Fraunhofer response and supercurrent spin switching in black phosphorus with strain and disorder

We develop theory models for both ballistic and disordered superconducting monolayer black phosphorus devices in the presence of magnetic exchange field and stress. The ballistic case is studied through a microscopic Bogoliubov–de Gennes formalism, while for the disordered case we formulate a quasiclassical model. Utilizing the two models, we theoretically study the response of supercurrent to an externally applied magnetic field in two-dimensional black phosphorus Josephson junctions. Our results demonstrate that the response of the supercurrent to a perpendicular magnetic field in ballistic samples can deviate from the standard Fraunhofer interference pattern when the Fermi level and mechanical stress are varied. This finding suggests the combination of chemical potential and strain is an efficient external knob to control the current response in highly sensitive strain-effect transistors and superconducting quantum interference devices. We also study the supercurrent in a superconductor-ferromagnet-ferromagnet-superconductor junction where the magnetizations of the two adjacent magnetized regions are uniform with misaligned orientations. We show that the magnetization misalignment can control the excitation of harmonics higher than the first harmonic $sin\phi$ (in which $\phi$ is the phase difference between the superconductors) in supercurrent and constitutes a full-spin-switching current element. Finally, we discuss possible experimental implementations of our findings. We foresee our models and discussions could provide guidelines to experimentalists in designing devices and future investigations.

Figure 1

Schematic of the monolayer black phosphorus Josephson junctions. The two-dimensional system resides in the $xy$ plane, so that the interfaces are along the $x$ direction, and the $y$ axis is normal to the junctions. The uniaxial/biaxial stress is exerted into the plane of the black phosphorus sheet, and an externally applied magnetic field $\mathit{H}$ is directed along $z$, perpendicular to the junction plane. The two $s$-wave superconducting electrodes (S), with different macroscopic phases ${\phi }_{l,r}$, are proximity coupled to black phosphorus. The chemical potentials of superconducting parts and nonmagnetized, and magnetized regions are marked by ${\mu }_S$, ${\mu }_N$, and ${\mu }_{F1},{\mu }_{F2}$. (a) The junction area is nonmagnetized BP of length $d$ and width $W$. (b) The junction area is magnetized through proximity coupling to ferromagnets (${\mathrm{F}}_1$ and ${\mathrm{F}}_2$). The ferromagnets possess different lengths $d_{F1}\ne d_{F2}$ with identical widths $W$. The magnetization orientation in each F region is determined through its angle with respect to the $z$ axis, i.e., ${\theta }_{1,2}$.

Figure 2

The response of critical current to an external magnetic field in a ballistic SNS BP Josephson junction. (a) The chemical potentials of both superconducting sides are varied, ${\mu }_S=\mu$, while the middle normal BP is undoped, ${\mu }_N=0$. The critical supercurrent is plotted as a function of magnetic flux passing through the junction area $\mathrm{\Phi }$ (normalized by the magnetic flux quantum ${\mathrm{\Phi }}_0=h/2e$) for different values of ${\mu }_S=\mu$ inside the S regions. The circles surround kinks where supercurrent reversals occur in cases with $\mu =0.4$ and 0.8 eV. (b) The chemical potential throughout the junction is considered to be the same, ${\mu }_S={\mu }_N=\mu$, and the critical current is plotted for various values of $\mu$, $\mu =0.4,0.8,1.2,1.6,2.0$ eV. In (c), we show the current phase relation at a current reversal point that the external magnetic field induces. We consider a representative case, which is $\mu =0.4$ eV in (a). The arrows indicate the change in maximum supercurrent when varying the external magnetic field from 0 to $\sim 3.0{\mathrm{\Phi }}_0$. The compression in (a)–(c) is kept fixed at ${\varepsilon }_{xx}=0.0,{\varepsilon }_{yy}=-0.2$. In (d1)–(e2) we consider an undoped middle normal region ${\mu }_N=0$ and set differing scenarios for applied stress. In (d1) and (d2) we set ${\mu }_S=\mu =1.5$ and 3.0 eV in the superconducting regions and uniaxial stress ${\varepsilon }_{xx}=0.0,{\varepsilon }_{yy}=-0.12,-0.16,-2.0,-2.4,-2.8$. In (e1) and (e2), we have increased $\mu$ to 2.0 and 4.0 eV and set equal biaxial compression components in both the $x$ and $y$ directions: ${\varepsilon }_{xx}={\varepsilon }_{yy}=-0.12,-0.16,-2.0,-2.4,-2.8$.

Figure 3

Current phase relation $I\left(\phi \right)$ in a ${\mathrm{SF}}_1{\mathrm{F}}_2\mathrm{S}$ BP Josephson junction. The current is plotted with increasing magnetization misalignment angle ${\theta }_1=0.1\pi –1.0\pi$ with a step of $0.1\pi$. The magnetization in ${\mathrm{F}}_2$ is kept fixed along the $z$ direction, ${\theta }_2=0$. Hence, ${\theta }_1=\pi /2$ is equivalent to perpendicular magnetizations, while ${\theta }_1=\pi$ means antiparallel alignment of magnetizations. In (a) and (b) we set equal magnetization strengths in both ${\mathrm{F}}_1$ and ${\mathrm{F}}_2$, i.e., $|{\mathbf{h}}_1|=|{\mathbf{h}}_2|=h_0$, and change it as (a) $h_0=0.1$ eV and (b) $h_0=0.2$ eV.

Figure 4

The critical current in a diffusive SNS BP Josephson junction as a function of external magnetic flux $\mathrm{\Phi }$. Here we set $\eta =1$, $\gamma =0.5$ and vary ${\chi }_y=0.1,0.2,0.3$.