Critical charge dynamics of superconducting ${\text{La}}_{2-x}{\text{Sr}}_x{\text{CuO}}_4$ thin films probed by complex microwave spectroscopy: Anomalous changes of the universality class by hole doping

We study the critical charge dynamics of the superconducting to the normal-state transition for ${\text{La}}_{2-x}{\text{Sr}}_x{\text{CuO}}_4$ (LSCO) thin films with a wide range of the Sr concentration by measuring the frequency-dependent excess parts of the complex microwave conductivity, which is induced by the superconducting fluctuations. We present a dynamic scaling analysis of the complex fluctuation conductivity, which includes the information on the universality class and the dimensionality of the critical charge dynamics as a function of the Sr concentration, the film thickness, and the magnetic field. In our previous study [H. Kitano et al., Phys. Rev. B 73, 092504 (2006)], the two-dimensional $\left(2\text{D}\right)\text{−}XY$ critical dynamics for underdoped LSCO and the three-dimensional $\left(3\text{D}\right)\text{−}XY$ critical dynamics for optimally doped LSCO were reported. In this study, we observed a two-dimensional unknown critical charge dynamics for overdoped thin films from $x=0.17$ to 0.20, which is clearly distinguished from the $2\text{D-}XY$ critical dynamics. Through the systematic measurements by changing the film thickness or by applying small magnetic field, it was confirmed that this unusual behavior, which is referred as 2D-“U” below, was not induced by the finite-size effect but was intrinsic to the overdoped LSCO. Thus, it was found that the critical behavior in the phase diagram of LSCO is classified into the following three types: (i) $2\text{D-}XY$ for underdoped region, (ii) $3\text{D-}XY$ for optimally doped region, and (iii) 2D-“U” for overdoped region. In other words, the dimensionality in the critical charge dynamics is changed twice with hole doping. We discuss possible origins of such anomalous dimensional crossovers with hole doping, including an interpretation based on the possible existence of a hidden quantum critical point near the optimally doped region.

Figure 1

Temperature dependence of the dc resistivity for LSCO thin films with $x=0.07–0.20$.

Figure 2

(a) Frequency dependence of ${\sigma }_1$ for LSCO thin films with $x=0.07–0.20$ at $T=100\text{K}$. (b) Frequency dependence of ${\sigma }_2$ normalized by ${\sigma }_1$ at 0.1 GHz for the same films. For all films, ${\sigma }_1$ is independent of frequency and ${\sigma }_2$ is much smaller than ${\sigma }_1$, suggesting that the charge dynamics in the normal state can be safely regarded as in the Hagen-Rubens limit of the Drude conductivity.

Figure 3

Frequency dependence of (a) ${\sigma }_1$ and (b) ${\sigma }_2$ of the $x=0.16$ film at $T=36.0–37.6\text{K}$ and (c) ${\sigma }_1$ and (d) ${\sigma }_2$ for the $x=0.20$ film at $T=33.2–36.5\text{K}$. All temperatures are just above $T_c$. Both ${\sigma }_1$ and ${\sigma }_2$ showed a remarkable enhancement in the low-frequency limit as temperature approaches $T_c$. Insets show the temperature dependence of dc resistivity. Each circle represents the temperature of each data in the main panels.

Figure 4

Frequency dependence of (a) the phase, ${\varphi }_{\sigma }$, and (b) the magnitude, $\left|\sigma \right|$, of the fluctuation-induced complex conductivity for the $x=0.16$ film and (c) ${\varphi }_{\sigma }$ and (d) $\left|\sigma \right|$ for the $x=0.20$ film.

Figure 5

(Color online) Scaled data of the phase (upper panels) and the magnitude (lower panes) of ${\sigma }_{\text{fl}\left(\omega \right)}$ for the LSCO films with various carrier concentrations. The phase and the magnitude are normalized by $\pi /2$ and the scaling parameter, ${\sigma }_0$, respectively. Difference in colors represents the measured data at different temperatures. The temperature ranges of the scaled data for each film are from $T=32.4$ to $35.0\text{K}\left(x=0.15\right)$, from $T=36.0$ to $37.6\text{K}\left(x=0.16\right)$, from $T=28.2$ to $31.0\text{K}\left(x=0.18\right)$, from $T=27.0$ to $29.0\text{K}\left(x=0.19\right)$, and from $T=33.2$ to $36.5\text{K}\left(x=0.20\right)$, respectively. Solid (dashed) lines are the 2D (3D) Gaussian scaling functions.

Figure 6

(Color online) Temperature dependence of the obtained scaling parameters: ${\omega }_0{\sigma }_0$ (top), ${\omega }_0$ (middle), and ${\sigma }_0$ (bottom). Left (Right) panels are for the optimally doped (the overdoped) LSCO. Error bars along the vertical and horizontal directions were estimated by the methods described in the Appendix and by the ambiguity of $T_c^{\text{scale}}$, respectively. Solid lines in the left panels are calculations given by Eqs. (2, 3) with $\nu =0.67$, $z=2$, and $d=3$, suggesting the $3\text{D-}XY$ critical charge dynamics. Three dashed lines in the right panels are similar calculations with $d=2$ and $\nu z=1.3,1.5,1.7$, respectively.

Figure 7

Plots of the highest (or the lowest) temperatures where one of the possible universality classes was clearly observed as a function of the Sr concentration (hatched area). Difference in symbols represents the difference in the observed critical behavior, which is classified into three groups of $2\text{D-}XY$ (squares), $3\text{D-}XY$ (circles), and 2D-“U” (triangles). Below the dashed line, the Nernst signal was observed (Ref. 49).

Figure 8

(Color online) Scaled data of ${\varphi }_{\sigma }$ for the $x=0.17$ films with different thicknesses. They are plotted in a temperature range from $T=29.0$ to 30.8 K for the $t=115\text{nm}$ film and from $T=35.0$ to 36.6 K for the $t=140\text{nm}$ film. Solid (dashed) lines are the 2D (3D) Gaussian scaling function. Insets: Temperature dependence of dc resistivity of the corresponding films.

Figure 9

(Color online) Scaled data of ${\varphi }_{\sigma }$ for the $x=0.18$ films with (a) 60 nm thick and (b) 240 nm thick. They are plotted in a temperature range from $T=28.2$ to 31.0 K for the $t=60\text{nm}$ films and from $T=35.0$ to 37.0 K for the $t=240\text{nm}$ films. Insets: Temperature dependence of dc resistivity of the corresponding films.

Figure 10

Dashed (solid) thick line represents a critical divergence of $\xi \left(T\right)={\xi }_0{\left(T/T_c-1\right)}^{0.67}$ in the $3\text{D-}XY$ critical charge dynamics with ${\xi }_0=3.5\text{nm}\left(20\text{nm}\right)$. Solid arrows show temperature ranges where the critical behavior was observed for the two LSCO ($x=0.16$ and $x=0.18$, respectively). Thin solid lines show the thicknesses, $t$, of the two films. Note that the $3\text{D-}XY$ critical charge dynamics of the $x=0.16$ film was not cut off by the film thickness $\left(t=140\text{nm}\right)$ in the observed temperature range. This suggests that ${\xi }_0$ is smaller than 3.5 nm. On the other hand, if the 2D critical behavior of the $x=0.18$ film in the observed temperature range was attributed to the $3\text{D-}XY$ critical behavior, which was cut off by the film thickness $\left(t=240\text{nm}\right)$, ${\xi }_0$ must be larger than 20 nm.

Figure 11

(Color online) Scaled data of ${\varphi }_{\sigma }$ (upper panels) and $\left|\sigma \right|$ (lower panes) for the $x=0.07$ film in zero and finite magnetic fields.

Figure 12

(Color online) Temperature dependence of (a) ${\omega }_0$, (b) ${\sigma }_0$, and (c) $\xi \propto 1/\sqrt{{\omega }_0}$ for the $x=0.07$ film. Error bars of ${\omega }_0$, ${\sigma }_0$, and $\xi$ were estimated by using the methods described in the Appendix. Solid lines in each panel are calculations given by Eq. (9) with ${\xi }_0=3\text{nm}$.

Figure 13

(Color online) Scaled data of ${\varphi }_{\sigma }$ (upper panels) and $\left|\sigma \right|$ (lower panes) for the $x=0.16$ film in zero and finite magnetic fields.

Figure 14

(Color online) Temperature dependences of the obtained scaling parameters, (a) ${\omega }_0{\sigma }_0$, (b) ${\omega }_0$, and (c) ${\sigma }_0$, for the $x=0.16$ film in zero and finite magnetic fields. Error bars were estimated by using the methods described in the Appendix. Solid lines are calculations given by Eqs. (2, 3) with $\nu =0.67$, $z=2$, and $d=3$, suggesting the $3\text{D-}XY$ critical charge dynamics.

Figure 15

(Color online) Scaled data of ${\varphi }_{\sigma }$ (upper panels) and $\left|\sigma \right|$ (lower panes) for the $x=0.18$ film in zero and finite magnetic fields.

Figure 16

(Color online) Temperature dependences of the obtained scaling parameters, (a) ${\omega }_0{\sigma }_0$, (b) ${\omega }_0$, and (c) ${\sigma }_0$, for the $x=0.18$ film in zero and finite magnetic fields. Error bars along the vertical and horizontal directions were estimated by the methods described in the Appendix and by the ambiguity of $T_c^{\text{scale}}$, respectively. Solid lines are calculations given by Eqs. (2, 3) with $\nu z=1.5$ and $d=2$.

Figure 17

Plots of the applied magnetic field for the optimally doped LSCO $\left(x=0.16\right)$ and the overdoped LSCO $\left(x=0.18\right)$ as a function of $1-T_c\left(B\right)/T_c\left(0\right)$. Error bars were estimated by the ambiguity of $T_c^{\text{scale}}$. Solid and dashed lines are calculations given by Eq. (10) with $\nu =0.67$ and $\nu =0.9$, respectively. Inset: Plots of the applied magnetic field, $B$, as a function of the critical temperature, $T_c\left(B\right)$, for each film, which was determined through the dynamic scaling analysis at each magnetic field. If vortex dynamics can be neglected, this plot agrees with the temperature dependence of $B_{c2}\left(T\right)$.

Figure 18

(Color online) Scaled data of ${\varphi }_{\sigma }$ (upper panels) and $\left|\sigma \right|$ (lower panels) for the $x=0.17$ film in zero and finite magnetic fields. Solid (dashed) lines are the 2D (3D) Gaussian scaling functions.

Figure 19

(Color online) Temperature dependences of the obtained scaling parameters, (a) ${\omega }_0{\sigma }_0$, (b) ${\omega }_0$, and (c) ${\sigma }_0$, for the $x=0.17$ film in zero and finite magnetic fields. Error bars along the vertical and horizontal directions were estimated by the methods described in the Appendix and by the ambiguity of $T_c^{\text{scale}}$, respectively. Solid (dashed) lines are calculations given by Eqs. (2, 3) with $\nu z=1.5$ and $d=2$ ($\nu =0.67$, $z=2$, and $d=3$).

Figure 20

Schematic drawings of (a) independent vortex-antivortex pair and (b) vortex ring in a layered superconductor.

Figure 21

(Color online) Schematic phase diagram of the 2D quantum antiferromagnet. Lines (a)–(c) represent ${\xi }_Q$ in each region.

Figure 22

Schematic phase diagram of LSCO together with our results about the critical charge dynamics. Solid circles are the phenomenologically well-established QCPs. An open circle near the optimal doping is a hidden QCP assumed to explain our findings. The region between two dashed lines is the quantum critical region. Thick solid line at $T=0$ is a long range ordered region.

Figure 23

(Color online) An example of the breakdown of the dynamic scaling for the ${\varphi }_{\sigma }$ of the thicker film $\left(t=120\text{nm}\right)$ with $x=0.18$. Inset: Temperature dependence of dc resistivity of the film compared to that of the thinner film $\left(t=60\text{nm}\right)$.

Figure 24

(Color online) Temperature dependence of ${\omega }_0$ with $\xi$ diverging at $T_c-\delta$ for several values of $\nu z$ and $\delta$, where $\delta$ represents the width of the smeared $T_c$. A vertical dotted line in each panel represents a temperature, $T_L$, above which the power-law behavior of ${\omega }_0$ was observed in the zero magnetic field measurement for $x=0.18$.

Figure 25

Temperature dependence of $1/\tau$, where $\tau$ is the correlation time. Solid symbols are data of LSCO films $\left(x=0.07,0.16,0.18\right)$ in this work and open symbols are data of NbN films $\left(t=50\text{nm},150\text{nm}\right)$ in Ref. 24. Error bars were estimated through those for ${\omega }_0$, which were estimated by using the methods described in the Appendix. A solid straight line is $1/{\tau }_{\text{GL}}$ given by Eq. (12).

Figure 26

(Color online) (a) The first method (method I) and (b) the second method (method II) to check data collapses and to evaluate their error bars.

Figure 27

(Color online) Comparison among three methods of the scaling procedures which were performed on the data of ${\varphi }_{\sigma }$ (upper panels) and $\mid \sigma \mid$ (lower panels) for $x=0.16$ in zero magnetic field. Here, the plots of scaled data were obtained by eyes (a1 and a2), by method I (b1 and b2), and method II (c1 and c2), respectively.

Figure 28

(Color online) Temperature dependence of the scaling parameters of ${\omega }_0$ (upper panel) and ${\sigma }_0$ (lower panel), which were obtained for $x=0.16$ in zero magnetic field.