In-plane optical spectral weight transfer in optimally doped ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{Ca}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{10}$

We examine the redistribution of the in-plane optical spectral weight in the normal and superconducting state in trilayer ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{Ca}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{10}$ (Bi2223) near optimal doping $(T_c=110\mathrm{K})$ on a single crystal via infrared reflectivity and spectroscopic ellipsometry. We report the temperature dependence of the low-frequency integrated spectral weight $W\left({\Omega }_c\right)$ for different values of the cutoff energy ${\Omega }_c$. Two different model-independent analyses consistently show that for ${\Omega }_c=1\mathrm{eV}$, which is below the charge transfer gap, $W\left({\Omega }_c\right)$ increases below $T_c$, implying the lowering of the kinetic energy of the holes. This is opposite to the BCS scenario, but it follows the same trend observed in the bilayer compound ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_8$ (Bi2212). The size of this effect is larger in Bi2223 than in Bi2212, approximately scaling with the critical temperature. In the normal state, the temperature dependence of $W\left({\Omega }_c\right)$ is close to $T^2$ up to $300\mathrm{K}$.

Figure 1

In-plane reflectivity spectra of optimally doped Bi2223 for selected temperatures.

Figure 2

Bi2223 $ab$-plane dielectric function at selected temperature.

Figure 3

$c$-axis optical spectra of Bi2223 at selected temperatures. Top panel: normal incidence reflectivity, middle panel: ${\sigma }_1\left(\omega \right)$, bottom panel: ${ϵ}_1\left(\omega \right)$.

Figure 4

In-plane optical conductivity of Bi2223 at selected temperatures. The inset displays the low energy part of the spectrum.

Figure 5

Optical constants of Bi2223 at selected photon energies as a function of temperature. Left panel: normal incidence reflectivity at 0.027, 0.14, and $0.56\mathrm{eV}$. Right panel: real and imaginary parts of the complex dielectric function at 0.8, 1.24, and $2.05\mathrm{eV}$. The photon energies are chosen close to the borders and in the middle of the experimental range.

Figure 6

Integrated spectral weight as a function of temperature (plotted vs $T^2$ for the reasons described in the text). The results corresponding to different ranges of integration are shown: (a) from 0 to $0.3\mathrm{eV}$, (b) from 0 to $1\mathrm{eV}$, (c) from 0 to 2.5, (d) from 0.3 to 1, and (e) from 1 to $2.5\mathrm{eV}$.

Figure 7

Spectral weight $W({\Omega }_c,T)$ at cut of frequency of $1.25\mathrm{eV}$ plotted as a function of $T^2$. The error bar is the maximum spread of the values of $W({\Omega }_c,T)$, obtained using three different low energy extrapolations shown in the panels of Fig. 8.

Figure 8

Illustration of the numerical test described in the text: reflectivity curves down to zero frequency for different positions of the low energy peak, from left to right 0, 60, and $90{\mathrm{cm}}^{-1}$.

Figure 9

The value of $(\pi ∕2){\sigma }_2\left(\omega \right)\omega$ plotted vs frequency. The zero energy extrapolation of this value represents the joint spectral weight of the condensate and the low-lying quasiparticle peak.

Figure 10

Difference spectra of the in-plane optical conductivity and of the integrated spectral weight. Top panel: ${\sigma }_1(\omega ,25\mathrm{K})-{\sigma }_1(\omega ,110\mathrm{K})$ (solid line) and ${\sigma }_1(\omega ,110\mathrm{K})-{\sigma }_1(\omega ,280\mathrm{K})$ (dashed-dotted line). Bottom panel: $W(\omega ,25\mathrm{K})-W(\omega ,110\mathrm{K})$ (solid line), $W(\omega ,110\mathrm{K})-W(\omega ,280\mathrm{K})$ (dashed-dotted line) and $W_{\mathrm{SC}}-W_{\mathrm{N}}$, corrected for the normal state temperature dependence as described in the text (dotted line).

Figure 11

Slope-differential spectra of the complex dielectric function, obtained experimentally (open symbols) together with the multi-oscillator fitting curves (solid lines) as described in the text.

Figure 12

The demonstration that ${\Delta }_{s}W\left({\Omega }_c\right)$ is well defined by the available data $({\Omega }_c=1\mathrm{eV})$. The mean squared error ${\chi }^2$ for the total fit of ${\Delta }_{s}R\left(\omega \right)$, ${\Delta }_s{ϵ}_1\left(\omega \right)$, and ${\Delta }_s{\sigma }_1\left(\omega \right)$ (shown in Fig. 11) as a function of ${\Delta }_{s}W({\Omega }_c{)}_{\mathrm{imposed}}$. More description is given in the text.

Figure 13

Top: the image of the $ac$ plane after polishing. Bottom: two geometries of the ellipsometry experiment.

Figure 14

Real and imaginary parts of the sensitivity functions $\partial {ϵ}_{\mathrm{pseudo}}∕\partial {ϵ}_{\mathrm{ab}}$ and $\partial {ϵ}_{\mathrm{pseudo}}∕\partial {ϵ}_c$ defined in the text.

Figure 15

Spectral weight as a function of temperature (for the cutoff of $1\mathrm{eV}$) obtained when a temperature dependent $c$-axis data were used to correct the $ab$-plane pseudodielectric function for the admixture of the $c$-axis component. The temperature resolution in this plot is lower than in Fig. 6 because the $c$-axis temperature dependent data have a resolution of $5\mathrm{K}$.