2.3.2 Nonlinear Effects in a d_x²-y²-Wave Superconductor

For a d_x²-y²-wave superconductor, the presence of nodes on the Fermi surface means that the supercurrent response to a weak applied magnetic field will be nonlinear even at $T \! = \! 0$. This is clearly seen in Fig. 2.4, where, due to the nodes, quasiparticle excitations will result from even a small displacement of the Fermi cylinder. For a given shift, the precise number of quasiparticle excitations will depend on the slope of the energy gap function $\Delta_{\bf \hat{k}}$ at the nodes and the direction of ${\bf v_s}$. The excited quasiparticles located in a narrow wedge at the nodes produce a current density which flows in a direction opposite to that of the superfluid.

 

For the case in which ${\bf v_s}$ is directed along a node, as shown in Fig. 2.4(a), the supercurrent-velocity relation is

$${\bf J_s} = -e n_s^{\prime} v_s \left( 1 - \frac{ \vert {\bf v_s} \vert }{v_0} \right) \, ,$$ (47)

where $v_0 \! = \! (d\vert\Delta(\theta)\vert/d \theta)/v_F^*$, such that $(d\vert\Delta(\theta)\vert/d \theta)$, is the angular slope of the energy gap at the node, and v_F^* is the Fermi velocity at the node. For the case in which ${\bf v_s}$ is directed along an antinode, as shown in Fig. 2.4(b), the supercurrent density is

$${\bf J_s} = -e n_s^{\prime} v_s \left( 1 - \frac{1}{\sqrt{2}} \frac{\vert {\bf v_s} \vert }{v_0} \right) \, .$$ (48)

The additional factor of $1/\sqrt{2}$ is easy to understand by comparing the angular size of the wedges at the nodes in Fig. 2.4(b) to the angular size of the wedge in Fig. 2.4(a). Due to the anisotropy of the nonlinear response, Yip and Sauls [30] proposed that the field dependence of the penetration depth in the Meissner state could be used to resolve the structure of the energy gap in a superconductor. The magnetic penetration depth can be derived using the expressions for ${\bf J_s}$. The result is that $\lambda$ changes linearly with H at low T [30,35,36]  

$$\frac{\lambda(H,T)}{\lambda(0,T)}=1+\beta_2(T) \left[ \frac{H}{H_0 (T)} \right] \, ,$$ (49)

where $H_{0} = 3 c v_0/2 e \lambda$ and $\beta_2 (T)$is a temperature dependent coefficient which remains finite at $T \! = \! 0$ due to the nodes in the gap. The actual value of $\beta_2 (T)$ will of course depend on the direction of ${\bf v_s}$. The definition of $\lambda$ in Eq. (2.50) is the same as that in Eq. (2.47) (i.e. it is related to the initial decay rate of the field).

As the temperature is increased, there is eventually a crossover to a situation in which thermal excitation of quasiparticles also occurs away from the nodes. Below this crossover temperature T^*(H), $\lambda (H,T)$ is linear in H but quadratic in T, whereas above T^*(H), $\lambda (H,T)$ is quadratic in H and linear in T [36]. The first evidence for a linear H-dependence accompanied with a T²-dependence in a high-T_c material, was obtained by Maeda et al. [37] for measurements of the in-plane magnetic penetration depth $\lambda_{ab}$ in Bi₂Sr₂CaCu₂O_y. Similar results have since been reported in YBa₂Cu₃O$_{7-\delta}$ and Tl₂Ba₂CaCu₂O_y [38]. However, the results of these experiments are suspect because of the large demagnetization effect (associated with the shape of the sample) which arises from applying a magnetic field perpendicular to the flat $\hat{a}$-$\hat{b}$ plane. Early measurements of $\lambda (H,T)$ in a single crystal of YBa₂Cu₃O_6.95 found a large H² term [39], but the sample had a reduced T_c indicating there may have been extrinsic effects due to impurities.