Temperature-modulation analysis of superconductivity-induced transfer of in-plane spectral weight in ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\mathrm{Cu}}_2{\mathrm{O}}_8$

We examine the superconductivity-induced redistribution of optical spectral weight in ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\mathrm{Cu}}_2{\mathrm{O}}_8$ near optimal doping using a detailed Kramers-Kronig consistency analysis of the kink (slope change) at $T_c$ of the temperature-dependent optical spectra, published earlier [H. J. A. Molegraaf et al., Science 295, 2239 (2002)]. We demonstrate that the temperature dependence of the complex dielectric function at high frequencies (above $0.75\mathrm{eV}$) imposes the most stringent limits on the possible changes of the low-frequency integrated spectral weight. The presented calculations provide additional arguments, supporting the previous conclusion about a superconductivity-induced increase of the integrated low-frequency spectral weight below $T_c$. The Ferrell-Glover-Tinkham sum rule is not satisfied well above $2.5\mathrm{eV}$, which indicates that this increase is caused by the transfer of spectral weight from the interband to the intraband region and only partially by the narrowing of the Drude peak.

Figure 1

(Color online) Temperature-dependent reflectivity of Bi2212 close to optimal doping $(T_c=88\mathrm{K})$ for selected frequencies in the infrared range. The dots are the data points, and the blue (red) solid curves are polynomial fits to the temperature dependence below (above) $T_c$, used to produce the kink (slope difference) values. The transition temperature is marked by the vertical dotted line.

Figure 2

(Color online) Temperature-dependent ellipsometrically measured ${ϵ}_1\left(T\right)$ (a) and ${ϵ}_2\left(T\right)$ (b) of Bi2212 close to optimal doping $(T_c=88\mathrm{K})$ for selected frequencies. The dots are the data points, and the blue (red) solid curves are polynomial fits to the temperature dependence below (above) $T_c$, used to produce the kink (slope difference) values. Not all temperature data points are shown. The transition temperature is marked by the vertical dotted line.

Figure 3

(Color online) The spectral dependence of the slope difference ${\Delta }_{s}R\left(\omega \right)$ (a) and ${\Delta }_s{ϵ}_1\left(\omega \right)$ and ${\Delta }_s{ϵ}_2\left(\omega \right)$ (b). Data points show the kink value, derived from the temperature-dependent curves (Figs. 1, 2) as described in Appendix . Only a few data points of ${\Delta }_{s}R\left(\omega \right)$ out of all used in the analysis are shown. The line show different fits, as described in the text. The solid green curve indicates the best fit of ${\Delta }_{s}R\left(\omega \right)$, ${\Delta }_s{ϵ}_1\left(\omega \right)$, and ${\Delta }_s{ϵ}_2\left(\omega \right)$ simultaneously, giving ${\Delta }_{s}W\left({\Omega }_c\right)\approx +770{\Omega }^{-1}{\mathrm{cm}}^{-2}{\mathrm{K}}^{-1}$ [the SC-induced increase of $W\left({\Omega }_c\right)$]. The dashed blue and dotted red curves are the best fits of the same data with an artificial constraint ${\Delta }_{s}W\left({\Omega }_c\right)=0$ and $-770{\Omega }^{-1}{\mathrm{cm}}^{-2}{\mathrm{K}}^{-1}$, respectively. One can see that both constraints are incompatible with the data. The green dash-dotted line is the best fit of ${\Delta }_s{ϵ}_1\left(\omega \right)$ by the formula $-A{\omega }^{-2}+B$. The integrated spectral weight ${\Delta }_{s}W\left(\omega \right)$ (c) for all three fits is shown by the same colors. The inset of panel (a) shows ${\Delta }_{s}R\left(\omega \right)$ calculated from the ellipsometrically measured dielectric function from $0.75\text{to}2.75\mathrm{eV}$ on an expanded vertical scale.

Figure 4

(Color online) The continuation of the model slope-difference curves ${\Delta }_s{ϵ}_1\left(\omega \right)$ (a) and ${\Delta }_s{ϵ}_2\left(\omega \right)$ (b) of Fig. 3b to the midinfrared region, where only the reflectivity was measured. The colors and types of curves correspond to the ones in Fig. 3 (namely, the solid green curve is the best fit of all data, without any assumptions about the integrated spectral weight, while the dashed blue and dotted red curves are the fits with artificial constraints on the spectral weight). Note that the error bars, which are estimated as described in the text, are significantly increasing as the frequency is lowered.

Figure 5

(a) Slope-difference conductivity ${\Delta }_s{\sigma }_1\left(\omega \right)$ of optimally doped Bi2212. (b) The corresponding integrated spectral weight ${\Delta }_{s}W\left(\omega \right)$. The dashed lines show a model which mimics the narrowing of the Drude peak.

Figure 6

The “sensitivity functions” $(\partial R∕\partial {ϵ}_1)\left(\omega \right)$ and $(\partial R∕\partial {ϵ}_2)\left(\omega \right)$ in the near and far infrared, obtained for Bi2212 near optimal doping at $T_c=88\mathrm{K}$ as described in Appendix .