Superconductivity coexisting with ferromagnetism in a quasi-one dimensional non-centrosymmetric (TaSe$_4$)$_3$I

Low-dimensional materials with broken inversion symmetry and strong spin-orbit coupling can give rise to fascinating quantum phases and phase transitions. Here we report coexistence of superconductivity and ferromagnetism below 2.5\,K in the quasi-one dimensional crystals of non-centrosymmetric (TaSe$_4$)$_3$I (space group: $P\bar{4}2_1c$). The unique phase is a direct consequence of inversion symmetry breaking as the same material also stabilizes in a centro-symmetric structure (space group: $P4/mnc$) where it behaves like a non-magnetic insulator. The coexistence here upfront contradicts the popular belief that superconductivity and ferromagnetism are two apparently antagonistic phenomena. Notably, here, for the first time, we have clearly detected Meissner effect in the superconducting state despite the coexisting ferromagnetic order. The coexistence of superconductivity and ferromagnetism projects non-centrosymmetric (TaSe$_4$)$_3$I as a host for complex ground states of quantum matter including possible unconventional superconductivity with elusive spin-triplet pairing.

Breaking of symmetry leads to diverse physical phases and phase transitions. For example, crystalline order in solids is obtained by breaking the translational symmetry of an amorphous phase, ferromagnetism may emerge via the breaking of the rotational symmetry of a paramagnetic phase, the spontaneous breaking of gauge symmetry facilitates superfluidity and superconductivity [13], and breaking of the time reversal symmetry lifts Kramer's degeneracy and has close connection to a number of exotic quantum phenomena, such as quantum anomalous Hall effect [14,15]. Even more complex varieties of physical properties like ferroelectricity, spin-orbit physics, Weyl physics, unconventional superconductivity, etc.
are often obtained by breaking of the inversion symmetry [16][17][18][19]. In a given family of compounds, keeping the chemical composition same, diverse electronic and magnetic states can be achieved simply by tuning the symmetry of the systems [20].
The family of linear-chain compounds (M Se 4 ) n I where M =Nb and Ta and n=2, 3, and 10/3, have attracted the attention of the community time and again because they display wide varieties of phase transitions and phase co-existences[1-4, [21][22][23][24][25]. In this family of compounds, the MSe 4 chains, which are separated from each other by the iodine atoms, run along the c axis thereby giving rise to quasi-one-dimensional structure[1-4]. Tight-binding band structure calculations reveal that the electronic properties of these compounds depend on the value of n, which drives the electronic character by altering the degree of band filling [3,4]. Being quasi-one-dimensional conductors, they are prone to undergo a Peierls transition leading to a charge-density wave (CDW) phase transition at low temperature.
Recently, a member of the (MSe 4 ) n I family with n = 2, (TaSe 4 ) 2 I, was shown to host an exotic axionic CDW phase [21,22]. However, for n =3, (NbSe 4 ) 3 I and (TaSe 4 ) 3 I are known to form a centrosymmetric structure with space group P 4/mnc [2]. Here, we have stabilized the (TaSe 4 ) 3 I in a non-centrosymmetric phase (space group: P42 1 c) over a broad temperature range and hereafter, we refer it as n-TSI phase. In this phase, interestingly, the system is metallic and shows a CDW transition around 150 K. With further lowering the temperature, it undergoes a long-range magnetic transition around 10 K and eventually settles in a superconducting ground state coexisting with ferromagnetism below 2.5 K. This unique coexistence seemingly arises from inversion symmetry breaking as the centrosymmetric phase of the same compound (space group: P 4/mnc) is known to be a non-magnetic insulator[1-4].
Single crystals of n-TSI were grown by chemical vapor transport method (refer to method section). The crystals are formed in ribbon-like fibres of length ∼ few mm, width ∼ few micron (see supplemental information for more details - Figure S1-S4). Single crystal X-ray analysis at 100 K revealed that n-TSI has a simple tetragonal unit cell (space group P42 1 c, no. 114, CCDC entry 2055811). The lattice parameters are: a = b = 9.4358(5) Å, c = 19.0464(11) Å; α = β = γ = 90 • . The schematic representation of the structure is illustrated in Figure 1 a-c. As shown in Figure 1.d, atomic resolution imaging under a transmission electron microscope revealed linear TaSe 4 chain-like structure along the c-axis.
The distance between TaSe 4 chains is d inter = 6.677 Å and the diameter of each TaSe 4 chain To investigate the electronic properties of the needle-like crystals, we have employed usual four-probe resistivity measurements on a bunch of ribbons as well as on a single ribbon. As shown in Figure 1e (multi ribbon device), Resistance (R) decreases rapidly with decreasing temperature, indicating good metallic behavior. Furthermore, the linear nature of the R vs. T 2 plot (in the lower inset) in the temperature range of 5-25 K confirms that n-TSI is a clean Fermi liquid metal. R(T ) curve shows a slope change around 150 K which is known to arise for a charge density wave (CDW) transition due to Peierls-like instability of the linear chain structure [1, 2, 21-25]. Below 2.5 K, the resistance shows a sharp fall which resembles a superconducting transition. When the samples were cooled in a dilution refrigerator with a lower base temperature, the zero-resistance state was achieved below ∼1K, thereby confirming superconductivity (see the upper inset of Figure 1e). In the supplementary information, Figure S5 displays much sharper zero-resistance transition for a single ribbon device, indicating that superconductivity is indeed an intrinsic property of the quasi-one dimensional crystals.
With the application of a magnetic field parallel to the length of the needle-like crystals (H||I||c), the transition temperature (T c ) gradually decreases and goes beyond our measurement limit above a magnetic field of H ||c c2 2 Tesla, as shown in Figure 2a. This is expected for a superconducting phase transition. The zero-temperature critical field, H ||c c2 (0) 2.35 Tesla is estimated from an approximated linear extrapolation (this is approximate, as there is a weak non-linearity in H ||c c2 vs. T c data (inset of Figure 2a). Here, T c has been taken as the temperature where the resistance drops to 90% of the normal state resistance. Possibly due to an anisotropy in the superconducting state, originating either from the geometry of the crystals or from an intrinsic anisotropy, the critical field is observed to be smaller when the magnetic field is applied perpendicular to the c-axis. The estimated zero-temperature critical field in this case is H ⊥c c2 (0) 0.75 Tesla (Figure 2b). We also found that the critical current dominated feature disappears at temperature above 2.5 K (see Figure 2d). All the above observations collectively confirm beyond any ambiguity that the resistive transition discussed above is due to superconductivity. 8 As per the general understanding of the microscopic theory of superconductivity, due to the breaking of U (1) symmetry in the superconducting phase, superconductors are also expected to be strongly diamagnetic. In order to probe the magnetic state of n-TSI, detailed dc magnetization measurements were performed. As shown in Figure 3a, when the magnetic field is applied along the length of the needles (c-axis), magnetization (M ) in zero-fieldcooled (ZFC) cycle monotonically increases with decreasing temperature and displays a peak around 9 K. Such a peak is often seen in systems with a helimagnetic order [29,30], where a ferromagnetic component of the order is also expected. To gain further insight on the nature of magnetic ground state, the paramagnetic susceptibility has been fitted to the Curie-Weiss (CW) expression as shown in supplementary information ( Figure S12). The fit yields positive CW temperature (T CW ∼ 5.5 K) which suggests that magnetism in n-TSI is dominated by ferromagnetic interactions.
To note, when the magnetic field is applied perpendicular to the c-axis (Figure 3b), M is found to be significantly smaller and the anomaly due to magnetic transition is not clearly visible. This indicates that the hard magnetic axis is perpendicular to the c-axis (see Section S3, supplementary information for additional data regarding anisotropic behavior). The inset of Figure 3a shows that M drops sharply below 3 K and becomes negative (diamagnetic) for low excitation field ( 2 mT). Sharp drop in M has also been observed at higher fields. The diamagnetic signal is also observed when the field is applied perpendicular to the c-axis (inset of Figure 3b). Motivated by this, we attempted to detect the Meissner signal which should appear in the field-cooled (FC) mode. As shown in Figure 3c, at 2 mT, magnetization decreases sharply below 3 K, confirming strong diamagnetic contribution. However, a clear diamagnetic signal in FC mode is observed at low temperature for lower fields (inset of Figure 3c). This is remarkable because in ferromagnetic superconductors, the diamagnetic signal was never directly observed before [31][32][33][34][35][36]. Further, to detect the shielding effect, we have measured dc-field-dependent high-frequency susceptibility of another small piece of n-TSI using a two-coil method. As shown in Figure 3d, the superconducting transition along with its systematic field dependence was clearly observed that confirms the superconducting ground state of n-TSI. It may also be mentioned that the peak at 9 K becomes much sharper in FC mode (Figure 3c). Therefore, we have shown that n-TSI is metallic at high temperatures and undergoes a ferromagnetic transition below 10 K and superconducting transition below 2.5 K. All these observations can be summarized in a phase diagram, deduced directly from the experimental data, as shown in Figure 4. The presence of a ferromagnetically ordered state in n-TSI is surprising mainly because the material does not host any magnetic element. The observed weak ferromagnetic state [40,41] could be a consequence of spin-orbit coupling which appears ubiquitously in inversion-symmetry broken materials. Furthermore, the fact that the system undergoes multiple phase transitions just above the superconducting critical temperature hints to the possibility of a quantum critical state [42][43][44][45][46]. Finally, coexistence of ferromagnetism and superconductivity with the indication of anisotropic nature of superconductivity that we found here, warrants further research for probing the symmetry of the order parameter in n-TSI. Such a system is ideal for the existence of the hitherto elusive spin-triplet pairing symmetry [5][6][7][8].