geophysical science and research since they are the link between the highly ionic alkali halides and covalent III–V compounds. One impor-tant aspect that has attracted interest in these materials is the fact that some of them are oxygen based and this is of fundamental interest in geophysics because oxygen is the dominant anion in the earth [3, 4]. Transition-metal ni-trides (TMN) generally have good properties of high hard-ness, high melting point and wear resistance. They are widely used for cutting tools and hard coatings [5–7]. Bi-nary tetrahedral semiconductors (II–VI group and III–V group) have attracted the attention of physicists due to their application in photovoltaic devices, as electro-optic crys-

tals with ultrafast response time spectroscopy and recent realization of light-emitting diodes and the have zincblende crystal structures. Most of the semiconductors that are used in the modern microelectronic industry have the zincblende crystallographic structure. The crystals with zincblende structure range from raw iron and zinc minerals to man-made GaN and BN semiconductors. The particular omni-triangulated nature in atomic structure gives these materi-als unique physical properties [8–12]. The compounds of the type AIIBIVC2

V and AIBIIIC2VI have attracted considerable

attention because of their interesting semiconducting, elec-trical, structural, mechanical and optical properties. Com-pared 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. The ternary com-pounds are direct-gap semiconductors with tetragonal chalcopyrite crystal structure. Structurally, these com-pounds are derived from that of the binary sphalerite struc-ture (AIIIBV and AIIBVI) with a slight distortion. Therefore, like binary compounds they have a high nonlinear suscep-tibility. However, because of the presence of two types of bonds in chalcopyrites they become anisotropic. This ani-sotropy gives rise to high birefringence. High nonlinear

In this paper we present empirical models for ground-stateproperties such as bulk modulus and cohesive energy of rock-salt-, zincblende- and chalcopyrite-structured solids. The bulkmoduli and cohesive energy of these solids exhibit a linear re-lationship when plotted on a log–log scale versus the nearest-neighbor distance d (Å), but fall on different straight lines ac-

cording to the product of valence electrons of the solids. Wehave applied the proposed relations to alkali halides, alkaline-earth chalcogenides, binary and ternary tetrahedral semicon-ductors and found a better agreement with experimental dataas compared to the values evaluated by earlier researchers.

346 A. S. Verma: An empirical model for bulk modulus and cohesive energy

© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.pss-b.com

ph

ysic

ap s sstat

us

solid

i b

susceptibility coupled with high birefringence in these compounds makes them very useful for efficient second-harmonic generation and phase matching. Apart from this, the other important technological applications of these ma-terials are in light-emitting diodes, infrared detectors, infra-red oscillations, lasers, etc. [12–16]. An empirical model has yielded explanations for a vast number of solid-state properties. Condensed-matter theo-rists can predict crystal structures, lattice constants, phase diagrams and related properties very accurately. Fortu-nately, recently [17–23] there have been some significant breakthroughs in these and related areas. These improve-ments depend heavily on new developments in empirical techniques, and to a greater extent on the insights gained through close collaborations between theorists and experi-mentalists doing research on the solid-state properties of solids. Empirical concepts such as valence electron, em-pirical radii, electronegativity, ionicity and plasmon energy are then useful [17–20]. These concepts are directly asso-ciated with the character of the chemical bond and thus provide means for explaining and classifying many basic properties of molecules and solids. In previous work we have presented empirical relations for mechanical and optical properties of chalcopyrite semi-conductors in terms of plasmon energy [24]. In many cases empirical relations do not give highly accurate results for each specific material, but they can still be very useful. In particular, the simplicity of empirical relations allows a broader class of researchers to calculate useful properties, and often trends become more evident. In the present arti-cle, we review various models for bulk modulus and cohe-sive energy of rocksalt-, zincblende- and chalcopyrite-structured solids. 2 Relationship between the ground-state pro-perties and nearest-neighbor distance 2.1 Bulk modulus The bulk modulus is an important mechanical property of the materials and defines its resist-ance to volume change when compressed. First, Anderson and Nafe [25] have proposed an empirical relationship be-tween bulk modulus B0 at atmospheric pressure and spe-cific volume V0 of the form B0 ~ V0

–x. They find it to hold for a particular class of compounds. The value of x de-pends on the class of compounds. For alkali halides, fluo-rides, sulfides and telluride they find x to be 1 and for oxide compounds x is close to 4. Recently, [5, 9, 19, 21, 36–38] many theoretical approaches have been reported to determine the value of bulk modulus of solid-state com-pounds. Cohen [19] predicted that the zero-pressure iso-thermal bulk modulus B in terms of nearest-neighbor dis-tance d (in Å) for rocksalt-type crystal structure com-pounds (alkali halides) might be expressed as

B = 550/d 3 . (1)

The relation of bulk moduli and geometrical properties of diamond and ZB solids had previously been investigated by Cohen [19] and Lam et al. [26]. Based on the Phillips

and Van Vechten scheme [18] and theoretical analysis of bond geometry of covalent ZB solids, Cohen proposed the following empirical relation:

B = 1761/d 3.5 , (2)

where d is the nearest-neighbor distance in Å and B is in GPa. Lam et al. [26] deduced an analytic relation of bulk moduli to lattice parameters within the local-density for-malism and the pseudopotential as approach as

B = 1971d–3.5 – 408 (ΔZ)2 d –4 , (3)

where ΔZ = 1 and 2 for III–V and II–VI semiconductors. Kumar et al. [27], have developed a relation based on the plasma oscillations theory of solids for the calculation of bulk modulus (B) of binary tetrahedral semiconductors. According to them the bulk modulus of these semiconduc-tors may be expressed as

B = K1(�ωp) – K2 , (4)

where �ωp is plasmon energy and ‘K1’ and ‘K2’ are con-stants. The numerical values of the constants ‘K1’ and ‘K2’ are 12.47 and 124.33, respectively, for AIIBVI and 13.54 and 130.33, respectively, for AIIIBV semiconductors. The energy of a quantum of plasma oscillations of the valence electrons in both metal and compound is given by the relation [20]

�ωp = 28.8 ( / )Z Wσ , (5)

where Z is the effective number of valence electrons taking part in the plasma oscillations, σ is the specific gravity and W is the molecular weight. Al-Douri et al. [9] studied the bulk modulus (B) of IV, III–V and II–VI semiconductors and proposed an empiri-cal relation for bulk modulus (B) in terms of lattice con-stant. According to them the bulk modulus of these semi-conductors may be expressed as

B = [3000 – (λ100)] (a/2)−3.5 , (6)

where a (in Å) is the lattice constant and λ is an empirical parameter appropriate for the group-IV (λ = 0), III–V (λ = 1) and II–VI (λ = 2) semiconductors. Recently, Al-Douri et al. [21] studied the bulk modulus of IV, III–V and II–VI semiconductors and proposed an empirical relation for bulk modulus in terms of transition pressure (Pt). According to them the bulk modulus of these semiconductors may be expressed as

B = [99 – (λ + 79)] (10Pt)1/3 , (7)

where Pt is the transition pressure in GPa from ZB to β-Sn and λ is a parameter appropriate for group-IV (λ = 1), III–V (λ = 5) and II–VI (λ = 8) semiconductors. 2.2 Cohesive energy According to Sohn et al. [28], the bond-streching potential acting on the nearest neighbors, the cohesive energy (Ecoh) per unit cell, can be

Phys. Status Solidi B 246, No. 2 (2009) 347

www.pss-b.com © 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Original

Paper

described as a sum of the Madelung energy and the Morse potential:

Ecoh = –(AZ*2e

2/r) + ND exp [–2β (r – rc)]

– 2 ND 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 po-tential. 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 con-stant β can be determined from the cohesive energy of group IV crystals that consists of only the Morse potential in Eq. (8):

Ecoh (IV) = ND exp [–2β (r – rc)]

– 2 ND exp [–β (r – rc)] , (9)

where N = 4. The bulk modulus B is related to the cohesive energy per unit cell volume V:

B = V (dr/dV)2 (d2Ecoh/dr

2)r = rc . (10)

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. [29] have studied the cohesive energy of zincblende solids and proposed an empirical relation for cohesive energy in terms of nearest-neighbor distance d. According to them the cohesive energy may be express 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 atomi-zation 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:

k(R) = C exp (–Z1/2/4) , (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 [30–32] many theoretical approaches have been reported to determine the value of cohesive energy of solid-state compounds. Schlosser [33, 34], has studied the cohesive-energy trends in rocksalt structure in terms of nearest-neighbor distance using the following relation,

Ecoh = constant/d . (14)

3 Concept of valence-electron theory Molecules of chemical substances are made up of two or more atoms joined together by some force acting between them. A chemical bond is formed when the atoms with incomplete

valence shells combine. There are the following main types of bonds: 1. ionic or electrovalent bond 2. covalent bond 3. coordinate bond 4. metallic bond. The valence electrons refer to the electrons that take part in chemical bonding. These electrons reside in the outermost electron shell of the atom. The participation of valence-shell electrons in chemical bonding may be explained on the basis of the following: (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. Krishnan–Roy theory [35], Jayaraman et al. [36] and Sirdeshmukh and Subhadra [37] found that substantially reduced ionic charges must be used to get better agreement with experimental values. Ionic charge depends on the number of valence electrons. Thus, there must be a correla-tion between valence electrons and the ground-state prop-erties of solids. For getting better agreement with experi-mental and theoretical data for rocksalt-, zincblende- and chalcopyrite-type crystal structure compounds, relation (1) may be extended as follows: – for binary (AB) solids,

bulk modulus (GPa) = S1 (VAVB)M/d 3 , (15)

– for complex (ABC2) solids,

bulk modulus (GPa) = 3200 (VAVBVC)0.22/d 5 . (16)

Similarly, relation (14) may be extended as follows: – for rocksalt-structured solids,

Ecoh = S2 (VAVB)N/d , (17)

– for zincblende-structured solids,

Ecoh = S2 (VAVB)N/d 2.5 , (18)

where VA, VB and VC are the valence electron of A, B and C2, respectively, and ‘S1’ and ‘M’ are constants, which de-pend upon crystal structure. For rocksalt crystal structure compounds values of constants ‘S1’ and ‘M’ are equal to 11 and 2, respectively, and ‘S2’ and ‘N’ are equal to 3.43 and 2.58, respectively. For zincblende crystal structure com-pounds values of constants ‘S1’ and ‘M’ are equal to 161.5 and 0.73, respectively, and ‘S2’ and ‘N’ are equal to 34 and 1.45, respectively. The ternary chalcopyrite semiconduc-tors have two types of bonds (A–C and B–C), hence two nearest-neighbor distances dAC and dBC, respectively. The effective nearest-neighbor distance (d) of the compound has been evaluated by taking the geometric mean of the two.

348 A. S. Verma: An empirical model for bulk modulus and cohesive energy

© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.pss-b.com

ph

ysic

ap s sstat

us

solid

i b

Figure 1 Plot of log B (bulk modulus) (GPa) against log d 3 (d = nearest-neighbor distance) for monovalent (alkali halides), divalent (alkaline-earth oxides) and trivalent (transition-metal nitrides) compounds. In the plots of log B and log d 3, the divalent and trivalent compounds lie on lines nearly parallel to the line for the alkali halides. In this figure all values are taken from experimental data [32, 34, 38–41, 44]. Only TMN values are theoretical.

4 Curves between the ground-state properties and nearest-neighbor distance Jayaraman et al. [36] have examined the bulk-modulus–volume relationship in rocksalt-structured solids. They observe that in the plots of log B0 and log V0; the divalent and trivalent chalcogenides lie on lines nearly parallel to the line for the alkali halides. Sirdeshmukh and Subhadra [37] have examined the bulk modulus of eight divalent oxides. They observe that in the plot of bulk modulus and unit-cell volume the divalent chalcogenides lie on a line nearly parallel to the line for the alkali halide and eight divalent oxides. We have plotted log B vs. log d 3 and log E vs. log d curves for alkali halides, divalent and trivalent chalcogenides and oxides, which are

presented in Figs. 1 and 2. For zincblende crystal structures, we have plotted log B vs. log d

3 and log E vs. log d2.5

curves for group III–V and group II–VI semiconductors, which are presented in Figs. 3 and 4. For chalcopyrite crys-tal structures, we have plotted log B vs. log d

5 curve for AIIBIVC2

V and AIBIIIC2VI semiconductors, which are pre-

sented in Fig. 5. We observe that in the plot of bulk modulus and nearest-neighbor distance, the divalent and trivalent chalcogenides lie on lines nearly parallel to the line for the alkali halides and in the plot of cohesive energy and nearest-neighbor distance, the alkaline-earth chalco-genides lie on a line nearly parallel to the line for the alkali halides. In Fig. 3, we observe that in the plot of bulk

Figure 2 Plot of log B (bulk modulus) (GPa) against log d 3 (d = nearest-neighbor distance) for group III–V and group II–VI semiconductors. In the plots of log B and log d

3, group III–V semiconductors lie on a line nearly parallel to the group II–VIs. In this figure all values are taken from experimental data [8, 19, 45].

Phys. Status Solidi B 246, No. 2 (2009) 349

www.pss-b.com © 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Original

Paper

Figure 3 Plot of log B (bulk modulus) (GPa) against log d 5 (d = nearest-neighbor distance) for AIBIIIC2

VI and AIIBIVC2V chalcopyrite

semiconductors. In the plots of log B and log d 5, AIBIIIC2

VI chal-copyrites lie on a line nearly parallel to the line for AIIBIVC2

V chal-copyrites. In this plot all data are taken from experimental results [50]. In this figure Series 1 and Series 2 represent the AIBIIIC2

VI and AIIBIVC2

V chalcopyrites, respectively. modulus and nearest-neighbor distance; the group III–V semiconductors lie on a line nearly parallel to the line for the group II–VI semiconductors and in the plot of cohe-sive energy and nearest-neighbor distance, the group III–V semiconductors lie on a line nearly parallel to the line for the group II–VI semiconductors. Similarly, in Fig. 5, we observe that in the plot of bulk modulus and nearest-

Figure 5 In the plot of log (Ecoh in kcal/mol) and log d 2.5, AIIIBV semiconductors lie on a line nearly parallel to the line for AIIBVI semiconductors, which is dependent upon the product of valence electrons. In this figure all experimental cohesive energy values are taken from Ref. [13]. neighbor distance; the group AIIBIVC2

V semiconductors lie on a line nearly parallel to the line for the group AIBIIIC2

VI semiconductors. From these figures it is quite obvious that the bulk modulus and cohesive energy trends in these com-pounds decreases with increased nearest-neighbor distance and fall on straight lines according to the valence-electron product of the compounds.

Figure 4 Plot of log E (cohesive energy) (kcal/mol) against log d (d = nearest-neighbor distance) for alkaline-earth chalcogenides and alkali halides. (In this plot all data are taken from experimental data). In the plots of log E and log d, the alkaline-earth chalcogenides lie on a line nearly parallel to the line for the alkali halides [32, 34, 38–41, 44].

350 A. S. Verma: An empirical model for bulk modulus and cohesive energy

© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.pss-b.com

ph

ysic

ap s sstat

us

solid

i b

Table 1 Values of bulk modulus B (GPa) and cohesive energy (Ecoh in kcal/mol) for alkali halides and alkaline-earth chalcogenides (rocksalt structure).

solids d (Å) VA VB B [theoretical] B [experimental] B [this work] Ecoh [theoretical] Ecoh [experimental] Ecoh [this work]

MgO 2.105e 2 6 162g 165e 170 1002a 932a,b 992 CaO 2.405b 2 6 114c 111g 114 871a 839a,b 868 SrO 2.580b 2 6 88c 89g 92 811a 796a,b 809 BaO 2.761b 2 6 61c 75f, 74g 75 756a 786a,b 756 MnO 2.222b 2 6 147c 144 911b 939 FeO 2.139c 2 6 163c 162 938b 976 CoO 2.133b 2 6 181c 163 955b 979 NiO 2.084b 2 6 173c 175 976g 1002 MgS 2.540a 2 6 97 816a 896a 822 CaS 2.830a 2 6 70 715a 764a 738 SrS 3.010b 2 6 58 652a 720a 694 BaS 3.180a 2 6 49 632a 679a 656 PbS 2.968c 2 6 62c 61 703 MgSe 2.720a 2 6 79 726a 780a 767 CaSe 2.960a 2 6 61 664a 726a 705 SrSe 3.100a 2 6 53 625a 693a 673 BaSe 3.300a 2 6 40i 44 589a 660a 633 CaTe 3.170a 2 6 50 616a 679a 658 PbSe 3.062c 2 6 54c 55 682 SrTe 3.330a 2 6 43 587a 667a 627 BaTe 3.500a 2 6 34i 37 553a 629a 596 PbTe 3.227c 2 6 46c 47 647 LiF 2.014a 1 7 63j 63d 66 266a 247a 258 LiCl 2.570h 1 7 33j, 30A 32d 32 205a 203a 202 LiBr 2.750h 1 7 24h 24d 26 191a 194a 189 LiI 3.012h 1 7 17h 17j 20 172a 180a 173 NaF 2.317h 1 7 47h, 49i 47d 43 233a 220a 224 NaCl 2.820h 1 7 25i, 24A 23d 24 189a 187a 184 NaBr 2.987h 1 7 21i, 20A 20d 20 180a 179a 174 NaI 3.236h 1 7 16i, 15A 15d 16 162a 167a 161 KF 2.674h 1 7 30h, 30A 31B 28 204a 194a 194 KCl 3.146h 1 7 17h, 17A 18B 17 172a 170a 165 KBr 3.300h 1 7 15h, 15A 15B 15 165a 163a 157 KI 3.533h 1 7 12h, 12A 12d 12 152a 154a 147 RbF 2.815a 1 7 27h, 27A 26B 24 200a 186a 185 RbCl 3.291h 1 7 16h, 16A 16B 15 166a 164a 158 RbBr 3.445h 1 7 14i, 13A 13B 13 157a 157a 151 RbI 3.671h 1 7 11i, 11A 11B 11 149a 149a 142 aRef. [38],

bRef. [34],

cRef. [37],

dRef. [39],

eRef. [40],

fRef. [32],

gRef. [41],

hRef. [33],

iRef. [36],

jRef. [42],

ARef. [43],

BRef. [44].

Table 2 Values of bulk modulus B (GPa) and cohesive energy (Ecoh in kcal/mol) for zincblende-structured solids.

solid d (Å) VA VB B [theoretical] B [experimental] B [this work] Ecoh [theoretical] Ecoh [experimental] Ecoh [this work]

ZnS 2.340u 2 6 90u 77u 77 151t 147t 149 ZnSe 2.460u 2 6 75u 62u 67 124t 125t 132 ZnTe 2.640u 2 6 59u 51u 54 109t 106t 110 CdS 2.520u 2 6 69u 62u 62 135t 132t 124 CdSe 2.620u 2 6 60u 53u 55 110t 114t 112 CdTe 2.810u 2 6 47u 42u 45 95t 96t 94 HgS 2.530v 2 6 69v 61 123 HgSe 2.630v 2 6 58v 55 111 HgTe 2.800v 2 6 48v 45 95 AlN 1.87x 3 5 186y 178 261 AlP 2.36u 3 5 89x 86x 89 197t 198t 202 AlAs 2.43u 3 5 75x 82x 81 177t 179t 187 AlSb 2.66u 3 5 63* 58x 62 162t 165t 150 GaN 1.88x 3 5 173z 190y 175

356

Phys. Status Solidi B 246, No. 2 (2009) 351

www.pss-b.com © 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Original

Paper

Table 2 (Continued)

solid d (Å) VA VB B [theoretical] B [experimental] B [this work] Ecoh [theoretical] Ecoh [experimental] Ecoh [this work]

GaP 2.36u 3 5 92x 89u 89 173t 174t 202 GaAs 2.45u 3 5 76x 77x 79 155t 155t 184 GaSb 2.65u 3 5 57x 57u 63 141t 139t 151 InN 2.08x 3 5 137z 130 277 InP 2.54u 3 5 74x 71u 71 159t 159t 168 InAs 2.61u 3 5 70* 60u 66 142t 144t 157 InSb 2.81u 3 5 40* 47u 53 128t 129t 130 BN 1.55x 3 5 321y 369y 313 577 BP 1.94x 3 5 173x 173x 160 329 BAs 2.04x 3 5 146x 137 290 BSb 2.24x 3 5 110x 104 230 TiN 2.11x 3 5 141x 124 267 TiP 2.49x 3 5 71x 76 176 TiAs 2.58x 3 5 59x 68 161 TiSb 2.75x 3 5 46x 56 138 uRef. [19],

vRef. [47],

xRef. [8],

yRef. [45],

zRef. [46],

*Ref. [21],

tRef. [29].

Table 3 Values of bulk modulus B (GPa) for transition-metal nitride (rocksalt structure) and AIIBIVC2

V and AIBIIIC2

VI (chalcopyrite solids).

solid d (in Å) VA VB VC B theoretical [48, 49] B experimental [50] B [this work]

TiN 2.125m 3 5 – 288m, 304–331l 258 VN 2.019k 3 5 – 346k, 338–383l 301 CrN 1.990k 3 5 – 364k 314 MnN 1.972k 3 5 – 374k 323 FeN 1.965k 3 5 – 368k 326 CoN 1.965k 3 5 – 358k 326 NiN 1.991k 3 5 – 311k 314 ZrN 2.310l 3 5 – 251l 201 CuAlS2 2.29 1 3 6 94 94 96 CuAlSe2 2.4 1 3 6 69 76 CuAlTe2 2.58 1 3 6 45 53 CuGaS2 2.30 1 3 6 93 94, 96 94 CuGaSe2 2.42 1 3 6 68 71 73 CuGaTe2 2.60 1 3 6 43 44 51 CuInS2 2.40 1 3 6 71 76 CuInSe2 2.51 1 3 6 54 62 61 CuInTe2 2.68 1 3 6 36 44 AgAlS2 2.40 1 3 6 73 76 AgAlSe2 2.51 1 3 6 55 61 AgAlTe2 2.68 1 3 6 36 44 AgGaS2 2.42 1 3 6 70 67 73 AgGaSe2 2.53 1 3 6 53 58 AgGaTe2 2.69 1 3 6 35 43 AgInS2 2.49 1 3 6 56 63 AgInSe2 2.61 1 3 6 42 50 AgInTe2 2.78 1 3 6 28 36 ZnSiP2 2.31 2 4 5 120 110 ZnGeP2 2.35 2 4 5 108 101 ZnSnP2 2.45 2 4 5 84 82 ZnSiAs2 2.41 2 4 5 93 89 ZnGeAs2 2.44 2 4 5 86 83 ZnSnAs2 2.53 2 4 5 67 70 CdSiP2 2.40 2 4 5 97 97 91 CdGeP2 2.44 2 4 5 86 83 CdSnP2 2.54 2 4 5 67 68 CdSiAs2 2.49 2 4 5 77 75 CdGeAs2 2.53 2 4 5 70 70 70 CdSnAs2 2.62 2 4 5 55 58 kRef. [5],

lRef. [6],

mRef. [7].

352 A. S. Verma: An empirical model for bulk modulus and cohesive energy

© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.pss-b.com

ph

ysic

ap s sstat

us

solid

i b

Table 4 Comparison between calculated and experimental data.

solid bulk modulus [experimental]

bulk modulus [this work]

% error

MgO 165 170 3 CaO 111 114 2.7 SrO 89 92 3.4 BaO 75 75 0 LiF 63 66 4.8 LiCl 32 32 0 LiBr 24 26 8.3 LiI 17 20 18 NaF 47 43 8.5 NaCl 23 24 4.3 NaBr 20 20 0 NaI 15 16 6.7 KF 31 28 9.7 KCl 18 17 5.6 KBr 15 15 0 KI 12 12 0 RbF 26 24 7.7 RbCl 16 15 6.3 RbBr 13 13 0 RbI 11 11 0 ZnS 77 77 0 ZnSe 62 67 8.1 ZnTe 51 54 5.9 CdS 62 62 0 CdSe 53 55 3.8 CdTe 42 45 7.1 AlP 86 89 3.5 AlAs 82 81 1.2 AlSb 58 62 6.9 GaN 190 175 7.9 GaP 89 89 0 GaAs 77 79 2.6 GaSb 57 63 11 InP 71 71 0 InAs 60 66 10 InSb 47 53 13 BN 369 313 15 BP 173 160 7.5 CuAlS2 94 96 2.1 CuGaS2 94 94 0 CuGaSe2 71 73 2.8 CuGaTe2 44 51 16 CuInSe2 62 61 1.6 AgGaS2 67 73 9 CdSiP2 97 91 6.2 CdGeAs2 70 70 0

5 Comparision between calculated and ex-perimental values The proposed empirical relations (15)–(18) have been applied to evaluate bulk modulus val-ues for alkali halides, alkaline-earth chalcogenides, transi-tion-metal nitrides (TMN), group AIIIBV, group AIIBVI and chalcopyrite semiconductors and the cohesive energy for alkali halides, alkaline-earth chalcogenides, group III–V

Table 5 Comparison between calculated and experimental data.

solid cohesive energy [experimental]

cohesive energy [this work]

% error

MgO 932 992 6.4 CaO 839 868 3.5 SrO 796 809 1.6 BaO 786 756 3.8 MgS 896 822 8.3 CaS 764 738 3.4 SrS 720 694 3.6 BaS 679 656 3.4 MgSe 780 767 1.7 CaSe 726 705 2.9 SrSe 693 673 2.9 BaSe 660 633 4.1 CaTe 679 658 3.1 SrTe 667 627 6 BaTe 629 596 5.2 LiF 247 258 4.5 LiCl 203 202 0.5 LiBr 194 189 2.6 LiI 180 173 3.9 NaF 220 224 1.8 NaCl 187 184 1.6 NaBr 179 174 2.8 NaI 167 161 3.6 KF 194 194 0 KCl 170 165 2.9 KBr 163 157 3.7 KI 154 147 4.5 RbF 186 185 0.5 RbCl 164 158 3.7 RbBr 157 151 3.8 RbI 149 142 4.7 ZnS 147 149 1.4 ZnSe 125 132 5.6 ZnTe 106 110 3.8 CdS 132 124 6.1 CdSe 114 112 1.8 CdTe 96 94 2.1 AlP 198 202 2 AlAs 179 187 4.5 AlSb 165 150 9.1 GaP 174 202 16 GaAs 155 184 19 GaSb 139 151 8.6 InP 159 168 5.7 InAs 144 157 9 InSb 129 130 0.8

and group II–VI semiconductors. The values so obtained are presented in Tables 1, 2 and 3 compared with the ex-perimental data (in Tables 4 and 5). We note that the evaluated values of bulk modulus and cohesive energy by the proposed relations are in close agreement with the ex-perimental data as compared to the values reported by pre-vious researchers so far. These results shows that our cur-

Phys. Status Solidi B 246, No. 2 (2009) 353

www.pss-b.com © 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Original

Paper

rent method is quite reasonable and can give us a useful guide in calculating and predicting the bulk modulus and cohesive energy of the binary and more complex class of ternary chalcopyrite semiconductors. 6 Conclusions There are several methods to deter-mine the bulk modulus and cohesive energy of solids, but due to the small changes of the unit-cell dimensions, the accuracy of determining these parameters has always been unpredictable. In the proposed models, calculations are simple, fast and more accurate; with regard to the applica-tions point of view it can be highly dependent. The only in-formation needed for calculating ground-state properties by the proposed relations are nearest-neighbor distance and the product of valence electron. The evaluated values are presented in Tables 1, 2 and 3. We come to the conclusion that the product of valence electrons of any compound is a very important parameter for calculating ground-state properties. It is also to be noted that the proposed empirical relations are simpler, widely applicable and values are in better agreement with experimental data as compared to empirical relations proposed by previous researchers, since we have been reasonably successful in calculating these parameters using the product of valence electrons and nearest-neighbor distance of the materials for rocksalt, zincblende and chalcopyrite solids. It is natural to say that this model can easily be extended to other more complex crystals for which work is in progress and will be appear-ing in a forthcoming paper.

References

[1] W. A. Harrison, Electronic Structure and the Properties of Solids, The Physics of the Chemical Bond (W. H. Freeman, San Francisco, 1980).

[2] R. S. Chouhan, K. Lal, and C. P. Singh, Physica B 396, 211 (2007).

[3] B. P. Singh, S. K. Srivastava, and K. Dinesh, Physica B 349, 401 (2004).

[4] A. K. Koh, J. Phys. Chem. Solids 58, 467 (1997). [5] C. Binli, M. Kaili, F. Q. Liu, and X. J. Fan, Mod. Phys. Lett.

B 18, 281 (2004). [6] A. Simunck and J. Vackar, Phys. Rev. B 64, 235115

(2001). [7] L. M. Liu, S. Q. Wang, and H. Q. Ye, J. Phys.: Condens.

Matter 15, 8103 (2003). [8] S. Q. Wang and H. Q. Ye, Phys. Rev. B 66, 235111 (2002). [9] Y. Al-Douri, H. Abid, and A. Aourag, Mater. Lett. 59, 2032

(2005). [10] M. S. Omar, Mater. Res. Bull. 42, 319 (2007). [11] S. Q. Wang and H. Q. Ye, J. Phys.: Condens. Matter 17,

4475 (2005). [12] M. Anani, C. Mathieu, S. Lebid, Y. Amar, Z. Chama, and

H. Abid, Comput. Mater. Sci. 41, 570 (2008). [13] A. H. Reshak, Physica B 369, 243 (2005). [14] L. K. Samanta and S. Chatterjee, Infrared Phys. Technol.

46, 370 (2005). [15] F. Chiker, B. Abbar, A. Tadjer, S. Bresson, B. Khelifa, and

C. Mathieu, Physica B 349, 181 (2004).

[16] A. Chahed, O. Benhelal, H. Rozale, S. Laksari, and N. Ab-bouni, Phys. Status Solidi B 244, 629 (2007).

[17] L. Pauling, The Nature of the Chemical Bond, 3rd ed. (Cor-nell University Press, Ithaca, 1960).

[18] J. C. Phillips, Bonds and Bands in Semiconductors (Aca-demic Press, New York, 1973).

[19] M. L. Cohen, Phys. Rev. B 32, 7988 (1985). [20] V. Kumar and B. S. R. Sastry, J. Phys. Chem. Solids 66, 99

(2005). [21] Y. Al-Douri, H. Abid, and H. Aourag, Mater. Lett. 59, 2032

(2005). [22] R. L. Moreira and A. Dias, J. Phys. Chem. Solids 68, 1617

(2007). [23] L. Q. Jiang, J. K. Guo, H. B. Liu, M. Zhu, X. Zhou, P. Wu,

and C. H. Li, J. Phys. Chem. Solids 67, 1531 (2006). [24] A. S. Verma, and S. R. Bhardwaj, Phys. Status Solidi B 243,

2858 (2006). [25] O. L. Anderson and J. E. Nafe, J. Geophys. Res. 70, 3951

(1965). [26] P. K. Lam, M. L. Cohen, and G. Martinez, Phys. Rev. B 35,

9190 (1987). [27] V. Kumar, G. M. Prasad, A. R. Chetal, and D. Chandra,

J. Phys. Chem. Solids 57, 503 (1996). [28] S. H. Sohn, D. G. Hyun, M. Noma, S. Hosomi, and Y. Ha-

makawa, J. Cryst. Growth 117, 907 (1992). [29] A. Aresti, L. Garbato, and A. Rucci, J. Phys. Chem. Solids

45, 361 (1984). [30] L. M. Liu, S. Q. Wang, and H. Q. Ye, J. Phys.: Condens.

Matter 17, 5335 (2005). [31] J. Haglund, G. Grimvall, T. Jarlborg, and A. F. Guillermet,

Phys. Rev. B 43, 14400 (1991). [32] R. C. Mota, S. C. Costa, P. S. Pizani, and J. P. Rino, Phys.

Rev. B 71, 224114 (2005). [33] H. Schlosser, J. Phys. Chem. Solids 53, 855 (1992). [34] H. Schlosser, Phys. Status Solidi B 179, K1 (1993). [35] K. S. Krishnan and S. K. Roy, Proc. R. Soc. 210, 481

(1952). [36] A. Jayaraman, B. Batlogg, R. G. Maines, and H. Bach,

Phys. Rev. B 26, 3347 (1982). [37] D. B. Sirdeshmukh and K. G. Subhadra, J. Appl. Phys. 59,

276 (1986). [38] V. P. Gupta and S. K. Sipani, Phys. Status Solidi B 111, 295

(1982). [39] Z. H. Fang and L. R. Chen, Phys. Status Solidi B 180, K5

(1993). [40] D. Alfe, Phys. Rev. Lett. 94, 235701 (2005). [41] B. R. K. Gupta and R. P. Goyal, Solid State Commun. 49,

559 (1984). [42] M. Kumari and N. Dass, J. Phys.: Condens. Matter 2, 7891

(1990). [43] L. Bosi, Phys. Status Solidi B 175, K47 (1993). [44] H. Neumann, Cryst. Res. Technol. 23, 531 (1988). [45] H. M. Tutuncu, S. Bagci, G. P. Srivastava, A. T. Albudak,

and G. Ugur, Phys. Rev. B 71, 195309 (2005). [46] K. Kim, W. R. L. Lambrecht, and B. Segall, Phys. Rev. B

53, 16310 (1996). [47] S. H. Wei and A. Zunger, Phys. Rev. B 37, 8958 (1988). [48] P. Grima Gallardo, Phys. Status Solidi B 182, K67

(1994). [49] H. Neumann, Cryst. Res. Technol. 23, 97 (1988). [50] H. Neumann, Phys. Status Solidi A 96, K121 (1986).