Fig. 4

B–T phase diagram of highly crystalline 2D superconductors. a Schematic image of quantum Griffiths state. Blue and yellow regions represent normal and quantum metallic state (rare regions), respectively. b Schematic image of quantum metallic state. In this state, vortices (blue circles) show creep motion in the superconducting region (yellow region). The quantum metallic phase in Fig. 4a is the same as that in Fig. 4b. c The plotted data based on Supplementary Figure 5a. T_c0 is the transition temperature determined by the thermal fluctuation theories (Aslamazov–Larkin and Maki–Thompson model), and true superconducting state (zero-resistance state) is realized below the Berezinskii–Kosterlitz–Thouless (BKT) transition temperature (T_BKT)⁴. Blue triangles show the crossover temperature T_cross between the thermal creep regime and the quantum creep regime (see Supplementary Figure 4). Orange squares, green diamonds, and pink circles show the crossing points B_c of R–B curves in Fig. 2b, d at neighboring temperatures, the superconducting onset T_onset (see Supplementary Figure 1), and the mean field upper critical field B_c2^MF derived from the Ullah–Dorsey (UD) scaling¹⁹, respectively. Error bars of green diamonds represent the ambiguity of T_onset defined by dR_sheet/dT = 0 due to the experimental resolution (see Supplementary Figure 1). Under a relatively low magnetic field, the finite resistance state occurs due to the thermal creep (pink region) and quantum creep (blue region) in the relatively high and low temperature region, respectively. B_SM is the hypothetical transition magnetic field from the zero-resistance state to the quantum metal³² (vortex liquid). The system eventually exhibits the quantum Griffiths state (orange region) up to the characteristic critical magnetic field B_c^*. The black solid and dashed curves show the B_c2^MF(T) based on the Wertharmer–Helfand–Hohenberg (WHH) model and, crossover curve between thermal fluctuation region and quantum fluctuation region, respectively