Ferromagnetic superconductor

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Ferromagnetic superconductors are materials that display intrinsic coexistence of ferromagnetism and superconductivity. To this date, ferromagnetic superconductivity has been experimentally observed in UGe $2$ , URhGe, and UCoGe [1]. Evidence of ferromagnetic superconductivity was also reported for ZrZn $2$ by Pfleiderer et al. in 2001, but later reports [2]cast shadows of doubt on these findings.

The nature of the superconducting state in ferromagnetic superconductors is currently under debate. Early investigations [3] studied the coexistence of conventional $s$ -wave superconductivity with itinerant ferromagnetism. However, the scenario of spin-triplet pairing soon gained the upper hand [4], [5]. Recently, a mean-field model for coexistence of spin-triplet pairing and ferromagnetism was developed by Nevidomskyy [6], and further studied by Linder and Sudbø [7].

It should be mentioned that we are here considering the situation of uniform coexistence of ferromagnetism and superconductivity. Another scenario where there is an interplay between magnetic and superconducting order in the same material is superconductors with spiral or helical magnetic order. Examples of such include ErRh $4$ B $4$ and HoMo $6$ S $8$ . In these cases, the superconducting and magnetic order parameters entwine each other in a spatially modulated pattern, which allows for their mutual coexistence, although it is no longer uniform. Even spin-singlet pairing may coexist with ferromagnetism in this manner.

Theory

In conventional superconductors, the electrons constituting the Cooper pair have opposite spin, forming so-called spin-singlet pairs. However, other types of pairings are also permitted by the governing Pauli-principle. In the presence of a magnetic field, spins tend to align themselves with the field, which means that a magnetic field is detrimental for the existence of spin-singlet Cooper pairs. A viable mean-field Hamiltonian for modelling itinerant ferromagnetism coexisting with a non-unitary spin-triplet state may after diagonalization be written as [8], [9]:

$H = H_0 + \sum_{\mathbf{k}\sigma} E_{\mathbf{k}\sigma}\gamma_{\mathbf{k}\sigma}^\dagger \gamma_{\mathbf{k}\sigma}$ ,

$H_0 = \frac{1}{2} \sum_{\mathbf{k}\sigma}(\xi_{\mathbf{k}\sigma} - E_{\mathbf{k}\sigma} - \Delta_{\mathbf{k}\sigma}^\dagger b_{\mathbf{k}\sigma}) + INM^2/2$ ,