Pressure-induced electronic phase separation of magnetism and superconductivity in CrAs

The recent discovery of pressure (p) induced superconductivity in the binary helimagnet CrAs has raised questions on how superconductivity emerges from the magnetic state and on the mechanism of the superconducting pairing. In the present work the suppression of magnetism and the occurrence of superconductivity in CrAs were studied by means of muon spin rotation. The magnetism remains bulk up to p  3.5 kbar while its volume fraction gradually decreases with increasing pressure until it vanishes at p  7 kbar. At 3.5 kbar superconductivity abruptly appears with its maximum Tc  1.2 K which decreases upon increasing the pressure. In the intermediate pressure region (3.5  p  7 kbar) the superconducting and the magnetic volume fractions are spatially phase separated and compete for phase volume. Our results indicate that the less conductive magnetic phase provides additional carriers (doping) to the superconducting parts of the CrAs sample thus leading to an increase of the transition temperature (Tc) and of the superfluid density (ρs). A scaling of ρs with as well as the phase separation between magnetism and superconductivity point to a conventional mechanism of the Cooper-pairing in CrAs.


ZF µSR experiments.
At ambient pressure CrAs is characterized by long-range helimagnetic order with a propagation vector k c = 0.3562(2) parallel to the c−axis and the magnetic moments lie in the ab plane. S2 The ordered magnetic moment per Cr is m ≃ 1.73 µ B . S2 Due to the incommensurability of the magnetic structure, a continuous set of local fields is expected to be seen at each particular muon stopping site. It was shown that such a magnetic structure leads to a field distribution given by: S1, S3, S4 and is characterised by two peaks due to the minimum (B min ) and maximum(B max ) cutoff fields (see the inset in Fig. 1 in the main text). Considering only one muon stopping site, the ZF muon-spin polarization for a powder sample would follow the relation: S4 Here λ T and λ L are the transverse and the longitudinal exponential relaxation rates, respectively. The occurrence of 2/3 oscillating and 1/3 non-oscillating µSR signal fractions originates from the spatial averaging in powder samples, where 2/3 of the magnetic field components are perpendicular to the muon-spin and cause a precession, while the 1/3 longitudinal field components do not. Figure 1 shows the dependence of the minimum B min and the maximum B max cutoff fields of CrAs on pressure. Points were obtained from the fit of ZF and wTF µSR data measured at T 5 K. Both B min and B max decrease with increasing pressure. Following Ref. S5 for a helical magnetic structure the upper and the lower cutoff fields should scale as 2m and m, respectively. Linear fits resulting in dB min /dp = −8.0(8) mT/kbar and dB max /dp = −15.3(5) mT/kbar thus confirm this statement. The decrease of B min and B max with increasing pressure implies a decrease of the ordered magnetic moment. By taking into account that the ambient pressure value of the ordered moment per Cr was found to be ≃ 1.73 µ B S2 our results would imply that with increasing pressure up to p ≃ 6.7 kbar Cr moments decrease down to ≃ 1.47 µ B .
µSR experiments under weak transverse field (wTF) applied perpendicular to the muon-spin polarization are a straightforward method to determine the onset of the magnetic transition and the magnetic volume fraction. In this case the contribution to the asymmetry from muons experiencing a vanishing internal spontaneous magnetisation can be accurately determined. Muons stopping in a non-magnetic environment produce long lived oscillations, which reflect the coherent muon precession around the external field B ex . Muons stopping in magnetically ordered parts of the sample give rise to a more complex, distinguishable signal, reflecting the vector combination of internal and external fields. The random orientation of the grains in a powder sample leads to a broad distribution of precession frequencies.
The situation is substantially simplified for B ex ≪ B int (weak transverse field regime). In this case one can neglect the influence of B ex on B int and the fitting function becomes: Here A nm (0) and A m are the initial non-magnetic and magnetic asymmetry, respectively, ϕ is the initial phase of the muon-spin ensemble, and σ nm is the temperature independent Gaussian relaxation rate caused by nuclear moments. P ZF (t) represent the ZF magnetic polarization and is described by Eq. (3).  Figure 2 represents the wTF µSR time spectra measured at ambient pressure above (T ≃ 300 K) and below (T ≃ 180 K) the magnetic transition (T N ≃ 265 K). The solid lines correspond to the fit of the first term on the right-hand side of Eq. (4) to the experimental data. The "magnetic term" [A m (0)P ZF (t)] vanishes within the first ∼ 0.1 µs and thus is not observed with the present data binning (≃ 0.063 µs). The "pressure cell" contribution is missing since experiments under ambient pressure were performed by using the sample outside of the cell on the low-background GPS spectrometer. Figure 3 demonstrates the dependence of the non-magnetic volume fraction f = A nm (0)/[A nm (0) + A m (0)] on temperature at various pressures.
TF µSR experiments Figure 4 shows the TF µSR time spectra measured at T = 0.24 K and 1.5 K at p = 5.8 kbar. The stronger damping at T = 0.24 K is due to inhomogeneous field distribution caused by formation of the flux line lattice (FLL) in the superconducting CrAs.
The TF µSR data were analyzed by using the following functional form: Here A nm (0), A m , ϕ , and σ nm have similar meanings as in Eq. (4), σ sc is the relaxation rate caused by FLL formation, and B is the magnetic field inside the sample. Due to the diamagnetism of the superconducting state B < B ex for T < T c and B ≃ B ex for T ≥ T c .

Comparison of T c (p) from the present study with the literature data
Wu et al. S6 have determined the transition temperature T c as the temperature where the resistivity reaches their close to zero value. The such determined T c continuously increases (from ≃ 0.5 K to ≃ 1.5 K) in the region where the superconductivity and magnetism coexists, reaches their maximum value (≃ 1.5 K) at p ≃ 11 kBar and further decreases with increasing pressure [see Fig. 5 would correspond to appearance of superconducting domains with relatively high transition temperature, while T c is determined by occurrence of Josephson type coupling between the superconducting domains.
The comparison of T c (p) data obtained within the present study with T onset c from Ref. S6 is shown in Fig. 5(b) and results in fare good agreements between two sets of the data. Note that the transition temperatures in Fig. 5(b) are normalized to their values at p ≃ 10 kbar.