Unified study of lattice dynamics of NiO

Journal of Physics and Chemistry of Solids 63 (2002) 127±133

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Uni®ed study of lattice dynamics of NiO K.S. Upadhyaya*, G.K. Upadhyaya, A.N. Pandey Physics Department, KN Govt. PG College, Gyanpur-221304, Sant Ravidas Bhadohi, UP, India Received 6 September 2000; accepted 21 March 2001

Abstract A uni®ed study of lattice dynamics of paramagnetic NiO has been studied by correcting the basic equations of the three-body force shell model for the valency of the cations and anions. The shell charge and core charge parameters are also modi®ed. This approach explains the complete lattice dynamics of NiO successfully only when both the ions are taken to be polarizable. There is good scope for fresh determination of positive ion polarizability and Debye temperature variation. q 2001 Elsevier Science Ltd. All rights reserved. Keywords: D. Lattice dynamics

1. Introduction The divalent transition metal oxides (TMO) form an important class of magnetic materials due to their electrical and magnetic properties of practical utility. Their importance from theoretical and purely scienti®c point of view is also revealed from the dif®culties which arise in conceiving suitable models to explain these properties. Nickelous oxide (NiO) is one of the antiferromagnetic TMO whose lattice dynamical study is of considerable interest to both experimental and theoretical workers. The information available regarding the structure of this oxide shows that it has simple rocksalt (NaCl) structure above the Neel temperature in its paramagnetic phase, and that it possesses two atoms per primitive unit cell with ionic bonding between them. The data on elastic and optical constants measured from the most modern techniques lead to the peculiar elastic and dielectric behaviour which is self-evident from the considerably large Cauchy discrepancy (C12 2 C44) with the strange relation: C44 < 2C12 and signi®cantly large dielectric constants (e 0 ˆ 11.90 and e 1 ˆ 5.70). The lattice dynamics of NiO has been reported in detail by many experimental [1] and theoretical workers [2±5] with moderate success. Some of the model parameters ®tted by them are physically unrealistic. This has motivated us to study the basic need of a lattice dynamical model for satisfactory description of its interesting properties. * Corresponding author.

The most extensively used models for the study of lattice dynamics of ionic crystals are the rigid shell model (RSM) [6], the breathing shell model (BSM) [7], the deformable shell model (DSM) [8] and the three-body force shell model (TSM) [9]. The RSM and BSM have been found to give better description of the dispersion, dielectric and thermal properties, but fail to account well for the observed Cauchy discrepancy. It has been pointed out by Sangster et al. [10] that the DSM is simply a BSM with slight changes in its theory. On the other hand, the TSM explains the elastic as well as the dielectric properties suf®ciently accurately. Although, as reviewed by Cochran [11] and Singh [12] independently, the TSM is approximately equivalent to the BSM, in its mathematical structure, there are de®nite differences in the physical contents of the two models. This can be realized from the fact that the radial deformation of electron shells has been introduced in the BSM in an arbitrary manner, which has no theoretical justi®cation, while the development of the TSM is based on the three-body interactions whose existence is very well founded by Lowdin [13,14] and Lundqvist [15±18]. In this way, the TSM predicts a satisfactory explanation of the dynamics of almost all the alkali halides [19±29]. It has also been recognized as a good working model by many workers in the ®eld. We have, therefore, chosen the TSM as the most suitable model and introduced a few corrections (shown in Section 2) in its theory for the satisfactory description of lattice dynamics of NiO. The computations and results are given in Section 3. The discussion and conclusion are presented in Section 4.

0022-3697/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 0022-369 7(01)00088-9

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K.S. Upadhyaya et al. / Journal of Physics and Chemistry of Solids 63 (2002) 127±133

2. Theory The TSM of Verma and Singh [9] seems to suffer from certain errors when it is applied to divalent metal oxides. Firstly, the basic equations of the TSM have been derived by considering the value of ionic charge Z ˆ ^1.0 and treating both the ions as polarizable. However, the solids under consideration possess Z ˆ ^2.0 and the positive ion polarizabilities for some of them are either negligibly small or have not been determined as yet. Secondly, the three-body coupling coef®cients evaluated by Ref. [9] are subject to minor errors due to omission of a point in the summation. We have evaluated the correct values of these coupling coef®cients. In an attempt to achieve the goal, the basic equations of the TSM have been corrected and are presented here in Sections 2.1 and 2.2 for both the ions polarizable model (BIPM) and the negative ion polarizable model (NIPM), respectively. 2.1. Both ions polarizable model (BIPM) The inclusion of the charge parameter Z in the theory of the TSM introduces a minor modi®cation in the long-range interaction matrix, which is equivalent to the replacement of the factor {1 1 12f(a)}0 by   12 Z2 1 1 f …a† Z 0 0

such that the matrix C of the TSM takes the form:   12 f …a† 1V C 0 ˆ CZ 2 1 1 Z 0

…1†

where C 0 is a matrix describing the electrostatic long-range Coulomb interactions between the unit charges as de®ned by Refs. [30,31] and V is the long-range three-body interaction matrix de®ned by Ref. [9]. The equations of motion of the TSM modi®ed in view of our Eq. (1) have been solved in the usual way in the longwavelength limit to obtain the following expressions for the elastic constants.

2aC11 ˆ

…Ze†2 ‰A 2 5:112j 1 9:3204j 0 Š v

…2†

2aC 12 ˆ

…Ze†2 ‰2B 1 0:226j 1 9:3204j 0 Š v

…3†

2aC 44 ˆ

…Ze†2 ‰B 1 2:556jŠ v

…4†

and for the optical vibration frequencies: …mv L2 †qˆ0 ˆ R 00 1

8p …Z 0 e†2 …j 1 6j 0 † 3 vfL

…5†

…mv T2 †qˆ0 ˆ R 00 2

4p …Z 0 e†2 …j† 3 vfT

…6†

where   12 jˆ 11 f …a† ; Z 0 R 00

ˆ R0 2 e

2

d12 d2 1 2 a1 a2

j0 ˆ

Z 0 ˆ Z 1 d1 2 d2 …Ze†2 …A 1 2B† v   8p a1 1 a2 fL ˆ 1 1 …j 1 6j 0 † 3 v R0 ˆ

and fT ˆ 1 2

  4p a1 1 a2 …j 0 † 3 v

0

…7†

…8†

…10† …11†

…12†

The remaining symbols have already been de®ned in the TSM. Due to the inclusion of three-body interactions in the dynamical matrix, the charge Ze has been modi®ed to Zj e. Therefore, the expression for Z given by Z ˆ X 1 Y

Model parameters Values

References

Parameters

Values

C11 (10 11 dyn cm 22) C12 (10 11 dyn cm 22) C44 (10 11 dyn cm 22) vL (THz) vT (THz) a 1 (10 224 cm 3) a 2 (10 224 cm 3) a (10 28 cm)

36.93 5.60 11.50 17.30 11.60 0.05 2.61 2.0885

±a [35] [35] [1] [1] ±a ±a [36,37]

A B f(a)0 …adf =da†0 d1 d2 Y1 Y2

7.0185 2 0.7944 0.0530 2 0.1056 0.0424 0.8082 2 1.5643 2 2.1388

Calculated to obtain reasonable values of the parameters.



…9†

Property

a

a df Z da

!

Table 1 Input data and model parameters for NiO (Z ˆ 2) using BIPM Input data



K.S. Upadhyaya et al. / Journal of Physics and Chemistry of Solids 63 (2002) 127±133

129

Fig. 1. Phonon dispersion curves for NiO.

is rewritten as Z j ˆ X j 1 Y j.Hence, the core and the shell charges should also be modi®ed by a factor j . Modi®ed equations for shell charges can now be expressed as Yk ˆ 2

ja k R0 ; e2 dk

k ˆ 1; 2

This modi®cation was not considered by Ref. [9] in their TSM. Thus, the corrected form of the both ions polarizable model (BIPM) contains only eight parameters: A, B, f(a)0, …adf =da†0 ; d1, d2, Y1 and Y2 which can be determined as follows. The ®rst four parameters are evaluated from the knowledge of the three elastic constants given by Eqs. (2)±(4) and the equilibrium condition B ˆ 21:165j

…13†

together with the ionic charge Ze ˆ ^2.0 e. The last four parameters are determined by the knowledge of electrical polarizability (a 1, a 2) and the zone center optical vibration frequencies (v L, v T) given by Eqs. (5), (6), (8), and (9). 2.2. Negative ion polarizable model (NIPM) In the case of certain systems of solids where the polarizability of the positive ion is either negligible or has not been determined as yet, we have modi®ed our BIPM. This reduces the number of parameters because a 1 ˆ 0 and a 2 ˆ a (a being the molecular polarizability). Also, d1 ˆ 0, d2 ˆ d (d being the total distortion or mechanical polarizability), Y1 ˆ 0 and Y2 ˆ Y. This substitution of the above values in the secular

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K.S. Upadhyaya et al. / Journal of Physics and Chemistry of Solids 63 (2002) 127±133

Fig. 2. Q D ±T curve of NiO (Debye temperature variation). Table 2 Comparison of observed peaks with our CDS peaks and their assignments from two-phonon Raman spectra using both neutron data and the present study (BIPM) for NiO Observed Raman peaks [34]

Assignments

Present study

Neutron study a

CDS peaks

350 480 540 580 700

LA(D) 1 TA(D) LA(X) 1 TA(X) LA(L) 1 TA(L) 2LA(X) TO(X) 1 LA(X) LO(X) 1 LA(X) 2TO(G ) 2LO(D) 2TO(D) 2TO(X) LO(X) 1 TO(X) LO(D) 1 TO(D) 2LO(X) 2LO(D) 2LO(L) 2LO(G )

330 457 531 544 705 746 751 774 800 866 906 938 946 1076 1080 1154

330 456 535 550 705 752 753 774 794 860 907 934 954 1074 1086 1154

333 447 550 560 ± 752 753 773 794 840 907 ± 954 ± ± ±

850 930 ^ 20 1140 ^ 50 1200 a

Calculated by us using neutron data.

equation and its solution in the long-wavelength limit leaves the elastic constants expressions unchanged with slight changes in the values of R 00 ˆ R0 2

e2 d 2 a

Z0 ˆ Z 2 d fL ˆ 1 1

…14† …15†

8p a {j 1 6j 0 } 3 v

…16†

4p a …j† 3 v

…17†

and fT ˆ 1 2

Thus, the corrected negative ions polarizable model (NIPM) contains only six parameters: A, B, f(a)0, …adf =da† 0 ; d and Y. Parameters B, f(a)0 and …adf =da†0 ; are determined by using the equilibrium condition and any one of the three elastic constants given by Eqs. (2)±(4). The remaining parameters A, d and Y are determined from the knowledge of the optical constants (v L, v T and a ). 3. Computations and results The dynamical behaviour of NiO has been studied using both the NIPM and the BIPM. The input data, their relevant references and the model parameters for the BIPM are given in Table 1. Out of six model

K.S. Upadhyaya et al. / Journal of Physics and Chemistry of Solids 63 (2002) 127±133

131

Fig. 3. Combined density of states curve for NiO ( # observed Raman peaks).

parameters of the NIPM, four [A, B, f(a)0 and (adf/da)0] remain the same (Table 1), while two (d ˆ 0.7658 and Y ˆ 24.6012) have been calculated using the molecular polarizability a ˆ 2.656 (estimated from the Clausius Mossotti equation). The computed phonon dispersion curves (PDC) along three symmetry directions [q00] [qq0] and [qqq] have been compared with the experimental data in Fig. 1. Our theoretical (Q D ±T ) curve using the BIPM is given in Fig. 2. The CDS curves have been calculated using the sampling method of Refs. [32,33] and are shown along with observed Raman scattering peaks [34] in Fig. 3. Table 2 shows the comparison of the observed Raman and our CDS peaks along with their

assignments using neutron data as well as the present study (BIPM).

Table 3 Comparison of values of Y1 and Y2 predicted by RSM ®t and the present study Parameters

RSM ®t [1]

Present study

Y1 Y2

1 3.22 2 3.30

2 1.5643 2 2.1388

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K.S. Upadhyaya et al. / Journal of Physics and Chemistry of Solids 63 (2002) 127±133

Table 4 Comparison of measured values of elastic constants with those predicted by RSM ®t and the present study Elastic constants

Experimental values

RSM ®t [1]

Present study

C11 (10 11 dyn cm 22) C12 (10 11 dyn cm 22) C44 (10 11 dyn cm 22)

31.60 5.60 11.50

38.90 12.60 11.90

36.93 5.60 11.50

4. Discussion and conclusion The measured data of Ref. [1] were well reproduced by the RSM ®t in their communication, but some of the model parameters remain physically unrealistic. Slightly better agreements have been reported by Refs. [3±5] by ®tting their model parameters to the neutron data, but only at the cost of their physical properties. An inspection of Fig. 1 clearly indicates that our BIPM explains the phonon dispersion data suf®ciently well, as compared to our NIPM, and all the parameters of our models are realistic. In the absence of measured data on the Debye temperature variation, our computed results (Fig. 2) will certainly help experimental workers to measure and compare their data in the future. A glance at the CDS curve in Fig. 3 shows that our peaks compare fairly well with the corresponding observed peaks. However, the ®ne structures of the spectra have not been adequately interpreted in terms of CDS peaks. This is not unexpected, because of the restrictions imposed by selection rules in the CDS approach [32]. The coarseness of the selection of wave-vectors within the Brillouin zone in the present calculations may also be responsible for this limitation. It is hoped that our BIPM will present a much better description if the Brillouin zone is divided into a ®ner mesh and more sophisticated ways [38,39] of computing the phonon density of states are adopted. However, our CDS peaks agree well with the more pronounced observed Raman scattering peaks (Fig. 3). Overall results of CDS peaks (Fig. 3) derived from the harmonic theory and a crude sampling method show suf®ciently good agreement with the experimental results for NiO. Our computed CDS peaks have been compared in Table 2 with their assignments from critical point analysis [40] for interpreting the ®ne structure of second order Raman spectra using neutron data and the BIPM of the present study. The excellent agreement obtained by us provides the next best test of our model. The basic aim of the detailed study of the two-phonon Raman spectra in the present work is to correlate the neutron and optical experimental results for NiO. It is expected that this study will be useful in deducing the values of individual phonon frequencies of the solid to explain the considerable coupling between the modes of vibration of their ions. Table 3 shows that the shell charge Y1 reported by us is negative, while the value predicted by RSM ®t [1] is positive, which leads to a physically unrealistic situation. Furthermore, a comparative chart of elastic constants

presented in Table 4 shows that the BIPM gives a much better description of elastic constants than those predicted by neutron data ®t [1]. The successful results obtained from our BIPM clearly indicate that NiO is strongly polarizable and possesses simple interatomic interactions. It behaves as an ionic crystal. The presence of magnetic and anharmonic interactions seems to have no effect on its lattice dynamics in the paramagnetic phase. There is good scope for the fresh determination of the polarizability of Ni 21 for a better description of its optical branches in PDC. To sum up, all the investigations carried out by the BIPM for NiO show that our model is quite capable of describing the uni®ed study of lattice dynamics of this solid only when both the ions are taken to be polarizable. A similar conclusion was also drawn by us when we applied our model to wustite [41] in the paramagnetic phase. Acknowledgements The authors are indebted to the principal, KN Govt. PG College, Gyanpur, for providing the necessary facilities and the computer center of Banaras Hindu University, Varanasi, for computational assistance. References [1] W. Reichardt, V. Vagner, W. Kress, J. Phys. C 8 (1975) 3955. [2] K.S. Upadhyaya, R.K. Singh, J. Phys. Chem. Solids 35 (1974) 1175. [3] B.R.K. Gupta, M.P. Verma, J. Phys. Chem. Solids 38 (1977) 929. [4] M.S. Kushwaha, Physica 112B (1982) 232. [5] S. Mohan, Mod. Phys. Lett. B (Singapore) 3 (1989) 115. [6] A.D.B. Woods, W. Cochran, B.N. Brockhouse, Phys. Rev. 119 (1960) 980. [7] V. Nusslein, U. Schroder, Phys. Stat. Sol. 21 (1967) 309. [8] A.N. Basu, S. Sengupta, Phys. Stat. Sol. 29 (1968) 367. [9] M.P. Verma, R.K. Singh, Phys. Stat. Sol. 33 (1969) 769. [10] M.L.J. Sangster, G. Peckham, D.H. Sanderson, J. Phys. C (Sol. State Phys.) 3 (1970) 1026. [11] W. Cochran, C.R.C. Critical Reviews in Solid State Sciences 2 (1971) 1. [12] R.K. Singh, Physics Reports 85 (1982) 259. [13] P.O. Lowdin, Ark. Mat. Astr. Fys. 35A (1947) 30. [14] P.O. Lowdin, Phil. Mag. Suppl. 5 (1954) 1. [15] S.O. Lundqvist, Ark. Fys. 6 (1952) 25.

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