Refractive indices, electronic polarizabilities and dielectric constants of alkali halides

Infrared PI~_w. Vol. 33, No. 5, pp. 389-393, 1992 Printed in Great Britain. All rights reserved

Copyright

OOZO-0891192 $5.00 + 0.00 sc 1992 Pergamon Press Ltd

REFRACTIVE INDICES, ELECTRONIC POLARIZABILITIES AND DIELECTRIC CONSTANTS OF ALKALI HALIDES R. R. REDDY, Department

of Physics, (Received

S. ANJANEYULU

Sri Krishnadevaraya 29 January

and T. V. R. RAO

University,

Anantapur

515 003, India

1992; in revised form 20 March 1992)

Abstract-New relations are proposed for the evaluation of refractive indices, electronic polarizabilities and optical dielectric constants of alkali halides. Refractive indices and optical dielectronic constants for alkali halides have been estimated based on the following relations: EBe’ 4n= 63.8 + 0.03~~ Eg et.& = 68.5 + 3.9~0, xc and t = 2

- 0.5E, + 5.9

where Eg, n, w, , co,x, and c are the energy gap (in eV), refractive index, equilibrium vibrational frequency (in cm-‘), vibrational anharmonicity constant term (in cm-‘) and optical dielectric constant, respectively. Strain derivative values of the electronic dielectric constant in ionic crystals are also evaluated using the energy gap. In all cases the calculated and the literature values are in good agreement with each other.

1.

INTRODUCTION

The evaluation of the refractive indices of a semi-conductor is of considerable importance for applications in integrated optics devices such as switches, filters and modulators etc. where the refractive index of a material is a key parameter for device design. (I) Among the various parameters controlling the refractive index, the lowest energy band gap plays a dominant role. MOSS(~) has pointed out that the two most interesting optical properties of a semi-conductor are the absorption edge & or optical energy gap, and the refractive index. It is natural to develop new relationships between refractive index and energy gap both from the point of view of fundamental interest and also as a technological aid in estimating the refractive index if only the energy gap is known. Moss (2)has given an excellent review on the relations between the refractive index and energy gap of a semi-conductor. Kumar et CZ~.‘~’ and Reddy et ~f.‘~.~’have proposed different relations for the evaluation of the refractive index (or) energy gap. After careful observation of the various relationships, Moss (2)has concluded that any relation between the index and the band energy must result from a close relation between the band edge and resonance energy. In view of the above the authors made an attempt to develop new relations between energy gap, refractive index, equilibrium vibration frequency (CO,) and vibrational anharmonicity constant term (w, x,). Several authors have studied W) different crystalline state properties of the salts using potential energy functions. The total energy of a diatomic molecule has been estimated using w, and CO,x, . Any relation which is proposed for the evaluation of the refractive index or energy gap involving CO, and W, x, is more appropriate because CO, and w,x, give the total energy of the molecule. In the present paper we study the dependence of dielectric constant on the energy gap parameter and the strain derivative values of the electronic dielectric constant in ionic crystals. 389

390

R. R. 2.

REDDY ef u/

THEORY

The refractive index is related to the energy gap by the well known Moss’*’ relation, and is given by .!Ql”= I73

(I)

where E, is the energy gap. The above relation is based on the fundamental principle that in a dielectric medium all energy levels are scaled down by a factor E’ or n4, where L is the dielectric constant. However the relation proposed by MOSS(~) is applicable to only semi-conductors of the zinc blend and diamond structure. In the present study, it is assumed that the following relations are suitable for evaluating the refractive index of alkali halides. E g e’ 4N= 63.8 + 0.030,

(2)

Eg e’.4n= 68.5 + 3.90, X,

(3)

where E,, n, w, and o,x, are the energy gap (in eV), refractive index, equilibrium vibrational frequency (in cm-‘) and vibrational anharmonicity constant (in cm-‘), respectively. The significance of w, and w,x, are explained clearly by Herzberg.“’ The numerical constants in equations (2) and (3) are the characteristic constants of alkali halides. Since w, and w,.x~ give the total energy of the molecule, the authors made an attempt to relate refractive index, energy gap and frequency. The exponential and linear forms involving n and o, or w, .xt have been observed to have almost the same values in the least square sense, so we concluded that these two forms are almost identical within tolerance limits and used these two functions to evaluate n given o, , co,x, and EC. Employing equations (2) and (3), n values are evaluated. These values are used in a LorentzLorenz relation, and the electronic polarizabilities are evaluated. According to the Penn’“’ mode1 (x -

I=

(hw,’) EE

(4)

where A is Planck’s constant divided by 27~.Here up and L, are plasma frequency and electronic dielectric constant respectively. Based on the relation given by Reddy ef aZ.@.‘)i.e. Eg = -9.12n

+ 22.2

(5)

the following relation is derived n* or c = &

- 0.5E, + 5.9

where C, Eg are the optical dielectric constant and energy gap (eV) respectively. It is observed in the work of Sharma et &‘9) that R de --= 6 dR

-(t

f 2>(t - 1) L

(7)

values disagree with the experimental values. (lo)In view of the above, the present authors propose the following relation R d6 -(L + 2)(c - l) + 1 4 --= . * (8) f t: dT In order to obtain R/c x dt/dR

value, 6 values from equation (6) are used in equation (8).

Alkali

Similar equations and Pauling

391

halides

are also proposed for the evaluation of (R/c) (dc /dR) using bond energy”’ (E,) (‘I) difference of the crystal (Ax). The proposed equations are of the

electronegativity

form

(6,+ 2)k - 1)+

R dc t dR

1.4

(9)

Es

and - 1) + 1 4

R dc _ (+, + 2)(+, L dR cEN

(10)

where cs and tEN are as follows n2 or t, =

(11)

-% 83.1 - OSE, + 4.2

and n2 or cEN = (3.2 - 0.3Ax2 + 0.003Ax4).

Equations

(11) and (12) are derived

based on the relations

(12)

given by Reddy

et aZ.‘4,5’

n = 2.049 - 0.11 E,

(13)

Es = 2 + 0.54(Ax2).

(14)

The constant 1.4 in equations (8) (9) and (10) is a correction factor where cEand cEN are derived from equations (13) and (14). The energy gap (E,), bond energy (E,) and Pauling electronegativity difference (Ax) are used for values are equal to 6 described by Sharma et .1.‘9’ the WdUatiOn Of 6, t, and trN . These 3. RESULTS

AND

DISCUSSION

Utilizing equations (2) and (3), refractive indices for various alkali halides are evaluated and presented in Table 1. The average percentage deviations are estimated and presented in the same Table

I. Refractive

indices,

electronic

polarizabiiities

and optical Electronic

Refractive

indices

dielectric

constants

polarizabilities

of alkali

halides

I O-24 cm3

(n)

~

Estimated x values Experiobtained from mental ~ Eq. (3) Ref. (3) Eq. (2)

Kumar ef u/. Ref. (3)

Experimental Ref. (3)

t values Eq. (6)

Ref. (15)

Eq. (2)

Eq. (3)

Kumar et ul. Ref. (3)

LiF LiCl LiBr LiI

1.487 1.543 1.635 1.876

1.497 1.541 1.624 1.880

1.442 1.510 1.572 1.723

I .391 I.662 I.784 I.952

1.122 2.561 3.558 6.927

1.140 2.551 3.508 6.950

1.032 2.428 3.269 6.010

0.887 2.916 4.120 6.145

1.94 2.76 3.18 3.81

1.96 2.78 3.17 3.80

NaF NaCl NaBr NaI

1.476 1.564 1.622 1.809

1.469 1.563 1.621 1.807

1.491 1.568 1.612 1.730

1.336 1.540 1.641 1.774

1.835 3.481 4.48 I 6.979

1.810 3.413 4.417 6.969

1.885 3.497 4.422 6.406

1.148 3.237 4.381 6.406

1.79 2.37 2.69 3.15

1.74 2.34 2.59 2.93

KF KC1 KBr KI

1.415 1.544 1.582 1.736

1.406 1.543 1.582 I .740

I.477 1.572 1.606 1.701

1.361 1.490 I.559 I.677

2.327 4.700 5.718 8.439

2.283 4.692 5.744 8.470

2.625 4.901 5.916 8.133

1.991 4.080 5.224 7.249

1.85 2.22 2.43 2.81

1.85 2.19 2.34 2.62

RbF RbCl RbBr RbI

1.432 1.551 1.575 1.730

1.422 1.544 1.580 1.742

1.495 1.586 1.612 1.708

I.398 I.493 1.553 1.649

3.016 5.461 6.464 9.460

2.955 5.398 6.513 9.573

3.393 5.146 6.795 9.244

2.537 4.626 5.770 7.795

1.96 2.23 2.41 2.72

1.96 2.19 2.34 2.59

CsF CSCI CsBr CSI

I.452 1.564 1.587 1.700

I.439 1.564 1.596 1.716

I.510 1.596 1.650 1.695

1.483 I.610 1.670 1.787

3.948 5.439 6.378 8.785

3.847 5.439 6.458 8.971

4.312 5.693 6.458 8.678

3.601 5.690 6.834 8.859

2.20 2.59 2.79 3.19

2.16 2.62 2.42 2.62

4.1

4.0

5.1

Alkali halides

Average percentage deviation

392

R. R.

REDDY

et al

table. The necessary experimental data (E, , co, and o, x,) required for the present work have been taken from the literature.“,‘2’ The calculated refractive indices are in reasonable agreement with those of Kumar et ~1.“’ as well as experimental values. From these studies it is concluded that there is a definite relation between refractive indices, band gaps, equilibrium vibrational frequency and the vibrational anharmonicity constant. Evaluated refractive indices from equations (2) and (3) are used in the well known Lorentz-Lorenz equation and electronic polarizabilities are estimated. For comparison purposes experimental and literature values are also presented in the same table. To analyse the diatomic alkali halide crystals, equations (2) and (3) prove very useful. Previously, many authors”” have estimated equilibrium vibrational frequency (o,), vibrational anharmonic constant (w, x,), rotational vibrational for the alkali halides. The Penn”’ nearly-free

constant

electron

(a,) and dissociation

model illustrates

energy using potential

the relation

between

energy functions

optical frequency

dielectric

constant, energy gap and plasma frequency. In the present study, the authors propose equation (6) for the evaluation of the optical dielectric constant. Relation (6) is interesting and significant because it relates the band gap and optical dielectric constant. Recently Fernandez and Sarker”4’ evaluated (c, - 1) values. Our evaluated t, , is in good agreement with the values cited by Ashcrof and Mermin(‘5’ and Fernandez and Sarkar.‘14’ Burstein and Smith”@ obtained (R/c)(dc/dR) values for ionic and MgO crystals. These values are not in good agreement with the experimental values. In view of the above, Sharma et al.‘“’ derived an equation taking into account the variation of polarizabilities of ions with strain for the evaluation of (R/c)(dc/dR). Moreover it is important to note that the Burstein and Smith”” relation yields negative values of (R/t)(dc/dR) even for MgO, whereas the experimental data suggest a positive value for this crystal. Different relations are proposed by the authors using the energy gap, bond energy and electronegativity differences [Equations (8) (9) and (lo)] for the evaluation of strain derivation of the electronic dielectric constant. In Table 2, we report the values of (R/c)(dc/dR) estimated from equations (8), (9) and (lo), along with the experimental values for alkali halides and MgO. Utilizing equation (10) the evaluated strain derivative values for all the fluoride molecules listed in Table 2 are positive. Several explanations have been given for the derivation of fluoride from other halides. “‘.‘Q Our equations (8) and (10) show their significance because they yield negative values of (R/c)(dc/dR) for alkali halides [except fluorides with equation (lo)] and a positive value for MgO. It is in good agreement with the observations of Sharma et ~1.‘~)and Patterson

and Subba

Swamy.“”

Table From I%. (8)

From Eq. (9)

From Eq. (10)

LiF LiCl LiBr LiI

-0.51 - 1.64 -2.16 -2.88

-0.54 - 1.04 - 1.27 - 1.57

f0.55 - 1.07 - 1.29 - 1.58

NaF NaCl NaBr Nal

-0.28 - 1.14 - 1.55 -2.12

-0.81 -1.31 -1.54 - 1.71

+0.79 -0.95 - 1.18 - 1.49

KF KC1 KBr KI

-0.38 -0.93 -1.21 - 1.71

-0.93 -1.26 - 1.46 - 1.72

+ 1.057 -0.82 - I .07 - I .39

RbF RbCl RbBr RbI

-0.54 -0.94 -1.18 - 1.58

-0.96 - 1.26 - 1.46 - 1.74

+ 1.05 -0.82 - 1.07 - 1.39

WO

+6.99

Alkali halides

-

+2.05

2. Values

of (R/c) (dc/dR)

Experimental Ref. (10)

Sharma et al. Ref. (9)

Ref. (16)

-0.57

-0.57 -1.11 - 1.54 -2.14

- 1.89 - 3.02 -3.53 -4.27

-0.55 ._

-0.61 -0.91 _ 1.21 - I .96

- 1.59 -2.47 -2.83 -3.35

-0.84 -0.98 _ 1.35

- 1.33 - I .09 - 1.20 - 1.39

- 1.77 2.55 -2.51 -2.90

~

-

- 1.89 -2.26 - 2.49 - 2.80

- I .38 - I .54 - I .57

-0.95

-0.93 - 1.35 - 1.57 - 1.44

+ I .07

1.74 1.30 1.34 1.47

+0.46

-3.39

Ref. (19)

1.22 1.09 1.29 1.47

Alkali 4.

halides

393

CONCLUSIONS

From these studies, it is concluded that the optical dielectric constant, energy gap, bond energy, electronegativity and strain derivatives are interrelated. In crystals where sufficient experimental data is not available, some of these studies are helpful in studying the photoelastic behaviour of crystals. Ackno~~ledgemenr-The

authors

are thankful

to Professor

T. S. Moss for his valuable suggestions

REFERENCES I. 2. 3. 4. 5. 6. 7. 8. 9. 10. 1 I. 12. 13. 14. 15. 16. 17. 18. 19.

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(1979).