## Ac Conductivity

For the frequency and temperature dependent dielectric constants (e) and ac conductivity [(o-ac(w))] measurements, the two polished faces of each of the samples (thickness 0.25-0.30 mm) were gold plated by sputtering and then annealed at about 150 °C for half an hour before conductivity measurement. This was done for better electrical contact with the sample surface. Conductivity measurements were made in the ohmic region as determined from the study of current-voltage (I-V) method.

It is well known that the frequency dependent ac conductivity data of amorphous or powdered semiconductors follow the relation [13, 14]

where A is a constant weakly dependent on temperature and s is the frequency exponent, generally less than unity. The ac conductivity was calculated from the total conductivity o~t(w) measured at frequency w and at a fixed temperature. Both dc conductivity (ffdc) and total conductivity, o~t(w), are measured independently and then ac conductivity [o"ac(w)] is estimated from the relation [13-14]

In the glassy FGNC system, the ac and dc conductivities arise due to completely different processes [81]. In other words, ac conductivity represents the dc conductivity in the limit [81] w ^ 0. Analyses of the ac conductivity [o"ac(w) = o-t(w) — o"dc] data of the semiconducting TMO glasses are generally made in the framework of quantum mechanical tunneling (QMT) [82], correlated barrier hopping (CBH) [12], and overlapping large polaron tunneling (OLPT) [83] models. In the following section, the temperature dependent ac conductivity data and the frequency exponent (s) of the nanoparticle dispersed glasses have been analyzed in terms of the OLPT model which is found to be the most appropriate for these nanocrystal-glass composite systems. Long [82] proposed the polaron tunneling model where the potential wells of two sites overlap thereby reducing the value of polaron hopping energy [84, 85] due to the long-range nature of the dominant Coulomb interaction. The polaron hopping energy has the form [81] WH = WHO(1 — rp/R) where WHO is the polaron hopping between two sites at infinite distance. The ac conductivity for the OLPT model [83] is given by o-ac (c) = (K4/12)e2 (kBT)2

where Rc = (1/2a)[ln(1/ln(cT))] is the optimum hopping length at a frequency c calculated by the quadratic equation + [fiWHc + lnCo)]RC - pWHcr'p = 0 (where RM = 2aRM, r'p = 2arp, and ft = 1/hBT). The frequency exponent (s) of oac(c) in this model is calculated from the relation s = 1 - (4 + 6ftWHorp/R'c)/(1 + ftWHorp/K)2/K (11)

Thus, the OLPT model [Eq. (11)] predicts that the exponent^) [in Eq. (9a)] should be both temperature and frequency dependent.

At high temperatures (above 0D/2), the temperature dependence of both odc and ot(c) were strong and consequently the measured oac (c), at all frequencies, coincided with odc in the high temperature region. Figure 11 represents the plots of log oac of a typical glass nano-composite [50] as a function of inverse temperature and frequency, viz. log c. The solid lines are obtained by a least-square fitting procedure. It is evident that oac (c) obeys the universal relation oac(c) = AcS, suggesting that the loss mechanism should have a distribution of relaxation times. The plots of exponents s (calculated from the slopes of the curves) as a function of temperature are shown in Figure 12 (for two typical FGNC samples with x = 10 and 20 wt%) which indicated that the exponent (s) decreased with increasing temperature and then exhibited a minimum at a temperature around 300 K and subsequently increased. This typical behavior of s suggested that the OLPT model [Eq. (11)] was appropriate for these FGNC systems. In the intermediate temperature range (i.e., below 300 K), the values of s resided around the theoretical curves (solid lines in Fig. 12) for various values of the normalized polaron radius r'p (shown in Table 4). In the high-temperature regime (above 300 K), an increase in s with increasing temperature was observed which was consistent with the behavior of the OLPT model. The best fit to the experimental points has been observed for the values of WHO and r'p as shown in Table 4. Both WHO and s decrease with increasing TiO2 content in the VPTI type FGNCs (i.e., with increase of nano-crystal size and concentration).

Ac conductivity of the FGNC system increased linearly with increasing temperature [i.e., oac(c)aTn with n = 1] over a limited range of temperature below 175 K as predicted by the QMT model [82]. Hence the experimental values of oac could also be fitted with the QMT model that predicted a linear temperature dependence of oac (c) due to weak ac conductivity and the corresponding expression for the oac (c) can be written as

where k is a constant factor and varies slightly between different treatments, and Rc [=(1/2a)ln(1/cTo)] is the characteristic tunneling distance and the corresponding frequency exponent has the form s = 1 + 4/ln(cTo). Therefore, according to the QMT model, the exponent s was

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