Spontaneous symmetry breaking induced thermospin effect in superconducting tunnel junctions

We discuss the charge and the spin tunneling currents between two Bardeen-Cooper-Schrieffer (BCS) superconductors, where one density of states is spin-split by the proximity of a ferromagnetic insulator. In the presence of a large temperature bias across the junction, we predict the generation of a spin-polarized thermoelectric current. This thermospin effect is the result of a spontaneous particle-hole symmetry breaking in the absence of any polarizing tunnel barrier. The two spin components, which move in opposite directions, generate a spin current larger than the purely polarized case when the thermoactive component dominates over the dissipative one.

Figure 1

(a) Top: $S–I–S_m$ junction building blocks. The top hot (bottom cold) superconductor is indicated with $S$ ($S_m$), and the exchange field is present in $S_m$ only. Bottom: Energy band diagrams of the superconductors are shown in the absence ($h_{\mathrm{exc}}=0$, left panel) and presence ($h_{\mathrm{exc}}=0.4{\mathrm{\Delta }}_0$, right panel) of an exchange field ($h_{\mathrm{exc}}$). For $h_{\mathrm{exc}}=0$, the DoS of $S_m$ exhibits two peaks. For $h_{\mathrm{exc}}=0.4{\mathrm{\Delta }}_0$, the spin-split DoS (blue $\uparrow$ and green $\downarrow$ for the up and down spin components, respectively) shows four peaks. The difference between the chemical potential of the two leads is ${\mu }_S-{\mu }_{S_m}=\text{eV}$. The temperature of the $S$ ($S_m$) superconductor is set at $T_S=0.7T_c$ ($T_{S_m}=0.01T_c$). (b) Quasiparticle current-voltage characteristics for $h_{\mathrm{exc}}=0$ (black) and $h_{\mathrm{exc}}=0.4{\mathrm{\Delta }}_0$ (aquamarine) display two and four matching peaks, respectively. Inset: Blow-up of the charge ($I_q$, solid lines) and spin currents ($I_s$, dashed lines) at low bias voltage.

Figure 2

(a) Quasiparticle current-voltage $I_q\left(V\right)$ characteristics are shown for different values of $h_{\mathrm{exc}}/{\mathrm{\Delta }}_0$. In the linear regime, the IV curves can be approximated by the linear $I_q=G_{0}V$ of Eq. (3) (dashed violet line), while, in the nonlinear regime in bias, the $G^{\text{max}}$ evaluated at the thermoelectric peaks (dotted aquamarine line). The Seebeck voltages ($\pm V_S$) (orange circles) and the matching peaks values ($\pm V_p^{\pm }$) are reported for $h_{ex}=0.4{\mathrm{\Delta }}_0$. (b) In the linear regime, the zero-bias conductance ($G_0$) as a function of $h_{\mathrm{exc}}$ and $T_S$ is shown, distinguishing thermoactive areas (blue tones) from dissipative ones (red tones). (c) Absolute value of the Seebeck voltage as a function of $h_{\mathrm{exc}}$ is shown for different values of the thermal bias ($T_S$). (d) Maximum conductance evaluated at the matching peak ${V_p}^{-}$ is displayed as a function of $T_S$ and $h_{\mathrm{exc}}$ in the nonlinear regime. The blue tones are linked to the negative conductance (thermoelectricity), while the red area is referred to the positive one (dissipation).

Figure 3

(a) The charge current ($I_q$) is separated into up ($I_{\uparrow }$) and down ($I_{\downarrow }$) spin components for $h_{\mathrm{exc}}=0.4{\mathrm{\Delta }}_0$. Only one spin component ($I_{\downarrow }$ for $V>0$, $I_{\uparrow }$ for $V<0$) generates a thermoactive peak. The spin current ($I_s$) is reported in the dashed orange line. (b) The IV characteristics of the charge current ($I_q$) and spin current ($I_s$) and the curve of the spin current generation efficiency (SGE) are displayed for $h_{\mathrm{exc}}=0.4{\mathrm{\Delta }}_0$. The yellow area between the Seebeck voltages ($\pm V_s$) highlights the thermoactive spin current generation, while in the violet area the system is dissipative behaving as a spin filter. (c) The spin current evaluated at $+{V_p}^{-}$ is reported as a function of $T_S$ and $h_{\mathrm{exc}}$. Fixing $T_S$ (colored lines), the maximal value of $I_s$ is reached for the optimal value of $h_{\mathrm{exc}}$ such as ${\mathrm{\Delta }}_S={\mathrm{\Delta }}_{S_m}-h_{\mathrm{exc}}$ (dashed white line). With a black dashed line, we show the approximation of $h_{\mathrm{exc}}^t$. The violet transparent zone highlights the dissipative behavior of the system. (d) The maximum spin current, evaluated at the matching peak voltage ($+{V_p}^{-}$), corresponding to the colored cuts of the previous panel (c) (dashed lines for the dissipative regime and solid lines for the thermoelectric regime), is shown as a function of $h_{\mathrm{exc}}$ for different values of $T_S$.