Rigid ion model of lattice dynamics - A re-evaluation

Solid State Communications, Vol. 9, pp. 185—189, 1971. Pergamon Press.

RIGID ION MODEL OF LATTICE DYNAMICS

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Printed in Great Britain

A RE-EVALUATION

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K.V. Namjoshi and S.S. Mitra Department of Electrical Engineering, University of Rhode Island, Kingston, Rhode Island 02881 and J.F. Vetelino Department of Electrical Engineering, University of Maine, Orono, Maine 04473 (Received 19 October 1970; in revised form 16 November 1970 by E. Bursiein)

A rigid ion model using an effective ionic charge gives reasonable agreement with phonon dispersion of crystals with NaC1, CsC1 and zinc-blende structures. Model parameters are obtained from three elastic constants and two long-wavelength optic phonon frequencies. If the pressure dependence of these quantities is also available, the model may be readily used for the evaluation of mode Griineisen parameters, GrUneisen constant and coefficient of thermal expansion.

THE RIGID ion (R.I.) model was proposed by t in 1940 for the calculation o~lattice Kellermann dynamics of NaCl and has since been abandoned as an unrealistic approximation in place of a numbet of more complex models. The latter include the deformation dipole model~ the shell model3 and the breathing shell model,4 among others.5

ameters which are usually obtained by fitting extensi ye phonon dispersion data, available through

The purpose of the present communication is to point out that the R.I. model, when properly used, viz., with an appropriate effective ionic charge, is a good approximation for the lattice dynamics of crystals to predict phonon dispersion when extensive experimental data on the same are not available,

efficient of thermal expansion, without introducing additional model parameters. A modified rigid ion (M.R.I.) model, only slightly different from the one used by Kellermann,~ particularly in the manner of choosing model parameters will be shown to be a realistic approximation for predicting phonon dispersion of crystals. The M.R.I. model was developed7 out of necessity for the zinc-bleride crystals. Neutron scattering data are available for only a few crystals of zinc-blende structure, viz., GaAs~, GaP9, InSb1~ and the acoustic branches of cubic ZnS.” The purpose was to develop a simple model of lattice dynamics: (i) Which obtains all the model parameters from experimental data other than the phonon dispersion curves it intends

neutron scattering measurements. Such a model is usually unsuitable6 ~or crystals for which no such data are available. Furthermore, these complicated models are not easily adaptable to calculations of other thermodynamical properties, e.g., the co-

The most elaborate lattice dynamical model so far available for ionic, partially ionic and covalent crystals is the shell model proposed by Cochran and its various ramifications.3 However, a successful shell model calculation of lattice dynamics involves a large number of fitting par*

V~orksupported in part by U.S. Army Research Office under Grant No. DA-ARO-D31-124 G 1104. 185

to predict. Such a model is thus particularly useful for crystals for which extensive phonon dispersion

186

RIGID ION MODEL OF LATTICE DYNAMICS

data, measured by inelastic neutron scattering, are not available; (ii) A model which can be readily used to obtain pressure- or volume- derivatives of phonon frequencies. Since we have initiated a systematic experimental determination of pressure derivatives of critical point phonon frequencies of diatornic crystals by infrared and Raman techniques, such a procedure will become increasingly useful. The model utilizes as few reliable experimental all the model parameters werecrystal obtained three elastic constants of a cubic and from the two observables as possible as input data. In particular, Brillouin zone center optic phonon frequencies accurate data for which are generally available from infrared and/or Raman scattering measurements. Subsequently, it has been found that the model is also applicable to crystals of NaC1, CsC1 and fluorite structures. For the NaC1 structure, the M.R.I. model consists of the following parameters: (i) An effective ionic charge Ze, which is less than the formal charge of an ion; (ii) non-central first neighbor force constants, ~ and ~3; (iii) central second neighbor force constant ~ assumed to be equal for anion—anion and cation—cation interactions; and (iv) central third neighbor force constant

T’.

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A RE-EVALUATION

Vol. 9, No.3

S(,~SheI/

~

_______________________________________ 6rç~L

________

--

3

2L o~

-

025

~ f0O~

0

025 0.50

~ (110~

FIG. 1. Phonon dispersion for NaF. (a) Comparison with the experimental data (reference 18) with conventional R.I., M.R.I. and simple shell model calculations. (b) Comparison with eleven parameter deformation dipole model calculation (reference 19) The M.R.I. results are indistinguishable from a nine parameter shell model result (reference 18). Parameters used for M.R.I. model calculation: a = 34.6402, ~ - 2 9029, ~ = 0.1690, r /= — — 0.4315 (all in io~dyne/cm) and Z = 0.781.1

These five parameters are obtained from the following five equations. C.

=

Jza

4~ a

—

a

=

3/a

=

4-~ a

~-

—

—

2~/a

2~/a 4f~

~-

—

2.55604 (ez~/%~(1)

4~ia

-~

4~ a

-~

=

(2a

=

(2a ~ 4~3 8~) ~i

0.11298(ez)2 ~ (2) l.27802(ez)2. ~ (3)

8~-~ — (cz)2 ~ 3

8~).~

(4)

constants. On the other hand, for the zinz-blende 7 that only four parameters were sufficient, viz., (i) an effective type cr~~stals, it was found charge, (ii) two non-central first neighbor force constants, an~(iii) a central second neighbor force constant. Because of the imbalance of nurnbets between the four unknown parameter and five experimental quantities an invariance relation was obtained13 among the opticand mode frequencies and two threelong-wavelength elastic constants, was found to be obeyed well by all zinc-blende

47T —

—~-

(cz)2

\~

~

(5)

In these equations \~,(= 2a~) is the unit cell volume, a the nearest neighbor distance and ~ the reduced mass. For the CsCl structure it was found° that the M.N.l. model gave the best results if the following parameters were chosen: (i) An effective charge; (ii) two non-central first neighbor force constants, and (iii) two non-central second neighbor force

type crystals. Since the long-wavelength optic mooe frequenc]es and the elastic constants enter as input data in the M.R.l. model, it guarantees that the calculated phonon dispersion curves agree at least with (i) 0 optic mode frequencies and (ii) longwavelength acoustic velocities. The conventional rigid ion model, on the other hand, always overestimates the ~ ~ 0 LO_-TO splitting, thus failing to agree with the LST relationl4 known to be obe~ed

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RIGID ION MODEL OF LATTICE DYNAMICS

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A RE-EVALUATION

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15 if one uses e = 1. ent from the Szigeti relation And for most crystals the effective charge calculated MR

from (6) is not much different from the Szigeti efective charge. Relation (6a) is also well suited for the determination of pressure dependence of the effective

(o) NoC

_____________

c 2oL

charge for those crystals for which J~ 0 LO and TO phonon frequencies are known as functions of pressure. 16

r

X

40

-Shell

K

P~el1

r

~

L

Another interesting aspect of the M.R.I. model is that it predicts certain relationships for the

MRI

Brillouin zone boundary phonoii frequencies. For the CsCl and zinc-blende type crystals one finds: ~ ~

(~) 2

LO~

=

0:60!-

2O~

7

0

12

(7)

where LO~and LA,frequencies represent the longitudinal optic and acoustic at the X critical

/

___________________________________________ 02~ 05 075 0 0~ 05 025 0 025 050 — 1fo~

0

point (100) and rn1 ( m 2) and ~:: are the ato:ic r.asses. For the NaCl structure, a similar relation is also obtained but pertaining to the LCP /1

13 2. (a) Phonon dispersion for NaC1. Circles represent experimental data from G. Gaunio, L. Almquist and R. Stedman, Phys. Rev. 178, 1496 (1969). Line represents M.R.I. model calculations. Parameters used are: a = 19.1259, ~3 = —1.6884, = 0.1024, ~ = 0.3276 (all in dyne’cm) and Z = 0.74. (b) Phonon dispersion for MgO. circles represent experimental data from C. Peckham, Proc. R. Soc. (London) 90, 657 (1967). The solid line represents M.R.I. model calculations and the dashed line shell model calculations of G. Peckham. Parameters for M.R.I. model calculation are: a = 78.9404, /3 = —2.8628, ~ = 4.5158, TI = 1.2440 (all in 10~dyne ‘cm) and Z = 1.15.

2 2~

FIG.

LOL LAL

—

nI) (11:2

1 2

(8)

Relationship of the type (7) and (8) are indeed known to be followed by many crystals. 17 Some results of the present model are presented next. In Fig. 1 we compare the dispersion curves for NaF calculated by several models and the experimental data obtained by inelastic neutron scattering.le It may be noticed from Fig. is that the conventional RI. model does not predict the correct LO—TO split-

by all diatomic crystals.

ting. A simple shell model calculation using five parameters did not give good agreement either. The

Using the LST relation and equations (4) and (5), one may obtain the following expression for the effective ionic charge in the M.R.I. model: 2 2 1 2 ~ (6a) z = LO — ~TO ~ e

present M.R.l. model utilizing only five parameters gave excellent agreement with the experimental data and was indistinguishable from nine the results of 18 utilizing parameters, an elaborate shell model which were obtained by fitting the experimental dispersion curves in a least square sense. The parameter; :of the M.R.I. model were obtained from equations (1—5) without any recourse to the phonon

Z

—

:

—

(6b)

where ~ and are low and high frequency dielectric constants. Equation (6b) is not very differ-

dispersion 1(b) our results are compared with data. that ofIna Fig. recent eleven parameter deformation dipole model calculation.19 Again better agreement with experimental data is obtained utilizing

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RIGID ION MODEL OF LATTICE DYNAMICS

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A RE-EVALUATION

Vol. 9, No.3

the present model. In Fig. 2 results for NaCl and MgO are presented.

model, on the other hand, gives approximate dispersion for all directions due to the fact that only Brillouin zone center phonon frequencies and

The M.R.I. model thus seems to be eminently suitable for predicting phonon dispersion where such experimental data do not exist. In cases of crystals for which experimental data on phonon dispersion are available, the suitability of the model can be judged by application of equation (7) or (8), as the case may be. If the experimental data agree reasonably well with these equations,

elastic constants are used as input data. For example, excellent agreement with experiment was 12’13 for the temperature dependence of obtained Debye temperature of ZnS, GaAs and CsBr.

there is a fair chance that the M.R.I. model will be a good approximation to the phonon dispersion. The shell model gives good phonon dispersion in directions where neutron scattering data exist but it does not insure that the calculated frequencies in the other directions be accurate, thus introducting some error in the calculated phonon of states and associated specific heat. density The M.R.I.

An examination of equations (1)—(5) will reveal that if data on the pressure derivatives of C 11’ C 2’ C 44, ~ and were known, the model parameters for the M.R.I. model may be determined as functions of crystal volume, which enables one to calculate lattice dynamics as a function of crystal volume. Thus without introducing any additional assumptions one may also calculate the mode GrOneisen constant and the temperature dependence 2° of thermal expansion, as reported elsewhere.

REFERENCES 1.

KELLER~\1ANNE.W., Phil. Trans. Ro\ . Soc. (London), A238, 513 (1940).

2.

HARDY J.R.,

3.

DICK B.G. and OVERHAUSER A.W., Pius. Rev. 112, 90 (1958); COCHRAN IV., Proc. R. Soc. (London), A253, 260 (1959), WOODS A.D.B., COCHRAN W. and BROCKHOUSE B.N., Phvs. Rev. 119, 980 (1961) COWLEY R.A., Proc. R. 5oc. (London). A268, 121 (1962).

4.

SCHRODER V., Solid State Conuuuiu. 4, 347 (1966).

5.

MUSGRAVE M.J.P. and POPLE J.A,, Proc. R. Soc. (London), A268, 474 (1962).

Phil.

\ia~.4, 1275 (1959); 7, 315 (1962).

6.

KAPLAN H. and SULLIVAN

J.J.,

7.

\‘ETELINO J.F. and MITRA

S.S., Phvs. Rex. 178, 1349 (1969).

Phys. Rev. 130, 120 (1963).

8.

WAUGH J.L.T. and DOLLING G., Phvs. Rd. 132, 2410 (1963); DOLLING G. and WAUGH J.L.T., Lattice D\nonuics p. 19 (edited by \VALLIS R.F.), Pergamon Press, London (1965).

9.

YARNELL J.L., WARREN J.L., WENZEL PG. and DEAN P.J., In the fourth JILl Svrlzposiun: oIl \eutron Scattering, Vol. I, International Atomic Energy Agency, Vienna, 1968.

10.

JOUFFROY

11.

JOUFFROY J., Bull. Soc. France Mineral Crzst. 90, 498 (1967). FELDKAMP L.A., VENKATARAMAN G. and KING J.S., Solid State Cornrnun. 21, 1571 (1969).

12.

VETELINO J.F., MITRA S.S. and NA~\IJOSHIK.V., Phvs. Rev. (in press).

13.

VETELINO J.F. and MITRA S.S., Solid State Coniniun. 7, 1181 (1969).

14.

LYDDANE R.H., SACHS R.G. and TELLER E., Phys. Rex. 59, 673 (1941).

15.

SZIGETI B., Proc. R. Soc. (London), A204, 51(1950).

16.

MITRA S.S., BRAFMAN 0., DANIELS W.B. and CRAWFORD R.K., Phys. Rev. 186, 942 (1969);

17.

MITRA S.S., POSTMUS C. and FERRARO J.R., Ploys. Rev. Lett. 18, 455 (1967). KEYES R.W., I. Chen:. Ph\s., 37, 72 (1962); MITRA S.S., Ph~,s.Rev. 132, 986 (1963); MITRA S.S. and MARSHALL R., J. Chem. Phvs., 41, 3158 (1964). Also see ROWE J.E., CARDONA M. and SHAKLEE K., Solid State Conun;u’:. 7, 441 (1969).

J.,

C. R. hebd. sdanc. ~-Icad. Sci. Pari.s 265, 67(1967); PONS-CORBEAU

J.

and

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RIGID ION MODEL OF LATTICE DYNAMICS

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A RE-EVALUATION

18.

BUYERS W.J.L., Phys. Rev. 153, 923 (1967).

19.

K.~ROA.M. and HARDY J.R., Phys. Rev. 181, 1272 (1969).

20.

VETELINO J.F., NAMJOSHI K.V. and MITRA S.S., J. appi. Ph vs. (in press).

Em Modell von starren lonen mit Verwendung einer effektiven lonenladung gibt befriedigende Ubereinstimmung mit der Dispersion der Phononen in Kristallen der NaCl-, CsCl-, und Zincblende-Struktur. Man erhält die Parameter des Modells aus drei elastischen Konstanten und zwei Frequenzen langwelliger optischer Phononen. Falls die Druckabhangigheit dieser Grössen ebenfalls bekannt ist, kann dieses Modell benOtzt werden zur Berechnung der GrOneisen-Parameter der Moden, der Gruneisen-Konstanten und des Koeffizienten für thermische Expansion.

189