Vol.3, No.2, 154164 (2011) Natural Science http://dx.doi.org/10.4236/ns.2011.32022 Copyright © 2011 SciRes. OPEN ACCESS Thermal dependence of the properties of cubic boron nitride crystal Zainelabideen Y. Mijbil1, Mudar A. Abdulsattar2, Ahmed M. AbdulLettif3 1Basic Science Department, Veterinary Medicine College, University of Babylon, Hilla, Iraq 2Ministry of Sciences and Technology, Baghdad, Iraq 3Department of Physics, College of Science, University of Babylon, Hilla, Iraq; *Corresponding author: abdullettif@yahoo.com Received 20 November 2010; revised 23 December 2010; accepted 26 December 2010. ABSTRACT Lattice constant, total energy, cohesive energy, bulk modulus, speed of sound (υ), plasmon en ergy (Epl), valence charge distribution and en ergy bands of cubic boron nitride crystal have been calculated and studied as a function of temperature using selfconsistent field tight binding method with complete neglect of dif ferential overlap version 2 using 8atom large unit cell approach. Our results illustrate that the increase of temperature leads to an increase of lattice constant, cohesive energy, and valence charge distribution at the atoms, whereas a de crease is obtained for bulk modulus, energy band widths, valence charge distribution in the intratomic distance, speed of sound, and the plasmon energy. The comparison with experi mental and other theoretical results has showed an excellent agreement for the lattice constant, bulk modulus, cohesive energy, speed of sound, plasmon energy value and the valence band, whereas remarkable differences in charge dis tribution values, and the band gap are found. These differences are common in the results that depend on this type of calculation. Values for conduction band and speed of sound have not been found for comparison. Our relation for EplT fails but the υT relation is successful. Keywords: Electronic Structure; Temperature Dependence; Boron Nitride 1. INTRODUCTION The high account of hardness [1], melting point [2], high thermal conductivity [3] and band gap [4], resis tance for oxidation as temperature increases[5] with inert state [6] low electrical conductivity [7], and no iron re action [8]are all for sp3 [9] cubic boron nitride. So it has amazing aspects [10] of multiuse [11,12] such as in pol ishing, cutting [13] and protection [14], and it has been the core for many studies [1518] since its discovery in 1957 [19]. Methods of semiempirical calculations give a privi lege of reasonable computer execution time for getting the results due to the approximations which are involved [20,21], accordingly they are used in many studies [2225]. In this paper we have used linear combination of atomic orbitals (LCAO) approach as a starting point [26] which includes both 1) the CNDO, as one of the primary semiempirical methods, that is presented by Popel et al. [27] in which many electronelectron inter actions had been neglected. CNDO focused on both va lence electrons and nuclei as a core of the atom (with the other electrons) [21], and 2) LUC (Large Unit Cell) ap proximation or (Supercell) [28] is used in order to mini mize the size of Brillion Zone (BZ) and so to reduce the number of the wave vector (k) points in which the as pects of the crystal such as band structure could be cal culated [29,30]. The aim of this project is to study the cBN by CNDO/2. CNDO/2 is an old fashion method but here we try to see its merits and drawbacks through works and results due to the trend to couple between ab initio cal culations with semiempirical methods, specially if we know that the ab initio method is used for small systems, in comparison with the semiempirical calculations which were applied for large systems, so it is a competition between the accuracy of the former and the high speed of the later [31]. 2. METHODOLOGY According to LCAOLUC with eight atoms, and STO (Slater Type Orbital), the atomic orbital is of the form: *1 ,,exp , n rGr rY (1) where Ψ(r, θ,φ) is Slater type orbital. r, θ,φ are the polar spherical coordinates of the atom. G is the normalization
Z. Y. Mijbil et al. / Natural Science 3 (2011) 154164 Copyright © 2011 SciRes. OPEN ACCESS 155 constant. n* is the effective principal quantum number, and Y(θ,φ) is the spherical harmonics [32]. The Fock matrix element (Fμμ) and the total energy (E) can be cal culated as [21]: 10.5 1 2AAA BAB BA FIAPQ Q (2) 0.5 AB AB FSP (3) 1 0.5 ABAB AB EPHFZZR (4) where μ and ν represent atomic orbitals and here they are used either as s or p orbitals, the quantity –0.5 (Iμ +Aμ) is either EES or EEP for μ ≡ s or p respectively, Iμ and Aμ represent the ionization potential and the atomic electron affinity respectively, βAB is the bonding parameter, Pμμ is the density matrix, QB and QA are the net charge on atom A and B respectively, and γAB is the average electrostatic repulsion between any electron on atom A and any elec tron on atom B. The final expressions with LUC approaches [20] are: , 1 00.50 2 0.5 0 oo oz AA BAB Bz oz oz AB ozvAA zo z FIA Q SPfx (5) , 00.50 oz oz AB o zvAB zv SP fx (6) 1 0.5 000 0.5 ABAB AB EPFH ZZR (7) where f(x) it is Szymanski enhancement function, o is the central LUC and z is the other LUCs [33]. We have firstly studied how CNDO/2 parameters: the orbital exponent (ξ), bonding parameter (β), sshell elec tronigativity (EES) and pshell electronegativity (EEP) could be changed and how this change may affect the results. Our suggested parameters are shown in Table 1, and secondly we have calculated the lattice constant, cohesive energy, valence band width, conduction band width, direct band gap, according to these parameters as shown later. Table 1.The parameters of CNDO/2. Property β (eV) ξ (a.u.) EES (eV) EEP (eV) Reference B N B N B N B N [33] 7.452 13.95 1.531 1.9575 9.158 4.128 7.511 [21] 17 25 1.3 1.959.549 19.316 4.0017.375 Pres. 8.455 11.95 1.53 1.9528.416 34.766 14.666 22.728 The procedure of the present work can be summarized as follows: 1) We change the four parameters of the CNDO method ξ, βAB, EES and EEP in arbitrary way until we get a value of the lattice that is very close to the ex perimental value with the minimum value of the total energy of the crystal. 2) Then we measure the values of the cohesive energy and all the energies of the electronic band. 3) If the values of energies are in agreement with the experiments then the iterations are ended and we fix the parameters. 4) If the values of the energies are incorrect the pa rameters are changed until getting the closest values to the experiments. 3. CALCULATIONS 3.1. The Total Energy The total energy (E) is calculated when the difference between two successive iterations from the program is less than (0.0136 eV) and our result is fitted to Birch Murnaghan equation [34] as below: /1 11 o B o ooo o oo o VV BV BV EE BB B (8) where Eo, Vo, and Bo are the total energy, volume of the unit cell and the bulk modulus at the equilibrium point, zero temperaturepressure point, respectively, V is the volume at any temperature, and B’ o is the pressure de rivative of bulk modulus with a value of (3.79) [35]. 3.2. Cohesive Energy The cohesive energy (Ecoh) is calculated from the LUC total energy according to the equation [36]: 8 cohfree zp E EEE (9) Efree is the energy of (sp) orbitals for the free atom and equals to 169.164 eV [37], and Ezp is the zero point en ergy and equals to 0.165 eV [38]. 3.3. Volume For tetrahedral structure it is found that [39]: 0.021 m T (10) α is the linear thermal expansion and Tm is the melting temperature [39] which equals to 3000 K [40]. We have also from [40] that: 1 o a aT (11)
Z. Y. Mijbil et al. / Natural Science 3 (2011) 154164 Copyright © 2011 SciRes. OPEN ACCESS 156 T is the temperature, a is the lattice constant at any tem perature and ao is the lattice constant at zero temperature. If we take the initial temperature equals to (0 K), and combine Eq.10 to Eq.11 we get a relation for tetrahedral structure: 1 0.021 o m T aa T (12) Let [1 + 0.021 T/Tm] ≡ X, and we have V = a3 then we will get: 3 o VVX (13) 3.4. Bulk Modulus The bulk modulus has been calculated from the Cohen empirical formula [41]: 3.5 1970200 C o Bd (14) IC is the Cohen ionicity factor (= 0, 1 and 2 for IV, IIIV and IIV groups respectively) [35], and it is used instead of Pauling ionicity factor (IP) (Eq.15 [42]), d (= 0.433 a) is the intratomic distance [43]. 2 1exp 4 AB P I (15) where χA and χB are the atom A and B electronigativity respectively. Now by substituting Eq.13 in Eq.14 with some sim ple mathematical treatment, we get a general relation for tetrahedral structure: 3.5 o BBX (16) 3.5. Plasmon Energy and the Speed of Sound According to (Kumar and Sastry) [40] the plasmon energy (Epl) is defined as: 2 3 15.3 l dE (17) From the above relations we have d = do X, so the general relation for tetrahedral structure is: 3 2 15.3 o pl d EX (18) For the speed of sound (υ) we have [44]: B (19) ρ represents the density at any temperature and relates to ρo (= 3.47 g/cm according to [45]) the density at zero temperature with the relation: o o V V (20) So by gathering Eqs.13,16,19,20 we get: 0.25 oX (21) o o o B (22) 3.6. Energy Bands They are found by calculating the energy difference between the symmetric points where (Γ25VΓ1V) deter mines the valence band [46,47], (X1C Γ25V) for indirect band gap, (Γ15C Γ25V) for direct band gap [1,33], while (X5CX1C) for conduction band [48]. 3.7. Valence Charge Density The electronic charge distribution ρ(r) can be calcu lated at any point from the equation: ()rPrr (23) Pμν is the density matrix, and both øμ(r) and øν(r) are the basis functions. 4. RESULTS AND DISCUSSION The calculated properties of cBN at 300 K and 0 GPa are illustrated in Table 2. Table 2. The structural and electronic properties of cBN at 300 K and 0 GPa. Prop erty Pres. Exp. [33] [1] [49] [38][43] ao (A)3.6192 3.617 [50] 3.6 3.606 3.623 3.633.582 Bo (GPa) 367.73 368 [50]1132 367 365.3 360392.3 B’o 4.848 3.94 3.763 Ecoh (eV) 13.21 13.2 [1]8.131 14.3 6.711713.9 DBG (eV) 23.929 14.5 [51]16.01 8.6 VBW (eV) 20.121 20.6 [51]21.78 20.3 CBW (eV) 14.05 η 5.019 υ (m/s)10326.813 Epl (eV)30.5036616722.758 [52] 24.53 [52] 30.55 [40]
Z. Y. Mijbil et al. / Natural Science 3 (2011) 154164 Copyright © 2011 SciRes. OPEN ACCESS 157 4.1. Total Energy Figure 1 shows the curve of the total energy against the lattice constant, and from it we can see that when atoms come closer, the attractive and repulsive forces appear [53] so the atomic orbitals overlap to make bonds [54]. The overlap presents three states: 1) If a > ao; the attractive force (Fa) will be grater than the repulsive force (Fr) due to the existence of the elec trons in the region between the atoms. 2) If a < ao; Fa < Fr, this case is due to the electron electron and nucleinuclei repulsions, in addition to Pauli exclusion principle which prevents atoms to be closer than ao. 3) If a = ao; Fa = Fr, which is the balance case. 4.2. Cohesive Energy According to Eq.9 the cohesive energy mainly de pends upon the total energy and when the distance in creases by temperature the balance state will be shifted therefore the cohesive energy increases as shown in Figure 2. 4.3. Volume The relation between the volume and temperature, as shown in Figure 3, can be simply attributed to the fact that the atom interacts mainly with its neighbor [55] Figure 1. Variation of the total energy with the lattice constant. Figure 2. Effect of temperature on the cohesive energy. Figure 3. Effect of temperature on the volume. In the meanwhile, when temperature is increased the kinetic rgy (Ek) also increases according to [56]:
Z. Y. Mijbil et al. / Natural Science 3 (2011) 154164 Copyright © 2011 SciRes. OPEN ACCESS 158 3 2 kB EkT (24) Depending on Bornvon Kármán theory which con siders the lattice as a punch of coupled harmonic oscil lators [57]. We have from this view [58]: 2 1 2 k ECu (25) where C is force constant, <u2> is the mean square dis placement of the atom, so by combing the last two equa tions we get: 23B k uT C (26) The latter equation means that the increase in tem perature entails an increase in atomic vibration ampli tude which makes the bond weaker [59] and the volume will increase. 4.4. Bulk Modulus Bulk modulus represents the hardness [60] and deter mines the ability of matter to withstand the volume change [43], so it depends upon the bond length [4] and this bond is short and very strong in cBN [61]. Tem perature increases volume and kinetic energy [56] which entails a decrease in density, resistance to volume change and bond strength [62] which decreases the bulk modulus [63] as shown in Figure 4 because it depends upon the density directly [64]. This behavior is noticed for silicon [65], diamond [55], and also for other solids [6670]. 4.5. Plasmon Energy and Speed of Sound Here we have presented two relations: one of them is for plasmon energy and the other is for speed of sound with temperature in order to simplify the calculations. Our results for plasmon energy as in Figure 5 are in contrast with the experiment [71,72] because we have depended on a relation that does not take into account the effective ionic charge e* T which is directly related to the plasmon energy [73]: 1 2 *4 pl T n Ee M (27) where n is the charge density [74], M is the reduced mass of the two atoms and ε∞ is the infrared dielectric constant. Experimentally [71] and theoretically [75] e* T increases with d and vice verse, so the plasmon energy must also increase with d. As a result this equation fails. On the other hand the speed of sound as in Figure 6, Figure 4. Effect of temperature on the bulk modulus. Figure 5. Effect of temperature on the plasmon energy.
Z. Y. Mijbil et al. / Natural Science 3 (2011) 154164 Copyright © 2011 SciRes. OPEN ACCESS 159 Figure 6. Effect of temperature on the speed of sound. which also represents the average of phonon velocity [76] and is related to the average energy of phonon Ēph by Eq. 28, behaves consistently with other experimental calcu lations [71] for IIIV compounds and theoretical calcula tions for InP [44,72], Si and Diamond [77] 1 3 6 ph N EV (28) 4.6. Energy Bands The energy bands show the following points: 1) The value of direct band gap is greater than the ex perimental value as shown in Table 3. This behavior is also noticed in most calculations that use HartreeFock method [7880] due to its approximations which are: • The minimal basis sets include (s and p) orbitals only, these orbitals determine the valence and the conduction band accurately, while the outer orbi tals that have not been taken into account decrease the value of the band gap [81]. • We have not considered the correlation corrections which decrease the value of the indirect band gap [79]. • Both s and p orbitals are given the same value of ξ and β and the latter parameter determines the bon ding and the antibonding states. • Neglecting of core states which affect the outer va lence electrons distribution [36]. 2) All bands decrease with temperature as shown in Figures 7 and 8. Figure 7. Effect of temperature on the direct and indirect band gaps. Figure 8. Effect of temperature on valence and conduction bands.
Z. Y. Mijbil et al. / Natural Science 3 (2011) 154164 Copyright © 2011 SciRes. OPEN ACCESS 160 Table 3. The symmetry points value with their approximate structure in (eV). Points Approximate structure Pres. [33] Exp. [33] Γ1V Nitrogen bonding s orbitals −20.1218 −21.78 −20.06 Γ1C Boron antibonding s orbitals 35.792 16.38 Γ15V Nitrogen bonding p orbitals 0 0 0 Γ15C Boron antibonding p orbitals 23.932 16.06 8.19 X1V Nitrogen bonding s orbitals −13.0319 −10.88 X1C Boron antibonding s orbitals 21.951 12.34 6 Symmetric points are a combination of atomic orbitals to produce either bonding or antibonding orbitals [82]. Our calculations of the main points with their relativistic corrections are illustrated in Table 3 and compared with (Zunger and Freeman) [10] and Ferhat et al. [42] who both explained the structure of these points. Valence band is nearly determined by (2s and 2p) ni trogen bonding orbitals [82], while the conduction band is determined by (2s and 2p) boron antibonding orbitals [79,40]. Therefore, when the difference between these points increases, the energy bands will be wider. Our results are consistent with [1,10,83] for cBN. The effect of temperature is similar to the effect of tension pressure but it is thermal [84], while it directly affects the band gaps [85]. Due to the increase in tem perature which leads to increase in distance, the orbitals overlap will be decreased and the repulsion will also decrease, and the energy difference between bonding and antibonding states will be decreased accordingly. This means that the widths become narrower. Our results are consistent with Pässler’s results for semicoductors [86], and with measurements for band gab as reported by Olguín et al. [87] and Fan [88]. 4.7. Valence Charge Distribution First of all we need to mention the approximations used in CNDO method because this kind of results is sensitive to them, and they are: 1) Zero differential overlap which neglects the repul sion integrals and the electronelectron overlap on the same atom, and these change the Coulomb potentials that affect the distribution [89]. 2) The values of EES and EEP are very high, “two of the semiempirical method parameters used to get the results”, and both of them represent the electronigativ ity of the atom and determine the bond polarity [90]. 3) Correlation corrections have not been taken into account. Figures 911 and the high anharmonicity parameter (η) [91] in Table 2 show that the density of the charge is higher around nitrogen’s atoms than boron’s atoms, be cause the electrons can stay near one of them more than the another which leads to transfer of charge which is called charge transfer factor (ci). This dimensionless parameter reflects the ability of atom to lose electrons either for another atom or for the intratomic distance. (a) (b) Figure 9. Valence charge distribution of the plane (110): a  at (0K) and b  at (1000K).
Z. Y. Mijbil et al. / Natural Science 3 (2011) 154164 Copyright © 2011 SciRes. OPEN ACCESS 161 (a) (b) Figure 10. Valence charge distribution of the plane (001): a  at (0K) and b  at (1000K). (a) (b) Figure 11. Valence charge distribution of the plane (400): a  at (0K) and b  at (1000K). This case is obvious in polar compounds such as cBN which has (ci = 0.8) [92]. On the other hand, nitrogen pulls the electrons from boron [93] due to its high elec tronigativity with respect to boron [94], but as tempera ture increases the atoms try to separate and return to their original state, as free atoms [95]. 5. CONCLUSIONS 1) TBLUCCNDO/2 method is preferred for wide range of calculations such as the lattice constant, cohesive energy and valence band etc.. 2) In this method the energy bands can not be accu rately calculated all together. 3) The correlation corrections have remarkable ef fects upon the band gaps’ width and charge distri bution. 4) The cBN tends to be more conductive with tem perature increase. 5) Even with high temperature, cBN can withstand corrosions and abrasives. 6) The infrared absorption of cBN increases with the increase of temperature.
Z. Y. Mijbil et al. / Natural Science 3 (2011) 154164 Copyright © 2011 SciRes. OPEN ACCESS 162 REFERENCES [1] Wentzcovitch, R.M., Chang, K.J. and Cohen, M.L. (1986) Physical Review B, 34, 10711079. doi:10.1103/PhysRevB.34.1071 [2] Sun, H., Jhi, S.H., Roundy, D., Cohen, M.L., Louie, S.G. (2001) Physical Review B, 64, 094108/16. [3] Kádas, K., Kern, G. and Hafner, J. (2000) Surface Sci ence, 494497. doi:10.1016/S00396028(00)002545 [4] Mattesini, M. and Matar, S.F. (2001) Computational Materials Science, 20, 107119. doi:10.1016/S09270256(00)001324 [5] Major, B., Mróz, W., Jelinek, M., Kosydar, R., Kot, M., Major, Ł., Burdyńska, S. and Kustosz, R. (2006) Bulletin of the Polish Academy of Sciences: Technical Sciences, 54, pp. 181188. [6] Knittle, E., Kaner, R.B. Jeanloz, R. and Cohen, M. L. (1995) Physical Review B, 51, 1214912156. doi:10.1103/PhysRevB.51.12149 [7] Lambrecht, W.R.L. and Segall, B. (1989) Physical Re view B, 40, 99099919. [8] Watanabe, S., Wheeler, D.R., Abel, P.B., Murakawa, M. and Miyake, S. (1998) Surface chemistry, microstructure, and tribological properties of cubic boron nitride films. NASA Lewis Research Center. [9] Mãrlid, B. (2001) Theoretical modling of thin film growth in the BN system. Acta Universitais. [10] Zunger, A. and Freeman, A.J. (1978) Physical Review B, 17, 20302042. doi:10.1103/PhysRevB.17.2030 [11] Meng, Y., Mao, H., Eng, P.J., Trainor, T.P., Newville, M., Hu, M.Y., Kao, C., Shu, J., Hausermann, D. and Hemley, R. J. (2004) Nature Materials, 3, 111114. doi:10.1038/nmat1060 [12] Karlsson, J. and Larsson, K. (2010) Journal of Physical Chemistry C, 114, 35163521. doi:10.1021/jp907186a [13] Horiuchi, S., Huang, J.Y., He, L.L., Mao, J.F. (1998) Philosophical Magazine A, 78, 10651072. doi:10.1080/01418619808239974 [14] Huang, J.Y. and Zhu, Y.T. (2002) Chemistry of Materials, 14, 18731878. doi:10.1021/cm0109645 [15] HangSheng, Y., FaMin, Q. and AnMin, N. (2010) Journal of Inorganic Materials, 25, 748752. doi:10.3724/SP.J.1077.2010.00748 [16] Khavryuchenko, O.V., Alekseev, S.A., Beobide, A.S., Kandilioti, G., Voyiatzis, G.A., Lisnyak, V.V. (2010) The Journal of Physical Chemistry C, 114, 11021109. doi:10.1021/jp908711x [17] Zhao, Y. and Wang, M. (2009) Journal of Materials Processing Technology, 209, 355359. doi:10.1016/j.jmatprotec.2008.02.005 [18] Yallese, M.A., Chaoui, K., Zeghib, N., Boulanouar, L., Rigal, J.F., (2009) Journal of Materials Processing Technology, 209, 10921104. doi:10.1016/j.jmatprotec.2008.03.014 [19] Wentorf, R.H., Jr. (1962) Journal of Chemical Physics, 34, 809812. doi:10.1063/1.1731679 [20] Harker, A.H. and Larkins, F.P. (1979) Journal of Physics C: Solid State Physics, 12, 24872495. doi:10.1088/00223719/12/13/013 [21] Pople, J.A. and Beveridge, D.L. (1970) Approximate molecular orbital theory. McGrawHill. [22] Harker, A.H. and Larkins, F.P. (1979) Journal of Physics C: Solid State Physics, 12, 24972507. doi:10.1088/00223719/12/13/014 [23] Freund, H. and Hohlneicher, G. (1979) Theoretica chimica acta, 51, 145162. doi:10.1007/BF00554098 [24] Schmid, E. and Brodbe, E. (1985) Canadian Journal of Chemistry, 63, 13651371. doi:10.1139/v85233 [25] Vogl, P., Hjalmarsons, H.P. and Dow, J.d. (1983) Journal of Physics and Chemistry of Solids, 44, 365378. doi:10.1016/00223697(83)900641 [26] Slater, J.C. and Koster, G.F. (1954) Physical Review, 94, 14981529. doi:10.1103/PhysRev.94.1498 [27] Pople, J.A. Santry, D.P. and Segal, G.A. (1965) Journal of Chemical Physics, 43, 129S135. doi:10.1063/1.1701475 [28] Bredow, T., Evarestov, R.A. and Jug, K. (2000) Physica Status Solidi B, 222, 495514. doi:10.1002/15213951(200011)222:2<495::AIDPSSB4 95>3.0.CO;25 [29] Rogan, J. and Lagos, M. (2001) Philosophical Magazine B, 81, 551559. doi:10.1080/13642810108225450 [30] Chadi, D.J. and Cohen, M.L. (1973) Physical Review B, 8, 57475753. doi:10.1103/PhysRevB.8.5747 [31] Janesko, B.G. and Yaron, D. (2004) Journal of Chemical Physics, 121, 56355645. doi:10.1063/1.1785771 [32] Daudal, R., Leroy, G., Peeters, D. and Sana, M. (1983) Quantum chemistry. John Wiley and Sons. [33] Hassan, I.Z. (2001) Semiempirical selfconsistent field calculations of IIIV zincblende semiconductors. Ph. D. thesis. University of AlNahrain. [34] Murnaghan, F.D. (1944) Proceedings of the National Academy of Science, 30, 244. doi:10.1073/pnas.30.9.244 [35] Zheng, J.C., Huan, C.H.A., Wee, A.T.S., Wang, R.Z. and Zheng, Y.M. (1999) Journal of Physics: Condensed Mat ter, 11, 927935. doi:10.1088/09538984/11/3/030 [36] Abdulsattar, M.A. (1997) Self consistent field calcula tions of covalent semiconductors. Ph.D. Thesis, Univer sity of Baghdad. [37] Allen, C. (1976) Astrophysical quantities. Athlone Press. [38] Lee, S.H., Kang, J.H. and Kang, M.H.J. (1997) The Ko rean Physical Society, 31, 811814. [39] Omer, M.S. (2007) Materials Research Bulletin, 42, 319326. doi:10.1016/j.materresbull.2006.05.031 [40] Kumar, V. and Sastry, B.S.R. (2001) Crystal Research and Technology, 36, 565569. doi:10.1002/15214079(200107)36:6<565::AIDCRAT5 65>3.0.CO;2F [41] Lam, P.K., Cohen, M.L. and Martinez, G. (1987) Physi cal Review B, 35, 91909194. doi:10.1103/PhysRevB.35.9190 [42] Ferhat, M., Bouhafs, B., Zaoui, A. and Aourag, H. (1998) Journal of Physics: Condensed Matter, 10, 79958006. doi:10.1088/09538984/10/36/010 [43] Wang, S.Q. and Ye, H.Q. (2002) Physical Review B, 66, 235111/17. [44] Schlosser, H. and Ferrante, J. (1988) Physical Review B, Condensed Matter, 37, 43514357. [45] Soma, T., Sawaoka, S. and Saito, S. (1974) Materials Research Bulletin, 9, 755. doi:10.1016/00255408(74)90110X [46] Calzaferri, G. and Rytz, R. (1996) Journal of Physical Chemistry, 100, 1112211124. doi:10.1021/jp960840t [47] Levitas, V.I. and Shvedov, L.K. (2002) Physical Review
Z. Y. Mijbil et al. / Natural Science 3 (2011) 154164 Copyright © 2011 SciRes. OPEN ACCESS 163 B, 65, 104109/16. [48] Merdan, M.G. (2005) Selfconsistent field calculations of the effect of pressure and temperature on some properties of grey tin crystal. Master’s Thesis, Babylon University. [49] Sekkal, W., Bouhaf, B., Aourag, H. and Certier, M. (1998) Journal of Physics: Condensed Matter, 10, 49754984. doi:10.1021/jp960840t [50] Mujica, A., Rubio, A., Munoz, A. and Need, R.J. (2003) Reviews of Modern Physics, 75, 863. doi:10.1103/RevModPhys.75.863 [51] Jayawardane, D.N., Pickard, C.J., Brown, L.M. and Payne, M.C. (2001) Physical Review B, 64, 11517/14. [52] Reddy, R.R., Ahammed, Y.N., Abdul Azeem, P., Gopal, K.R., Devi, B.S. and Rao, T.V.R. (2003) Defence Science Journal, 53, 239248. [53] Murrel, J.N., Kettle, S.F.A. and Tedder, J.M. (1982) The chemical bond. 2nd Edition, Translated to Arabic by M. N. AlZakum, University of Basrah. [54] Ladd, M.F.C. (1983) Structure and bonding in solid state chemistry. Translated to Arabic by I. J. Selomi, Univer sity of Mosul. [55] Adamska, C.M., Sloma, P. and Tomaszeewski, J. (2006) Journal of Physics: Condensed Matter, 18, 751758. doi:10.1088/09538984/18/2/028 [56] Sibona, G.J. Schreiber, S., Hoppe, R.H.W., Strizker, B. and Revńic, A. (2003) Materials Science in Semicon ductor Processing, 6, 7176. doi:10.1016/S13698001(03)000738 [57] Phillips, J.C. (1956) Physical Review, 104, 12631277. [58] Kittel, C. (1996) Introduction to solid state physics. 7th Edition, John Wiley and Sons. doi:10.1103/PhysRev.104.1263 [59] Glazov, V.M. and Shchelikov, O.D. (1998) Semiconduc tor Equipment and Materials International, 32, 382384. [60] Olguín, D., Cantarero, A., Ulrich, C. and Syassen, K. (2003) Physica Status Solidi B, 235, 456463. doi:10.1002/pssb.200301602 [61] Audronis, M., Valiulis, A.V. and Silickas, P. (2004) Jour nal of Materials Science, 10, 152156. [62] Jiang, J.Z., Lindelov, H., Gerward, L., Ståhl, K., Recio, J. M., MoriSanchez, P., Carlson, S., Mezouar, M., Doory hee, E., Fitch, A. and Frost, D.J. (2002) Physical Review B, 65, 161202/14. [63] D’Evelyn, M.P. and Taniguchi, T. (1998) Elastic proper ties of translucent polycrystalline cubic boron nitride as characterized by the dynamic resonance method. GE Re search & Development Center, General Electric Com pany. [64] Wang, B.T., Zhang, P., Shi, H.L., Sun, B. and Li, W.D. (2010) The European Physical Journal B, 74, 303308. doi:10.1140/epjb/e201000081x [65] Hung, V.V., Jindo, K.M. and Haun, P.T.M. (2006) Jour nal of Physics: Condensed Matter, 18, 283293. doi:10.1088/09538984/18/1/021 [66] Mijbil, Z.Y. (2011) Journal of Bab. University, 19, Under Publishing. [67] Garai, J. and Laugier, A. (2007) Journal of Applied Physics, 101, 023514/14. [68] Anderson, O.L. (1966) Physical Review, 144, 553557. doi:10.1103/PhysRev.144.553 [69] Nandanpawar, M.L. and Rajagopalan, S. (1978) Journal of Applied Physics, 49, 39763979. doi:10.1063/1.325408 [70] Wachtman, J.B. Jr., Tefft, W.E., Lam, D.G., Jr., and Ap stein, C.S. (1961) Physical Review, 122, 17541759. doi:10.1103/PhysRev.122.1754 [71] Sanjurjo, J.A., LópezCruz, E., Vogl, P. and Cardona, M. (1983) Physical Review B, 28, 45794584. doi:10.1103/PhysRevB.28.4579 [72] Trommer, R., Müller, H., Cardona, M. and Vogl, P. (1980) Physical Review B, 21, 48694878. doi:10.1103/PhysRevB.21.4869 [73] Burstein, E., Perkowitz, S. and Brodsky, M.H. (1968) Journal de Physique, 29, C4/7883. [74] Miranda, L.C.M. and Haar, ter D. (1972) Revista Bra sileira de Ensino de Física, 2, 7786. [75] MolinàsMata, P.A., Shields, J. and Cardona, M. (1993) Physical Review B, 47, 18661875. doi:10.1103/PhysRevB.47.1866 [76] Raj, G.D. (2004) Solid state physics. 1st Edition, Anmol Pub. Pvt. Ltd.. [77] Xu, C.H., Wang, C.Z., Chan, C.T. and Ho, K.M. (1991) Physical Review B, 43, 50245027. doi:10.1103/PhysRevB.43.5024 [78] Kleinman, L. and Phillips, J.C. (1960) Physical Review, 117, 460464. doi:10.1103/PhysRev.117.460 [79] Yang, D., Su, Z., Du, Y., Ji, X., Yang, X., Gong, X. and Zhang, T. (2006) Chinese Physics Letters, 23, 1324 1326. doi:10.1088/0256307X/23/5/072 [80] Dovesi, R., Pisani, C., Roetti, C. and Dellarole, P. (1981) Physical Review B, 24, 41704176. doi:10.1103/PhysRevB.24.4170 [81] Chadi, D.J. (1977) Physical Review B, 16, 35723578. doi:10.1103/PhysRevB.16.3572 [82] Mori, Y., Ikai, T., Teranishi, R. and Takarabe, K. (2003) Physica Status Solidi B, 235, 302306. doi:10.1002/pssb.200301573 [83] Kim, K., Lambrecht, W.R.L. and Segall, B. (1996) Physical Review B, 53, 1631016326. doi:10.1103/PhysRevB.53.16310 [84] Jain, S.C., Willander, M. and Maes, H. (1996) Semicon ductor Science and Technology, 11, 641671. doi:10.1088/02681242/11/5/004 [85] Vurgaftmana, I., Meyer, J.R. and Mohan, L.R.R. (2001) Journal of Applied Physics, 89, 58155875. doi:10.1063/1.1368156 [86] Pässler, R. (1999) Physica Status Solidi B, 216, 9751007. doi:10.1002/(SICI)15213951(199912)216:2<975::AID PSSB975>3.0.CO;2N [87] Olguín, D., Cantarero, A. and Cardona, M. (2000) Physica Status Solidi B, 220, 3339. doi:10.1002/15213951(200007)220:1<33::AIDPSSB33 >3.0.CO;2J [88] Fan, H.Y. (1951) Physical Review, 82, 900905. doi:10.1103/PhysRev.82.900 [89] Raczkowski, D., Canning, A. and Wang, L.W. (2001) Physical Review B, 46, 121101/14. [90] Cotton, F.A. and Wilkinson, G. (1976) Basic inorganic chemistry. John Wiley & Sons. [91] Rose, J., Smith, J.R., Guinea, F. and Ferrante, J. (1984) Physical Review B, 29, 29632969. doi:10.1103/PhysRevB.29.2963 [92] Sánchez, P.M., Pendás, A.M. and Luaña, V. (2002) Jour nal of the American Chemical Society, 124, 14721
Z. Y. Mijbil et al. / Natural Science 3 (2011) 154164 Copyright © 2011 SciRes. OPEN ACCESS 164 14723. doi:10.1021/ja027708t [93] Bouhafs, B., Aourag, H. and Certier, M. (2000) Journal of Physics: Condensed Matter, 12, 56555668. doi:10.1088/09538984/12/26/312 [94] Wu, R.Q., Peng, G.W., Lui, L. and Feng, Y.P. (2006) Journal of Physics: Condensed Matter, 18, 569575. doi:10.1088/09538984/18/2/015 [95] AlJelawy, I.O.R. (2005) Effect of pressure and tempera ture on some properties of diamond crystal using har treefock method. Master Thesis, University of Babylon.
