Electrical transport measurements in the superconducting state of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mi mathvariant="normal">Bi</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi mathvariant="normal">Sr</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mi mathvariant="normal">Ca</mml:mi><mml:msub><mml:mi mathvariant="normal">Cu</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi mathvariant="normal">O</mml:mi><mml:mrow><mml:mn>8</mml:mn><mml:mo>+</mml:mo><mml:mi>x</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mi mathvariant="normal">Tl</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi mathvariant="normal">Ba</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mi mathvariant="normal">Cu</mml:mi><mml:msub><mml:mi mathvariant="normal">O</mml:mi><mml:mrow><mml:mn>6</mml:mn><mml:mo>+</mml:mo><mml:mi>δ</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>

Precise measurements of the in-plane microwave surface impedance of high-quality single crystals of ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}\mathrm{Ca}{\mathrm{Cu}}_{2}{\mathrm{O}}_{8+x}$ and ${\mathrm{Tl}}_{2}{\mathrm{Ba}}_{2}\mathrm{Cu}{\mathrm{O}}_{6+\ensuremath{\delta}}$ are used to probe the relaxation time of nodal quasiparticles in the $d$-wave superconducting state through a two-fluid analysis of the microwave conductivity. While this analysis requires us to posit a form for the frequency-dependent quasiparticle conductivity, we clearly demonstrate that the extraction of the relaxation rate is quite insensitive to the assumed shape of the quasiparticle spectrum. The robustness of the analysis is rooted in the oscillator-strength sum rule and the fact that we simultaneously measure the real and imaginary parts of the conductivity. In both ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}\mathrm{Ca}{\mathrm{Cu}}_{2}{\mathrm{O}}_{8+x}$ and ${\mathrm{Tl}}_{2}{\mathrm{Ba}}_{2}\mathrm{Cu}{\mathrm{O}}_{6+\ensuremath{\delta}}$ we infer a linear temperature dependence of the transport relaxation rate $1∕\ensuremath{\tau}$ and a small but finite zero-temperature intercept. The linear temperature dependence of $1∕\ensuremath{\tau}$ is in accord with expectations for weak elastic scattering in an unconventional superconductor with line nodes and a small residual density of states. The same analysis reveals an onset of inelastic scattering at higher temperatures similar to that seen in the $\mathrm{Y}{\mathrm{Ba}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{6+x}$ superconductors. Finally, we extrapolate the two-fluid model over a range of frequencies up to five times the measurement frequency, where the extrapolation predicts behavior that is qualitatively similar to terahertz conductivity data on ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}\mathrm{Ca}{\mathrm{Cu}}_{2}{\mathrm{O}}_{8+x}$ thin films. While relaxation rates in ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}\mathrm{Ca}{\mathrm{Cu}}_{2}{\mathrm{O}}_{8+x}$ and ${\mathrm{Tl}}_{2}{\mathrm{Ba}}_{2}\mathrm{Cu}{\mathrm{O}}_{6+\ensuremath{\delta}}$ are substantially higher than in $\mathrm{Y}{\mathrm{Ba}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{6+x}$ there are qualitative similarities between all three materials, and the differences can likely be attributed to varying levels of static disorder. We therefore conclude that a universal picture of quasiparticle scattering in the cuprates is emerging.

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