One of the most salient features of heavy fermion superconductivity is its coexistence with various competing orders. Superconductivity often emerges near or at the border of these exotic orders and their interplay may give rise to many interesting quantum phenomena. In this paper, we give a brief review of the various heavy fermion superconductors discovered so far and show there may exist an intimate connection between their superconducting pairing and quantum critical spin/charge/orbital fluctuations. We classify these superconductors into three categories: (A) CeM2X2, CenMmIn3n+2m, CePt3Si, CeMX3, CeNiGe3, Ce2Ni3Ge5 and CePd5Al2, in which superconductivity emerges at the border of antiferromagnetic phase;YbRh2Si2, in which superconductivity was very recently found inside the antiferromagnetic phase at 2 mK;UX2Al3 and UPt3, in which superconductivity occurs inside the antiferromagnetic phase;and UBe13 and U6Fe, in which the connection between magnetism and superconductivity is not yet clear. Among them, CePt3Si and CeMX3 are noncentrosymmetric, while UPt3 exhibits spin triplet pairing inside an antiferromagnetic phase. (B) UGe2, URhGe, UCoGe, UIr and U2PtC2, are spin triplet superconductors under the influence of ferromagnetic order or fluctuations. (C) URu2Si2, PrOs4Sb12, PrT2X20, Pu-115, NpPd5Al2 and β-YbAlB4, in which superconductivity may be related to other exotic quantum states or fluctuations such as hidden order, valence fluctuations and quadrupolar fluctuations. In these compounds, f-electrons may participate in both superconductivity and other competing orders and often behave simultaneously itinerant and localized. These could be described by a phenomenological two-fluid theory, in which two coexisting fluids—an itinerant heavy electron fluid (the Kondo liquid) and a spin liquid of unhybridized local f-moments—compete to give rise to the various low temperature orders as well as superconductivity. Combining the two-fluid picture and the idea of spin-fluctuation-induced superconducting pairing, a BCS-like formula is proposed for calculating the superconducting transition temperature, and the results are found to be in good agreement with the experimental data for Ce-115. This model can explain naturally the microscopic coexistence of superconductivity and antiferromagnetism in these materials, and provides a promising guidance to other heavy fermion superconductors to achieve a systematic examination of the interplay between superconductivity and other exotic orders.
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