Hc2(T) in C4KHg

H_c2(T) data for a T_c = 1.5 K C₄KHg are shown in Figs. 5 and 6. The data were taken with the applied field both parallel (Fig. 5) and perpendicular (Fig. 6) to the c-axis of the graphene planes. Two fits are plotted along with each set of data. With the data are plotted a linear and a quadratic fit. If the hypothesis discussed in the previous section about type I superconductivity for is correct, then the data should be approximately quadratic rather than linear. The residuals were calculated using Eq. 2, only now with the error of the ith data point, sigma_i, estimated to be 10% of the experimentally measured critical field. The linear temperature dependence gives a lower residual than the quadratic fit, but the difference between the two residuals is not statistically significant.

Another check on the fits is to look at the magnitude of T_c obtained from the extrapolation of H_c2 to zero. As expected, the T_c obtained from extrapolation of the linear fits usually agrees well with the T_c value measured in zero-field temperature sweeps. In contrast, extrapolation of the linear fits to H_c2 = 0 consistently gives K for specimens whose T_c was 1.5 K according to zero-field temperature sweeps.[10] On the other hand, extrapolation of the quadratic fits to = 0 always results in T_c about 1.5 K, in agreement with the zero-field sweeps. At this point the functional form of remains unresolved.

In Fig. 7, one of the fits is a simple straight line, while the other is calculated from the Helfand-Werthamer (HW) equation, which describes the temperature dependence of the upper critical field of a typical type II superconductor.[26,47] For the T_c = 1.5 K samples, the residuals for the linear fit are consistently at least a factor of 2 lower than those for the Helfand-Werthamer fit. Thus we conclude that H_c2(T) for the T_c = 1.5 K pink C₄KHg samples exhibits extended linearity. The question of extended linearity cannot be decided for the lower-T_c mixed-phase samples because their data (not shown) extends over a smaller reduced temperature range due to the limitations imposed by the cryogenic apparatus.

The most convincing demonstration of the deviation of the H_c2(T) data above the typical type II superconductor curve is displayed in Fig. 8. By plotting the reduced field, versus reduced temperature t (== T/ T_c), all the H_c2(T) data for 5 different C₄KHg samples can be displayed together. The residual index for the linear fit is 3/4 that of the HW fit, indicative of about a 90% probability that the linear fit describes the data better.[6] Figure 8 also shows that at (the lowest accessible reduced temperature) the data have already reached 0.7, the zero-temperature value of h^* calculated using the Helfand-Werthamer formalism.[26] Therefore, while it would be interesting to measure H_c2(t) to lower reduced temperatures and see larger deviations from the HW theory, the available data convincingly demonstrate a deviation from the HW functional form.

If the critical field is truly linear for both field orientations, then the anisotropy parameter epsilon should be a constant:

1/epsilon = dH_{c2, perp c}/dT/dH_{c2, par c}/dT

If epsilon is temperature-independent, the only temperature-dependent quantity left in Eq. 1 should be H_c2(0°). Therefore plots of H_c2(theta) / H_c2(0)° versus theta should lie directly on top of one another, since all temperature dependence has presumably been removed. A plot of H_c2(theta) / H_c2(0)° is shown in Fig. 9 for a T_c = 1.5 K C₄KHg sample at three different reduced temperatures. As the Figure shows, the curves for t = 0.76 and t = 0.55 do indeed lie on top of one another, implying a nearly temperature-independent anisotropy in this temperature range. However, the curve for t = 0.29 clearly lies above the other two curves near theta = 90°, which suggests a small increase in 1/epsilon at low temperatures. A similar plot of the H_c2(theta) data of Iye and Tanuma[30] on C₈KHg shows that epsilon is also temperature-dependent in the stage 2 KHg-GIC.[9,10] This increase in epsilon could be caused either by a slight upturn from linearity in (positive curvature) or a slight downturn from linearity in . Since is expected to be quadratic for type I behavior, the second alternative seems more consistent with our experience with the H_c2(theta) fits. It should be noted, however, that strong positive curvature of has been observed by Iye and Tanuma[30] in C₈RbHg.