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

TL;DR

This paper reviews recent advances in understanding non-equilibrium phenomena in spin-split superconductors, highlighting their transport properties and potential applications in thermoelectricity and spintronics.

Contribution

It provides a comprehensive theoretical framework based on quantum kinetic equations to describe non-equilibrium charge, spin, and energy distributions in hybrid superconducting structures.

Findings

Large thermoelectric effects observed in spin-split superconductors

Transport measurements reveal properties of non-equilibrium modes

Spin injection and diffusion are extensively characterized

Abstract

We review the recent progress in understanding the properties of spin-split superconductors under non-equilibrium 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. We discuss different experimental situations and present a theoretical framework based on quantum kinetic equations. Within this framework we provide an accurate description of the non-equilibrium distribution of charge, spin and energy, which are the relevant non-equilibrium modes, in different hybrid structures. We also review experiments on spin-split superconductors and show how transport measurements reveal the properties of the non-equilibrium modes and their mutual coupling. We discuss in detail spin injection and diffusion and very large thermoelectric effects in spin-split superconductors.

Figures

Tables

Table 1. Table 1: Magnetic properties of different ferromagnetic insulator-superconductor junctions used in experiments. Middle column shows the spin-filter efficiency characterized by the polarization P = ( G ↑ − G ↓ ) / ( G ↑ + G ↓ ) 𝑃 subscript 𝐺 ↑ subscript 𝐺 ↓ subscript 𝐺 ↑ subscript 𝐺 ↓ P=(G_{\uparrow}-G_{\downarrow})/(G_{\uparrow}+G_{\downarrow}) of the FI barrier (red) with normal-state conductance G σ subscript 𝐺 𝜎 G_{\sigma} for spin σ 𝜎 \sigma . The exchange splittings measured in the superconductor (blue) are listed in the right column. The data is extracted from 1 Tedrow  et al. ( 1986 ) ; 2 Moodera  et al. ( 1988 ) ; 3 Hao  et al. ( 1990 ) ; 4 Moodera  et al. ( 1993 ) ; 5 Senapati  et al. ( 2011 ) ; 6 Pal and Blamire ( 2015 ) . Note that μ B ⋅ \mu_{B}\cdot 1 T=58 μ 𝜇 \mu eV ≅ \cong 670 mK.

Material Combination	Barrier polarization	Exchange Splitting (applied field)
EuO/Al/AlO3/Al 1	no spin-filter barrier	1 T (0.1 T)-1.73 T(0.4 T)
Au/ EuS/Al 2	0.8	1.6 T (0 T)
Al/ EuS/Al 3	0.6-0.85	1.9-2.6 T (0T)
Ag/ EuSe/Al 4	$>$ 0.97	none at zero field
EuSe/Al/AlO3/Ag 4	no spin-filter barrier	4 T (0.6 T)
NbN/ GdN/NbN 5	0.75
NbN/ GdN/TiN 6	0.97	1.4 T (0T)

Equations

$$N=\frac{1}{2}\mathrm{Re}\frac{\epsilon +h_{\mathrm{eff}}}{\sqrt{{\left(\epsilon +h_{\mathrm{eff}}\right)}^2-{\Delta }^2}}+\frac{1}{2}\mathrm{Re}\frac{\epsilon -h_{\mathrm{eff}}}{\sqrt{{\left(\epsilon -h_{\mathrm{eff}}\right)}^2-{\Delta }^2}},$$

$${\check{g}}=\left(\begin{array}[]{cc}{\hat{g}^{R}}&{\hat{g}^{K}}\\ 0&{\hat{g}^{A}}\end{array}\right)\;.$$

$${\check{g}}^2=\check{1}.$$

$$D\nabla \cdot (\check{g}\nabla \check{g})+[i\epsilon {\tau }_3-i\mathbf{h}\cdot \mathbf{\sigma }{\tau }_3-\check{\Delta }-\check{\Sigma },\check{g}]=0.$$

$${\nabla }_k{j}_{kb}^a=H^{ab}+R^{ab}+I_{\mathrm{coll}}^{ab},$$

$$j_{kb}^a=\frac{1}{8}\mathrm{Tr}{\tau }_b{\sigma }_a(\check{g}{\nabla }_k\check{g}{)}^K.$$

$$J_k=\frac{{\sigma }_N}{2e}{\int }_{-\infty }^{\infty }d\epsilon j_{k3}^0,$$

$${\sigma }_N{j}_{\perp ,b}^a=-\frac{1}{8e{R}_{\square }}\mathrm{Tr}{\tau }_b{\sigma }_a{\left[\hat{\Gamma }{\check{g}}_2{\hat{\Gamma }}^{†},\check{g}\right]}^K,$$

$${\hat{g}}^K={\hat{g}}^R\hat{f}-\hat{f}{\hat{g}}^A.$$

$$\hat{f}=f_L\hat{1}+f_T{\tau }_3+(f_{T3}{\sigma }_3+f_{L3}{\sigma }_3{\tau }_3).$$

$${\hat{f}}_{eq}(\epsilon )=(1-2n_F)\hat{1}=\tanh (\epsilon /2T)\hat{1}.$$

$$\mu (\mathbf{r},t)$$

$${\mu }_{sa}(\mathbf{r},t)$$

$$q(\mathbf{r},t)$$

$$q_{sa}(\mathbf{r},t)$$

$$\mu =-\frac{1}{2}{\int }_{-\infty }^{\infty }d\epsilon (N_{+}{f}_T+N_{-}{f}_{L3})$$

$${\mu }_z=-\frac{1}{2}{\int }_{-\infty }^{\infty }d\epsilon [N_{+}{f}_{T3}+N_{-}(f_L-f_{\mathrm{eq}})],$$

$$q=\frac{1}{2}{\int }_{-\infty }^{\infty }d\epsilon \epsilon [N_{+}(f_L-f_{eq})+N_{-}{f}_{T3}]$$

$$q_{sa}=\frac{1}{2}{\int }_{-\infty }^{\infty }d\epsilon \epsilon [N_{-}(f_L-f_{\mathrm{eq}})+N_{+}{f}_{T3}].$$

$$\displaystyle f_{\mathrm{eq},L(T)}(E)=\frac{1}{2}[\tanh\bigl{(}\frac{E+eV}{2T}\bigr{)}+(-)\tanh\bigl{(}\frac{E-eV}{2T}\bigr{)}]\,.$$

$$g_{nl}=\frac{d{I}_{\det }}{d{V}_{\mathrm{inj}}}.$$

$$I_{\det }=(\mu +P_{\det }{\mu }_z)/R_{\det },$$

$$\nabla \cdot \left(\begin{array}{c}{j}_e\\ j_s\\ j_c\\ j_{se}\end{array}\right)=\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& S_{T3}\\ 0& 0& R_T& R_{L3}\\ 0& 0& R_{L3}& R_T+S_{L3}\end{array}\right)\left(\begin{array}{c}{f}_L\\ f_{T3}\\ f_T\\ f_{L3}\end{array}\right),$$

$$\left(\begin{array}{c}{j}_e\\ j_s\\ j_c\\ j_{se}\end{array}\right)=\left(\begin{array}{cccc}{D}_L& D_{T3}& 0& 0\\ D_{T3}& D_L& 0& 0\\ 0& 0& D_T& D_{L3}\\ 0& 0& D_{L3}& D_T\end{array}\right)\left(\begin{array}{c}\nabla f_L\\ \nabla f_{T3}\\ \nabla f_T\\ \nabla f_{L3}\end{array}\right).$$

$$\frac{\partial \mathbf{m}}{\partial t}+\nabla \cdot {\mathbf{j}}_s=\gamma \mathbf{m}\times {\mathbf{h}}_s-\mathbf{m}/{\tau }_S.$$

$$\left(\begin{array}[]{c}I\\ \dot{Q}\end{array}\right)=\left(\begin{array}[]{cc}G&\alpha\\ \alpha&G_{\rm th}T\end{array}\right)\left(\begin{array}[]{c}V\\ -\Delta T/T\end{array}\right).$$

$$ZT=\frac{{\alpha }^2}{G_{\mathrm{th}}GT-{\alpha }^2}=\frac{S^{2}GT}{{\tilde{G}}_{\mathrm{th}}},$$

$$\alpha ={\alpha }_{N}G(\Delta /T),G(x)=\frac{3}{2{\pi }^2}{\int }_x^{\infty }\frac{y^{2}dy}{{\cosh }^2(y/2)},$$

$$I$$

$${\dot{Q}}_i$$

$$G$$

$$G_{\mathrm{th}}$$

$$\alpha$$

$$S=\frac{\alpha }{GT}\approx \frac{P\Delta }{eT}[\tanh (\tilde{h})-h/\Delta ].$$

Full text

1. I Introduction

2. II Superconductor with an exchange field

3. II.1 Brief overview of the quasiclassical theory of diffusive superconductors

4. III Nonequilibrium modes in spin-split superconductors

5. III.1 Description of nonequilibrium modes in superconductors with spin splitting

6. III.2 Accumulations in terms of the non-equilibrium modes

7. IV Spin injection and diffusion in superconductors

8. IV.1 Detection of spin and charge imbalance: Non-local transport measurements

9. IV.2 Non-local conductance measurements in spin-split superconductors

10. IV.3 Spin Hanle effect

11. IV.4 Spin imbalance by ac excitation

12. V Thermoelectric effects

13. V.1 Charge and heat currents at a spin-polarized interface to a spin-split superconductor

14. V.2 Linear response: heat engine based on a superconductor/ferromagnet structure

15. V.3 Spin Seebeck effect

16. V.4 Thermophase in a S(FI)S contact

17. VI Summary and Outlook

I Introduction

Ferromagnetism and spin-singlet superconductivity are antagonist orders and hardly coexist in bulk materials. However, hybrid nanostructures allow for the possibility of combining the two phenomena via mutual proximity effects. The combination leads to the emergence of novel features not present in either system alone. We can make a distinction among those characteristics affecting the spectral properties of the materials, showing up when the probed systems are in equilibrium, and those related to nonequilibrium phenomena. The emphasis of our text is in the latter phenomena, especially related to steady-state currents or voltages applied across the structures.

Both superconductors and ferromagnets are examples of electron systems with spontaneously broken symmetries, and thereby characterized by order parameters. The order parameter for a conventional spin singlet superconductor is the amplitude of (Cooper) pairing between electrons in states with opposite spins and momenta Bardeen et al. (1957). The presence of this complex pairing amplitude $F$ leads to two characteristic features of conventional superconductivity Tinkham (1996); de Gennes (1999): An equilibrium supercurrent that is proportional to the gradient of the phase of $F$ and that can be excited without voltage, and to the quasiparticle spectrum exhibiting an energy gap proportional to the absolute value of $F$. The resulting density of states (DOS, Eq. (1) for $h_{\rm eff}=0$) is strongly energy dependent and results into a non-linear nonequilibrium response of superconductors.

The main defining feature of ferromagnets is the broken spin-rotation symmetry into the direction of magnetization, and the associated exchange energy $h$ that splits the spin up and down spectra. This also leads to a strong spin dependence (spin polarization) of the observables related to ferromagnets.

There are two mechanisms that prevent most of the ferromagnetic materials from becoming superconducting. One of them is the orbital effect due to the intrinsic magnetic field in ferromagnets. When this field exceeds a certain critical value, superconductivity is suppressed Ginzburg (1957). The second mechanism is the paramagnetic effect Chandrasekhar (1962); Clogston (1962); Saint-James et al. (1969). This is due to the intrinsic exchange field of the ferromagnet that shows up as a splitting of the energy levels of spin-up and spin-down electrons and hence prevents the formation of Cooper pairs. We focus here on the regime where this spin-splitting field is present, but not yet too large to kill superconductivity.

In superconductors the spin-splitting field can be generated either due to the Zeeman effect in magnetic field or as a result of the exchange interaction between the electrons forming Cooper pairs and those which determine the magnetic order. Such fields can lead to drastic modifications of the ground state of a spin-singlet superconductor. The best-known example is the formation of the spatially inhomogeneous superconducting state predicted by Fulde and Ferrell (1964) and Larkin and Ovchinnikov (1965) and dubbed as FFLO. Although extensively studied in the literature, the FFLO phase only takes place in a narrow parameter window and therefore its experimental realization is challenging.