Nonequilibrium spin-dependent phenomena in mesoscopic superconductor–normal metal tunnel structures

We analyze the broad range of spin-dependent nonequilibrium transport properties of hybrid systems composed of a normal region tunnel coupled to two superconductors with exchange fields induced by the proximity to thin ferromagnetic layers and highlight its functionalities. By calculating the quasiparticle distribution functions in the normal region, we find that they are spin dependent and strongly sensitive to the relative angle between exchange fields in the two superconductors. The impact of inelastic collisions on their properties is addressed. As a result, the electric current flowing through the system is found to be strongly dependent on the relative angle between exchange fields, giving rise to a huge value of magnetoresistance. Moreover, the current presents a complete spin polarization in a wide range of bias voltages, even in the quasiequilibrium case. In the nonequilibrium limit, we parametrize the distributions with an “effective” temperature, which turns out to be strongly spin dependent, though quite sensitive to inelastic collisions. By tunnel coupling the normal region to an additional superconducting electrode, we show that it is possible to implement a spin-polarized current source of both spin species, depending on the bias voltages applied.

Figure 1

(Color online) Sketch of two possible implementations of the FS-I-N-I-SF structure analyzed in this work. (a) Spin-valve setup consisting of a sequence of stacked metallic layers. (b) A planar structure. Ferromagnetic layers (F) induce in each superconductor, through the proximity effect, an exchange field $\left({\mathit{h}}_{1,2}\right)$ whose relative orientation can be controlled by an externally applied magnetic field. The F exchange fields are confined $\left({\mathrm{a}}^{\prime }\right)$ to the $y\text{−}z$ plane for the setup shown in (a) and $\left({\mathrm{b}}^{\prime }\right)$ to the $x\text{−}y$ plane for the setup shown in (b) and are misaligned by an angle $\varphi$. A voltage bias $V$, applied across the structure, allows to control the energy distributions in the N region. The structure is assumed quasi-one-dimensional.

Figure 2

(Color online) Spin-dependent quasiparticle distribution functions $f_{\sigma }\left(\epsilon \right)$ in the full nonequilibrium limit vs energy $\epsilon$ for several bias voltages at $\varphi =0$, $T_{\text{bath}}=0.1T_c$, and $h^{*}=0.2{\Delta }^{*}$. (a) $f_{+}\left(\epsilon \right)$ and (b) $f_{-}\left(\epsilon \right)$.

Figure 3

(Color online) Spin-dependent quasiparticle distribution functions $f_{\sigma }\left(\epsilon \right)$ in the full nonequilibrium limit vs energy $\epsilon$ for several bias voltages at $\varphi =\pi$, $T_{\text{bath}}=0.1T_c$, and $h^{*}=0.2{\Delta }^{*}$. (a) $f_{+}\left(\epsilon \right)$ and (b) $f_{-}\left(\epsilon \right)$.

Figure 4

(Color online) Spin-dependent quasiparticle distribution functions $f_{\sigma }\left(\epsilon \right)$ vs energy $\epsilon$ calculated for several ${\mathcal{K}}_{\mathrm{coll}}$ values at $\varphi =0$, $eV={\Delta }^{*}$, $T_{\text{bath}}=0.1T_c$, and $h^{*}=0.2{\Delta }^{*}$. (a) $f_{+}\left(\epsilon \right)$ and (b) $f_{-}\left(\epsilon \right)$.

Figure 5

(Color online) Spin-dependent quasiparticle distribution functions $f_{\sigma }\left(\epsilon \right)$ vs energy $\epsilon$ calculated for several ${\mathcal{K}}_{\mathrm{coll}}$ values at $\varphi =\pi$, $eV={\Delta }^{*}$, $T_{\text{bath}}=0.1T_c$, and $h^{*}=0.2{\Delta }^{*}$. (a) $f_{+}\left(\epsilon \right)$ and (b) $f_{-}\left(\epsilon \right)$.

Figure 6

(Color online) (a) Nonequilibrium current vs bias voltage $V$ for several angles $\varphi$ at $T_{\text{bath}}=0.1T_c$ and $h^{*}=0.2{\Delta }^{*}$. (b) Nonequilibrium differential conductance $G$ vs $V$ calculated for the same values as in (a).

Figure 7

(Color online) Idealized finite-temperature exchange field-split density of states ${\mathcal{N}}^{{\mathrm{S}}_{1,2}}$ of ${\mathrm{S}}_{1,2}$ for parallel spin species, at $\varphi =\pi$ and different bias voltages $V$. In particular, (b), (c), and (d) show how features in the tunneling current originate at $eV=2({\Delta }^{*}-h^{*})$, $eV=-2h^{*}$, and $eV=-2({\Delta }^{*}+h^{*})$, respectively. Antiparallel spin species gives rise to features at opposite voltages. Green-dashed lines represent the superconducting DOS in the absence of the exchange field.

Figure 8

(Color online) (a) Nonequilibrium tunnel magnetoresistance ratio ∣TMR∣ vs $V$ calculated for several angles $\varphi$ at $T_{\text{bath}}=0.1T_c$ and $h^{*}=0.2{\Delta }^{*}$. (b) ∣TMR∣ vs $V$ for different $h^{*}$ values at $T=0.1T_c$ and $\varphi =\pi$.

Figure 9

(Color online) (a) Nonequilibrium current polarization $P_I$ vs $V$ calculated for several angles $\varphi$ at $T_{\text{bath}}=0.1T_c$ and $h^{*}=0.2{\Delta }^{*}$. (b) $P_I$ vs $V$ for different $h^{*}$ values at $T_{\text{bath}}=0.1T_c$ and $\varphi =\pi$.

Figure 10

(Color online) (a) Nonequilibrium tunnel magnetoresistance ratio ∣TMR∣ vs $V$ calculated for several $\Gamma$ values at $T_{\text{bath}}=0.1T_c$, $h^{*}=0.2{\Delta }^{*}$, and $\varphi =\pi$. (b) Nonequilibrium $P_I$ vs $V$ calculated for the same values as in (a).

Figure 11

(Color online) (a) Tunnel magnetoresistance ratio ∣TMR∣ vs $V$ in nonequilibrium (solid line) and quasiequilibrium (dash-dotted line) at $T_{\text{bath}}=0.1T_c$, $h^{*}=0.2{\Delta }^{*}$, and $\varphi =\pi$. (b) Current polarization $P_I$ vs $V$ in nonequilibrium (solid line) and quasiequilibrium (dash-dotted line) calculated for the same values as in (a). [(c) and (d)] $P_I$ vs $V$ in the intervals of large polarization calculated for several ${\mathcal{K}}_{\mathrm{coll}}$ values and the same parameters as in (a).

Figure 12

(Color online) (a) Nonequilibrium spin-dependent electron effective temperature $T_{\sigma }^{\mathrm{eff}}$ vs $V$ calculated for $\varphi =0$ at $T_{\text{bath}}=0.1T_c$ and $h^{*}=0.2{\Delta }^{*}$. (b) $T_{\sigma }^{\mathrm{eff}}$ vs $V$ calculated for $\varphi =\pi$ at $T_{\text{bath}}=0.1T_c$ and $h^{*}=0.2{\Delta }^{*}$.

Figure 13

(Color online) (a) Difference of the nonequilibrium spin-dependent electron effective temperatures $\delta T^{\mathrm{eff}}$ vs $V$ calculated for different $h^{*}$ values at $T_{\text{bath}}=0.1T_c$ and $\varphi =\pi$. (b) $\delta T^{\mathrm{eff}}$ vs ${\mathcal{K}}_{\mathrm{coll}}$ calculated at $eV=1.0{\Delta }^{*}$ for $T_{\text{bath}}=0.1T_c$, $h^{*}=0.2{\Delta }^{*}$, and $\varphi =\pi$.

Figure 14

(Color) (a) Sketch of a possible setup for the implementation of a spin-polarized current source. An additional superconducting electrode $\left({\mathrm{S}}_3\right)$ is coupled to the N region through a tunnel junction of resistance $R_{\mathrm{p}}$. The exchange fields in ${\mathrm{S}}_{1,2}$ are arranged in the antiparallel configuration. Spin-polarized current can be extracted by biasing the ${\mathrm{S}}_3$ terminal with $V_{\text{bias}}$. (b) Contour plot of the nonequilibrium current polarization $P_I$ vs $V$ and $V_{\text{bias}}$ at $T_{\text{bath}}=0.1T_c$ and $h^{*}=0.2{\Delta }^{*}$. (c) $P_I$ vs ${\mathcal{K}}_{\mathrm{coll}}$ calculated $eV=1.0{\Delta }^{*}$ and $e{V}_{\text{bias}}=0.5{\Delta }^{*}$ for $T_{\text{bath}}=0.1T_c$, $h^{*}=0.2{\Delta }^{*}$, and $\varphi =\pi$.