Superconducting phase diagram and spin diode effect via spin accumulation

Spin-split superconductors offer new functionality compared to conventional superconductors such as diode effects and efficient thermoelectricity. The superconducting state can nevertheless only withstand a small amount of spin splitting. Here, we self-consistently determine the spin transport properties and the phase diagram of a spin-split superconductor in the presence of an injected spin accumulation. Energy and spin relaxation are accounted for in the relaxation time approximation via a single effective inelastic scattering parameter. We find that the spin-splitting field in the superconductor enables a spin diode effect. Moreover, we consider the superconducting phase diagram of a system in contact with a spin accumulation and in the presence of spin relaxation, and find that the inclusion of energy and spin relaxation alters the phase diagram qualitatively. In particular, these mechanisms turn out to induce a superconducting state in large parts of the phase diagram where a normal state would otherwise be the ground state. We identify an Fulde–Ferrel–Larkin–Ovchinnikkov-like state even in the presence of impurity scattering, which can be controllably turned on and off via the electrically induced spin accumulation. We explain the underlying physics from how the superconducting order parameter depends on the nonequilibrium modes in the system as well as the behavior of these modes in the presence of energy and spin relaxation when a spin-splitting field is present.

Figure 1

Suggested experimental setups for the predicted effects. (a) Setup A: Spin injection into a spin-split superconductor will be quantitatively different depending on the polarity of the injected spin, resulting in a spin diode effect. (b) Setup B: Interfacing a spin-split superconductor with two spin accumulations on each side that can be polarized parallel or oppositely strongly influences the superconducting phase diagram. The spin-split superconductor is achieved by taking a thin film superconductor and either applying an in-plane magnetic field or growing a ferromagnetic insulator under it. As for actual materials, the superconductor is a BCS superconductor such as Al, NbN, or Pb. The normal metal could, for example, be Cu, and the ferromagnets could, for example, be Co, Fe, or Ni. Examples of ferromagnetic insulators include EuS and GdN.

Figure 2

Spin currents in setup A for different inelastic scattering parameters $\delta /{\mathrm{\Delta }}_0$. The spin-splitting field in the superconductor is set to $m=0.32{\mathrm{\Delta }}_0$. (a) Spatial profile of the spin current for spin accumulation $\left|e\right|V_s/{\mathrm{\Delta }}_0=2$. (b) Spatially averaged spin current as a function of spin accumulation. The current is antisymmetric around $\left|e\right|V_s=0.32{\mathrm{\Delta }}_0$. The inset shows the dependence of the diode efficiency $\eta$ on the inelastic scattering parameter. The numerical method for determining $\left|e\right|V_s^{\pm }/{\mathrm{\Delta }}_0$ gave a small uncertainty $\mp 0.01$ in determining these critical spin accumulations. The thickness of the line showing $\eta \left(\delta \right)$ in the inset accounts for this uncertainty in $V_s^{\pm }$. The uncertainty is thus much smaller than the actual magnitude of $\eta \left(\delta \right)$.

Figure 3

Density of states and occupation number $n_{e\uparrow }$ for spin-$\uparrow$ electrons in setup A for $\left|e\right|V_s/{\mathrm{\Delta }}_0=0.52$ (top plots) and $\left|e\right|V_s/{\mathrm{\Delta }}_0=1.47$ (bottom plots). The dashed line shows the spatial profile of the gap $\left|\mathrm{\Delta }\right|/{\mathrm{\Delta }}_0$. The solid line shows the density of states at $x/\xi =0$. The inelastic scattering parameter is set to $\delta /{\mathrm{\Delta }}_0=0.01$.

Figure 4

Phase diagram for setup B with parallel spin accumulations showing the regions of superconductivity, bistability and the normal state for different spin accumulations and spin-splitting fields. In (a), inelastic scattering is not included in the kinetic equations, while in (b) it is included. The spatial profile of the superconducting gap for given parameter sets is given in (c) and (f)–(h). (d)–(e) show the contribution from the distribution function in the gap equation in the middle of the superconductor for given parameter sets, and the spatially averaged superconducting gap for given spin-splitting fields is shown in (i)–(k). In (d)–(e) and (i)–(k), the dashed lines originate from the lower branch in phase diagram (a), while the solid lines originate from the lower branch in phase diagram (b).

Figure 5

Phase diagram for setup B with antiparallel spin accumulations showing the regions of superconductivity (including the FFLO-like state), bistability and the normal state for different spin accumulations and spin-splitting fields. In (a), inelastic scattering is not included in the kinetic equations, while in (b) it is included. The spatial profile of the superconducting gap for given parameter sets is given in (c)–(d) and (g)–(h). (e)–(f) show the contribution from the distribution function in the gap equation in the middle of the superconductor for given parameter sets, and the spatially averaged superconducting gap for given spin-splitting fields is shown in (i)–(k). In (e)–(f) and (i)–(k), the dashed lines originate from the lower branch in phase diagram (a), while the solid lines originate from the lower branch in phase diagram (b).