Abdul Azeem, P

Devi, B Sasikala

Nazeer Ahammed, Y

Rama Gopal, K

Rao, T V R

Reddy, R R

IACS Institutional Repository

Idian J. Phys. 77A (3), 237-241 (2003)/U P  A  %Interatomic force constants, ionicity and m icrohardness o f binarytetrahedral sem iconductorsR R Reddy*, Y Nazxcr Ahammed, P Abdul Azeem, K itania Gopal, B Sasikala Devi and T V R RaoDepartment of Physics, Sri Krishnadevaraya University, |\namapiir-515 (K)3, Andhra Pradesh, Indiaand j S H Bcher^Department of Physics, Marathwada University, A«irungahad-431 004, Maharashtra, India E-mail : rajururreddy@yahoo cu uk. -1R eceived  29  D ecem ber 2000, a ccepted  5  February 2003Abstract , A simple relation between lattice energy (t/), bond stretching and bond bending force constants lonicity (/,) andmiLrohardness (H ) developed for and A'"B'' semiconductors. The results arc in excellent agreement with the values reported by differcnt investigatorsKeywords : Lattice energy, force constants, lonicity, microhardness, tetrahedral semiconductors.PACS Nos. . 62.20.Qp, 05.50.+q, 34.20.Cf1. IntroductionDuring the last few decades, a considerable amount of experimental and theoretical work has been done to understand different properties such as optical, electronic, clastic and thermal properties o f binary semiconductors. In the recent past, much attention has been given to the studybinary tetrahedral sem iconductors because of their potential applications in the field of linear and non-linear optics, solar cells, light emitting diodes, laser diodes and ‘ntcgraicd optical devices such as switches, modulators and lillers.In the valence-force-field model for sphalerite-structure compounds, Martin 11 ] has introduced two interatomic force constants like bond stretching (a) and bond-bending force constants (/J), respectively. The nearest-neighbour bond- ^ f^ctching central forces have been characterised by the parameters, aCNm"*) and next-nearest-neighbour bond bending non-ccntral forces by the parameters j3(Nm“‘), In analysis [ 1 ], the contribution of Coulomb force ’^oTcsponding Authorto the clastic constants has been described in terms of the macroscopic effective charge which is responsible for the splitting of transverse and longitudinal optical modes. Lucovsky et al \2] have pointed out that the Martin approach [II is incorrect. The ratio j3 /a  = 0.3 (I -* /)  measures the importance of covalent bonds in determining the stability of the tetrahedral structures. The overall trend of p /a  lends to zero if ionicity f, tends to unity, i,e. for purely ionic crystals. Neumann [3] has extended the Keating [4] model considering localized effective charge to account for long- range Coulom b force and dipole-dipole interaction in analysing the vibrational properties of binary and ternary compounds with a sphalerite-structure. Neumann [S-S] has introduced the experimental values o f bond length (d) and spectroscopic bond ionicity (f,) to determine the constant associated with the above theory. Recently, using plasma oscillations theory of solids, Kumar [91 have developed a simple relation between plasmon energy (/jcu,,) and bond length. Kumar et al [lOJ have established a connection© 2003 lACS238 R R Reddy et albetween a  and bond length and P is proportional to a . Thus* p  is connected by plasmon energy and the other force constant (a).  Reddy et al [11-16] and Kumar et al[10] have reported a simple correlation between lattice energy U (kcal/m ol), plasmon energy {h(Op) and microhardness H (OPa) for binary tetrahedral semiconductors. Since the optical* structural* stability and olher phy.sical properties of semiconducting compounds play important roles in device development and fabrication* it is worthwhile to develop different relations in terms of force constant* microhardness* ionicity and lattice energy. In the present paper* the correlation between lattice energy, bond stretching and bending force constants, ionicity and microhardness have been obtained for and A***B''semiconductors.2, TheoryThe nearest-neighbour bond-stretching central forces have been characterized by the parameter a ,  and niMi neighbour bond-bending non-ccntral forces by the patarneter /i fltesc parameters depend on interatomic distance obtained from lattice vibration data [10|. The lattice vibration data have been further obtained from various types of two-body intcracton potentials given in the literature. Such potentials have the advantage of keeping the repulsive and attractive forces in the same mathematical form. Neumann [5-8] and Harrison [17*18] have attempted to describe these type of potentials. On the basis of repulsive and attractive forces and on Born and Mayer cycle* Ladd and Lee [19] have proposed different potential functions to estimate lattice energies. Interatomic distance* lattice energy, Madelung constant and others are well related by Born-Mayer theory and other potential functions. Reddy et al [\2\ and Kumar et al [10] have proposed different relations between lattice energy and plasmon energy. The proposed relations arc successfully used for the estimation of lattice energy. The results are in good agreement with the experimental values. Kumar et al [20] have also established another relation between microhardness and plasmon energy and bond length and plasmon energy. The above discussion finally suggests that there must be a definite relation between a , j3, bond length or interatomic distance* lattice energy* plasmon energy* ionicity and microhardness.Kumar e t a l [ l 0} have developed the following quadratic relation between lattice and plasmon energiest/* fll+ fc l(ft0 )p )-c l(» (0 p )2 . 0 )The numerical values for a l , 61, c l are resepetively* 421.224, 27.940 and 0.178 for both II-VI and Ill-V  groups ofsemiconductors. Reddy et al [12] have derived an empirical linear relation between U and in the case of n.-vi group semiconductors asf / =  381.9 + 24.3 {h<o,,). (2)Recently, Kumar [9] have also proposed the following relations :a  = 0.398(ftdi^)>*70,where a  and are force constant and plasmon energy respectively. Kumar [9] have given another linear relation which describes the connection between microhardness H and plasmon energy h(Of, :H = -  Ki, (4)where K\ and Kj are constants.The above equations clearly demonstrate the correlation between lattice energy f/* force constant Of, microhardness and plasmon energy. To estimate U, a  and //, the required parameter is onlyBased on the above correlations* the following equation is proposed for the calculation of a  :a - m „ U - h a ,  (!))where nia and ha are constants, a  and U arc in Nm ' and kcal/mol. The significance and the meaning of a  and V are described earlier. In eq. (4), experimental lattice energies are used for the estimation of a . The estimated a  values from eq. (5) arc in excellent agreement with the lileiaiurc values and proved its linear nature.Neumann [5-8] has obtained a relation between bondstretching (a )  and bond bending (p )  force constants asP  =  P o O - f i ) cc, (6 )where Po = 0.28 and /  is the ionicity [21].Similar to eq. (6), the following relation is proposedP ^ m p U - b p ,  (7)where ntp and bp are constants.a  and P values obtained from cqs. (5) and (7) are substituted in eq. (6) and then ionicity /  can be written s'!f i ^ l - i p / a P o ) ,  (8)here p^ = 0.28.Based on the least-square fit o f the points between microhardness (fO and lattice energy ((/), the following relation has been obtainedH ^ m „ U - b H ,  where m// and hn are constans.Interatomic force constants, ionicity and microhardness etc 239In this model, only lattice energy data is required to estimate o t , p , f i  ^  values. The computed force constants (a,P),  ionicity and microhardness are in good agreement with the literature values [5-9,20,22-24].3 Results and discussionthe present paper reports different correlations between a p, fi ^  semiconductors. Therelevant numerical constants used in the above equations lU'c given in Table 1. The calculated values of the aboveTable 1. Numerical values of the constants m and h.Parameter SemiconductorsPHA»BV'AiiifivA"B'"A'^ 'B''A"BV'A"'BV0.22420.24220.02390.04910.017280.006139.24153.5915.45331.91111.57070.0112parameters from eqs. (5), (7), (8) and (9) are listed in Tables 2 and 3. A fairly good agreement has been observed between the calculated force constants, ionicity and microhardness with the experimental and literature values [5-9,20,22-24]. The mentioned theory clearly indicates the correlation betv\^en lattice energy and / ,  and H. Figures (1-4) sho>^ the trend between present approach and the values obta&ied by others. A slight deviation has been observed only^n the case of ZnSe, ZnTe cind CdTc, GaSh for a  and P Vijtth the results of Neumann [5-8]. This may be due to stroi^ electronegativity values for Se~ and Te. The main advi itage of the present model is the simplicity of the forniilae.ijsing dq. (8), the ionicity (J]) values for A^B'" and A*“Br semiconductors have been calculated. The results are presented in Table 3 together with the values obtained by Biswas et al [22] and Kumar et al [23]. These values are in consistence with the previous investigations. The calculatedTabic 2.CompoundsInteratomic force Lattice energy t/(kcal/mol) Ref. [10]constants of binary .semiconductorsa(Nm“ ) ^(Nnr‘)Eq. [5] Ref. (9] Ref. f3-61 Eq [7] Ref. t9j Ref. t3-'6]A"B'''BeS 899 62.648 63.18 - 6.018 6 14 -BeSe 875 56.967 57.89 - 5 445 5.49 -BcTe 825 45.755 46.02 - 4.250 4.22 -MgTc 754 29.834 30.07 2 555 2.65 -ZnO 939 71.318 73.66 - 6.973 7.3 -ZnS 838 48.67 49.09 44.73 4 561 4.54 4.36ZnSe 818 44.185 44.26 38.61 4 083 4.05 4.65ZnTe 795 39.028 39.02 32.04 3 534 3.52 4 47CdS 798 39.701 39.63 - 3.606 3.58 -CdSe 778 35.216 35.22 - 3.128 3.14CdTe 756 30.283 30.65 29.44 2.602 2.71 2.48A'»BVBN 999 88.357 89.58 ... 17.179 17.58BP 944 75.037 75.21 - 14.477 14.48BaS 911 67.044 66.94 - 12.8.55 12 74AIN 969 81.091 81.98 - 15.705 15.87 -AlP 837 49.122 48.78 - 9.219 9.07 -AlAs 817 44.278 44.1 - 8.236 8.15 -AlSb 771 33.137 33.77 35.74 5.975 6.15 6.63CaN 949 76.248 76.6 14.722 14.77Gap 834 48.395 48 48.57 9.071 8.91 10.4GaAs 808 42.098 42.04 43.34 7.794 7.74 8.88CaSb 763 31.2 32.08 34.42 5.,582 5.84 7.16InN 884 60.505 60.14 11.528 11.35InP 795 38.950 39.02 4.29 7.1515 7.15 -InAs 779 35.075 35.52 37.18 6.369 6.49 5 47InSb 748 27.567 28.88 30.44 4.845 5.23 4.73_  Average deviation 4.58 4.57 - 4.36 12.8 -240 R R Reddy et alTable 3. lonicity and microhardness of binary tetrahedral semiconductors.CompoundsLattice energy (/(kcal/mol) Ref. [lOJlonicity. /, Micro hardness, //(Gpa)Eq [81 Ref. (22.23] Eq. [9] Ref [20] Ref [24] Ref. [24)BeS K99 0.655 0.656 3.964 3.23 - -BeSe 875 0 658 0.661 3..549 2.75 -BcTe 825 0.667 0.672 2.685 1.8 - -MgTe 7.54 0.693 0.685 4.655 4.05 3.9-4.8 4 6ZnO 939 0 650 0.646 2.909 2.05 1.7, 2.8, 3.5 3..^ZnS 838 0.665 0.669 2.564 1.66 1.3--I.8 1 7ZnSc 818 0.669 0.673 2.166 1.23 0.8-1.1 I 0ZnTc 795 0 676 0.678 2.218 1.28 1.2 1 tCdS 798 0.675 0.677 1.873 0.91 0.7-1.2 08CdSc 778 0 682 0.681 I 492 0.53 0 4-0.64 0 5CdTc 756 0 692 0.684 1.458 0.48 - -A«»BVBN 999 0.305 0.299 43.389 26.90 .34 .3-73 0 45 8BP 944 0 311 0 312 23 434 20.7 ^ 31 4-40 32 3BaS 911 0.315 0 320 16.193 17.24 19 24 2AIN 969 0.308 0 306 .31 007 23 48 12.3 19 ^AlP 837 0.329 0 336 7.069 9.64 5.5 10 9AlAs 817 0-335 0.340 5 65 7.67 4.85 6 ^AlSb 771 0.356 0.349 3,375 3 23 4.0GaN 949 0.310 0311 24 78 21.32 - 10 9GaP 834 0.330 0.337 6.836 9.32 9 45 -GaA.s 808 0.338 0.342 5.109 6.79 7.50 -GaSh 763 0.361 0.350 3.086 2.48 4 48 -InN 884 0 319 0.326 11 96 14 40 - 5 3InP 795 0,344 0.345 4.416 5.50 4.10InAs 779 0.351 0.347 3.692 3.99 3 3 -ImSb 748 0-372 0.353 2.609 1.06 2.20 -Average deviation I 23 - - - - -values of H obtained from eq. (9) are listed in Tabic 3 and compared with the values reported by Kumar et al (20] andFigure 1. The relationships between the experimental values (Refs. (3-6]) of force constant.^ , a  of binary semiconductors and those calculated from lattice energy U  (eq. (5)) and plasmon energy, /ico,, (Ref. 19]). The line is of unit slope. Circle indicates eq. (5) and square indicates or(/iaj^) (Ref. [9]) of binary semiconductors.Garbato et al [24], These values are in good agreement with the reported values in literature [20,24]. In most of the cases.Figure 2. The relationships between the experimental values (3-^)) of force constants p  of binary semiconductors and those calculated from lattice energy U (eq, (7)) and plasmon energy h(Op (Ref. line is of unit slope. Circle indicates eq. (7) and square indicates (Ref. [9]) of binary semiconductors.Interatomic force constants, ionicity and microhardness etc 241the estimated values arc in good agreem ent with the work of Kumar et al f9). Thus, the proposed model has been verified and it agreed well for these sem iconductors. It isHjjurc 3. The relationships between the experimental values (Rets 122,2-^ I) lonicily /, of binary semiconductors and those calculated from kuiu’c energy U (eq. (8)) 1‘hc line is of unit slope Circle indicates eq (SI of binary semiconductors.figure 4. The relationships between the experimental values (Ref. [21)) of microhardness H  of binary semiconductors and those calculated Iroiii latiKc energy (eq (9)) and plasmon energy h(Op (Ref 124]) The line is of unit slope. Circle indicates eq. (9) and square indicates H { h o )^  (Kd (24|) of binary semiconductorsconcluded from the present study that one can predict the values of a .  j\ and H for unknown compounds belonging to these groups o f sem iconductors from their lattice energy.Average percentage deviation has been com puted by considering Neum ann [5-8] values as standard ones for the parameters a, P and /  and arc listed at the end o f  the Tables (2 - 3 ).M icrohardness v a lues qu o ted  in the lite ra tu re  arc inconsistent w ith each other. In m ost o f  the cases, our estimated values arc in fair agreem ent with Kumar et al [20].C om paring  the resu lts betw een K um ar et al | 2 0 ] and Neumann [5~8], it is noticed that H values arc deviating more in the case of ZnO, BN, BP, BaS, GaN and InN. M ore divergence has been observed with our results. In view of the above, average percen tage dev ia tion  has not been estimated. Some refinement in the nature o f proposed model majf yield better results.A d juow icdginen tsThe| support of this research by the U niversity  G rants Coijjlmission, New Delhi under UGC Career Award (RRR) is y a tcfu lly  acknowledged.KiT<|'cnccs(13 R M Martin Pliys R ev lU 4(X)S (1970)[2] G l.iicovsky. R M Martin and E Burstcin Phy.w R ev , B4 (1971)[3] H Neumann C n xt. R es Texhtwi 20 773 (1985)|4J P K Keating Ph\s Rev 145 637 (1966)15] H Neumann H elv Phys A rta  56 773 (1985)(61 H Neumann Cryst R es T echnol 24 325 (1989)[7] n Neumann C iyst Res Tei h n oi 24 619 (1989)|8] 11 Neumann Cryst. Res T eelm ol 21 1361 (1986)[9] V Kumar J. Phys. Chem  Solids 61 91 (2(KK))[10] V Kumar. G M Prasad and D Chandra /  Phys Chem  Solids 58 463 (1997)fl 1] R R Reddy, T V R Rao and R Viswanath P roc. Indian Natl. Sci A ca d  55A  901 (1989)[12] R R Reddy, M R Kumar and T V R Rao .A Phys Chem. Soltds 55 603 (1993)[13] R R Reddy. Y N Ahammcd. K R Gopal. P A A/eem. T V R Rao and P M Reddy Opt M ater 14 355 (2000)[14] R R Reddy. Y N Ahammcd, K R Gopal and I) V Raghuran^  O ptM ater 10 95 (1998)[15] R R Reddy. M R Kumar and T V R Rao Cry.st Res Technol. 28 97  ^ (1993)[16] R R Reddy, Y N Ahaimiicd, K R Gopal and T V R Rao Infrared  Phys Tech 39 55 (1998)[17] W A Harrison Phys Rev. B27 3592 (1983)[18] W A Hamson P/m. Rev. B34 2787 (1986)[19] M F C Ladd and W H Lee /  Client. P hys 60 1954 (1974)[20] V Kumar, G M Prasad, A R Chctal and D Chandra J. Phy.wChem. Solids 57 503*0996)[21] J C Philliphs Rev. M od. Phys 42 317 (1970)[22] R Biswas and D R Hatmnann Phy.w Rev. Lett. 55 2(X)I (1985)[23] V Kumar J. Phys. Chem. Solids 48 827 (1987)[24] L Garbalo and A Rucci Phil. Maf>. 35 1681 (1977)

Interatomic Force Constants, Ionicity and Microhardness of Binary Tetrahedral Semiconductors

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Interatomic Force Constants, Ionicity and Microhardness of Binary Tetrahedral Semiconductors

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