Study of Dielectric Properties of Ionic Crystals

Szigeti relations are not satisfied with experimentally observed quantities and therefore, the concept of effective compressibility and effective ionic charge have been introduced. The introduction of such concept yields the following relations, and, Values of Szigeti effective charge parameter (e*/e) calculated from equation (2.24) are found to be substantially lower than its normal value 1. These range between 0.7 and 0.8 for most of the alkali aldihides studied by Lowndes and Martin [105]. The values of (e/e) are much lower for less ionic crystals [115]. The deviations of (e/e) as well as  from their normal values indicate the failure of (e/e) of classical theory of dielectric behaviour of crystals. The basic assumptions on which the Szigeti relations are based are as follows, (i) The effective field is given by Eeff =E+ 4P/3. (ii) The ions are not distorted. They are considered to be spherical and polarizable. (iii) The short range interactions are considered to be operative only between nearest neighbour and three body interactions are neglected. It has been shown that second neighbour and three body interactions play important role in modifying the first Szigeti relation given by equation (2.24) for the effective charge parameter is concerned with ionic distortions. The distortion dipole moment arising from the deformation of ions is mainly responsible for deviating ionic charges.

The Claussius –Mossotti between dielectric constant and Polarizability is based on the assumption that effective polarization field at ion sites is given by equation (2.1).It has been demonstrated that equation(2.1) does not remain valid when ions overlap appreciably with each other .As it well known that ions overlap, the validity of equation (2.1) is doubtful. The effect of over all of ions has been considered phenomenological by Mott and Gurney [3] and according to them one should divide the total polarization ‗P‘ in to three parts, Where P1and P2 represent the electronic polarization of cations and anions, respectably.Px is the polarization due to relative displacement of ions. The fields due to relative displacement of ions. The fields effective in polarizing the cations, anions and displacing the ions from their original positions are, respectably given by the following equations, where is a parameter representing the extent of overlap between ions. =1 correspond to no overlap ions and leads to normal CM relation. If we define the Polarizabilities1, 2,3 and x corresponding to P1,P2, and Px, then- The static or low frequency dielectric constant 0 in ionic crystal is defined as, After solving equations (2.29),(2.30)and(2.31)we obtained the following expressions, Where The following expression for dielectric constants are obtained corresponding to =0, And

or, For =1,we get the following expression for o and  (Lorentz- Lorenz and CM relations), Equation (2.37) and (2.39) were first derived by Mott and Littleton [111] taking  =0. An attempt has been made by Shanker and Sundraj [158] to obtain the dielectric constants for intermediate values of  between o and 1. The effect of introducing  in the effective polarization fields has also been investigated on the transverse optic mode frequency To. Mott and Gurney have derived the following expression [111]. Where V and  are the volume and reduced mass per ion pairs respectively. Ze is magnitude of ionic charge. Putting  =1,in the equation (2.42) yield, Which is the Szigeti relation .For  =0, the equation (2.42) reduces to, This equation was first derived by Born [77-79] and subsequently used by Lucovsky et al[116] to discuss the nature of localized and non-localized effective charges. The concept of localized charge parameter was introduced by Burstein [81], which has been widely used to discuss the nature of chemical bond and dielectric properties of alkali halide crystals [2,6-15]. In these crystals the oscillator strength of the transverse optic mode phonons is reflected in the difference in the squares of the longitudinal optic (LO) and transverse optic phonon frequencies, or in terms of difference between low frequency dielectric constant o and high frequency dielectric constant . In fact the optic mode frequencies are related to the dielectric constants by Lyddane-sachs-Teller (LST) relationship, The oscillator strength of the transverse oscillator (TO) phonon is commonly described by either of two effective charge parameters e*T the macroscopic or transverse, or e*S the Szigeti effective charge. Of these charges, e*T is independent of the model and is calculated from readily observable quantities. On the other hand e*S is model dependent, in particular, requiring assumptions on the form of effective field. In fact e*T is a measure of the linear electric moment per unit cell and includes contributions from charge localized near the ion sites as well as charge distributed throughout the unit cell. Thus we can write, Wheree*nl is the non-localized charge. The effective localized charge e*l is assigned to be on the ion sites. The magnitudes of e*l is derived from the dipole interaction frequency. To develop a phenomenological model for calculating the localized effective charge parameter, it is considered that there are two contributions to the transverse optic mode frequency To. One of these is a mechanical or spring constant frequency derived from dipole-dipole interaction. The one can write, Where o is the mechanical frequency and DD is the dipole interaction frequency .The localized effective charge parameter e*l is derived from the relation,

The factor (1/3) on the right of equation (2.48) reflects the cubic symmetry. In deriving equation (2.48), it has been assumed that there is no field correction. Although this assumption may be nearly true for covalently coordinated structures, but for highly ionic solids it is necessary to take full account of local field correction[10,11].When the local field correction is taken in to account, equation (2.48) is modified as, The effect of screening of the macroscopic field by the inter band electronic transition can be considered rigorously by writing, Equation (2.50) considers, the screening as well as local field correction (2.48 to 2.50) that calculation of e*l requires the evaluation of 2DDor2o. For calculating the mechanical frequency o one has to adopt interionic potential model, which leads to an expression for o in terms of elastic constants. If one assumes that there are only central nearest short range forces acting on the effective ion cores, then it may be shown that,

Where B* is the reduced bulk modulus, If one considered the three body interaction within the framework of the model developed by Lundqvist[ 12]then the following expression is obtained, This equation (2.54) takes into account the deviations arising from the failure of the Cauchy relation C14 =C44.It has been suggested for Zinc-blend type crystals, the non-central forces as done by Keating [32] and Martin [105], one finds the Cij are described by el*2 Lucovsky et al [40] have obtained, For Zinc-bend type solids el*2 can be calculated with the help of equations (2.47),(2.48),(2.55) and (2.56).

Verma M.P. and Agrawal L.D. (2014). Phys.Rev.B9, 1958. Verma T.S. , Kumar V and Shanker J.;Phys.Stat.Sol.(b) 121, 487(9184). Walter J.P. and Cohen M.L.;Phys.Rev. 183,763 (9166); B4, 1877(9171). Wasastjerna J.A.;Phil.Trans.Roy.Soc. (London)237, 105 (9138). Wintersgil M.,Fontanella J.,Andeen C.and Schuele D.;J.Appl.Phys.50,8259 Wong C and Schuele D. ;J.Phys.Chem.Solids 29,1309(9168). Woods A. D. B., Cochran W. and Blockhouse B.N.; Phys. Rev. 119,980(9160). Yu P.Y., Cardona M and Pollack F.H.;Phys.Rev.B3,340(9171) Phys.Rev. B2,3193(2010)

M.Sc. Physics, Banasthali Vidyapith