Static and dynamical properties of II–VI and III–V group binary solids

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Phys. Scr. 85 (2012) 015701 (6pp) doi:10.1088/0031-8949/85/01/015701

Static and dynamical properties of II–VIand III–V group binary solidsD S Yadav1 and D V Singh2

1 Department of Physics, Ch Charan Singh PG College, Heonra, Etawah 206001, UP, India2 Department of Physics, Agra College, Agra 282002, UP, India

E-mail: dhirendra.867@rediffmail.com

Received 13 July 2011Accepted for publication 9 November 2011Published 7 December 2011Online at stacks.iop.org/PhysScr/85/015701

AbstractIn this paper, we extend to II–VI and III–V group binary solids of zinc blende (ZB) structurewith conduction d-electrons the calculation of static and dynamical properties such as bulkmodulus (B) and cohesive energy or total energy (Ecoh) using the plasma oscillation theory ofsolids formalism already employed for ternary chalcopyrite semiconductors. The presentmethod is not limited to tetrahedrally coordinated semiconductors and ternary chalcopyrites,but can be used for all semiconducting compounds. We have applied an extended formula onZB structured binary semiconductors and found better agreement with the experimental dataas compared to the values evaluated by previous researchers. The bulk modulus and cohesiveenergy of ZB-type structure compounds exhibit a linear relationship when plotted on a log–logscale against the plasmon energy h̄ωp (in eV), but fall on a straight line. The results for bulkmodulus differ from experimental values by the following amounts: ZnS 0.36%, ZnSe 10%,ZnTe 0.62%, CdS 1.8%, CdSe 7.4% and CdTe 1.6%, AlP 2.6%, AlAs 5.3%, AlSb 4.0%, GaP0%, AlAs 0%, AlS 4.4%, InP 0%, InAs 0% and InSb 2.1%; and the results for cohesive energydiffer from experimental values by the following amounts: ZnS 0.16%, ZnSe 0.73%, ZnTe0.6%, CdS 7.6%, CdSe 3.5%, CdTe 2.5%, AlP 2.0%, AlAs 3.0%, AlSb 11.1%, GaP 14.6%,AlAs 17.0%, AlSb 8.7%, InP 4.3%, InAs 5.5% and InSb 0.6%.

PACS numbers: 71.20.Nr, 71.15.Nc, 72.80.Ey

1. Introduction

In recent years, much attention has been paid to the studyof the binary tetrahedral semiconductors zinc, cadmiumand mercury chalcogenides and boron, aluminum, galliumand indium pnictides with zinc blende (ZB)-type structurebecause of their potential applications in the field of linearand nonlinear optics, solar cells, light-emitting diodes, laserdiodes and integrated optical devices such as switches,modulators and filters. Most of the physical world aroundus and a wide area of modern technology are based onsolid materials. Extensive research devoted to the physics andchemistry of solids during the last quarter century has led togreat advances in the understanding of the properties of solidsin general. So it is interesting to study the behavior and variousproperties of different types of solids. In recent years, therehas been considerable interest in theoretical and experimentalstudies of II–VI- and III–V-type crystals with ZB structure.It is attributed to their high symmetry and the simplicity of

their ionic bonding [1]. Almost all the AIIBVI and AIIIBV

compound semiconductors crystallize in either the ZB orwurtzite structure. A common and dominant feature of thesestructures is the tetrahedral bonding to four atoms of otherelements. Tetrahedral coordinated semiconductors with ZBstructure are arranged in a cubic-type structure, while they arein a hexagonal-type structure. Indeed, the centers of similartetrahedra are arranged in a face-centered cubic (fcc) array inthe former and in a hexagonal closed-packed (hcp) array inthe latter [2]. In these binary solids, InN, BN, ZnO and HgSrepresent two III–V and II–VI compounds crystallizing in ZBand wurtzite structures with semiconducting characteristicssuch as large band gaps and large valence band widths.These oxides and sulfides have been considered to betypical examples of the most fundamental materials for theindustrial sciences. This is because of their wide rangeof applications, ranging from catalysis to micro-electronics.For example, their catalytic properties are important forchemical engineering. In the last few decades, increasing

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Phys. Scr. 85 (2012) 015701 D S Yadav and D V Singh

attention has been paid to the study of ZB and wurtizestructure binary solids. Most of the semiconductors used in themodern micro-electronic industry have ZB crystallographicstructure. Crystals with ZB or sphalerite structure rangefrom raw iron and zinc materials to man-made GaN andBN semiconductors. The particular omni-triangulated natureof atomic structure gives these materials unique physicalproperties. Recently [3–7], II–VI semiconductors and theiralloys have attracted considerable interest due to theirapplication in photovoltaic devices, as electro-optic crystalswith ultrafast response time spectroscopy and the recentrealization of light-emitting diodes (based on ZeTe). Thishas motivated several theorists to model band offset [8],structural and thermo-dynamical properties in this familyand their alloys [9, 10] and to revise many fundamentalphysical, structural, electronic, optical and thermo-dynamicalproperties based on ab initio calculations [11]. From atheoretical point of view, the optical, structural and electronicproperties of II–VI and III–V group binary compounds havebeen determined [12–14] by using the full potential linearizedaugmented plane wave plus local orbital (FP-LAPW + LO)method within the density functional theory. In fact, the roleof metal d-states in II–VI semiconductors is very important.Wei and Zunger [16] have shown that p–d repulsion andhybridization (i) lower the band gaps, (ii) reduce the cohesiveenergy and (iii) increase the equilibrium lattice parametersand also influence other features.

A considerable amount of experimental and theoreticalwork has been performed over the last few years on thecohesive energy and bulk modulus of these semiconductorsand insulators. A modern computational method has madeit possible to study the structural, mechanical and opticalproperties of a wide variety of molecules and solids in greatdetail. There are, however, instances where this level of detaileither cannot be easily attained because of the complexity ofthe system or is not needed, as when studying broad trends inthe behavior of a large set of the systems. Empirical conceptssuch as valence, empirical radii, electro-negativity, ionicityand plasmon energy are then useful [16, 17]. These conceptsare directly associated with the character of the chemical bondand thus provide a means of explaining and classifying manybasic properties of molecules and solids. We therefore feel thatit would be of interest to give an alternative explanation forthe bulk modulus and cohesive energies of II–VI and III–Vgroup tetrahedrally coordinated semiconductors with ZB andwurtzite structure. In this paper, we propose a method basedon a plasma oscillation theory of solids for the calculationof the static and dynamical properties such as bulk modulus(B in GPa) and cohesive energy (ECoh in kcal mole−1) ofthese compound semiconductors. It is now well establishedthat the plasmon energy of a metal changes [18, 19], when itundergoes a chemical combination and forms a compound. Aplasmon is a collective excitation of the conduction electronsin a metal with energy h̄ωp, which depends on the densityof electrons in the conduction band. This is due to the factthat the plasmon energy depends on the density of conductionelectrons and the effective number of valence electrons, whichchanges when a metal forms a compound.

Many researchers [20–29] have developed varioustheories and calculated the micro-hardness, heat of formation,

bulk modulus, cohesive energy, lattice thermal conductivity,average energy gap, ionic gap, ionicity, dielectric constants,electronic polarizability and optical susceptibility for II–VIand III–V group binary solids. Recently, we [30] calculatedthe electronic properties such as average energy gap, ionicgap, ionicity, electronic polarizability, bulk modulus, cohesiveenergy, micro-hardness and static dielectric constants foraluminum, gallium and indium pnictides using Phillips andVan Vechten’s dielectric theory of solids. Therefore, we havecalculated the bulk modulus (B in GPa) and cohesive energy(ECoh in kcal mole−1) of II–VI and III–V group binary solidsin ZB structured using the plasma oscillation theory of solids.

The paper is organized as follows. In section 2,we describe the theoretical concepts; the results anddiscussion concerning static and dynamical properties suchas bulk modulus (B in GPa) and cohesive energy (ECoh inkcal mole−1) are given in sections 3; section 4 presents theconclusions.

2. Theoretical concepts

A metallic crystal may be described as an assembly ofimmediate positive ion cores and conduction electrons thatare nearly free to move over the whole of the crystal. Thecondition of charge neutrality is maintained because of thebalance struck between the negative charge on electrons inthe conduction band and an equal concentration of positivecharge on ion cores. Thus a metal serves as a good example ofa plasma. A random motion may be a momentary fluctuationin the equilibrium position of an electron, caused by theaverage electrostatic field of all other electrons. The positionfluctuation would create a charge imbalance in the region ofthat electron, as a result of which other electrons would rushinto that region in order to restore the condition of chargeneutrality. At any finite temperature, electrons, being verylight particles, move with a fairly high speed relative to ions,which we consider to be at rest. The electrons rushing intothe region of the electron that suffered a fluctuation in itsequilibrium position are unable to stop at the desired positionsand overshoot their mark on account of the large kineticenergy, which by the way represents the total energy there. Assoon as the energy becomes totally electrostatic, electrons turnaround and again attempt to approach the wanted locationsin the region of the misbehaving electron. The repetition ofthis process constitutes the collective oscillatory motion. Theenergy of a quantum of plasma oscillations of the valenceelectrons in both the metal and the compound is given by therelation [31]

h̄ωp = 28.8

√Neffd

M, (1)

where Neff is the number of effective electrons taking part inplasma oscillations, d the density and M the molecular weight.Equation (1) is valid for free electrons but is also applicablefor semiconductors and insulators, up to a first approximation.Philipp and Ehrenreich [32] and Raether [33] have shownthat the plasmon energy for semiconductors and insulators isgiven by

h̄ωpd = h̄ωp/√

1 − δε0, (2)

where δε0 is a very small correction term to the free-electronplasmon energy h̄ωp and can be neglected to a first

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Phys. Scr. 85 (2012) 015701 D S Yadav and D V Singh

approximation. Philipp and Ehrenreich [32] have shown thatthe calculated values of h̄ωp and h̄ωpd are in fair agreementwith their observed values of plasma energy in dielectrics.It has also been pointed out by Kittel [34] that the plasmonoscillations in dielectrics are physically the same as inmetals. Previously, Srivastava [35–37] studied the electronicproperties of rock salt, CsCl, diamond and ZB structuredsolids in terms of the plasmon energy h̄ωp (in eV) by thefollowing relation:

d(in Å) = K (h̄ωp)−S, (3)

where d is the nearest-neighbor distance between atoms A andB in units of Å, and K and S are constants that depend onthe structure of the compound semiconductors. In this paper,we extend the calculation of the bulk modulus and cohesiveenergy reported earlier in the case of ZB structured solids.The values found for these parameters are in better agreementwith the values reported by previous researchers.

2.1. Bulk modulus

Anderson and Nafe [38] first proposed an empiricalrelationship between the bulk modulus B0 at atmosphericpressure and specific volume V0 of the form B0 ≈ V −x

0 . Theyfound it to hold for a particular class of compounds; the valueof x depends on the class of compounds. For alkali halides,fluoride, sulfide and telluride, they find x to be 1 and for oxidecompounds x is close to 4. Jayaraman et al [39] predictedthat the bulk modulus directly depends on the product ofionic charges. According to them, for rock-salt-type crystalstructure compounds, the bulk modulus may be expressed as

B0 = Z1 Z2

{(r0

ρ

)− 2

}r−4

0 . (4)

In recent years [22–24, 28], many theoretical approacheshave been reported for determining the value of the bulkmodulus of solid-state compounds. Cohen [40] predicted thatthe zero-pressure isothermal bulk modulus B in terms of thenearest-neighbor distance d (in Å) for rock-salt-type crystalstructure compounds might be expressed as

B =const

d3. (5)

The relationship of bulk moduli and geometrical propertiesof diamond and zinc-blend solids has previously beeninvestigated by Cohen [40] and Lam et al [41]. Based onPhillips and Van Vechten’s scheme [42] and a theoreticalanalysis of the bond geometry of covalent ZB structuredsolids, Cohen [40] proposed the following empirical relation:

B =const

d3.5, (6)

where d is in Å and B is in GPa. Lam et al [41] deducedthe analytic relationship of bulk moduli to lattice parameterswithin the local-density formalism and the pseudopotentialapproach:

B = 1971d−3.5− 408(1Z)2d−4, (7)

where 1Z = 1 and 2 for III–V and II–VI semiconductors. Togain better agreement between experimental and theoretical

data for ZB structured compound semiconductors, usingthe plasma oscillation theory of solids [35–37], the Cohenrelation (6) may be extended to

B = D(h̄ωp)S, (8)

where D and S are constants that depend on the crystalstructure. For ZB structured compounds, the values ofconstants D and S turn out to be equal to 0.1257 and 2.333,respectively.

2.2. Cohesive energy

Agrawal et al [21] have used van der Waal’s coefficients tocalculate the cohesive energy per mole for II–VI group binarysolids with rock-salt structure. The cohesive energy of an ioniccrystal can be expressed as

Ec = 8C + 8R + 8M, (9)

where 8C, 8R and 8M are the potential terms due toelectrostatic energy, the repulsive potential and the three-bodypotential, respectively. Aresti et al [43] have studied thecohesive energy of ZB solids and proposed an empiricalrelation for cohesive energy in terms of the nearest-neighbordistance (d) as follows:

ECoh = ECoh(IV) − B(d, R){

1 −

∑ECoh(I)/ECoh(IV)

},

(10)where ECoh(IV) is the cohesive energy of purely covalentcrystals and B(d, R) is a new parameter depending on dand R:

B(d, R) = ECoh(IV) −k(R)d(B)

d,

k(R) = C exp

(−Z0.5

4

),

where C is a constant, which depends on the rows, andZ = Z(A) + Z(B) is the atomic number of atoms A and B.Previously [22–26, 44–46], many theoretical approaches havebeen reported to determine the value of the cohesive energyof the solid-state compounds. Schlosser [47, 48] has studiedthe cohesive energy trends in the rock-salt structure in termsof the nearest-neighbor distance using the following relation:

ECoh =const

d. (11)

Recently, Verma et al [26] studied the cohesive energy of ZBstructure solids and proposed an empirical relation in terms ofthe product of ionic charge and the nearest-neighbor distanced (in Å) as follows:

ECoh = const(Z1 Z2)Ad−S, (12)

where A and S are constants, which depend on the structure ofsolids. To obtain better agreement between the experimentaland theoretical data for ZB-type crystal structure compounds,we may extend the relation (12) in terms of plasma energy inthe following form:

ECoh = D∗(h̄ωp)S∗

, (13)

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Phys. Scr. 85 (2012) 015701 D S Yadav and D V Singh

Table 1. The present calculated values of bulk modulus and cohesive energy of II–VI group binary solids.

II–VI h̄ωp Bulk modulus (B in GPa) Cohesive energy (ECoh in kcal mole−1)solids equation (1) Calc. Expt Theor. Calc. Expt Theor. Theor.

equation (8) [52, 53] [40] equation (13) [43] [43] [26]

ZnO 21.50 161.403 – – 223.79 – – –ZnS 16.66 89.023 88.70 90 146.32 146.6 151.3 147.6ZnSe 15.78 78.436 71.10 75 133.67 124.5 123.7 130.2ZnTe 13.80 57.368 57.80 59 106.92 106.3 108.6 109.2CdO 18.46 113.097 – – 173.60 – – –CdS 14.88 68.394 69.44 69 121.22 131.6 134.6 122.6CdSe 14.02 60.120 55.94 60 109.77 113.6 110.3 111.3CdTe 12.72 47.435 48.20 47 93.34 95.8 95.4 93.4HgS 14.84 67.965 – – 120.67 – – 121.4HgSe 13.99 59.228 – – 109.38 – – 110.2HgTe 12.85 48.574 – – 94.94 – – 94.2

Table 2. The present calculated values of bulk modulus and cohesive energy of III–V group binary solids.

III–V h̄ωp Bulk modulus (B in GPa) Cohesive energy (ECoh in kcal mole−1)solids [19, 30] Calc. Expt Theor. Calc. Expt Theor. Theor.

equation (8) [52, 54] [28, 40] equation (13) [43] [43] [26]

BN 24.53 219.53 – 202 385.81 – – –BP 21.71 165.10 – 164 314.79 – – –BAs 20.12 138.26 – 142 277.32 – – 287.7BSb 17.85 104.57 – – 227.18 – – 227.7AlN 22.97 188.33 – 186 345.81 – – –AlP 16.64 88.77 86.50 89 202.11 198.0 197.0 199.8AlAs 15.75 78.09 74.10 79 184.42 178.9 177.2 185.8AlSb 13.72 56.88 59.30 57 146.54 165.0 162.4 148.2GaN 21.98 169.93 – 167 321.34 – – –GaP 16.51 87.04 87.20 87 199.28 173.8 173.2 199.8GaAs 15.61 76.48 76.50 76 181.69 154.7 154.6 181.9GaSb 13.95 58.83 56.30 58 150.66 138.6 140.5 149.6InN 18.82 118.31 – 119 248.12 – – –InP 14.76 67.11 67.10 67 165.51 158.6 159.3 166.3InAs 14.07 60.02 60.20 61 152.82 144.3 141.7 155.4InSb 12.73 47.52 46.50 47 129.35 128.5 128.3 129.2

where D∗ and S∗ are constants and h̄ωp is the plasma energyof the materials in eV. The constant D∗ has the values1.350 and 1.867 for II–VI and III–V group binary compoundsemiconductors, respectively, and S∗

= 1.665 for both typesof compound semiconductors. A detailed discussion ofcohesive energy for these compound semiconductors has beengiven elsewhere [26, 44–48] and will not be presented here.

3. Results and discussion

The bulk modulus and cohesive energy are importantstatic and dynamical properties of a material: the bulkmodulus defines its resistance to change in its volume whencompressed. Both experimental and theoretical results suggestthat the bulk modulus is a critical single-material propertyto indicate hardness. The solid-state physics of an atom incombination with O, S, Se and Te depends primarily onits electronic configuration. Jayaraman et al [39] predictedthat bulk modulus directly depends on the product of ioniccharges. Any change in the crystallographic environment of anatom is related to core electrons via the valence electrons. Thechange in wave function that occurs for the outer electronsusually means a displacement of electric charge in the valenceshell so that the interaction between valence, shell and coreelectrons is changed. This leads to a change in binding

energy of inner electrons and to a shift in the position of theabsorption edge. Plasmon energy also depends on the effectivenumber of valence electrons and the density of conductionelectrons, which changes when the metal forms a compound.

According to this idea, we have proposed the empiricalrelations (8) and (13) for the calculation of the bulk modulusand cohesive energy of II–VI and III–V group binary solids.The values so obtained are presented in tables 1 and 2compared with the experimental and theoretical data reportedso far. We note that the values of bulk modulus and cohesiveenergy calculated from our proposed relations are in closeragreement with the experimental data than values reported sofar by previous researchers. For example, the results for bulkmodulus differ from experimental values by the followingamounts: ZnS 0.36%, ZnSe 10%, ZnTe 0.62%, CdS 1.8%,CdSe 7.4%, and CdTe 1.6%, AlP 2.6%, AlAs 5.3%, AlSb4.0%, GaP 0%, AlAs 0%, AlSb 4.4%, InP 0%, InAs 0% andInSb 2.1%; and the results for cohesive energy differ fromexperimental values by the following amounts: ZnS 0.16%,ZnSe 0.73%, ZnTe 0.6%, CdS 7.6%, CdSe 3.5%, CdTe 2.5%,AlP 2.0%, AlAs 3.0%, AlSb 11.1%, GaP 14.6%, AlAs 17.0%,AlSb 8.7%, InP 4.3%, InAs 5.5% and InSb 0.6% in thepresent study. These results show that our approach proposedin this paper is quite reasonable and can give us a usefulguide for calculating and predicting the static and dynamical

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Phys. Scr. 85 (2012) 015701 D S Yadav and D V Singh

Figure 1. Plot of bulk modulus versus plasmon energy of II–VIgroup binary solids.

Figure 2. Plot of cohesive energy versus plasmon energy of II–VIgroup binary solids.

properties of these materials. We have plotted the graph ofbulk modulus and cohesive energy against plasmon energy(h̄ωp) for II–VI and III–V group binary solids in figures 1–4.From figures 1–4, it is quite obvious that the bulk modulusand cohesive energy trends in the compounds increase onincreasing the plasma energy (h̄ωp) of these materials. Thus, itis possible to predict the order of bulk modulus and cohesiveenergies of semiconducting compounds from their plasmonenergies. The present approach for the calculation of staticand dynamical properties of binary solids can be used forsemiconducting compounds without having any knowledgeof crystal ionicity, transition pressure, cell volume, meltingtemperature and micro-hardness, while the model given byNeumann [49, 50] and Al-Douri et al [51] requires the valuesof these parameters to calculate the bulk modulus of solidsand is limited to simple binary compounds.

4. Conclusions

From the above results and discussion obtained by using theproposed approach, it is quite obvious that parameters suchas bulk modulus (B in GPa) and cohesive energy (ECoh inkcal mole−1) reflecting the static and dynamical propertiescan be expressed in terms of plasmon energies of thesematerials, which is definitely a surprising phenomenon. Thecalculated values of these parameters are presented in tables 1and 2. We come to the conclusion that plasmon energyof any compound is a key parameter for calculating thestatic and dynamic properties of ZB structure binary solids.Furthermore, we found that in the compounds investigatedhere, the bulk modulus of II–VI and III–V group binarysemiconductors exhibits a linear relationship when plotted

Figure 3. Plot of bulk modulus versus plasmon energy of III–Vgroup binary solids.

Figure 4. Plot of cohesive energy versus plasmon energy of III–Vgroup binary solids.

on the log–log scale against the plasma energy (h̄ωp), butfall on the same line. From figures 1–4, we observed thatthe data points for II–VI and III–V group compounds lie ona straight line because bulk modulus and cohesive energyare linearly dependent on their plasma energy. According tothis idea, we may evaluate all important properties of ZBstructure semiconductors using their plasmon energies, whichare basic parameters. Thus this theory can be easily extendedto binary solids. Good agreement was found between ourcalculated and experimental values of the bulk modulus andcohesive energies for these materials as compared to thetheoretical values reported by different previous researchers.But in the case of boron, aluminum, gallium and indiumatoms linked with nitride, the experimental values are notavailable for comparison with the calculated values so far. It isalso noteworthy that the proposed approaches are simple andwidely applicable.

Acknowledgments

The authors are grateful to the referee for his useful commentsand valuable suggestions, which were very helpful in revisingthe manuscript. DSY is grateful to Professor R S Yadav(Principal, Ch Charan Singh PG College Heonra, Etawah,India) for continuous encouragement during this study. Theauthors are grateful to Dr R K Vishnoi for his help in mailingand preparing the manuscript.

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Phys. Scr. 85 (2012) 015701 D S Yadav and D V Singh

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