标题： 磁性双层约瑟夫森结中的量子化共振隧穿效应 显示英文标题

标题： Quantized resonant tunneling effect in Josephson junctions with ferromagnetic bilayers

摘要： 我们研究SF$_1$F$_2$S结中的约瑟夫森效应，该结由常规的s波超导体(S)通过两个铁磁体(F$_1$和F$_2$)连接而成。 在低温下，当铁磁层 F$_1$和 F$_2$的交换场（$h_1$，$h_2$）和厚度（$d_1$，$d_2$）变化时，若 F$_1$/F$_2$界面存在势垒，则约瑟夫森临界电流会表现出周期性共振峰。 这些共振峰在量化条件$Q_{1(2)}d_{1(2)}=\left(n_{1(2)}+1/2\right)\pi$下出现。 这里，$Q_{1(2)} = 2h_{1(2)}/(\hbar v_F)$表示库珀对携带的质心动量，其中$v_F$是费米速度，$n_{1(2)} = 0, 1, 2, \cdots$。 这种临界电流特征来源于自旋三重态对在磁化轴方向上自旋投影为零时引起的共振隧穿效应。 在共振时，约瑟夫森电流主要来自于平行和反平行磁化配置中的第一谐波，而在垂直配置中第二谐波变得更为显著。 在两个铁磁层具有相同交换场和厚度的情况下，势垒会选择性地抑制0态中的电流，同时在平行配置中的$\pi$-态中保持电流。 相反，在反平行配置中，0态中的电流始终被保留。 显示英文摘要

摘要： We study the Josephson effect in SF$_1$F$_2$S junctions, which consist of conventional s-wave superconductors (S) connected by two ferromagnets (F$_1$ and F$_2$). At low temperatures, the Josephson critical current displays periodic resonance peaks as exchange fields ($h_1$, $h_2$) and thicknesses ($d_1$, $d_2$) of the F$_1$ and F$_2$ layers vary, provided a potential barrier exists at the F$_1$/F$_2$ interface. These resonance peaks emerge under the quantization conditions $Q_{1(2)}d_{1(2)}=\left(n_{1(2)}+1/2\right)\pi$. Here, $Q_{1(2)} = 2h_{1(2)}/(\hbar v_F)$ represents a center-of-mass momentum carried by Cooper pairs, where $v_F$ is the Fermi velocity and $n_{1(2)} = 0, 1, 2, \cdots$. This critical current feature arises from a resonant tunneling effect induced by spin-triplet pairs with zero spin projection along the magnetization axis. At resonance, the Josephson current primarily originates from the first harmonic in both parallel and antiparallel magnetization configurations, whereas the second harmonic becomes more significant in perpendicular configurations. In cases where both ferromagnetic layers have identical exchange fields and thicknesses, the potential barrier selectively suppresses the current in the 0-state while maintaining it in the $\pi$-state for parallel configurations. Conversely, in antiparallel configurations, the current in the 0-state is consistently preserved.