2.5 Measuring the Characteristic Length Scales

In the Meissner state, $\lambda$ can be measured by excluded volume techniques (such as microwave cavity perturbation), inductive methods and far infrared reflectivity. The latter is a surface measurement which can determine the absolute value of $\lambda$.In the vortex state, $\mu$SR, nuclear magnetic resonance (NMR) or small angle neutron scattering can be used to measure the magnetic field inhomogeneity due to the vortex lattice in the bulk of the sample, from which an absolute value of $\lambda$ can be obtained.

In contrast, there are few direct measurements of $\xi$.Estimates of its magnitude can be obtained from the contribution of fluctuations to measured quantities such as the specific heat, susceptibility or conductivity. Scanning tunneling microscopy (STM ) can be used to measure the vortex-core radius r₀ at the sample surface, which provides an estimate of $\xi$ . However, the coherence length is most often determined indirectly from measurements of the upper critical field, H_c2. At this field the vortices begin to overlap and the superconductor undergoes a first order phase transition into the normal state. Since the radius of a normal vortex core is about the size of the coherence length, then at H_c2 there is a direct relationship with $\xi$. In particular, from GL theory  

$$\xi (T) = \sqrt{\frac{\Phi_0}{2 \pi H_{c2} (T)}} \, .$$ (69)

In the high-T_c materials, H_c2 is extremely large (e.g. on the order of 10² T in YBa₂Cu₃O₇ at $T \! = \! 0$) and is therefore difficult to measure accurately. Measurements are generally limited to temperatures near T_c where H_c2 is considerably smaller ($H_{c2} \! \rightarrow \! 0$ as $T \! \rightarrow \! T_c$). However, near T_c thermal fluctuations of the vortex lines can depin or melt the solid 3D vortex-lattice into a vortex liquid phase. Rather than H_c2, what is often measured is the transition of the ordered 3D vortex solid into a vortex fluid phase, in which many of the vortices are free to move independently of each other. The phenomenon is analogous to the way in which thermal vibrations of water molecules cause ice to melt into water. The problem of thermal fluctuations is most serious in short coherence length superconductors with high transition temperatures. This is because fluctuation effects become important when the thermal energy k_B T exceeds the condensation energy $\xi_{a} \xi_{b} \xi_c H_c^2/ 8 \pi \! \propto \! (T \! - \! T_c)^{1/2}$[43]. Here, $\xi_{a} (T) \xi_{b} (T) \xi_c (T)$ is the minimum volume occupied by the fluctuation and H_c is the thermodynamic critical field. In the high-T_c materials, thermal fluctuations are mainly responsible for the large variation in reported values of $\xi$ determined from H_c2 measurements. A more appropriate way to determine $\xi$ is to perform measurements deep in the superconducting state, well away from the strong fluctuation regime. This can be achieved with $\mu$SR, a bulk technique which is described next.