M398_18b 223..226

The scaling function S(q/­) is constrained by the physics of the highand low-frequency limits. As q=­ ! `, S must approach i­/q in order for j to assume its superconducting form, equation (1). At low frequencies, S approaches a real constant S1(0) which characterizes the d.c. conductivity of the normal state. By comparing the measured complex conductivity to equation (2), we can extract both the phase stiffness and correlation time at each temperature. To analyse the experimental data in terms of equation (2), we note that the phase angle of the complex conductivity, J [ tan 2 1…j2=j1†, equals the phase angle of S(q/­). Therefore J depends only on the single parameter ­, and is independent of Tu. With the appropriate choice of ­(T), all the measured values of J should collapse to a single curve when plotted as a function of the normalized frequency q/­. Knowing ­(T), Tu is obtained from a collapse of the normalized conductivity magnitude, (~­=kBT0v†jj…q†j=jQ, to jS…q=­†j. Figure 3 shows the collapse of the data to the phase angle and magnitude of S. As anticipated, S approaches a real constant in the limit q=­ ! 0, and approaches i­/q as q=­ ! `. When analysed further, the data reveal a con®rmation of thermal generation of vortices in the normal state. In the KTB picture we expect that the d.c. conductivity will equal kBT/nfD© 2 0, which is the ` ̄ux̄ow' conductivity of nf free vortices with quantized ̄ux ©0, and diffusivity D (ref. 16). Together with equation (2), this implies that ­ is a linear function of nf, that is, ­ ˆ ­0nf avc=£, where avc is the area of a vortex core, £ [ T=Tu is the reduced temperature, and ­0 [ p 2S1…0†D=avc. Moreover, we expect that nf will be a thermally activated function, except for T very close to TKTB. The activation energy is simply CkBT 0 u, where C is a non-universal constant of order unity. It follows that the ̄uctuation frequency depends exponentially on the reciprocal of the reduced temperature, ­ ˆ …­0=£†exp…2 2C=£†. The inset to Fig. 3 is a plot of log(£­) versus 1/£ which shows that the exponential relation is observed over nearly four orders of magnitude. This is direct evidence that vanishing of phase coherence in our samples re ̄ects the dynamics of thermally generated vortices. From the slope and intercept of a straight-line ®t we obtain C ˆ 2:23 and ­0 ˆ 1:14 3 10 14 s 2 . In Fig. 4 we present the behaviour of the bare stiffness and phasecorrelation time obtained from our measurement and modelling of j(q). The main panel contrasts Tu with the dynamical stiffness Tu(q) measured at 150 and 400 GHz. The inset shows t as a function of temperature together with hatching that highlights the region where t , ~=kBT. The parameters displayed in Fig. 4 suggest that while phase correlations indeed persist above Tc, they vanish well below T . The loss of coherence is driven by the decrease of Tu with increasing temperature, which renders the system increasingly defenceless to the proliferation of free vortices. This decrease of Tu is consistent with a phenomenological description of a d-wave superconductor derived from a Mott insulator. In this picture, the phase stiffness of an underdoped copper oxide superconductor is undermined below T by the thermal generation of normal electrons very near the points in momentum space where the superconducting gap vanishes. The pairing which remains in other regions of momentum space appears to contribute little to the overall phase stiffness. Near 95 K, t falls to its minimum detectable value of ~/kBT, which is the electron mean free time. Beyond this point, superconductivity becomes indistinguishable from the ballistic dynamics of normal electrons and the recovery of the incoherent normal state is complete. M