Quantum Griffiths singularity in three-dimensional MoTiN superconducting films

Quantum Griffiths singularity (QGS) has been experimentally observed in a range of two-dimensional (2D) superconducting systems. Although it is theoretically suggested that the QGS also exists in three-dimensional (3D) superconductors, there is almost no experimental support to the theoretical prediction. In the present paper, we observe the occurrence of QGS in a series of $\sim$80-nm-thick Mo$_{0.8}$Ti$_{0.2}$N$_x$ ($0.84 \lesssim x \lesssim 1.12$) superconducting films near the field-driven superconductor-metal transition (SMT). These films have a NaCl structure and are 3D with respect to the superconductivity. For each film, the low-temperature magnetoresistance isotherms, measured at magnetic fields being perpendicular or parallel to the film plane, do not cross at a single point but at a clear wide region. The dynamical critical exponents $z\nu_{\perp}$ (for perpendicular field) and $z\nu_{\parallel}$ (for parallel field) obtained by analyzing the related magnetoresistance isotherms increase with decreasing temperature and tend to diverge as $T\rightarrow 0$ K. In addition, the effective resistivity data for the perpendicular and parallel field in the vicinity of the SMTs both obey an activated scaling based on the random transverse-field Ising model. We also fabricate a $\sim$80-nm-thick (Mo$_{0.8}$Ti$_{0.2}$)$_2$N$_{1.06}$ superconducting film with face-centered cubic structure at low nitrogen partial pressure. It is found that the low-temperature magnetoresistance isotherms for the perpendicular (parallel) field cross at a single point and the resistivity data for the perpendicular (parallel) field in the vicinity of the field-induced SMT obey the power-law scaling deduced from the dirty-boson model. Our results provide unambigous experimental evidence for the existence of QGS in 3D superconductors.