Interatomic force constants of semiconductors

Journal of Physics and Chemistry of Solids 61 (2000) 91–94 www.elsevier.nl/locate/jpcs

Interatomic force constants of semiconductors V. Kumar* Department of Electronics and Instrumentation, Indian School of Mines, Dhanbad 826 004, India Received 8 December 1998; accepted 28 May 1999

Abstract Interatomic force constants of A IIB VI and A IIIB V semiconductors have been calculated using plasma oscillations theory of solids. Two simple equations relating the bond-stretching force constant (a ) and plasmon energy have been proposed. The calculated values of a have been used in calculating the values of bond-bending force constant (b ). Our calculated values of a and b are in excellent agreement with the values reported by different workers. q 1999 Elsevier Science Ltd. All rights reserved. Keywords: Bond-bending force constant

1. Introduction During the last few decades a considerable amount of experimental and theoretical work has been done to understand the structural properties such as optical, electronic, elastic, and thermal of binary semiconductors. Interatomic force constants of A IIB VI and A IIIB V semiconductors have been an important parameter to study because these semiconductors have potential applications in a variety of optoelectronic devices such as integrated circuits, detectors, lasers, light emitting diodes, modulators and filters. Using the valence-force-field model of Keating [1], the elastic properties of the A IIB VI and A IIIB V semiconductors with a sphalerite-structure have been analysed by Martin [2] and several other workers [3–5]. A considerable amount of discrepancies have been obtained between theory and experiment in evaluating vibrational modes on the basis of the model parameters derived from elastic constant data. Nowadays more reliable elastic constant data are available which differ partially from those obtained by Martin [2]. In the Martin analysis the contribution of Coulomb force to the elastic 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. [6] has pointed out that the Martin approach is incorrect and that the contribution of Coulomb forces to the elastic constants and the transverse optical frequencies must be * Tel.: 1 91-326-822273; fax: 1 91-326-832040.

described in terms of the localized effective charge which differs from the macroscopic effective charge. Neumann [7– 11] has extended the Keating model considering localized effective charge to account for long-range Coulomb force and dipole–dipole interaction in analysing the vibrational properties of binary and ternary compounds with a sphalerite-structure. In this model, Neumann [7] has taken experimental values of bond length (d) and spectroscopic bond ionicity (fi) [3] to determine the constant associated with the equations. Recently the ab initio calculations for lattice dynamic for BN and AlN semiconductors have been given by Karch and Bechstedt [12]. In all the above investigations, it has been established that a is only function of bond length. Further it has been found that b is proportional to a i.e. b , …1 2 f i †a and therefore, shows the same dependence on the bond length. In fact, the force constant a has an exponential relation with the bond length. Several workers [13–17] have given some explanation for this exponential relation and the constant associated with the equation. Recently, using plasma oscillations theory of solids, the authors [18–21] have developed a simple relation between plasmon energy and bond length. This is based on the fact that the plasmon energy, "vp ˆ "…4pne2 =m†1=2 ; is related to the effective number of valence electrons (n) in a compound. The bond length also depends on n. Thus the plasmon energy and bond length has been found to have a correlation between them. The force constant (a ) also depends on bond length. This shows that there must be a correlation between force constant and

0022-3697/00/$ - see front matter q 1999 Elsevier Science Ltd. All rights reserved. PII: S0022-369 7(99)00238-3

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V. Kumar / Journal of Physics and Chemistry of Solids 61 (2000) 91–94

Table 1 Interatomic force constants of binary semiconductors Compounds

1 A IIB VI BeS BeSe BeTe BePo a MgTe ZnO ZnS ZnSe ZnTe ZnPo a CdS CdSe CdTe CdPo a HgS HgSe HgTe a A IIIB V BN BP BAs AlN AlP AlAs AlSb GaN GaP GaAs GaSb InN InP InAs InSb a

"vp (eV) [18,20]

a (N m 21)

fi [17,19]

b (N m 21) from Eq. (8) with a from

Eq. (6)

Eq. (7)

Neumann [7,9]

Eq. (6)

Eq. (7)

Neumann [7,9]

2

3

4

5

6

7

8

9

19.52 18.39 16.12 14.91 12.97 21.48 16.71 15.78 14.76 12.36 14.88 14.01 13.09 11.43 14.85 12.98 11.43

62.18 56.19 44.91 39.34 31.03 73.17 47.74 43.31 38.66 28.63 39.22 35.38 31.52 25.03 39.06 31.08 25.03

63.18 57.89 46.02 39.78 30.07 73.66 49.09 44.26 39.02 27.13 39.63 35.22 30.65 22.79 39.47 30.12 22.79

– – – – – – 44.73 38.61 32.04 – – – 29.44 – – 37.43 29.32

0.656 0.661 0.672 0.621 0.685 0.646 0.669 0.673 0.678 0.631 0.677 0.681 0.684 0.633 0.621 0.629 0.634

5.99 5.33 4.12 4.17 2.73 7.25 4.42 3.96 3.48 2.96 3.54 3.16 2.78 2.57 4.14 3.23 2.56

6.14 5.49 4.22 4.22 2.65 7.30 4.54 4.05 3.52 2.80 3.58 3.14 2.71 2.34 4.18 3.13 2.34

– – – – – – 4.36 4.65 4.47 – – – 2.48 – – 2.37 2.54

24.53 21.71 20.12 22.97 16.65 15.75 13.72 21.98 16.50 15.35 13.38 18.82 14.76 14.07 12.73

91.69 74.50 65.47 82.01 47.45 43.17 34.15 76.09 46.73 41.33 32.72 58.44 38.66 35.64 30.06

89.58 75.21 66.94 81.68 48.78 44.10 33.77 76.60 48.00 42.04 32.08 60.14 39.02 35.52 28.88

– – – – – – 35.74 – 48.57 43.34 34.42 – 44.29 37.18 30.44

0.299 0.312 0.320 0.306 0.336 0.340 0.349 0.311 0.337 0.342 0.350 0.326 0.345 0.347 0.353

17.99 14.35 12.46 15.93 8.82 7.97 6.22 14.68 8.67 7.61 5.95 11.03 7.09 6.51 5.44

17.58 14.48 12.74 15.87 9.07 8.15 6.15 14.77 8.91 7.74 5.84 11.35 7.15 6.49 5.23

– – – – – – 6.63 – 10.40 8.88 7.16 – 6.26 5.47 4.73

Values of plasmon energy and ionicity of these compounds are reported for the first time in this paper.

plasmon energy. In the paper, we have proposed two equations relating the bond-stretching force constant (a ) and plasmon energy for the A IIB VI and A IIIB V semiconductors. The calculated values of a from these equations have been further used to calculate the values of b . Our calculated values of a and b from both the equations are in fair agreement with the values reported by Neumann [7,9].

2. Theory and calculation The nearest-neighbour bond-stretching central forces have been characterised by the parameter a , and next-neighbour bond-bending non-central forces by the parameter b .

These parameters depend on interatomic distance obtained from lattice vibration data. The lattice vibration data have been further obtained from various types of two-body interatomic potential given in the literature. Such potentials have the advantage of keeping the repulsive and attractive forces in the same mathematical form. The simplest form of interatomic potential has been described by Neumann [9,10] and Harrison [14,15] in which it has been assumed that both the repulsive and attractive parts of interatomic potential are described by the power law of interatomic distance (r). This form of potential for the total energy per pair of atom can be written as [9] V1 …r† ˆ C=rm 2 D=rn

…1†

V. Kumar / Journal of Physics and Chemistry of Solids 61 (2000) 91–94

93

Fig. 1. Interatomic force constants of binary semiconductors.

where C, D, m and n are the constants. These parameters have been estimated for equilibrium condition when the repulsion is half of the attraction i.e. m ˆ 2n [9] and the following equation has been obtained

Eq. (3) for equilibrium condition a ˆ 2b; the following equation has been obtained [9]

a ˆ a0 d 2x

where a 1 and b are constants. The values of a1 ˆ 704:3 N m21 and b ˆ 1:138 × 1010 m21 have been obtained [9] in the case of the A IIB VI and A IIIB V semiconductors based on the best fit data. This again requires the experimental values of the bond length, lattice constant and ionicity of the semiconductors. Based on our previous publication [19], following the relation between bond length and plasmon energy for the A IIB VI and A IIIB V groups of semiconductors can be written as

…2†

where a 0 and x are constants. Plotting the double-logarithmic curve of a against the experimental values of bondlength (d), Neumann [7] has obtained a0 ˆ 1:183 and x ˆ 2:56 for A IIB VI and A IIIB V semiconductors if d is in nm and a in N/m. This value of x ˆ 2:56 has been obtained if the A IIB VI and A IIIB V semiconductors are considered together and if these semiconductors are considered separately x ˆ 2:30 and 2.72 have been obtained, respectively, for the A IIB VI and A IIIB V groups of semiconductors [9]. The other form of potential is based on Morse potential. In this type of potential both the repulsive and attractive terms are described by exponential functions of interatomic distance. The general form of Morse potential is given by [9] V2 …r† ˆ A exp…2ar† 2 exp…2br†

…3†

where A, B, a and b are constants. The above equation has been further used to describe the two-body interaction in total energy calculation of Si [16]. Neumann [8–11] has also extended it to ternary chalcopyrites. Solving above

a ˆ a1 exp…2bd†

d…nm† ˆ 1:53…"vp †22=3

…4†

…"vp in eV†

…5†

From the above Eqs. (2), (4) and (5) we get following two relations between a and "vp for the A IIB VI and A IIIB V semiconductors.

a ˆ 0:398…"vp †1:70

…6†

and

a ˆ 704:3 exp{ 2 17:41…"vp †22=3 }

…7†

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V. Kumar / Journal of Physics and Chemistry of Solids 61 (2000) 91–94

In the above Eqs. (6) and (7), a is in N m 21 and "vp in eV. According to Martin [2] the bond-bending force constant (b ) follows the proportionality relation b / …1 2 fi †a; where fi is the ionicity of the A–B bond in the A IIB VI and A IIIB V semiconductors. Using the reported values of fi [22,23], Neumann [7] has plotted a curve between b=a and …1 2 f i † and a linear relation has been obtained between them. Based on the least-square fit of the data points the following relation has been obtained [7].

b ˆ b0 …1 2 fi †a

…8†

where b0 ˆ 0:28 ^ 0:01 [7] is the proportionality constant. If we compare this value of b0 ˆ 0:28 for fi ˆ 0 with the values b=a ˆ 0:285 and 0.294 found in Si and Ge, respectively [2], a good agreement has been obtained by Neumann [7]. In the present calculation, we have taken the value of b0 ˆ 0:28: The values of "vp and fi have been taken from our previous publications [18,20] in all the semiconductors except BePo, ZnPo, CdPo and HgTe semiconductors which are reported for the first time in this paper. The values of a have been calculated using both Eqs. (6) and (7). The calculated values of a from these equations have been used to calculate the values of b from Eq. (8).

equation, the data obtained from Eq. (6) are very close to the data obtained from Eq. (7). The curve further shows that b is proportional to a . For the comparison, limited number of known data of a and b are reported in the literature [7,9] for these sets of binary semiconductors. Thus the model has been verified for some of the A IB IIIC2VI and A IIB IVC2V semiconductors and it has been found that the proposed model is valid well with the ternary semiconductors too. Hence, it is possible to calculate the values of interatomic force constants a and b of the A IIB VI and A IIIB V groups of semiconductors from their plasmon energy even if the experimental data of lattice vibration and bond length are unknown. Thus, one can predict the values of a and b for unknown compounds belonging to these groups of semiconductors from their plasmon energy. Acknowledgements The author is grateful to Prof. D.K. Paul, Director and Prof. Dinesh Chandra, Head, Department of Electronics and Instrumentation, Indian School of Mines, Dhanbad for their continuous inspiration and encouragement in conducting this work.

References 3. Conclusion The values of bond-stretching and bond-bending force constants of the A IIB VI and A IIIB V semiconductors have been calculated using Eqs. (6)–(8). The calculated values are listed in Table 1 and compared with the values reported by Neumann [7,9]. Our calculated values of a from both the Eqs. (6) and (7), and b from Eq. (8) are in good agreement with the values reported by Neumann [7,9]. In most of the semiconductors the deviation between calculated and reported values is below 10%. However, in some cases it is more than 10%, which may be due to uncertainties in the values of ionicity [22–25]. The main advantage of the present model is the simplicity of the formulae, which do not require any experimental data except the plasmon energy of the semiconductors. In contrast, the previous models require the experimental values of lattice parameter or bond length of the semiconductors. Neumann [7,9] has reported the values of a and b for 13 semiconductors of these groups with a sphalerite-structure while we have calculated these values for 28 sphalerite and 4 wurtzite (MgTe, AlN, GaN and InN) structure semiconductors. The trends of force constants and plasmon energy have been shown in Fig. 1. This curve shows that a is exponential in nature as shown by Eq. (7) which is of the form Y ˆ K1 e2k2 X : In Eq. (6), if we approximate the power of "vp ; which is 1.70, equals 2, it will be an exact parabola of the form Y ˆ K3 X 2 : Thus we can say that Eq. (6) is an approximate parabola. Since the parabola is a part of an exponential

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