Phenomenology of $\widehat{a}$-axis and $\widehat{b}$-axis charge dynamics from microwave spectroscopy of highly ordered $\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.50}$ and $\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.993}$

Extensive measurements of the microwave conductivity of highly pure and oxygen-ordered $\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+y}$ single crystals have been performed as a means of exploring the intrinsic charge dynamics of a $d$-wave superconductor. Broadband and fixed-frequency microwave apparatus together provide a very clear picture of the electrodynamics of the superconducting condensate and its thermally excited nodal quasiparticles. The measurements reveal the existence of very long-lived excitations deep in the superconducting state, as evidenced by sharp cusplike conductivity spectra with widths that fall well within our experimental bandwidth. We present a phenomenological model of the microwave conductivity that captures the physics of energy-dependent quasiparticle dynamics in a $d$-wave superconductor which, in turn, allows us to examine the scattering rate and oscillator strength of the thermally excited quasiparticles as functions of temperature. Our results are in close agreement with the Ferrell-Glover-Tinkham sum rule, giving confidence in both our experiments and the phenomenological model. Separate experiments for currents along the $\widehat{a}$ and $\widehat{b}$ directions of detwinned crystals allow us to isolate the role of the CuO chain layers in $\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+y}$, and a model is presented that incorporates both one-dimensional conduction from the chain electrons and two-dimensional transport associated with the $\mathrm{Cu}{\mathrm{O}}_2$ plane layers.

Figure 1

(Color online) Comparison of surface resistance measurements made on the same sample of $\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.50}$ using the broadband experiment (solid symbols) with those from five microwave resonators (open symbols). The agreement between methods is excellent. The data are plotted as $R_s\left(\omega \right)∕{\omega }^2$ to remove the frequency dependence associated with the superfluid screening.

Figure 2

(Color online) The surface resistance $R_s\left(T\right)$ for ortho-II ordered $\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.50}$ measured with currents in the $\widehat{a}$ (top panel) and $\widehat{b}$ directions (bottom panel).

Figure 3

(Color online) Broadband absorption measurements of $R_s$ for ortho-II ordered $\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.50}$ with screening currents parallel to the crystal $\widehat{a}$ (top panel) and $\widehat{b}$ axes (bottom panel). Plotting the data as $R_s∕f^2$ gives a quick means of estimating the conductivity spectrum from the raw data [see Eq. (2)]. Here we emphasize that the frequency dependence does not follow $1∕f^2$ as it would for a Drude conductivity, but rather the power is closer to $1∕f^{1.5}$ over most of the range.

Figure 4

(Color online) Representative curves of the quasiparticle conductivity ${\sigma }_1(\omega ,T)$ for ortho-II ordered $\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.50}$ measured with currents in the $\widehat{a}$ (top panel) and $\widehat{b}$ directions (bottom panel). The fits shown use the phenomenological model of Eq. (10). The low temperature cusplike shape and high frequency power law ($\sim 1∕f^y$ with $y\sim 1.5$) are features expected for a $d$-wave superconductor in the presence of weak-limit impurity scattering, and the spectrum broadens quickly as other scattering mechanisms are introduced at higher temperatures.

Figure 5

(Color online) Representative curves of the quasiparticle conductivity ${\sigma }_1(\omega ,T)$ for sample No. 1 of ortho-I ordered $\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.993}$ measured with currents in the $\widehat{a}$ (top panel) and $\widehat{b}$ directions (bottom panel). The solid lines are fits to the data with Eqs. (10, 13) for the $\widehat{a}$ and $\widehat{b}$ axes, respectively. At the lowest temperature the cusplike spectrum is evident, however, a more Drude-like spectrum is recovered at higher temperatures, consistent with our earlier findings for this ortho-I ordered $\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.993}$.

Figure 6

(Color online) The low temperature quasiparticle conductivity ${\sigma }_1(\omega ,T)$ for sample No. 2 of ortho-I ordered $\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.993}$ measured using the bolometric method with currents in the $\widehat{a}$ (top panel) and $\widehat{b}$ directions (bottom panel). The spectrum is very similar to that of sample No. 1, although some differences in the fit parameters will be discussed.

Figure 7

(Color online) The exponent $y$ of Eqs. (10, 13) obtained from fitting to broadband measurements of the absorption spectrum for $\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.993}$ and $\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.50}$ samples. The exponent $y$ describes the high frequency decay of the quasiparticle conductivity curves shown in Figs. 4, 5, 6. The dashed lines indicate the asymptotic values of 1.45 (ortho-II) and 1.75 (sample No. 1, ortho-I) subsequently used for fitting to the higher temperature spectra obtained with the fixed-frequency resonators. Sample No. 2 of the ortho-I material shows larger values of $y$, corresponding to broader ${\sigma }_1\left(\omega \right)$ curves, which we attribute to a slightly increased level of oxygen disorder in that sample. Despite this, the value of $y$ is still inconsistent with the Drude value of $y=2$.

Figure 8

(Color online) The temperature evolution of the quasiparticle conductivity ${\sigma }_1\left(T\right)$ for sample No. 1 of overdoped $\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.993}$ measured with currents in the $\widehat{a}$ (top panel) and $\widehat{b}$ directions (bottom panel). This data has been processed using the new absolute values of $\lambda (T\to 0)$ given in Table resulting in a modification of the absolute magnitude of the curves from their previously published (Ref. 15) form.

Figure 9

(Color online) The temperature evolution of the quasiparticle conductivity ${\sigma }_1\left(T\right)$ for ortho-II ordered $\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.50}$ measured with currents in the $\widehat{a}$ direction (top panel) and $\widehat{b}$ direction (bottom panel).

Figure 10

(Color online) The spectral width parameter $1∕\Lambda \left(T\right)$ for both in-plane orientations and both oxygen dopings obtained from modeling the quasiparticle conductivity spectra shown in Figs. 4, 5, 6 with Eqs. (10, 13). The open symbols are from fits to the cavity perturbation data and the solid symbols are from fits to the bolometry data. The upper panel details the low temperature behavior.

Figure 11

(Color online) Normal fluid oscillator strength for fully doped $\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.993}$ associated with the two-dimensional $\mathrm{Cu}{\mathrm{O}}_2$ planar bands. Measurements of the real and imaginary parts of the conductivity allow this to be determined by two methods: from integrating the ${\sigma }_1\left(\omega \right)$ data [see Eq. (10) and Fig. 5], and from the disappearance of oscillator strength in the superfluid response, as determined by measurements of $\Delta \lambda \left(T\right)$ at $1.1\mathrm{GHz}$ [see Eq. (17)]. The close agreement in slopes for the $\widehat{a}$ axis indicates that the conductivity sum rule is obeyed. This is not the case for the $\widehat{b}$ axis where a considerable fraction of the total spectral weight resides in the broad spectral feature in ${\sigma }_1\left(\omega \right)$ associated with the one-dimensional CuO chains. The difference in the $\widehat{b}$-axis case between sample No. 1 and 2 indicates that this parameter is very sensitive to oxygen content.

Figure 12

(Color online) Normal fluid oscillator strength for ortho-II ordered $\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.50}$ associated with the two-dimensional $\mathrm{Cu}{\mathrm{O}}_2$ planar bands. As for the fully doped case, the $\widehat{a}$-axis data closely obey the sum rule and the $\widehat{b}$-axis data are more complicated. Note that only integration of ${\sigma }_1\left(\omega \right)$ gives the correct $T\to 0$ offset since our calculation of $n_n{e}^2∕m^{*}$ from $1∕{\lambda }^2$ assumes there is no residual quasiparticle oscillator strength. In all cases we observe a fraction of the total spectral weight that remains uncondensed (see Table ) similar to observations in other cuprates.

Figure 13

(Color online) Temperature dependence of the broad spectral feature in the $\widehat{b}$-axis conductivity associated with the quasi-one-dimensional CuO chain bands for both ortho-I and ortho-II $\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+y}$. The conductivity component ${\sigma }_1^{1\mathrm{D}}(\omega \to 0,T)$ in Eq. (13) is frequency independent in our phenomenological model since it is known to possess a width much greater than our highest measurement frequency.