Capacitance measurements on grain boundaries in ${\mathrm{Y}}_{1-x}{\mathrm{Ca}}_x{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{7-\delta }$

The capacitance of $24°\left[001\right]$ tilt calcium doped ${\mathrm{Y}}_{1-x}{\mathrm{Ca}}_x{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{7-\delta }$ grain boundaries has been measured for thin films with $x$ in the range $0.0–0.3$. The capacitance was determined from the hysteresis in the $I\text{−}V$ characteristic. By measuring the capacitance as a function of the voltage across the junctions it was possible to observe the contribution of both parasitic substrate capacitance and heating to the hysteresis. These effects enable the determination of the intrinsic capacitance of the grain boundaries. The effect of thermal noise on the measurement is also assessed, and found to be much less than the observed changes in the capacitance. The capacitance is found to increase as the calcium doping increases: from $0.2{\mathrm{Fm}}^{-2}$ for $x=0.0$ to a maximum of $1.2{\mathrm{Fm}}^{-2}$ for $x=0.3$. The changes in the capacitance per unit area are observed to be inversely proportional to the corresponding changes in the resistance area product.

Figure 1

(a) Current density-voltage characteristics for $3\mu \mathrm{m}$ wide junctions of each of the 3 dopings, measured at $4.2\mathrm{K}$ with the sample immersed in liquid helium. (b) Detail of the hysteresis for the 30% doped $2\mu \mathrm{m}$ wide junction. Ten consecutive measurements are shown to give an indication of the noise level.

Figure 2

Plot of the McCumber parameter, ${\beta }_c$, against the critical current, $I_c$ for the 30% calcium doped $2\mu \mathrm{m}$ junction at two different temperatures. Equation (1) predicts a linear relationship between ${\beta }_c$ and $I_c$ for a constant capacitance. Lines of constant capacitance per unit area, spaced at intervals of $0.1{\mathrm{Fm}}^{-2}$ are also shown.

Figure 3

(a) Diagram illustrating the effect of a large substrate capacitance on the electric field distribution in the vicinity of the grain boundary. The substrate contributes parasitically to the overall capacitance. (b) Schematic temperature and frequency dependence of the dielectric constant of single domain ${\mathrm{SrTiO}}_3$. The equivalent Josephson voltage for a given frequency is shown as an alternative $x$-axis (based on Neville, Hoeneisen, and Mead, Ref. 20).

Figure 4

Measured capacitance per unit area, $C∕A$, shown against the return voltage. At $4.2\mathrm{K}$ the sample is immersed in liquid helium, at $15\mathrm{K}$ it is held in helium vapor. The Josephson frequency is shown as an alternative $x$-axis and is effectively the frequency at which the capacitance is measured. The sharp rise in the capacitance at low frequencies is due to parasitic components from the substrate. At higher frequencies the capacitance is constant and temperature independent.

Figure 5

Fiske resonance dispersion relations for undoped junctions from Tarte et al. (Ref. 15) and for the 30% doped junctions from this study.

Figure 6

(a) Temperature dependence of the zero field capacitance of a number of junctions in the study. The capacitance is determined from the hysteresis in the $I\text{−}V$ curve with the sample in helium vapor, or immersed in liquid helium at $4.2\mathrm{K}$. In most cases the capacitance is independent of temperature to within the error associated with the measurement (approximately 10% if the systematic errors are ignored). (b) Return voltage of these junctions as a function of temperature. With the exception of the $3\mu \mathrm{m}$ 30% calcium doped junction the return voltages are all above the substrate dominated regime at $4.2\mathrm{K}$.

Figure 7

Junction capacitance at $4.2\mathrm{K}$ shown against width of the device for the different samples measured. The straight lines serve as a guide to the eye.

Figure 8

(a) Distribution of critical currents for the 30% doped $2\mu \mathrm{m}$ junction at $4.2\mathrm{K}$. The continuous line shows a fit corresponding to a noise temperature of $54\mathrm{K}$. (b) $\ln (1∕\omega \tau )$ versus $E∕k$ [see Eq. (3)]. The gradient of this graph is used to extract the noise temperature of the measurement and the condition of zero intercept can be used to extract the zero noise critical current. The linear relationship indicates a thermal distribution.

Figure 9

Dependence of (a) the normal resistance area product, (b) the capacitance per unit area, (c) the critical current density, and (d) the critical temperature, on the amount of calcium doping. (a)–(c) were measured with the samples immersed in liquid helium at atmospheric pressure. The different symbols correspond to junctions of different widths. Experimental errors are approximately 15% of the measured value in (a) and (c) and $1\mathrm{K}$ in (d). The errors in (b) are discussed in more detail in Secs. 3, 4.

Figure 10

Capacitance per unit area shown against resistance area product at $4.2\mathrm{K}$ for all the junctions measured in this study. The solid line shows the relation $C∕A\propto 1∕R_{n}A$.