The first- and second-order strain derivatives of electronic dielectric constants in alkali halides

Pergamon

0021~3697(93)EOO22-T

J. Phys. Chem. Solids Vol. 55. No. 3, pp. 237-242. 1994 Copyright Q 1994 El&r Science Ltd Printed in Great Britain. All rights reserved 0022-3697/W $7.00 + 0.00

THE FIRST- AND SECOND-ORDER STRAIN DERIVATIVES OF ELECTRONIC DIELECTRIC CONSTANTS IN ALKALI HALIDES W. KUCHARCZYK~ Department of Physics, University of Natal, P.O. Box 375, Pietermaritzburg 3200, South Africa (Received 2 September

1993; accepted

15 December

1993)

Abstract-An analysis of the first- and second-order strain derivatives of the electronic dielectric constants of alkali halide crystals is performed in terms of a bond-polarizability approach and Penns’ model. The derivatives V(dc, /dV) and V*(d%,/d I”) are related to the second- and third-order nonlinear polarizabilities of crystal bonds. VZ(d2t,/dV2) is considered in connection with the quadratic electro-optic effect. Results obtained suggest small changes in the third-order polarizability of the crystal bonds in different crystals. The assumption of the approximate constancy of the third-order polarizability provides a simple link of nonlinear phenomena with the crystal structure. Keywords: Higher-order volume derivatives of optical frequency dielectric constants, quadratic electrooptic effect, alkali halides.

Raman scattering [8]. In view of the fact that the strain derivatives of the electronic dielectric constant are related to a variety of phenomena their investigation is of great interest. The aims of this work are to calculate the firstand second-order strain derivatives of cm in terms of the Penn model and bond-polarizability approach, as well as to discuss changes in the magnitude of the lattice contribution to the third-order nonlinear electronic susceptibility in different crystals.

1. INTRODUCTION There exist many experimental data related to the first-order strain derivative V(dt,/dV) of the electronic dielectric constant E, in alkali halide crystals. The second-order effect is much smaller and because of the difficulties in measuring it, no experimental values of V2(d%,/dV2) have yet been reported. To date it is only the second-order strain derivative V2(d2E,/dV2) of the static dielectric constant 6r in alkali halides that has been measured [l]. In investigations of interionic forces and potentials a knowledge of the strain derivatives of the electronic dielectric constant cm is useful from the point of view of understanding the relative contributions to the derivatives of c3 of the electronic polarizability and the lattice term arising from the displacement of ions. A fact of more importance, however, is that V(dc,/d I’) and VZ(d2E,/dV2) are due to the same anharmonicities as a number of nonlinear phenomena. It is known (see, e.g. [2-6]) that in the alkali halides the first-order strain derivative can be related to the photoelastic effect and density derivative of the refractive index. One can show that V(dE,/dV) can also be linked to the volume dependence of the effective Szigeti charge [6, 7]. The nonlinearities from which V*(d*c, /dV*) originates are responsible for the quadratic electro-optic effect and second-order

2. THE FIRST-ORDER STRAIN DERIVATIVE

OF L, In the Penn model [9] the optical macroscopic susceptibility of a crystal originates from transitions from the valence to the conduction band or to the exciton state. Accordingly, traditional individual ionic polarizabilities have no well-defined meaning [lo]. However, Weber has shown that in the alkali halides, with the exception of the Li compounds, excitations of cation cores cannot be neglected and also that the optical susceptibility of these crystals should be decomposed into two parts [ 11, 121 x = x’ + x0.

t On leave from the Institute of Physics, Technical University of Lodi, Poland.

(1)

The term 1’ describes that part of the optical frequency susceptibility 1 which is interionic in nature, being due to transitions from the valence to the conducton band. The contribution x” accounts

237

W. KUCHARCZYK

238

for excitations of the cation cores and represents the intraionic part. According to this picture the optical susceptibility of the crystals stems from excitations which are both interionic and intraionic. Weber has assumed that the intraionic part satisfies the Clausius-Mosotti relationship

the ThomasFermi screening wave number, and b is introduced to account for the deficiency of the free electron model [15, 161. It seems, however, that phenomena related to second-neighbour interactions are also implicity included in b [8, 171. The homopolar term E,, depends on the nearest-neighbour distance as E,,=AhR-‘,

where N, is the cation density and cc,its polarizability. The interionic term can be described by Penn’s relation

In eqn (3) Es is the effective energy gap, A is a factor related to the oscillator strength and o+, is the plasma frequency given by

where e and m are the electron charge and mass, respectively, N is the density of valence electrons and Q is the permittivity of free space. In this paper we have employed the expression for A given by Sharma and Auluck [ 131,

A =k

(1 +A2)tan-];-A

(5)

where s x 2.48. Because of the high ionicity of the alkali halide crystals, the use of a somewhat different value of s [I81 does not affect the results significantly. We have calculated the factor A, employing the refractive data of those purely covalent crystals for which d-state cores do not contribute to the optical susceptibility, i.e. diamond and silicon. The use of eqns (3)-(7) and (9) leads, when E,, is given in eV and R in A, to A, = 35.33. Previously, Johnson et al. have computed ~la~zabilities of alkali halides in a localdensity approximation scheme [ 191. The calculations have heen based on two approaches: the completely self-consistent ab initio method and a method of pseudo~tential. Both the methods lead to close values of the cation polarizabilities. In order to decompose the total electronic polarizability into x’ and 2” we have employed a, derived by the pseudopotential method along with the long wavelen~h values of em reported by Lowndes and Martin [20]. The components x’ and x” obtained are listed in Table 1. In Table 1 the values of the ionicity parameter X defined in terms of the Phillips-Van Vechten theory [15] as

where A = E,/4E, with EF the Fermi energy given by f;= E = fr2(3n’N)2’3 F 2m ’ According to the Phillips-Van Vechten dielectric theory [14,15] the effective energy gap can be decomposed into two parts, E:=C=+E;.

(7)

The heterpolar term C is related to the ionic binding. In the Phillips-Van Vechten theory C is given by C = be2($

- ?)exp(F),

(8)

where, r, and r, are the cation and anion radii, respectively, 2, and Z, are the numbers of their valence electrons, R = r, + r, is the bond length, k, is

0g2* 8

(10)

Table 1. Values of the ionicity parameter 1;. and the interionic

x’ and

intraionic x* parts susceptibility

LiF LiCl LiBr LiI NaF NaCl NaBr NaI KF KC1 KBr KI

RbF RbCl RbBr RbI

2.143 2.786 0.640 1.265 1.542 I .962 0.490 0.934 1.145 1.467 0.411 0.837 1.027 1.316

0.033 0.019 0.017 0.014 0.100 0.065 0.058 0.048 0.360 0.236 0.215 0.183 0.519 0.343 0.313 0.264

of the

electronic

0.914 0.902 0.895 0.888 0.950 0.937 0.933 0.928 0.970 0.961 0.957 0.953 0.976 0.967 0.964 0.960

Strain derivatives of electronic dielectric constants

and derived employing the factor A of Sharma and Auluck f 19 are also included. Employing eqns (1x10) one can obtain

v(f$+[

-22 -6+U+sf,)(2fz)

+~~-~~)I -2$‘(x”+3) ]

(11)

with

A z=,tan-it.

cw

Equation (11) was obtained on the assumption that the excitation of the core electrons is not greatly affected by the compression. In theories treating the photoelastic effect as arising from the polarizabilities of inde~ndently polarizable ions, the con~bution of changes in cation polarizabilities is also often neglected (see, e.g. [2]) or shown to be small [S]. Previously, we have used the values of x ‘ and x’ given by Weber [I 11 and different expressions for A [9,21] to calculate by means of the Penn model and the P~llip~Van Vechten theory the photoelastic effect [4,6], the ionic radii as related to refractive data [6,22,23], and the pressure dependence of the Szigeti effective charge [6,7]. We have found that both the coefficients b and db/dR increase with the crystal ionicity. Values of the ionicity parameter obtained using the factor A given by Sharma and Auluck 1131 are slightly different from the previous values [6, 14-161. However, the ionicity parameter is not strongly sensitive to A. Employing the relationship between the parameter b and ionic radii given in 122,231 and using the experimental ionic radii determined for LiF, LiCl, NaCl, KCl, KBr and RbCl from X-ray scattering measurements, as listed in [22,23], we have found the best agreement with experiment when the dependence of b on x is described by b = 4.45fi”.

(13)

Because of the significant spread within the experimental values of V(de,/dV) used to determine the dependence of db/dR on the crystal ionicity, we have employed the averaged values of the photofor elastic coefficients P,, it and p1,22 r~ommend~ Iz = 0.55/rm by the Landolt-Bornstein tables [24,25]. V(dc,/dV) is related to the photoelastic coefficients by

= -f”4hl

f2P,l22).

(14)

239

Applying eqns (11) and (14) for LiF, NaCl, KC1 and KBr, for which the experimental ionic radii are known and the averaged photoelastic coefficients are reported, we have found the best fit with the experimental data using the following dependence db dR = 2.47j-;.4*.

(15)

Theoretical values of V(dc, /d V) calculated from eqn (11) for ii = 0.59 pm are shown in Table 2 where they are compared with experimental data determined at wavelengths 0.55-0.65 ,um. The experimental results were derived by eqn (14) from the photoelastic coefficients pi,,, and pllZZ as well as from the reported values for the pressure dependence of the refractive index employing the following relation

(16) In Table 2 some other theoretical values of V(dr,/dV) are also listed. The second column presents values obtained from (R/c,)(dc,/dR) computed from the first principles within a local density approximation scheme [5], while in the third column the values were calculated on the basis of Ruffa’s theory [26].

3. THE SECOND-ORDER STRAIN DERIVATIVE OF t,

There have been several attempts to calculate Y*(d%, /dV2) for the alkali halide crystals [3,8,26,34]. It has been shown [S] that V2(d2c,/dV2) can be related to the second- and third-order nonlinear polarizabilities of the bonds in these crystals. In the bond-polarizability approach the macroscopic optical su~eptibility is given by [I 1,35-391

(17) where /?L*Tdenotes the longitudinal or transverse component of the axially symmetric bond polarizability tensor, aij is the relevant direction cosine, and the summation is taken over all bonds in the volume V. According to eqn (17) the linear bond polarizability jl is defined in the way which includes local field effects. The lattice contribution to the second- and third-order nonlinear ~la~~bilities of

W. KUCHARCZYK

240

Table 2. Comparison of theoretica

and experimental values of the first-order strain derivative of t,

Experiment

Theory Ref. 5

This work

LiF LiCl LiBr LiI NaF

-0.38 -0.85 -1.11 -1.57 -0.43

-0.30 -1.00 -1.31 -1.94 -0.36

-0.41 -1.23 -1.65 -2.53 -0.49

NaCl NaBr NaI KF KC1 KBr KI

-0.81 -1.01 -1.35 -0.73 -0.93 - 1.09 - 1.36

-0.90 -1.15 -1.61 -0.61 -0.90 -1.08 -0.93

-0.90 - I .03 - 1.85 -0.79 -0.86 -0.99 -1.25

RbF RbCl

-0.91 -1.05

-0.83 -1.03

-0.97 -0.95

RbBr

-1.18

-1.16

- 1.05

Rbi

-1.41

-1.44

-1.26

the bond can be described by the factors f IT and FL,T, respectively,

Ref.

Ref. 26

Crystal

-0.360

25

- 0.420 -0.291 -0.388 -0.824

27 28 29 25

-0.858 - 1.095 -1.421 - 1.389 - I .476 -1.460

24 25

-

33 29 33 29 33

:: 29 32

I .0.50 - 1.066 - 1.285 - I .254 - I .463

defined as follows f 2- cr” + W’) (1+2c) ’

(18) and

F=(FL+2~FT) (I +a)

(19) In terms of the noniinear polar&abilities the firstand the second-order strain derivatives of L, are given by [S]

V $$ =f[X’(f-“3)-X”X”+3)] ( >

(20)

.

In eqns (22) and (23) the factor c = j?*/flt describes the anisotropy of the linear polarizability of the bond. The coefficient F can be evaluated from the quadratic electro-optic coefficients g, ,,, and g,,22 [8,39]. In order to calculate F the quadratic electrooptic coefficient should be decomposed into the primary (true) part Si$j, the photoelastic-electrostrictive term, and the contribution related to the piezo-optic effect and the stress aOPk,due to the electrostatic attraction of eiectrodes

and

q,,op uo,kl gijkl =

v2 $ (

=%[x’(F-4f+g)+X”(X”+3)2],

(21)

gijkl f

pijmn dmnkl +

EE

Y k

(24)

1

)

where the factorsfand F stand for the averaged ionic contribution to the second- and third-order nonlinear ~la~~bilities of the bond, respectively, and are

where Pijmn, d,,,“k.,and qijop are the photoelastic, quadratic electrostrictive, and piezo-optic coefficients, respectively. The primary quadratic electrooptic coefficient should be next divided into the ionic

Strain derivatives of electronic dielectric constants

Table 3. Comparison of values of the second-order strain derivative oft, obtained in this work with values calculated in 26. V2(d2c,/dV2) for LiF has been derived in this paper using experimental values of the quadratic electrooptic coefficientsg,, ,, and g,,= from [40,42,43]

and purely electronic parts

The electronic part of gijjucan be evaluated employing the third-order optical susceptibility xijL,o, which is measured, for example, in the third-harmonic generation, four-wave mixing or nonlinear birefringence experiments. In terms of the bond-polarizability approach the ionic contributions to the quadratic electro-optic coefficients g,, ,1and g, ,rr in the rock-salt structure crystals can be written as [8,39]

-2n4R2c1@‘-

&?iYil=

+‘) 2cj

[2FL+4(l

-c)]Q2

+_LWL-

l>Q,

(26)

[(2Fr - 4)c - 2]Q2,

(27)

and

&%2

=

-zn4~2(1@’

-

+‘) zc)

241

where Q stands for the relative displacement of the ions induced by the applied static or low-frequency electric field [36]. The factor f,

describes the electronic contribution to the secondorder nonlinear polarizability, i.e. a change in the bond polarizability due to the electric field with the lattice being fixed .[8,35]. To date it is only for LiF that the experimental values of the coefficients g,,i, and g,,,, have been determined [40]. For other alkali halides with the exception for LiI the relative quadratic electro-optic coefficient g,,,,-g,,,, is known [41] but this is not sufficient to evaluate both the components FL and FT necessary to obtain the coefficient F as in eqn (22). To obtain V2(d2c, /d V2) in LiF we have employed the primary quadratic electro-optic coefficients g,,,, and g, ,22[42] obtained from experimental coefficients gi,,, and g,,,, measured at 1 = 0.63pm [40] and corrected according to the results given in [43] by taking into account the appropriate sign and values of the quadratic electrostrictive coefficients. The purely electronic parts of g,,,, and gllz2 have been obtained using the experimental values of x,,,,(~) and refractive index x1122 O) derived from the nonlinear [44]. The coefficient c was found in the way proposed by Weber [12,37] employing the photoelastic data from [25]. On the basis of eqns (26) and (27) we have found F = 11.5 and V2(d2c,/dV2) = 2.67.

Crystal

This work

Ref. 26

LiF LiCl LiBr LiI NaF NaCl NaBr NaI KF KC1 KBr KI RbF RbCl RbBr RW

2.67 5.50 6.98 9.43 2.29 4.50 5.55 7.27 2.69 4.12 4.94 6.27 3.02 4.20 4.92 6.09

1.01 2.84 3.99 6.73 1.14 2.07 2.55 4.78 2.14 2.14 2.44 3.03 2.21 2.33 2.68 3.18

One of the goals of this paper is to discuss changes in the magnitude of the lattice contribution to the third-order nonlinear polarizability of bonds in different crystals. The validity of the predictions for F in the alkali halides can be tested by a comparison of our trial results for V2(d%,/dV2) with values calculated on the basis of the classical concept of an individual ionic polarizability. Recently, V2(d2e,/dV2) has been obtained by Sipani and Gupta [26] within the framework of the Sharma and Shanker approach [3] using the Lorentz-Lorenz relation and the theory of Ruffa. In the analysis presented in this paper we have found that the assumption of approximate constancy of the factor F leads to good agreement with V2(d2c,/dV2) calculated by Sipani and Gupta. Values of V2(d2c,/dV2) obtained employing F = 11.5 and eqn (21) are compared with those of [26] in Table 3. Our calculated values listed in Table 3 indicate that the electronic contribution is responsible for nearly 10% of V2(d2.z,/dV2) reported by Fontanella et al. [l]. 4. DISCUSSION The first-order strain derivative of t, which we evaluated theoretically for the alkali halides are in good agreement with experiment, as Table 2 indicates. This serves to support the approach used. Because of the lack of experimental data the second-order derivatives of c, calculated from the quadratic electro-optic coefficients can be related only to theoretical results. A comparison of our values with those obtained employing the classical model of

W. KUCHARCZYK

242

the independently polarizable ions [26] shows that the both sets have the same sign and comparable magnitude. Taking into account the experimental uncertainties made

of the data used and also the simplifications in the

theories,

the

agreement

is very

good.

Trends

in the values shown in the first and second of Table 3 are similar. Our results are consistently larger then those of [26] by a factor of

columns

about 2. Despite the fact that KDP is known for its strongly nonlinear optical properties, it was found recently for KDP that F z 15 [8], which is not significantly different from F = 11.5 obtained in this work for LiF. This finding and the observed agreement

between

the two sets of V2(d2c,/dV2)

in

Table 3 support the assumption of the approximate constancy of the third-order bond polarizabihty in different crystals. The assumption

of the approximate

constancy

of F

with the bond length and polarizabilities of the ions participating in the given bond allows one to estimate V2(d2C, /dV2) for other crystals. From a crystallographic point of view the assumption of small changes in the third-order polarizability of bonds in different

crystals

is very

attractive.

It provides

a

simple link between a crystal structure and macroscopic nonlinear optical phenomena. In the alkali halides the experimental data on g,, ,i-g,,,, allow one to calculate the two components FL and FT so making such estimations more accurate. Then by a generahzation of the approaches previously proposed [39,45], it is possible to predict the quadratic electro-optic coefficients polarizabilities order Raman

and

also

the values

which are responsible scattering.

of the

nonlinear

for the second-

Acknowledgements-The author thanks Prof. for his valuable comments. Research support acknowledged by the University of Natal.

R. E. Raab is gratefully

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