Optical and thermodynamic properties of the high-temperature superconductor $\mathrm{Hg}{\mathrm{Ba}}_2\mathrm{Cu}{\mathrm{O}}_{4+\delta }$

In- and out-of-plane optical spectra and specific heat measurements for the single layer cuprate superconductor $\mathrm{Hg}{\mathrm{Ba}}_2\mathrm{Cu}{\mathrm{O}}_{4+\delta }$ (Hg-1201) at optimal doping $(T_c=97\mathrm{K})$ are presented. Both the in-plane and out-of-plane superfluid density agree well with a recently proposed scaling relation ${\rho }_s\propto {\sigma }_{\mathit{dc}}{T}_c$. It is shown that there is a superconductivity induced increase of the in-plane low frequency spectral weight which follows the trend found in underdoped and optimally doped ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\mathrm{Cu}}_2{\mathrm{O}}_{8+\delta }$ (Bi-2212) and optimally doped ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{Ca}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{10+\delta }$ (Bi-2223). We observe an increase of optical spectral weight which corresponds to a change in kinetic energy $\Delta W\approx 0.5\mathrm{meV}∕\mathrm{Cu}$ which is more than enough to explain the condensation energy. The specific heat anomaly is 10 times smaller than in $\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+\delta }$ (YBCO) and 3 times smaller than in Bi-2212. The shape of the anomaly is similar to the one observed in YBCO showing that the superconducting transition is governed by thermal fluctuations.

Figure 1

(Color online) (a) In-plane reflectivity for selected temperatures. The reflectivity above $6000{\mathrm{cm}}^{-1}$ has been calculated from the measured dielectric function. The thin red curve is the reflectivity at room temperature calculated from the pseudodielectric function. The inset shows the far infrared reflectivity. (b) In-plane optical conductivity ${\sigma }_1\left(\omega \right)$ for selected temperatures. The inset shows the low frequency part on an expanded scale. Some sharp structures in the region below $700{\mathrm{cm}}^{-1}$ are due to the remnants of the $c$-axis phonons. The temperatures are the same in all panels and are indicated in the inset in (b).

Figure 2

(Color online) (a) Far infrared $c$-axis reflectivity at selected temperatures. Also shown is the in-plane reflectivity at $20\mathrm{K}$ measured with (red) and without polarizer (black). (b) The $c$-axis optical conductivity ${\sigma }_1\left(\omega \right)$. The temperatures are the same in both panels and are indicated in (b).

Figure 3

(Color online) Dielectric function at selected temperatures corrected for the $c$-axis contribution using the method explained in the text. The pseudodielectric function at room temperature is indicated by the broken thin red curves.

Figure 4

(Color online) Reflectivity plotted versus ${(\omega ∕2\pi c)}^{1∕2}$ and the fitted extrapolations (dashed) to zero frequency.

Figure 5

(Color online) (a) Specific heat data for magnetic fields of 0 T and 14 T. (b) Difference of the specific heat data $\Delta C∕T=[C(B,T)-C(14T,T)]∕T$ for a number of magnetic fields $B$ showing the anomaly at the superconducting transition.

Figure 6

(Color online) The frequency dependent scattering rate and effective mass of Hg-1201. The small peak in $m^{*}(\omega ,T)∕m_b$ at $\approx 620{\mathrm{cm}}^{-1}$ for temperatures above $T_c$ is due to the remnant phonon structure.

Figure 7

Temperature dependence of $1∕\tau (\omega ,T)$ and $m^{*}(\omega ,T)∕m_b$ for selected energies. Both quantities show a clear departure from the normal state trend at $T_c$.

Figure 8

(Color online) $-{\omega }^2{ϵ}_1\left(\omega \right)$ plotted at $20\mathrm{K}$ and $200\mathrm{K}$. Extrapolating the $20\mathrm{K}$ curve to $\omega =0$ agrees with the value ${\omega }_{p,s}=9600\pm 400{\mathrm{cm}}^{-1}$ found from the Drude-Lorentz fit of the spectrum indicated by the square at $\omega =0$.

Figure 9

Screened plasma frequency ${\omega }_p^{*}$ versus $T^2$. The relative error bars on this quantity are comparable to the size of the points. The inset shows the derivative.

Figure 10

Integrated spectral weight for a cut off ${\Omega }_c=1\mathrm{eV}$. The error bar indicates the estimated error due to the extrapolations used.

Figure 11

(Color online) Temperature dependencies of the in-plane reflectivity and dielectric function for selected energies. The dashed lines indicate $T_c$.

Figure 12

(Color online) Spectral dependence of (a) ${\Delta }_{s}R$ (squares), (b) ${\Delta }_s{ϵ}_{1,2}$ (squares). Solid lines in (a) and (b) are the fits of the ${\Delta }_{s}R$ and ${\Delta }_s{ϵ}_{1,2}$ spectra. The inset in (a) shows the excellent match between the directly measured ${\Delta }_{s}R$ (squares) and the ${\Delta }_{s}R$ calculated from ${\Delta }_s{ϵ}_{1,2}$ (circles). (c) ${\Delta }_{s}W$ calculated from the best fit to ${\Delta }_{s}R$ and ${\Delta }_s{ϵ}_{1,2}$ using Eq. (10). The observed spectral weight increase is ${\Delta }_{s}W\left({\Omega }_c\right)=+540{\Omega }^{-1}{\mathrm{cm}}^{-2}{\mathrm{K}}^{-1}$. The dashed line indicates the cutoff frequency used to calculate $W({\Omega }_c,T)$.

Figure 13

(Color online) Comparison between the kinetic energy relative to $20\mathrm{K}$, $\delta W\left(T\right)=W({\Omega }_c,T=20\mathrm{K})-W({\Omega }_c,T)$ on the left-hand side and the internal energy $\Delta U\left(T\right)$ relative to $60\mathrm{K}$ on the right-hand side. The red line in the left-hand panel is $\Delta U\left(T\right)$ plotted on the same scale as $\delta W\left(T\right)$.