Diamagnetism of YBa${}_2$Cu${}_3$O${}_{6+x}$ crystals above $T_c$: Evidence for Gaussian fluctuations

The magnetization of three high-quality single crystals of YBa${}_2$Cu${}_3$O${}_{6+x}$, from slightly overdoped to heavily underdoped, has been measured using torque magnetometry. Striking effects in the angular dependence of the torque for the two underdoped crystals, a few degrees above the superconducting transition temperature ($T_c$), are described well by the theory of Gaussian superconducting fluctuations using a single adjustable parameter. The data at higher temperatures ($T$) are consistent with a strong cutoff in the fluctuations for $T\gtrsim 1.1T_c$. Numerical estimates suggest that inelastic scattering could be responsible for this cutoff.

Figure 1

Angular dependence of the torque density for the UD57 YBa${}_2$Cu${}_3$O${}_{6.5}$ crystal in 10 T at $T=58.1$, 60.3, 61.5, 66.9, and 72.2 K after correcting for a fixed instrumental offset of 10${}^{\circ }$ and subtracting the gravitational term (Ref. 12). The solid lines show single-parameter fits to the formula for 2D GF derived from Eq. (2) plus ${\chi }_D^N\left(T\right)$ shown in Fig. 2a. Note the $\sin 2\theta$ behavior at higher $T$.

Figure 2

(a) Main: ${\chi }_D\left(T\right)$ for the three crystals; solid lines show fits to ${\chi }_c^{\text{FL}}\left(T\right)+{\chi }_D^N\left(T\right)$ for OD89 and UD57, and dashed lines show ${\chi }_D^N\left(T\right)$. Inset: Symbols show $M$ calculated for various values of $\epsilon$, using Eq. (2), when the anisotropy parameter $r\equiv {[2{\xi }_c\left(0\right)/s]}^2=0$. For $r=0.13$ symbols show $M$ given by the 2D-3D form of Eq. (2), which contains $r$ and an extra integral (Ref. 2). The lines show formulas used (Ref. 23) to represent these values of $M$ when fitting $\tau \left(\theta \right)$ data. (b)–(d) Plots of $1/|{\chi }_c^{\text{FL}}\left(T\right)|$ vs $T$ for the three crystals. GF fits based on Eq. (1) are shown by short dashed lines, without a cutoff, and by solid lines, with a strong cutoff (Ref. 21). Red triangles for UD57 show ${\xi }_{ab}{\left(0\right)}^2/\epsilon$ obtained by fitting $\tau \left(\theta \right)$ to the full 2D GF formula when $M\left(B\right)$ is nonlinear, and converted to $1/|{\chi }_c^{\text{FL}}\left(T\right)|$ using Eq. (1). For UD22 the full GF formula was used for all the points shown in (b).

Figure 3

Magnetic field dependence of the magnetization obtained from the torque data for UD57 at $T=58.1$, 60.3, 61.5, 66.9, and 72.2 K. Solid lines show fits to the 2D GF formula for $M$ plus the same normal state contribution used in Figs. 1, 2a, and 2c.