We consider superconducting nanowire proximity coupled to superconductor /ferromagnet / superconductor junction, where the magnetization penetrates intosuperconducting segment in nanowire decaying as $\sim\exp[-\frac{\mid n\mid}{\xi}]$ with site index $n$ and the decay length $\xi$. We tune chemicalpotential and spin-orbit coupling so that topological superconducting regimehosting Majorana fermion is realized for long $\xi$. We find that when $\xi$becomes shorter, zero energy state at the interface between superconductor andferromagnet splits into two away from zero energy. Accordingly, the behavior ofJosephson current is drastically changed due to this "zero mode-non-zero modecrossover". By tuning the model parameters, we find an almost second-harmoniccurrent-phase relation, $\sin2\varphi$, with phase difference $\varphi$. Basedon the analysis of Andreev bound state (ABS), we clarify that current-phaserelation is determined by coupling of the states within the energy gap. We findthat the emergence of crossing points of ABS is a key ingredient to generate$\sin2\varphi$ dependence in current-phase relation. We further study both theenergy and $\varphi$ dependence of pair amplitudes in the ferromagnetic region.For long $\xi$, odd-frequency spin-triplet $s$-wave component is dominant. Themagnitude of the odd-frequency pair amplitude is enhanced at the energy levelof ABS.
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机译:我们考虑耦合到超导体/铁磁体/超导体结的超导纳米线邻近,其中磁化强度渗透到纳米线中的超导段中,衰减为$ \ sim \ exp [-\ frac {\ mid n \ mid} {\ xi}] $,且具有站点索引$ n $和衰减长度$ \ xi $。我们对化学势和自旋轨道耦合进行了调整,从而使拓扑超导态势承载马约拉那费米子的时间更长。我们发现,当x \ xi $变得更短时,超导体和铁磁体之间的界面处的零能量状态会从零能量分裂为两个。因此,由于这种“零模式-非零模式转换”,约瑟夫森电流的行为被彻底改变。通过调整模型参数,我们发现一个几乎为二次谐波的电流-相位关系,\\ sin2 \ varphi $,具有相位差$ \ varphi $。基于对Andreev束缚态(ABS)的分析,我们阐明了电流-相位关系是由能隙内的状态耦合确定的。我们发现ABS交叉点的出现是在电流-相位关系中产生$ \ sin2 \ varphi $依赖性的关键因素。我们进一步研究了铁磁区域中线对振幅的能量依赖性和$ \ varphi $依赖性。对于较长的$ \ xi $来说,奇频自旋三联体$ s $波分量占主导地位。在ABS的能级上,奇数频率对幅度的幅度增大。
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