Spin-flip reflection at the normal metal-spin superconductor interface

We study spin transport through a normal metal-spin superconductor junction. A spin-flip reflection is demonstrated at the interface, where a spin-up electron incident from the normal metal can be reflected as a spin-down electron and the spin $2\times \hslash /2$ will be injected into the spin superconductor. When the (spin) voltage is smaller than the gap of the spin superconductor, the spin-flip reflection determines the transport properties of the junction. We consider both graphene-based (linear-dispersion-relation) and quadratic-dispersion-relation normal metal-spin superconductor junctions in detail. For the two-dimensional graphene-based junction, the spin-flip reflected electron can be along the specular direction (retro-direction) when the incident and reflected electron locates in the same band (different bands). A perfect spin-flip reflection can occur when the incident electron is normal to the interface, and the reflection coefficient is slightly suppressed for the oblique incident case. As a comparison, for the one-dimensional quadratic-dispersion-relation junction, the spin-flip reflection coefficient can reach 1 at certain incident energies. In addition, both the charge current and the spin current under a charge (spin) voltage are studied. The spin conductance is proportional to the spin-flip reflection coefficient when the spin voltage is less than the gap of the spin superconductor. These results will help us get a better understanding of spin transport through the normal metal-spin superconductor junction.

Figure 1

Band structures of a graphene with small magnetic momentum (FMG) and of a spin superconductor (SSC). In the graphene side, the red line denotes the dispersion spectrum of spin-up electrons and the blue line represents that of spin-down ones. A spin-up electron i with wave vector $q^{+}$ is incident from the graphene side to the spin superconductor. Spin-flip reflected electron A, normal reflected electron B, normal transmitted electron C, and spin-flip transmitted electron D are represented by the filled circles, with the direction of motion indicated by the arrows. There are two kinds of incident energies as shown in the figure. For one kind of incident energies, the incident electron and the spin-flip reflected electron locate at the same band, whereas for the other kind of incident energies, they locate at the conduction and valence bands.

Figure 2

Transmission and reflection coefficients of a spin-up electron incident from the left side with $M_L=0,\mathrm{\Delta }=1$, and $M=-10\mathrm{\Delta }$. The incident angle ${\alpha }^{+}$ is 0, $\pi /8$, and $\pi /4$ for the first column [(a), (d), and (g)], for the second column [(b), (e), and (h)], and for the third column [(c), (f), and (i)], respectively. The potential energy is $V=0$ for the first row (a)–(c), $V=-5\mathrm{\Delta }$ for the second row (d)–(f), and $V=-15\mathrm{\Delta }$ for the third row (g)–(i).

Figure 3

(a)–(c) Transmission and reflection coefficients of the graphene-based normal-spin superconductor junction as a function of the incident energy $E$, with (a) ${\alpha }^{+}=0$, (b) ${\alpha }^{+}=\pi /32$, and (c) ${\alpha }^{+}=\pi /16$. The other parameters are $\mathrm{\Delta }=1,M_L=-0.5\mathrm{\Delta },M=-10\mathrm{\Delta }$, and $V=0$. (d) A 2D plot of spin-flip reflection coefficient $R_{\downarrow \uparrow }$ vs ${\alpha }^{+}$ and $E$. The parameters are the same as (a)–(c).

Figure 4

(a) and (b) Cross sections of the energy spectrum of the graphene with $M_L\ne 0$ in Fig. 1. The red lines denote the dispersion spectrum of the spin-up states and the blue lines represent the spin-down ones. The direction of motion of the states is indicated by the arrows. A spin-up electron in the conduction band incident from the graphene can be either spin-flip retroreflected (a) or spin-flip specular reflected (b), depending on whether the reflected electron locates in the valence band (a) or the conduction band (b). (c) Trajectories of an incident electron i, the normal reflected electron B, and the spin-flip reflected electron A, for different incident energies $E$ by fixing the incident angle. By increasing the incident energy $E$ from 0, although the normal reflected angle remains the same, the spin-flip angle rotates anticlockwise that it disappears and reappears when the spin-flip reflected state is shifted from the valence band to the conduction one.

Figure 5

Transmission and reflection coefficients of a spin-up electron incident from the left side with $\mathrm{\Delta }=1,M_L=-0.5\mathrm{\Delta }$, and $M=-10\mathrm{\Delta }$. The incident angle ${\alpha }^{+}$ is $\pi /32,\pi /8$, and $\pi /4$ for the first column [(a), (d), and (g)], for the second column [(b), (e), and (h)], and for the third column [(c), (f), and (i)], respectively. The potential energy is $V=-0.2\mathrm{\Delta }$ for the first row (a)–(c), $V=-2\mathrm{\Delta }$ for the second row (d)–(f), and $V=-10\mathrm{\Delta }$ for the third row (g)–(i).

Figure 6

Differential charge conductance $G_e$ and differential spin conductance $G_s$ vs the charge voltage $e{V}_s$ (a) and vs the spin voltage $e{V}_s$ (b). The parameters are $M_L=0,V=0,\mathrm{\Delta }=1$, and $M=-10\mathrm{\Delta }$.

Figure 7

Band structures of the normal metal (N) and the spin superconductor (SSC) for the quadratic dispersion case. In the normal metal, the spin degree of freedom is degenerate, and the spin-up states are denoted by the red lines and the spin-down states are denoted by the blue line. A spin-up electron i with wave vector $q$ is incident from the normal metal to the spin superconductor. Spin-flip reflection A, normal reflection B, normal transmission C, and spin-flip transmission D are indicated by the arrows.

Figure 8

Transmission and reflection coefficients versus the incident energy $E$ for the normal metal-spin superconductor junction with the quadratic dispersion case. The Schottky potential strength is $Z=0$ (a), 0.3 (b), 1.0 (c), and 3.0 (d). The other parameters are $\mathrm{\Delta }=1,V=-15\mathrm{\Delta }$, and $M=-10\mathrm{\Delta }$.

Figure 9

Differential charge conductance $G_e$ versus $e{V}_s$ and spin conductance $G_s$ vs $e{V}_s$ under a charge voltage [(a) and (b)] and under a spin voltage [(c) and (d)]. $Z=0.3$ in (a) and (c) and $Z=1.0$ in (b) and (d). The other parameters are the same as those in Fig. 8.