2.1.3 Penetration Depth for a d_x²-y²-Wave Superconductor

If there are line or point nodes in the energy gap function $\Delta_{\bf k}$, a power law dependence is expected, where $\lambda (T) \! - \! \lambda (0) \! \propto \! T^n$ ($n \! = \!$ 1, 2, 3 or 4) [11,12]. The presence of the nodes on the Fermi surface allows quasiparticle excitations to occur for an infinitesimal amount of thermal energy. In particular, for the high-T_c superconductors there is strong evidence that the energy gap has d_x²-y² symmetry. In this case the energy gap function is given by

$$\Delta_{\bf \hat{k}} = \Delta_0 (\hat{k}_x^2-\hat{k}_y^2) = \Delta_0 \cos (2 \theta)$$ (27)

where $\Delta_0$ is the maximum value of the energy gap. For this symmetry there are line nodes on the Fermi surface. At low temperatures an approximate analytical expression can be obtained for the magnetic penetration depth [13]

$$\lambda (T) - \lambda (0) \simeq \lambda (0) C \frac{T}{\Delta (0)} \, ,$$ (28)

where $C \! = \! \ln (2)$ for a circular Fermi surface. There are now many published measurements of the penetration depth in high-T_c compounds, which support this linear-T behaviour. Recent measurements in the Meissner state of high quality single crystals of YBa₂Cu₃O$_{7-\delta}$ [14,15,16], Bi₂Sr₂CaCu₂O$_{8+\delta}$ [17,18,19,20] and magnetically aligned powders of crystalline HgBa₂Ca₂Cu₃O$_{8+\delta}$ [21] show a strong linear-T dependence for $\lambda(T) \! - \! \lambda(0)$at low T. Muon spin rotation ($\mu$SR) measurements have determined that $\lambda (T)$also changes linearly as a function of T in the vortex state of YBa₂Cu₃O$_{7-\delta}$ [2,3,5] and La_1.85Sr_0.15CuO₄ [22]. Experiments on the high-T_c materials which show T dependences other than linear are often explained in terms of extrinsic effects. For instance, in a d_x²-y²-wave superconductor, impurity scattering can change the low-temperature behaviour of $\lambda (T)$ from a T to T² dependence [13,23]. This has been verified experimentally by substituting small quantities of the non-magnetic impurity Zn²⁺ for Cu²⁺ in YBa₂Cu₃O$_{7-\delta}$ [24,25]. An exception in the high-T_c family appears to be the electron-doped superconductors. For instance, measurements of $\lambda_{ab}(T)$ in Nd_1.85Ce_0.15CuO₄ single crystals are consistent with s-wave BCS theory and show no evidence of a linear-T term [26,27]. However, the large rare earth moments which are present in the electron-doped high-T_c cuprates may have a large effect on the measured $\lambda_{ab}(T)$. This extrinsic effect may mask the intrinsic nature of the superfluid which is common to the hole-doped high-T_c materials.