Measurements and analysis of the upper critical field $H_{c2}$ of underdoped and overdoped ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x\mathrm{Cu}{\mathrm{O}}_4$ series of compounds

The upper critical field $H_{c2}$ is one of the many nonconventional properties of high-$T_c$ cuprates. It is possible that the $H_{c2}\left(T\right)$ anomalies are due to the presence of inhomogeneities in the local charge carrier density $\rho$ of the $\mathrm{Cu}{\mathrm{O}}_2$ planes. In order to study this point, we have prepared good quality samples of polycrystalline ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x\mathrm{Cu}{\mathrm{O}}_4$ using the wet-chemical method, which has been demonstrated to produce samples with a good cation distribution. In particular, we have studied the temperature dependence of the upper critical field $H_{c2}\left(T\right)$ through magnetization measurements on two samples equidistant from the optimal doping on the phase diagram and nearly equal critical temperatures, namely, ${\rho }_m=0.08$ (underdoped) and ${\rho }_m=0.25$ (overdoped). The results close to $T_c$ do not follow the usual Ginzburg-Landau theory and are interpreted by a theory which takes into account the influence of the inhomogeneities.

Figure 1

X-ray diffraction spectra at room temperature for both samples used in the experiment. The principal peaks were identified according to Rietveld analysis.

Figure 2

(Color online) The zero-field-cooled magnetic moment at some applied magnetic fields as a function of temperature for the sample with 25% of strontium. The inset shows how the values of $T_c$ were determined for two values of the applied field.

Figure 3

(Color online) Values of the calculated $T_c\left(H\right)$, as shown in the inset for Fig. 2 (for ${\mathrm{La}}_{1.75}{\mathrm{Sr}}_{0.25}\mathrm{Cu}{\mathrm{O}}_4$). The value of $T_c(H_{c2}=0)$ is taken where the curve extrapolates to zero. In this case, we get $T_c(H_{c2}=0)=24.1\mathrm{K}$, a value that will be used further for Fig. 7.

Figure 4

Resistivity as a function of temperature. The inset shows the first derivative of the data and the maximum is taken as the superconductivity transition temperature for this sample at $T_c(25\%\mathrm{Sr})=19.9\mathrm{K}$.

Figure 5

(Color online) Temperature evolution of the local pairing amplitude $\Delta (i,T)$ at each stripe on a square of $14\times 14$, that is, with 196 sites $i$, for different samples. Because of the inhomogeneities, the sites in the left have $\rho \left(i\right)\approx 0$ and the ones in the right have $\rho \left(i\right)\approx 2{\rho }_m$. As the systems are cooled down, more regions become superconducting and they percolate at $T_c\left({\rho }_m\right)$. These percolation threshold temperatures for each compound are indicated in their respective panels. The panel with ${\rho }_m=0.22$ does not have the stripe structure, because it has a random distribution of doping values, which are sorted following a Gaussian distribution.

Figure 6

(Color online) To explain the concept of local superconducting temperature and especially how certain regions develop a nonvanishing pairing amplitude, we plot the $\Delta (i,T)$ or $\Delta (l,T)$ for each site $i$ of our two dimensional $14\times 14$ array. The onset temperature values for which these pairing amplitudes develop for each stripe are clearly indicated in the figure, and the highest value, namely, $T=90\mathrm{K}$, is taken as the lower pseudogap temperature of this compound.

Figure 7

Experimental points (connected by a thin line) against the reduced temperature $t=T∕T_c$ and the calculated curve (thick line) of $H_{c2}$ considering inhomogeneous samples with a stripe distribution of local superconducting temperature $T_c\left(l\right)$ for $\rho =0.08$ and a similar calculation with a Gaussian distribution for $\rho =0.25$.