Crossover from impurity-controlled to granular superconductivity in ${\left(\mathbf{TMTSF}\right)}_2{\mathbf{ClO}}_4$

Using a proper cooling procedure, a controllable amount of nonmagnetic structural disorder can be introduced at low temperature in ${\left(\mathrm{TMTSF}\right)}_2{\mathrm{ClO}}_4$. Here we performed simultaneous measurements of transport and magnetic properties of ${\left(\mathrm{TMTSF}\right)}_2{\mathrm{ClO}}_4$ in its normal and superconducting states, while finely covering three orders of magnitude of the cooling rate around the anion ordering temperature. Our result reveals, with increasing density of disorder, the existence of a crossover between homogeneous defect-controlled $d$-wave superconductivity and granular superconductivity. At slow cooling rates, with small amount of disorder, the evolution of superconducting properties is well described with the Abrikosov-Gorkov theory, providing further confirmation of non-$s$-wave pairing in this compound. In contrast, at fast cooling rates, zero resistance and diamagnetic shielding are achieved through a randomly distributed network of superconducting puddles embedded in a normal conducting background and interconnected by proximity effect coupling. The temperature dependence of the ac complex susceptibility reveals features typical for a network of granular superconductors. This makes ${\left(\mathrm{TMTSF}\right)}_2{\mathrm{ClO}}_4$ a model system for granular superconductivity where the grain size and their concentration are tunable within the same sample.

Figure 1

Normal and superconducting properties of ${\left(\mathrm{TMTSF}\right)}_2{\mathrm{ClO}}_4$ for sample no. 1 at various cooling rates. (a) Interlayer resistivity ${\rho }_{c^{*}}$ vs $T$ at 0.020 K/min showing the drop of the elastic resistivity $\mathrm{\Delta }{\rho }_{c^{*}}=0.23\mathrm{\Omega }\text{cm}$ at $T_{\mathrm{AO}}$ as indicated by the red arrow. (b) Change in the imaginary part of the ac susceptibility. (c) Change in the real part of the ac susceptibility. (d) ${\rho }_{c^{*}}\left(T\right)$ simultaneously measured with the data in (b) and (c). The inset presents ${\rho }_{c^{*}}\left(T\right)$ on a broader temperature range for 0.020, 0.52, 2.5, 7.6, and 18 K/min. For the 0.020- and 18-K/min data, the results of the fitting, performed using the procedure explained in the text, are presented with the broken curves.

Figure 2

Cooling-rate dependence of superconducting properties for sample no. 1, and its comparison to that of sample no. 2. (a) Cooling-rate dependence of the various critical temperatures: the blue circles correspond to $T_{\mathrm{c}\chi }$, the purple closed squares to $T_{\mathrm{c}\rho }$, and the purple open squares to $T_{\mathrm{c},\rho =0}$. The red circles indicate $T_{\mathrm{c}\chi }$ for sample no. 2. (b) Cooling-rate dependence of the relative shielded volume fraction $v_{\mathrm{shield}}$ for samples no. 1 (blue) and no. 2 (red). (c) Cooling-rate dependence of the residual resistivity ${\rho }_{c^{*}0}$. (d) Low-temperature dissipation, as measured by $\mathrm{\Delta }{\chi }_{\mathrm{ac}}^{\prime \prime }$ for sample no. 1. The $\mathrm{\Delta }{\chi }_{\mathrm{ac}}^{\prime \prime }$ data have been averaged in the temperature range $0.1\text{K}<T<0.3\text{K}$.

Figure 3

Dependence of $T_{\mathrm{c}}$'s and relative shielded volume fraction $v_{\mathrm{shield}}$ on ${\rho }_{c^{*}0}$ for sample no. 1. (a) $T_{\mathrm{c}\rho }$ as a function of ${\rho }_{c^{*}0}$, compared with $T_{\mathrm{c},\rho =0}$ and $T_{\mathrm{c}\chi }$. The dotted curve presents results of fittings of $T_{\mathrm{c}\rho }$ with the AG formula [Eq. (2)]. The fitting has been performed in the range $0<{\rho }_{c^{*}0}<0.05\mathrm{\Omega }\text{cm}$. The inset shows the same data in a wider range. (b) $v_{\mathrm{shield}}$ as a function of ${\rho }_{c^{*}0}$. The inset shows the same data in a wider range.

Figure 4

Volume fraction of the anion-ordered domain derived from resistivity ($p$, obtained from the effective-medium theory in this work, purple circles) and from high-resolution x rays ($v_{\mathrm{XRD}}$, from Ref. [55], blue crosses), compared with the shielded fraction evaluated from the ac susceptibility ($v_{\mathrm{shield}}$, black squares). The broken arrow shows the estimate for the ordered volume from x-ray measurements at 100 K/min [53].

Figure 5

(a) $T_{\mathrm{c}\rho }$, (b) ${\rho }_{c^{*}0}$, and (c) low-temperature dissipation as measured by $\mathrm{\Delta }{\chi }_{\mathrm{ac}}^{\prime \prime }$, plotted against the disordered volume fraction $1-p$ derived through Eq. (1). Deviation from the linear dependence (broken lines; fitted in the range $1-p<20\%$ in all panels) occurs when the first finite disordered cluster emerges. At complete disorder (not accessible before the onset of the SDW ground state above 18 K/min), the resistivity would amount to ${\rho }_{\mathrm{Max}}=0.26\mathrm{\Omega }\text{cm}$, as indicated by the open circle. The $\mathrm{\Delta }{\chi }_{\mathrm{ac}}^{\prime \prime }$ data in (c) have been averaged in the temperature range $0.1\text{K}<T<0.3\text{K}$.

Figure 6

Dependencies of $T_{\mathrm{c}\rho }$ (purple circles) and $T_{\mathrm{c}\chi }$ (blue circles) on the disordered volume fraction $1-p$ for sample no. 1. Note that $T_{\mathrm{c}\chi }$ is practically equivalent to $T_{\mathrm{c},\rho =0}$. For comparison, we plot $T_{\mathrm{c}}$ from the specific-heat study [51], with $1-p$ values estimated from the curves in Fig. 4. For these data, the upper error bar corresponds to the onset of the transition, and the lower error bar to the peak in the electronic specific heat. In the light-blue region, superconductivity develops within individual puddles. On the border between the light-blue and light-red regions, the proximity effect starts to induce SC coherence between puddles. The large scale phase-coherent superconductivity establishes progressively in the light-red region as a result of the interpuddle proximity effect coupling. The broken curves indicate the region used for the analysis based on the proximity effect: $T_{\mathrm{c}\rho }$ is assumed to be constant (1.27 K) as shown with the magenta curve and $T_{\mathrm{c}\chi }$ is calculated by solving Eq. (6). The red curve is obtained by fitting the theory to the $T_{\mathrm{c}\chi }$ data in the range $1-p>40\%$. The resulting fitting parameters are $\mathcal{C}=5.2{\mathrm{K}}^{3/2}$ and $\alpha =100{\mathrm{nm}/\mathrm{K}}^{1/2}$.