Fate of quasiparticles in the superconducting state

Quasiparticle properties in the superconducting state are masked by the superfluid and are not directly accessible to infrared spectroscopy. We show how one can use a Kramers–Kronig transformation to separate the quasiparticle from superfluid response and extract intrinsic quasiparticle properties in the superconducting state. We also address the issue of a narrow quasiparticle peak observed in microwave measurements, and demonstrate how it can be combined with infrared measurements to obtain a unified picture of electrodynamic properties of cuprate superconductors.

Figure 1

Extended-Drude analysis of optimally doped Bi2212 with $T_c=92$ K. (a) Real part of memory function $M_1\left(\omega \right)$ obtained from Eq. (4). (b) Imaginary part of memory function $M_2\left(\omega \right)$ from Eq. (4). (c) Quasiparticle effective mass $m^{*}\left(\omega \right)/m_b$ calculated from Eq. (2). The values of plasma frequency ${\omega }_p$ used for each temperature are shown in Fig. 4.

Figure 2

(a) Real part of the optical conductivity ${\sigma }_1\left(\omega \right)$ of a BCS model with $T_c=90$ K. (b) Imaginary part of the optical conductivity ${\sigma }_2\left(\omega \right)$. Thick lines represent the model function, whereas thin lines of the same color represent Kramers-Kronig-corrected ${\sigma }_2^{qp}\left(\omega \right)$. (c) Temperature dependence of plasma frequency ${\omega }_p$, superconducting plasma frequency ${\omega }_s$, and the total plasma frequency $\left({\omega }_p^2+{\omega }_s^2\right){}^{1/2}$.

Figure 3

Real and imaginary parts of the memory function, $M_1\left(\omega \right)$ and $M_2\left(\omega \right)$, for the model shown in Fig. 2. The results are shown before (blue lines) and after (red lines) Kramers-Kronig correction.

Figure 4

(a) Real part of the optical conductivity ${\sigma }_1\left(\omega \right)$ of optimally doped Bi2212 with $T_c=92$ K. Thick lines represent the experimental data, thin dashed lines of the same color represent Drude extrapolations, and the circles represent microwave data at 34.7 GHz [11]. (b) Imaginary part of the optical conductivity ${\sigma }_2\left(\omega \right)$. Thick lines represent the experimental data, thin dashed lines of the same color represent Drude extrapolations, and thin lines represent Kramers-Kronig-corrected ${\sigma }_2^{qp}\left(\omega \right)$. (c) Temperature dependence of plasma frequency ${\omega }_p$, superconducting plasma frequency ${\omega }_s$, and the total plasma frequency ${({\omega }_p^2+{\omega }_s^2)}^{1/2}$. Open squares represent the values of superconducting plasma frequency ${\omega }_s$ obtained without Kramers-Kronig correction.

Figure 5

Real and imaginary parts of the memory function, $M_1\left(\omega \right)$ (top panels) and $M_2\left(\omega \right)$ (middle panels), as well as quasiparticle effective mass $m^{*}/m_b$ (bottom panels) for the Bi2212 data shown in Fig. 4. The results are shown before (blue lines) and after (red lines) Kramers-Kronig correction.

Figure 6

(a) Temperature dependence of scattering rate $1/\tau (\omega =30{\text{cm}}^{-1})$ before and after Kramers-Kronig correction. (b) Temperature dependence of effective mass $m^{*}\left(\omega \right)/m_b$ before and after Kramers-Kronig correction. These values are extracted as explained in the text. The effective mass without Kramers-Kronig correction appears to decrease in the superconducting state. However, with Kramers-Kronig correction the effective mass continues to increase below $T_c$, which indicates that the quasiparticles are more correlated than in the normal state.