Long forgotten fields dv4/28/2023 Moreover, the spin-dependent properties of quasiparticles and their non-equilibrium behavior also change in the presence of SOC. Recently, theory and experiments have demonstrated singlet-to-triplet pair conversion via SOC in $s$-wave superconducting structures with or without magnetic layers. It is established that certain forms of magnetic inhomogeneity at an $s$-wave superconductor interface with a ferromagnet can transform spin-singlet Cooper pairs into a spin-polarized spin-triplet Cooper pairs, enabling transformative concepts for cryogenic computing. One such area relates to the generation and interplay of spin-polarized spin-triplet Cooper pairs in superconducting structures with magnetic interfaces. Understanding the emergent phenomena in such systems is an important aim in condensed matter physics. Even with well-understood materials, conventional $s$-wave superconducting hybrid structures with SOC provide a platform for realizing exotic phenomena and counterparts in the normal state. The interplay of SOC with superconductivity has attracted significant interest over the past decade and understanding has substantially progressed, both experimentally and theoretically. Spin-orbit coupling (SOC) describes the interaction between an electron's motion and its spin, and is ubiquitous in condensed matter systems. This work provides evidence that odd-frequency pairing may form intrinsically within relativistic superconductors. The relativistic correction to the anomalous Green function to first order in the boost parameter is completely odd in frequency. In the boosted frame the order parameter contains terms which are both even and odd in frequency. ![]() The odd-frequency pairing emerges dynamically as a result of the boost. We choose an ansatz for the gap function in a particular frame which is even-frequency and analyze the effects on the anomalous Green function under a boost into a relativistic frame. The form of the relativistically invariant pairing term is chosen such that it reduces to BCS form in the non-relativistic limit. Solving the resulting Gor'kov equations yields expressions for relativistic normal and anomalous Green functions. ![]() To see this, we consider a Dirac model with quartic potential and perform a mean-field analysis to obtain a relativistic Bogoliubov-de Gennes system. Here we show that an inherent odd-frequency mode emerges dynamically under application of a Lorentz transformation of the anomalous Green function with the general frequency-dependent gap function. Odd-frequency superconductivity is an exotic superconducting state in which the symmetry of the gap function is odd in frequency.
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