Determination of the optical properties of La${}_{2-x}$Ba${}_x$CuO${}_4$ for several dopings, including the anomalous $x=\frac{1}{8}$ phase

The optical properties of single crystals of the high-temperature superconductor La${}_{2-x}$Ba${}_x$CuO${}_4$ have been measured over a wide frequency and temperature range for light polarized in the a-b planes and along the $c$ axis. Three different Ba concentrations have been examined, $x=0.095$ with a critical temperature $T_c=32$ K, $x=0.125$ where the superconductivity is dramatically weakened with $T_c\simeq 2.4$ K, and $x=0.145$ with $T_c\simeq 24$ K. The in-plane behavior of the optical conductivity for these materials at high temperature is described by a Drude-like response with a scattering rate that decreases with temperature. Below $T_c$ in the $x=0.095$ and 0.145 materials there is a clear signature of the formation of a superconducting state in the optical properties allowing the superfluid density (${\rho }_{s0}$) and the penetration depth to be determined. In the anomalous 1/8 phase, some spectral weight shifts from lower to higher frequency ($\gtrsim$300 cm${}^{-1}$) on cooling below the spin-ordering temperature $T_{\mathrm{so}}\simeq 42$ K, associated with the onset of spin-stripe order; we discuss alternative interpretations in terms of a conventional density-wave gap versus the response to pair-density-wave superconductivity. The two dopings for which a superconducting response is observed both fall on the universal scaling line ${\rho }_{s0}/8\simeq 4.4{\sigma }_{\text{dc}}{T}_c$, which is consistent with the observation of strong dissipation within the a-b planes. The optical properties for light polarized along the $c$ axis reveal an insulating character dominated by lattice vibrations, superimposed on a weak electronic background. No Josephson plasma edge is observed in the low-frequency reflectance along the $c$ axis for $x=1/8$; however, sharp plasma edges are observed for $x=0.095$ and 0.145 below $T_c$.

Figure 1

(a) Temperature vs hole doping phase diagram of La${}_{2-x}$Ba${}_x$CuO${}_4$ single crystals (reproduced from Ref. 46). Onset temperatures: $T_c$ for bulk superconductivity (SC), $T_{\mathrm{co}}$ for charge stripe order (CO), $T_{\mathrm{so}}$ for spin stripe order (SO), and $T_{\mathrm{LT}}$ for the low-temperature structural phases LTT and LTLO. The unit cell of La${}_{2-x}$Ba${}_x$CuO${}_4$ is shown in the (b) high-temperature tetragonal (HTT), (c) low-temperature orthorhombic (LTO), and (d) low-temperature tetragonal (LTT) phases.

Figure 2

(a) The temperature dependence of the $ab$-plane (solid line) and $c$-axis (dashed line) resistivities for $x=0.095$ (Ref. 34). In the normal state, ${\rho }_{ab}$ decreases linearly with decreasing temperature, while ${\rho }_c$ is observed to increase until just above $T_c$, below which both decrease dramatically and a weak anomaly is observed in ${\rho }_{ab}$. (b) The $ab$-plane and $c$-axis resistivities for the $x=0.125$ material (see Refs. 28 and 29). Weak anomalies are observed in ${\rho }_{ab}$ and ${\rho }_c$ at $T_{\mathrm{co}}$; ${\rho }_{ab}$ continues to decrease until $T_{\mathrm{so}}$, below which it falls dramatically before once again adopting a linear-temperature dependence. Along the $c$ axis, ${\rho }_c$ continues to increase below $T_{\mathrm{co}}$ until just below $T_{\mathrm{so}}$ where it begins to decrease quickly until $\simeq$25 K, below which it decreases less rapidly. Note different scales for ${\rho }_{ab}$.

Figure 3

A photomicrograph of the cleaved $ab$-plane surface of La${}_{1.875}$Ba${}_{0.125}$CuO${}_4$ revealing a number of striations and cleavage steps; the total area shown is roughly $2\mathrm{mm}\times 2\mathrm{mm}$.

Figure 4

The reflectance of La${}_{1.905}$Ba${}_{0.095}$CuO${}_4$ at 295 K in the far-infrared region for light polarized in the a-b planes. (a) The reflectance from a cleaved surface. There are two prominent antiresonances at $\simeq$460 and 570 cm${}^{-1}$, which are associated with $c$-axis longitudinal optic modes that become infrared-active due to a surface misorientation. This effect can be seen in the $c$-axis loss function (dashed line). (b) The reflectance from the polished surface of the same sample with a measured surface misorientation of $4^{\circ }$; the antiresonances are still prominent. (c) The reflectance from the polished surface with a measured surface misorientation of $\lesssim$$1^{\circ }$; the antiresonances are substantially weaker.

Figure 5

The temperature dependence of the reflectance of La${}_{2-x}$Ba${}_x$CuO${}_4$ for light polarized in the a-b planes over a wide frequency range for (a) a polished sample with $x=0.095$, (b) a cleaved sample for $x=0.125$, and (c) a polished sample with $x=0.145$. Insets: the temperature dependence of the reflectance in the far-infrared region.

Figure 6

The real part of the optical conductivity of La${}_{1.905}$Ba${}_{0.095}$CuO${}_4$ for light polarized in the a-b planes. (a) The infrared region for several temperatures above $T_c=32$ K. The antiresonances observed in the reflectance in Fig. 4c are indicated by asterisks, while the in-plane infrared-active lattice modes are indicated by the arrows. Inset: the Drude-Lorentz fit to the data at 295 K. The Drude component (dashed line) and Lorentz oscillators (dash-dot lines) combine (long-dashed line) to reproduce the data (solid line) quite well. (b) The optical conductivity just above and well below $T_c$ in the infrared region illustrating the transfer of spectral weight into the condensate. Inset: the Drude-Lorentz fit to the data at 35 K.

Figure 7

The real part of the optical conductivity of La${}_{1.855}$Ba${}_{0.145}$CuO${}_4$ for light polarized in the a-b planes measured just above and well below the critical temperature. The asterisks denote structure due to surface misorientation, while the arrows indicate the in-plane infrared-active lattice modes. Inset: the temperature dependence of the optical conductivity in the normal state.

Figure 8

(a) The real part of the optical conductivity La${}_{1.875}$Ba${}_{0.125}$CuO${}_4$ for light polarized in the a-b planes at several temperatures above $T_{\mathrm{co}}$ and $T_{\mathrm{so}}$. Inset: the infrared reflectance for the same temperatures. (b) The conductivity for several temperatures below $T_{\mathrm{co}}$ and $T_{\mathrm{so}}$ in the same region showing the transfer and loss of spectral weight occurring mainly below $T_{\mathrm{so}}$. Inset: the infrared reflectance for the same temperatures contrasting the different responses at low and high frequency. The asterisks denote structure due to surface misorientation, while the arrows indicate the in-plane infrared-active lattice modes.

Figure 9

The temperature dependence of the in-plane frequency-dependent scattering rate in La${}_{2-x}$Ba${}_x$CuO${}_4$ (with the phonons removed) in the far-infrared region for (a) $x=0.095$, (b) the 1/8 phase, and (c) $x=0.145$. (d) Detailed temperature dependence of the scattering rate for the 1/8 phase below 60 K. The values for $1/{\tau }_D$ determined from the Drude fits are indicated by the symbols at the origin. The dashed line indicates $1/\tau \left(\omega \right)=\omega$.

Figure 10

The critical temperature $T_c$ vs the product of the dc resistivity measured just above $T_c$ and the spectral weight of the condensate, ${\rho }_{\mathrm{dc}}{\rho }_{s0}/8$, in the copper-oxygen planes for a variety of electron and hole-doped cuprates, compared with the a-b plane results for La${}_{2-x}$Ba${}_x$CuO${}_4$ for $x=0.095$ and 0.145 (highlighted within the circles). The dashed line corresponds to the general result for the cuprates $T_c\simeq 0.23({\rho }_{\mathrm{dc}}{\rho }_{s0}/8)$.

Figure 11

The temperature dependence of the reflectance of La${}_{2-x}$Ba${}_x$CuO${}_4$ for light polarized along the $c$ axis for (a) $x=0.095$, (b) $x=0.125$, and (c) $x=0.145$. The wave number (photon energy) scale is linear up to 60 cm${}^{-1}$; above this, it is logarithmic.

Figure 12

The temperature dependence of the real part of the optical conductivity of La${}_{2-x}$Ba${}_x$CuO${}_4$ for light polarized along the poorly conducting $c$ axis for (a) $x=0.095$, (b) $x=0.125$, and (c) $x=0.145$.

Figure 13

The temperature dependence of the superfluid density along the $c$ axis normalized to an extrapolated zero-temperature value vs the reduced temperature, $t=T/T_c$, in La${}_{2-x}$Ba${}_x$CuO${}_4$ for $x=0.095$ ($•$) and 0.145 ($\blacksquare$) and optimally-doped La${}_{1.85}$Sr${}_{0.15}$CuO${}_4$ ($◊$) (see Ref. 126). The dashed line is a guide to the eye.

Figure 14

The log-log plot of the spectral weight of the superfluid density $N_c\equiv {\rho }_{s0}/8$ vs ${\sigma }_{\mathrm{dc}}{T}_c$ along the $c$ axis for a variety of electron and hole-doped cuprates compared with La${}_{2-x}$Ba${}_x$CuO${}_4$ for $x=0.095$ and 0.145 (highlighted within the circles). The superscripts next to the symbols for La${}_{2-x}$Sr${}_x$CuO${}_4$ and YBa${}_2$Cu${}_3$O${}_{6+y}$ refer to different chemical dopings.[71] The dashed line corresponds to the general result for the cuprates ${\rho }_{s0}/8\simeq 4.4{\sigma }_{\mathrm{dc}}{T}_c$.

Figure 15

The temperature dependence of the smoothed ratio of the real part of the optical conductivity with the 60 K data. The phonons and antiresonances have been subtracted. The arrow denotes a kink attributed to the formation of a gap on the Fermi arc below $T_{\mathrm{so}}$. Inset: the ratio over a wide frequency range.