Spin-polarized currents and noise in normal-metal/superconductor junctions with Yu-Shiba-Rusinov impurities

Conventional superconductors disordered by magnetic impurities demonstrate physical properties that are drastically different from their pristine counterparts. In our previous work [D. Persson et al., Phys. Rev. B 92, 245430 (2015)], we explored the spectral and thermodynamic properties of such systems for two extreme cases: completely random and ferromagnetically aligned impurity magnetic moments. Here we consider the transport properties of these systems and show that they have a potential to be used in superconducting spintronic devices. Each magnetic impurity contributes a Yu-Shiba-Rusinov (YSR) bound state to the spectrum, residing at subgap energies. Provided the YSR states form metallic bands, we demonstrate that the tunneling current carried by these states can be highly spin polarized when the impurities are ferromagnetically ordered. The spin polarization can be switched by tuning the bias voltage. Moreover, even when the impurity spins are completely uncorrelated, one can still achieve almost $100\%$ spin polarization of the current, if the tunnel interface is spin active. We compute electric current and noise, varying parameters of the interface between tunneling and fully transparent regimes, and analyze the relative role of single-particle and Andreev reflection processes.

Figure 1

Superconducting thin film (S) is deposited on a substrate and magnetic impurities are homogeneously distributed within the sample. The magnetic moments of the impurities are either (a) completely unpolarized or (b) ferromagnetically ordered (by the applied magnetic field $B$) in the plane of the film. The film is contacted by a normal-metal probe (N) for transport measurements.

Figure 2

Diagram illustrating all possible elementary processes occurring at an NS interface. The full black (red dashed) lines denote electronlike (holelike) quasiparticle trajectories. The arcs connecting full and dashed lines show the possibility of particle-hole branch conversion (Andreev reflection). The coherence functions $\gamma$ and $\tilde{\gamma }$ have a meaning of probability amplitude of $h\to e$ and $e\to h$ conversions, respectively [28]. To construct arbitrary amplitude, one starts with an incoming (arrow pointing towards the interface) particle or hole line, traces all possible ways to arrive at the desired outgoing line, and takes a superposition of them.

Figure 3

Scattering amplitudes at the NS interface for spin-up quasiparticles, calculated for different transparencies; see Eq. (22). Impurity spins are assumed ferromagnetically aligned with $\alpha >0$, and other parameters are taken as $\mathrm{\Gamma }/2\pi k_B{T}_{c0}=0.01,u_{\mathrm{S}}=5,E_Z/2\pi k_B{T}_{c0}=0.02$, and $T=0.01T_{c0}$. Here, $T_{c0}$ is the clean-limit critical temperature of the superconductor (at $\mathrm{\Gamma }=0$). The black full line is the probability of normal reflection, $R_{ee,\uparrow }$, while the red long-dashed line corresponds to Andreev reflection, $R_{he,\uparrow }$. The green short-dashed and the blue dash-dotted lines are the normal transmission, $T_{ee,\uparrow }(1-|{\tilde{\gamma }}_{\uparrow }{|}^2)$, and the transmission with branch conversion, $T_{he,\uparrow }(1-|{\gamma }_{\uparrow }{|}^2)$, respectively. The inset in (d) is a zoom on energies around the YSR impurity band. The interface is assumed spin inactive here ($\mathrm{\Theta }=0$ and ${\mathcal{D}}_{\uparrow }={\mathcal{D}}_{\downarrow }$) and the transmission probabilities independent of the quasiparticle incidence angle $\theta$; see Eq. (20).

Figure 4

Differential conductance for the case of randomly oriented impurity spins, given in units of normal-state conductance $G_N$, where $G_N=R_N^{-1}=2e^2{N}_F{A}_c\mathcal{D}{\langle v_{Fz}\rangle }_{{\mathbf{p}}_F·{\mathbf{e}}_z<0}$. Here, ${\langle v_{Fz}\rangle }_{{\mathbf{p}}_F·{\mathbf{e}}_z<0}$ is the average velocity of quasiparticles with momenta pointing towards S; see Fig. 2. (a) The effect of increasing $u_{\mathrm{S}}=2,4,8$ with $\mathrm{\Gamma }/2\pi k_B{T}_{c0}=0.01$. (b) The effect of increasing $\mathrm{\Gamma }/2\pi k_B{T}_{c0}=0.05,0.1,0.2$ with $u_{\mathrm{S}}=5$. The interface parameters are $\mathrm{\Theta }=0$ and ${\mathcal{D}}_{\uparrow }={\mathcal{D}}_{\downarrow }=0.01$. Transmission probabilities are assumed independent of the incidence angle $\theta$; see Eq. (20). The temperature is $T=0.01T_{c0}$, where $T_{c0}$ is the clean-limit critical temperature of the superconductor (at $\mathrm{\Gamma }=0$).

Figure 5

(a) Charge and (b) spin differential conductance as a function of bias. (c) Their ratio $P$ [see Eq. (34)], providing information on the spin polarization of current. Impurity spins are assumed to be ferromagnetically aligned. Black full lines correspond to $\alpha >0$ and red dashed ones to $\alpha <0$. The system parameters are $u_{\mathrm{S}}=5,\mathrm{\Gamma }/2\pi k_B{T}_{c0}=0.005$, and $E_Z/2\pi k_B{T}_{c0}=0.02$. The interface parameters and temperature are the same as in Fig. 4.

Figure 6

Differential Fano factor for the case of (a) ferromagnetically ordered and (b) randomly oriented impurity spins. Black full lines correspond to $\alpha >0$, while red dashed ones to $\alpha <0$. The impurity strength is $u_{\mathrm{S}}=4$ and the other parameters are the same as in Fig. 5.

Figure 7

Differential charge conductance and noise for the case of magnetic impurity spins being (a), (d) randomly oriented, (b), (e) ferromagnetically aligned with $\alpha >0$, and (c), (f) ferromagnetically aligned with $\alpha >0$ and spin-dependent transparency. The system parameters are $\mathrm{\Gamma }/2\pi k_B{T}_{c0}=0.005,E_Z/2\pi k_B{T}_{c0}=0.02,u_{\mathrm{S}}=4,\mathrm{\Theta }=0$, and $T=0.01T_{c0}$. Transmission probabilities are assumed independent of the incidence angle $\theta$; see Eq. (20).

Figure 8

Spatial profile of the order parameter for a clean NS junction (no magnetic impurities) with a spin-active interface; see Fig. 2. (a) Effect of the interface transparency, ${\mathcal{D}}_{0\uparrow ,\downarrow }={\mathcal{D}}_0$ [see Eq. (20)] and $\mathrm{\Theta }=0$. (b) Effect of the spin-mixing angle, $\mathrm{\Theta }\ne 0$ and ${\mathcal{D}}_{0\uparrow ,\downarrow }=0$. In both plots, the temperature is $T=0.01T_{c0}$.

Figure 9

Local density of states in the superconductor for spin-up quasiparticles, assuming that magnetic impurities have randomly oriented spins. The left column shows heat maps of the density of states as a function of energy and distance from the interface; see Fig. 2. From top to bottom, the spin-mixing angle is $\mathrm{\Theta }=0,\pi /4,\pi$. Arrows indicate positions of the YSR bands in the bulk. The right column shows cuts of the heat maps at given values of distance from the interface, indicated by the dashed lines. For all plots, we take $\mathrm{\Gamma }/2\pi k_B{T}_{c0}=0.001,u_{\mathrm{S}}=7,T=0.01T_{c0}$, and ${\mathcal{D}}_{0\uparrow ,\downarrow }={10}^{-4}$ [see Eq. (20)].

Figure 10

Interplay of the ABS and YSR impurity bands. Impurities are assumed to have randomly oriented magnetic moments. The system parameters are $\mathrm{\Gamma }/2\pi k_B{T}_{c0}=0.001,u_{\mathrm{S}}=7,T=0.01T_{c0}$, and the spin-mixing angle is $\mathrm{\Theta }=\pi /4$. The panels read (a), (d) charge conductance, (b) spin conductance, (c) differential noise, (e) polarization, and (f) differential Fano factor. Transparency of the interface is ${\mathcal{D}}_{0\uparrow ,\downarrow }={10}^{-4}$ [see Eq. (20)] for the black full lines, and ${\mathcal{D}}_{0\uparrow ,\downarrow }=0.1$ for the red dashed ones.