Co-appearance of superconductivity and ferromagnetism in a Ca2RuO4 nanofilm crystal

By tuning the physical and chemical pressures of layered perovskite materials we can realize the quantum states of both superconductors and insulators. By reducing the thickness of a layered crystal to a nanometer level, a nanofilm crystal can provide novel quantum states that have not previously been found in bulk crystals. Here we report the realization of high-temperature superconductivity in Ca2RuO4 nanofilm single crystals. Ca2RuO4 thin film with the highest transition temperature Tc (midpoint) of 64 K exhibits zero resistance in electric transport measurements. The superconducting critical current exhibited a logarithmic dependence on temperature and was enhanced by an external magnetic field. Magnetic measurements revealed a ferromagnetic transition at 180 K and diamagnetic magnetization due to superconductivity. Our results suggest the co-appearance of superconductivity and ferromagnetism in Ca2RuO4 nanofilm crystals. We also found that the induced bias current and the tuned film thickness caused a superconductor-insulator transition. The fabrication of micro-nanocrystals made of layered material enables us to discuss rich superconducting phenomena in ruthenates.

j ≈ 1.6 × 10 4 A/cm 2 , respectively. Sample 6 exhibits a supercurrent below I c = ±200 nA and j c = 1.6 × 10 4 A/cm 2 at 5 K (Fig. S2e). Table S1 summarizes the current densities obtained from our samples. We have observed the BKT behavior of α = 3 at 5 K even in 17 T as shown in Fig. S2e. The critical magnetic field is expected to be larger than 17 T due to its short coherence length.
The dependence of R □/layer on temperature is qualitatively different from the thermal activation type exponential temperature dependence in bulk Ca 2 RuO 4 . We found that the R □/layer at 500 nA was characterized by a 2D variable-range-hopping (VRH), R(T ) =  Table S1. Summary of j and electronic states of Ca 2 RuO 4 R 0 exp(T 0 /T ) 1/3 , in the 10 − 65 K range, where T 0 becomes 1280 K as shown in Fig. S1.
The result suggests the 2D conduction. Figure S2a shows the dependence of the sheet resistance R □/layer = ρ/d of sample 4 on temperature for different bias currents I, where the interlayer distance of Ca 2 RuO 4 d = 6.13 A , and ρ = 2.0 × 10 −4 Ω·cm is the resistivity at 290 K. The resistivity of sample 4 is much smaller than that in the ab-plane ρ ab ∼ 6 Ω·cm in bulk Ca 2 RuO 4 and is more like the resistivity under hydrostatic pressure above 0.5 GPa [29]. At a low bias current of I = 20 nA, the sheet resistance decreases to zero within the accuracy of our measurement. Although the onset temperature is rather high at 52 K, this broad transition behaviour occurs within a wide temperature range of 43 K. Figure S2b shows the I − V characteristics of sample 4 for several selected temperatures. The result clearly shows supercurrents of ∼ ±100 nA for the low bias current region, which strongly suggests the presence of superconductivity in a Ca 2 RuO 4 flake.
The typical behaviour for a 2D superconductor is shown in Fig. S2. We analyse the broad transition based on the theory proposed by Berezinskii, Kosterlitz and Thouless [47,48]. BKT theory predicts that a quasi-long-range order will be established in 2D systems below the BKT transition temperature T BKT . Above T BKT , the resistance resulting from the free motion of vortices and antivortices is expressed as Halperin-Nelson equation [74], , where b is a constant of the order of unity, and the meanfield transition temperature T c0 . In contrast, below T BKT a vortex and an antivortex form a pair. Here we determined T c0 by analyzing the resistance data through the 2D Aslamasov-Larkin (AL) model [75], , where R N is the normal sheet resistance.
The sheet resistance at T onset Another important feature of the BKT transition is the presence of a universal jump in the pairs. The exponent α jumps abruptly from 1 to 3 with decreasing temperature, and this has been reported in various 2D superconducting systems such as a Hg-Xe alloy [76], an ultrathin , and a superconducting wire network [77]. The d log(V )/d log(I) plot shows that α jumps from 1 to more than 3 at a low temperature as shown in the upper and lower insets of Fig. S2b. The plot are well fitted at three orders of magnitude.
T BKT is estimated to be 9.5 K, which is consistent with T BKT = 9 K determined from the resistance data at I = 20 nA for sample 4 in Fig. S2a. We have also confirmed the BKT transition in different Ca 2 RuO 4 thin films as shown in Fig. S2e. On the basis of the above standard analysis, we suggest that a Ca 2 RuO 4 thin film shows a typical BKT transition for a 2D superconductor, namely a topological phase transition. Recently, however, a role of superconducting inhomogeneity in nonlinear I − V characteristics of 2D superconductors has been reported in SrTiO 3 interfaces [78]. The inhomogeneity is important in our results.
So, we need to investigate further the relationship between the BKT physics and emrgent inhomogeneity in a layered Ca 2 RuO 4 .

Difference of transport properties in clean and dirty systems.
Samples 2 and 4 have approximately the same thickness at 10 and 12 nm, respectively.
The resistivity for sample 2 exceeds that of sample 4 at 280 K. This means that the sample 2 contains more disorder or intrinsic inhomogeneity than sample 4. Sample 4 showed the insulating behavior at 500 nA. We found that the resistance was characterized by a 2D VRH in the 10 − 65 K range, where T 0 becomes 1280 K. In the "clean" system (sample 4),   1 and 2). Moreover, as shown in Fig. S6, a precursor superconducting phenomenon may be observed at a rather high temperature T * = 215 K in sample 1 because the bias-current dependence of the resistance appears below T * . In Ref.
[34], the magnetism under 5 mA begins to decrease below 220 K. This is close to T * = 215 K observed in sample 1. In addition to these results, according to arguments related to the pseudogap in cuprate superconductors, T ∼ 200 K may indicate the temperature for preforming Cooper pairs. The resistance behavior at 3000 nA for sample 1 is similar to the behavior of a ferromagnetic metal under pressure in bulk [29].

First-principles calculations.
We performed first-principles calculations based on spin-polarized density function theory, using a local spin-density approximation (LSDA), which is implemented in the plane-wave and projector augmented wave method the Vienna Ab-initio Simulation Package (VASP 5.4.1) [80][81][82][83][84][85]. We adopted the LSDA+U scheme [86] with a U ef f of 2.5 eV, which has been reported to reproduce the band gap of bulk Ca 2 RuO 4 [87]. We used a supercell consisting of 28 atoms for a bulk system. We also considered a 1 × 1 monolayer, a bilayer and a trilayer of Ca 2 RuO 4 with a 15Å vacuum layer to model a thin film. We applied a 500 eV cutoff for the plane-wave basis set and a Gaussian smearing model of σ = 0.05 eV. 4 × 4 × 2 and 4 × 4 × 2 Monkhorst-Pack special k-point grids [88] for the first Brillouin zone sampling were used, respectively, for a bulk system and a slab model. All the atoms were relaxed until the force on each atom was less than 0.02 eV/Å.
We also performed the first-principles calculation with simple LSDA and HSE06 hybrid functionals. The results with these functionals also show the same trends, such as 5 a decreasing tilting angle, through monolayer exfoliation. In addition, LSDA+U provides similar results to those obtained with hybrid HSE06 functionals. These points validate the LSDA+U methods.