In non-centrosymmetric superconductors, where the crystal structure lacks a centre of inversion, parity is no longer a good quantum number and an electronic antisymmetric spin-orbit coupling (ASOC) is allowed to exist by symmetry. If this ASOC is sufficiently large, it has profound consequences on the superconducting state. For example, it generally leads to a superconducting pairing state which is a mixture of spin-singlet and spin-triplet components. The possibility of such novel pairing states, as well as the potential for observing a variety of unusual behaviors, led to intensive theoretical and experimental investigations. Here we review the experimental and theoretical results for superconducting systems lacking inversion symmetry. Firstly we give a conceptual overview of the key theoretical results. We then review the experimental properties of both strongly and weakly correlated bulk materials, as well as two dimensional systems. Here the focus is on evaluating the effects of ASOC on the superconducting properties and the extent to which there is evidence for singlet-triplet mixing. This is followed by a more detailed overview of theoretical aspects of non-centrosymmetric superconductivity. This includes the effects of the ASOC on the pairing symmetry and the superconducting magnetic response, magneto-electric effects, superconducting finite momentum pairing states, and the potential for non-centrosymmetric superconductors to display topological superconductivity.

fruitful discussions, Guanghan Cao and Zhicheng Wang for assisting with 3 He-SQUID measurements, and Xiaoyan Xiao for assistance with single crystal x-ray diffraction.

To trace the origin of time-reversal symmetry breaking (TRSB) in Re-based superconductors, we performed comparative muon-spin rotation/relaxation (µSR) studies of superconducting noncentrosymmetric Re 0.82 Nb 0.18 (T c = 8.8 K) and centrosymmetric Re (T c = 2.7 K). In Re 0.82 Nb 0.18 , the low-temperature superfluid density and the electronic specific heat evidence a fully-gapped superconducting state, whose enhanced gap magnitude and specific-heat discontinuity suggest a moderately strong electron-phonon coupling. In both Re 0.82 Nb 0.18 and pure Re, the spontaneous magnetic fields revealed by zero-field µSR below T c indicate time-reversal symmetry breaking and thus unconventional superconductivity. The concomitant occurrence of TRSB in centrosymmetric Re and noncentrosymmetric ReT (T = transition metal), yet its preservation in the isostructural noncentrosymmetric superconductors Mg 10 Ir 19 B 16 and Nb 0.5 Os 0.5 , strongly suggests that the local electronic structure of Re is crucial for understanding the TRSB superconducting state in Re and ReT . We discuss the superconducting order parameter symmetries that are compatible with the observations. Time reversal and spatial inversion are two key symmetries which influence at a fundamental level the electron pairing in the superconducting state: on the one hand, a number of unconventional superconductors exhibit spontaneous time-reversal symmetry breaking (TRSB) on entering the superconducting state; on the other hand, the absence of inversion symmetry above T c leads to an antisymmetric spin-orbit coupling (SOC), lifting the degeneracy of the conduction-band electrons and potentially giving rise to a mixed-parity superconducting state [1,2]. Some noncentrosymmetric superconductors (NCSC), such as CePt 3 Si [3], CeIrSi 3 [4], Li 2 Pt 3 B [5, 6], and K 2 Cr 3 As 3 [7, 8], exhibit line nodes in the gap, while others such as LaNiC 2 [9] and (La,Y) 2 C 3 [10], show multiple nodeless superconducting gaps. In addition, due to the strong influence of SOC, their upper critical field can greatly exceed the Pauli limit, as has been found in CePt 3 Si [11] and very recently in (Ta,Nb)Rh 2 B 2 [12]. In general, TRSB below T c and a lack of spatial-inversion symmetry of the crystal structure are independent events. Yet, in a few cases, such as in LaNiC 2 [13], La 7 Ir 3 [14], and, in particular, in the Re-based compounds Re 6 Zr [15], Re 6 Hf [16], Re 6 Ti [17], and Re 24 Ti 5 [18], TRSB below T c is concomitant with an existing lack of crystal inversion symmetry. Such an unusually frequent occurrence of TRSB among the superconducting ReT binary alloys (T = transition metal) is rather puzzling. Its persistence independent of the particular transition metal, points to a key role played by Re. To test such a hypothesis, and to ascertain the possible relevance of the noncentrosymmetric structure to TRSB in Re-based NCSC, we proceeded with a twofold study. On one hand we synthesized and investigated an-other Re-based NCSC, Re 0.82 Nb 0.18 . On the other hand, we considered the ...

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