Colloquium: Nonequilibrium effects in superconductors with a spin-splitting field

This Colloquium discusses the recent progress in understanding the properties of spin-split superconductors under nonequilibrium conditions. Recent experiments and theories demonstrate a rich variety of transport phenomena occurring in devices based on such materials that suggest direct applications in thermoelectricity, low-dissipative spintronics, radiation detection, and sensing. This text discusses different experimental situations and presents a theoretical framework based on quantum kinetic equations. This framework provides an accurate description of the nonequilibrium distribution of charge, spin, and energy, which are the relevant nonequilibrium modes, in different hybrid structures. This Colloquium also reviews experiments on spin-split superconductors and shows how transport measurements reveal the properties of the nonequilibrium modes and their mutual coupling. In particular, the emphasis of the text is on spin injection and diffusion and very large thermoelectric effects in spin-split superconductors.

Figure 1

Central panel: Quasiparticle spectrum and density of states in a superconductor with spin splitting, $N_0$ is the normal metal DOS. (a)–(d) Schematic of various nonequilibrium phenomena occurring at normal metal/insulator/superconductor (NM/I/S) and ferromagnetic metal/insulator/superconductor (FM/I/S) interfaces discussed in this review. For clarity we show the limit of half-metallic FM with $N_{\uparrow }=0$. (a) Spin-resolved tunneling from a ferromagnetic metal to a spin-split superconductor that leads to the spin-valve effect, i.e., the charge current in the parallel magnetic configuration is different from that in the antiparallel one. (c) Creation of spin and charge accumulation in the voltage-biased FM/S junction. (b), (d) Schematic of thermally excited currents in NM/S and FM/S junctions with a spin-split superconductor. (b) Spin Seebeck effect in NM/S junction: A pure spin current is generated by the temperature bias between a spin-split superconductor at temperature $T_{\mathrm{SC}}$ and a normal metal at temperature $T_{\mathrm{NM}}>T_{\mathrm{SC}}$. (d) Thermoelectric effect in a FM/I/S junction: Here the spin current is partially converted to the charge current due to the spin-dependent density of states in the ferromagnet.

Figure 2

Measured differential conductance $dI/dV$ of a $\mathrm{EuS}/\mathrm{Al}/{\mathrm{Al}}_2{\mathrm{O}}_3/\mathrm{Al}$ junction as a function of the applied voltage and external magnetic field. $H_{\mathrm{co}}$ denotes the coercive field of the EuS layer when the magnetization switches. Adapted from [174].

Figure 3

Calculated density of states of a thin superconducting film at $T\to 0$. We show the DOS for only one of the spin species $N_{\uparrow }$. Shown by dashed red lines is the DOS in the absence of relaxation ${\tau }_{\mathrm{s}\mathrm{n}}=1/({\tau }_{\mathrm{sf}}^{-1}+{\tau }_{\mathrm{s}\mathrm{o}}^{-1})=\infty$ and zero exchange field $h=0$ resulting into a gap ${\mathrm{\Delta }}_0$. Other curves are plotted for $h=0.4{\mathrm{\Delta }}_0$ and different spin-relaxation rates. (a) Spin-flip relaxation $\beta =({\tau }_{\mathrm{so}}-{\tau }_{\mathrm{sf}})/({\tau }_{\mathrm{so}}+{\tau }_{\mathrm{sf}})=1$; curves from top to bottom correspond to an increasing $({\tau }_{\mathrm{s}\mathrm{n}}{\mathrm{\Delta }}_0{)}^{-1}$, varying equidistantly from 0 by 0.04 steps. (b) Spin-orbit relaxation $\beta =-1$; curves from top to bottom correspond to an increasing $({\tau }_{\mathrm{s}\mathrm{n}}{\mathrm{\Delta }}_0{)}^{-1}$, varying equidistantly from 0 by steps of 3.4. For clarity the curves are shifted along the vertical axis.

Figure 4

Schematic of the quasiparticle distribution function components in a superconductor with spin splitting $2{\mu }_{B}B$. The occupied states are represented by filled circles. (a) Equilibrium distribution, (b) electron-hole imbalance $f_T$, (c) spin imbalance $f_{T3}$, (d) spin-energy imbalance $f_{L3}$, and (e) energy imbalance $f_L$. The dashed and dotted arrows show elastic processes which lead to the formation, and the reverse processes to the relaxation, of a particular nonequilibrium mode. In (c) and (d) the dashed black lines show particle-hole branch transitions while the dotted blue lines correspond to the spin-flip processes.

Figure 5

Scanning electron microscopy image of the lateral structure. From [84].

Figure 6

(a) Nonlocal conductance measured as a function of the injecting voltage $g_{\mathrm{nl}}(V_{\mathrm{inj}})$. Adapted from [84]. (b) The same quantity calculated using the kinetic theory for ${\alpha }_{\mathrm{orb}}=1.33$, $\beta =0.5$, $({\tau }_{\mathrm{s}\mathrm{n}}{T}_{c0}{)}^{-1}=0.2$, $T=0.05T_{c0}$, effective inelastic relaxation length $L=20{\mathrm{\lambda }}_{\mathrm{s}\mathrm{n}}$, $d=5{\lambda }_{\mathrm{s}\mathrm{n}}$, where ${\lambda }_{\mathrm{s}\mathrm{n}}=\sqrt{D{\tau }_{\mathrm{s}\mathrm{n}}}$ is the normal-state spin relaxation length. Solid black and dash-dotted red curves correspond to the injection from nonferromagnetic ($P_{\mathrm{inj}}=0$) and ferromagnetic ($P_{\mathrm{inj}}=0.5$) electrodes, respectively, at the spin splitting $h=0.2{\mathrm{\Delta }}_0$. The dashed blue line corresponds to $h=0$. The conductance is normalized to $g_0=R_{\xi }/R_{\mathrm{inj}}{R}_{\det }$, where $R_{\xi }=\xi /A_s{\sigma }_N$ is the normal-state resistance of the wire with length $\xi$ and cross section $A_s$.

Figure 7

Cooling power from reservoir $R$ vs voltage for different values of the exchange field $h$, assuming a unit polarization $P=1$ at the temperature $k_{B}T=0.3\mathrm{\Delta }$ close to that yielding optimal cooling for $P=h=0$. The exchange fields are given in units of $\mathrm{\Delta }$. Changing the sign of $P$ or $h$ inverts the voltage dependence with respect to $V=0$.

Figure 8

(a) Schematic setup for measuring the thermoelectrically induced current, used by [104]. S, F, and N stand for a superconductor, a ferromagnet, and a normal metal, whereas FI is a ferromagnetic insulator. (b) Setup used for a direct measurement of the Seebeck effect. (c) Heat engine realized in a lateral setup with “$n$-doped” and “$p$-doped” junctions using a FNF trilayer with antiparallel magnetization directions. To disregard spin accumulation, the island has to be large compared to the spin-relaxation length. (d) Heat engine with a spin-split superconducting island. The ferromagnets can also be replaced by a normal metal if the interfaces to the superconductor contain a ferromagnetic insulator. (c), (d) The heating power $P_{\mathrm{heat}}$ is partially converted to “useful” work $P_{\mathrm{work}}$ dissipated on the load.

Figure 9

Thermoelectric current as a function of the applied magnetic field, measured in [104]. The circles show the measurement values, and the solid lines show a comparison to Eq. (29). The three solid lines correspond to slightly different temperature differences.

Figure 10

Figure of merit in a N-FI-S-FI-N heat engine as a function of temperature for different polarizations $P$ of the junction. The figure has been calculated with $h=0.5\mathrm{\Delta }$ and $g=1000$, without calculating $\mathrm{\Delta }$ self-consistently. The solid lines correspond to $\mathrm{\Gamma }={10}^{-6}\mathrm{\Delta }$ and the dashed lines to $\mathrm{\Gamma }={10}^{-4}\mathrm{\Delta }$. The figure of merit at low temperatures reaches very close to $P^2/(1-P^2)$ unless $P$ is very close to unity, but the exact temperature scale where this happens depends on the value of polarization. At the lowest temperatures $ZT$ is limited by another spurious heat conduction process, due to nonzero density of states inside the gap, described here by the Dynes $\mathrm{\Gamma }$ parameter.