Scaling Analysis of Conductivity-Related Quantities of YBCO Using Ac Loss Peak Temperature as Scaling Parameter

Abstract

The ac response of polycrystalline YBCO sample in the frequency range 10–2500 kHz has been measured by using a contactless method. For this purpose, two parameters S(ω,T) and S′(ω,T) have been derived from the measured impedance of the system. The parameters S(ω,T) and S′(ω,T) are related to the real and imaginary parts of ac conductivity, respectively. Both the real and the imaginary parts of ac conductivity show frequency-dependent nature. All the observed features have been interpreted by considering the flux flow losses. Due to the flux penetration, S(ω,T) exhibits a peak below the transition temperature (T _C ), which could be explained on the basis of Anderson’s thermally activated flux creep model. Finally, we have applied a model-independent scaling technique to scale all the measured data and obtained the scaling functions by curve fitting method. The same scaling analysis has been found to be also useful for ac susceptibility data.

Appendix

If Z _s is the impedance of the system in presence of the sample and Z _v is that of absence of the sample, then $Z_{s} - Z_{v} = i\omega [(\varepsilon_{s}^{ *} - \varepsilon_{v}^{ *} )C_{e} - (\mu_{s} - \mu_{v})C_{m}]$, where C _e and C _m are geometrical factors.

Now, in vacuum or in air, the relative quantities ε _v and μ _v can be taken as 1, and the above equation takes the form

$$ Z_{s} - Z_{v} = i\omega \bigl[\bigl(\varepsilon_{s}^{ *} - 1\bigr)C_{e} - (\mu_{s} - 1)C_{m}\bigr]. $$

(3)

Now for a simple practical cylindrical coil, the impedance can be written as

$$\begin{aligned} Z = R + i\omega L = R + i\omega L_{0}\mu, \end{aligned}$$

where the symbols have their usual meaning [37, 38]. If we consider R _v to be the sum of the coil resistance and the extra resistance due to the connecting wire, μ _s and μ _v are core permeability with and without the sample in the vacuum environment, then the impedance of the coil with and without the sample can be written as

$$ Z_{s} = R_{v} + i\omega L_{0}\mu_{s} $$

(4)

and

$$ Z_{v} = R_{v} + i\omega L_{0} \mu_{v} = R_{v} + i\omega L_{0}\quad (\mbox{as}\ \mu_{v} \approx 1). $$

(5)

Now, Z _s −Z _v =iωL ₀(μ _s −1), and, therefore,

$$ (\mu_{s} - 1) = \frac{Z_{s} - Z_{v}}{i\omega L_{0}}. $$

(6)

Now, using relation (6) in relation (3), we have

$$\begin{aligned} &(Z_{s} - Z_{v}) = i\omega \biggl[\bigl( \varepsilon_{s}^{ *} - 1\bigr)C_{e} - \frac{Z_{s} - Z_{v}}{i\omega L_{0}}C_{m}\biggr] \\ &\quad {}\Rightarrow \quad \frac{Z_{s} - Z_{v}}{\omega L_{0}} = \frac{i}{L_{0}}\biggl[\bigl\{ ( \varepsilon_{1} - i\varepsilon_{2}) - 1\bigr\} C_{e} - \frac{Z_{s} - Z_{v}}{i\omega L_{0}}C_{m}\biggr] \\ &\quad {}\Rightarrow \quad \frac{Z_{s} - Z_{v}}{\omega} \biggl\{ 1 + \frac{C_{m}}{L_{0}} \biggr\} = \bigl\{ \varepsilon_{2} + i(\varepsilon_{1} - 1) \bigr\} C_{e}. \end{aligned}$$

(7)

Now using the relation $\varepsilon (\omega ) = 1 + \frac{4\pi i}{\omega} \sigma (\omega )$, we can write

$$\begin{aligned} &\varepsilon_{1} = 1 - \frac{4\pi \sigma_{2}}{\omega} \\ &\quad {} \Rightarrow\quad (\varepsilon_{1} - 1) = - \frac{4\pi \sigma_{2}}{\omega}\quad \mbox{and}\quad \varepsilon_{2} = \frac{4\pi \sigma_{1}}{\omega}. \end{aligned}$$

Using these two relations in Eq. (7), we have

$$\begin{aligned} &\frac{Z_{s} - Z_{v}}{\omega} \biggl\{ 1 + \frac{C_{m}}{L_{0}}\biggr\} = \biggl[ \frac{4\pi \sigma_{1}}{\omega} - i\frac{4\pi \sigma_{2}}{\omega} \biggr]C_{e} \\ &\quad {} \Rightarrow\quad \frac{Z_{s} - Z_{v}}{\omega L_{0}}\biggl\{ 1 + \frac{C_{m}}{L_{0}} \biggr\} \frac{L_{0}}{C_{e}} = \frac{4\pi \sigma_{1}}{\omega} - i\frac{4\pi \sigma_{2}}{\omega} \\ &\quad {} \Rightarrow \quad \frac{Z_{s} - Z_{v}}{L_{0}}Y = 4\pi \sigma_{1} - i4 \pi \sigma_{2} \\ &\phantom{\quad {} \Rightarrow \qquad}\mbox{where } Y = \biggl(1 + \frac{C_{m}}{L_{0}} \biggr)\frac{L_{0}}{C_{e}} \\ &\quad {}\Rightarrow \quad \frac{Z_{s} - Z_{v}}{L_{0}} = \frac{4\pi}{Y}\sigma_{1} - i\frac{4\pi}{ Y}\sigma_{2} . \end{aligned}$$

Now we define two quantities S(ω) and S′(ω) for constructing a complex quantity to equate with the $\frac{Z_{s} - Z_{v}}{L_{0}}$, i.e., $S + iS' = ( \frac{Z{}_{s} - Z_{v}}{L_{0}} )$. Then $S(\omega ) = \frac{4\pi}{Y}\sigma_{1}$ and $S' (\omega ) = - \frac{4\pi}{Y}\sigma_{2}$.