Introduction to unconventional superconductivity in non-centrosymmetric metals

These lecture notes are an extension of my previous notes [1] presented in this lecture series and are concerned with the recently emerging research field of unconventional superconductivity in non-centrosymmetric metals. Inversion symmetry together with time reversal symmetry represent key symmetries for the formation of Cooper pairs in superconductors and allows to distinguish between even-parity spin-singlet and odd-parity spin-triplet pairing. The absence of at least one of two symmetries leads to the spin-splitting of the electronic states, through Zeeman fields (loss of time reversal symmetry) and through antisymmetric spin-orbit coupling (loss of inversion symmetry), which has a strong influence on the Cooper pairing states possible. Anderson’s theorems show the basic symmetry requirements for the Cooper pair formation. The meaning of these theorems can be demonstrated in a perturbative analysis of the superconducting instability. The structure of the pairing states are derived for systems without inversion and time reversal symmetry, and are shown to be non-unitary. In the case of non-centrosymmetric materials the pairing interaction displays interesting spin-orbit coupling-induced features which are analyzed within a toy model for the superconductivity in CePt3Si, one of the non-centrosymmetric heavy Fermion superconductors, in order to give a catalogue of possible pairing states in this material. A further important point is the essentially universal behavior of the spin susceptibility in the superconducting phase of a non-centrosymmetric materials. This behavior is spectacularly manifested in the upper critical field of CeRhSi3 and CeIrSi3. Magneto-electric effects represent one of the most extraordinary parts in the phenomenology of non-centrosymmetric superconductors. Two examples of magneto-electric behaviors are discussed: (1) the helical phase in the mixed superconducting state and (2) relation between supercurrent and the spin magnetization. Eventually also the possibility of surface Andreev bound states is discussed and it is shown that such states can carry spin currents.