We present magnetic penetration depth and electrical transport data in single crystals of quasi-one-dimensional (q1D) ${\mathrm{Tl}}_2{\mathrm{Mo}}_6{\mathrm{Se}}_6$, which reveal a $1\mathrm{D}\to 3\mathrm{D}$ superconducting dimensional crossover. The $c$-axis penetration depth shows the onset of superconducting fluctuations below $T_{1D}^{ons}=6.7$ K, whereas signatures of superconductivity in the $ab$-plane penetration depth (a uniquely sensitive probe of the transverse phase stiffness) only emerge below $T_{3D}^{ons}=4.9$ K. An anomalously low superfluid density persists down to $\sim 3$ K before rising steeply, in agreement with a theoretical model for crossovers in q1D superconductors. Our data analysis suggests that a sequence of pairing and phase fluctuation regimes controls the unusually broad superconducting transition. In particular, the electrical resistivity below $T_{3D}^{ons}$ is quantitatively consistent with the establishment of phase coherence through gradual binding of Josephson vortex strings to form 3D loops. This dimensional crossover within the superconducting state occurs despite the relatively large transverse hopping predicted from the band structure. Our results have important consequences for the low-temperature normal state in ${\mathrm{Tl}}_2{\mathrm{Mo}}_6{\mathrm{Se}}_6$ and similar q1D metals, which may retain one-dimensional behavior to lower temperatures than expected from theory.

• Received 29 September 2017
• Revised 26 June 2018

DOI:https://doi.org/10.1103/PhysRevB.98.054507