Spin-pumping in superconductor-antiferromagnetic insulator bilayers
SSpin-pumping in superconductor-antiferromagnetic insulator bilayers
Eirik Holm Fyhn and Jacob Linder Center for Quantum Spintronics, Department of Physics, NorwegianUniversity of Science and Technology, NO-7491 Trondheim, Norway (Dated: February 18, 2021)We study theoretically spin pumping in bilayers consisting of superconductors and antiferromagnetic insulators.We consider both compensated and uncompensated interfaces and include both the regular scattering channeland the Umklapp scattering channel. We find that at temperatures close to the critical temperatures andprecession frequencies much lower than the gap, the spin-current is enhanced in superconductors as comparedto normal metals. Otherwise, the spin-current is suppressed. The relevant precession frequencies where thespin-current in SC/AFI is enhanced compared to NM/AFI is much lower than the typical resonance frequenciesof antiferromagnets, which makes the detection of this effect experimentally challenging. A possible solution liesin the shifting of the resonance frequency by a static magnetic field.
I. INTRODUCTION
Both superconductors (SC) and antiferromagnets (AF) are ofparticular interest in the context of spintronics. Antiferromag-nets disturb neighbouring components less than ferromagneticor ferrimagnetic materials, because they produce no net strayfield [1]. This means that antiferromagnetic components can bepacked more tightly and are more robust against external mag-netic fields than their ferromagnetic counterparts. Additionally,antiferromagnets operate at THz frequencies, which are muchfaster than the GHz frequencies of ferromagnets (F). This canallow for ultrafast information processing when working withantiferromagnets.Superconductivity is a type of order that normally com-petes with magnetism. However, the discovery of spin-tripletsuperconductivity has shown that complete synergy betweensuperconductivity and magnetism is possible [2–6], and super-conductors are now an integral part of spintronics research. Inaddition to the potential for minimal Joule heating that comeswith superconductivity, superconductors are interesting from aspintronics perspective because of spin-charge separation [7, 8],which allows spin- and charge-imbalances to decay over differ-ent length scales. It has been observed that the spin relaxationtime can be considerably longer than the charge relaxationtime [9].Since both superconductors and antiferromagnets are usefulas building blocks in spintronic devices, it is of interest tostudy spin-transport in hybrid superconductor-antiferromagnetdevices. Despite this, SC/AF structures are largely unex-plored compared to superconductor-ferromagnetic structures.Here, we study theoretically spin-pumping in superconductor-antiferromagnetic insulator (SC/AFI) bilayers. This refers tothe injection of a spin-current in the superconductor by theapplication of a precessing magnetic field in the AFI [10]. Spinpumping has been observed in F/SC structures [11–13] and in-vestigated theoretically in F/SC structures by calculations basedon the local dynamic spin susceptibility in the SC [14, 15] andquasiclassical theory [16, 17]. The theoretical works found anenhanced spin current in superconductors compared to normalmetals (NMs) below the transition temperatures [14, 15].While spin-pumping in SC/AF structures has, to our knowl-edge, not been explored, some important work has been done with normal metal-antiferromagnetic systems. It has beenfound theoretically that spin-pumping is of a similar magni-tude as in the ferromagnetic case [18, 19], and more recentlymeasurements of the inverse spin-Hall voltage demonstratedthe spin-pumping effect in MnF /Pt [20]. Combining thedemonstration of AF/NM spin-pumping with the above men-tioned evidence of F/SC spin-pumping, AF/SC spin-pumpingis feasible and merits further study.We mainly follow the methodology presented in [15], butmodified for a superconductor-antiferromagnetic insulator bi-layer. In particular, the staggered magnetic order of the AFIgives rise to two different scattering channels [21–23], and thetwo different sublattices can be coupled to the superconductorin a symmetric or asymmetric way. To capture this we will notapproximate the interaction Hamiltonian by a uniform scatter-ing amplitude, as in [15], but instead model the interaction withan exchange coupling between itinerant electrons in the SCand the localized spins in the AFI. Using this coupling, it turnsout that the relevant quantity is not the local dynamic spin sus-ceptibility, as in [14, 15], but instead the planar dynamic spinsusceptibility. Using the planar dynamic spin susceptibilitywe find that the spin-pumping into superconductors from anti-ferromagnets is enhanced as compared to spin-pumping intonormal metals when the temperature is close to the transitiontemperature and the precession frequency is small compared tothe energy gap. Otherwise the spin-current in the superconduc-tor is suppressed. This is similar to the results obtained fromferromagnets. However, unlike in the case of ferromagnets,the resonance frequency in antiferromagnets is typically toolarge for spin-pumping with frequencies below the gap to beexperimentally detectable. One possible solution is to apply astatic magnetic field, which we discuss in Section VI. II. MODEL
The system depicted in Fig. 1 is modelled by the Hamiltonian 𝐻 = 𝐻 SC + 𝐻 AFI + 𝐻 int , (1)where the Bogoliubov-de Gennes Hamiltonian, 𝐻 SC = ∑︁ 𝒌 ∈ (cid:3) (cid:16) 𝑐 † 𝒌 , ↑ 𝑐 − 𝒌 , ↓ (cid:17) (cid:18) 𝜉 𝒌 ΔΔ ∗ − 𝜉 𝒌 (cid:19) (cid:18) 𝑐 𝒌 , ↑ 𝑐 †− 𝒌 , ↓ (cid:19) , (2) a r X i v : . [ c ond - m a t . s up r- c on ] F e b h ( t ) SC AFI
FIG. 1: Sketch of a Superconductor (SC)-antiferromagnetic insulator(AFI) bilayer with a precessing external magnetic field 𝒉 ( 𝑡 ) . where (cid:3) is the first Brillouin zone (1BZ) in the superconductor,gives a mean-field description of superconductivity. Theantiferromagnetic insulator Hamiltonian is given by 𝐻 AFI = 𝐽 ∑︁ (cid:104) 𝑖, 𝑗 (cid:105) 𝑺 𝑖 · 𝑺 𝑗 − 𝐾 ∑︁ 𝑖 𝑆 𝑖,𝑧 − 𝛾 ∑︁ 𝑖 𝑺 𝑖 · 𝒉 . (3)where (cid:104) 𝑖, 𝑗 (cid:105) means that the sum goes over nearest neighboursand (cid:205) 𝑖 goes over lattice points in the AFI. The exchangecoupling at the interface is given by 𝐻 int = − ∑︁ 𝑖 𝐽 𝑖 (cid:16) 𝑐 † 𝑖, ↑ 𝑐 † 𝑖, ↓ (cid:17) 𝝈 (cid:18) 𝑐 𝑖, ↑ 𝑐 𝑖, ↓ (cid:19) · 𝑺 𝑖 , (4)where the sum goes over the lattice points in the interface. Here, 𝜉 𝒌 , is the kinetic energy measured relative to the chemicalpotential 𝜇 , 𝑐 𝒌 ,𝜎 is the annihilation operator for electrons withspin 𝜎 and wavevector 𝒌 , 𝐽 is the antiferromagnetic exchangeparameter, 𝐾 is the easy-axis anisotropy, 𝑺 𝑖 is the spin at latticesite 𝑖 in the AFI and 𝛾 gives the coupling strength to the externalmagnetic field 𝒉 . The vector of Pauli matrices is given by 𝝈 ,and 𝐽 𝑖 = 𝐽 𝐴 ( 𝐽 𝑖 = 𝐽 𝐵 ) when 𝑖 belongs to the 𝐴 ( 𝐵 ) sublattice.Also, Δ is the superconducting gap parameter, which we assumereal and satisfies1 = 𝜆 ∫ 𝜔 𝐷 tanh (cid:16) √ 𝜀 + Δ / 𝑇 (cid:17) √ 𝜀 + Δ , (5)where 𝑇 is the temperature, which we assume to be the samefor the superconductor and AFI, and 𝜔 𝐷 and 𝜆 are material-specific parameters that determine the critical temperature 𝑇 𝑐 and the zero-temperature gap Δ (cid:66) Δ ( ) .In order diagonalize 𝐻 AFI we can do a Holstein-Primakofftransformation followed by a Fourier transform and a Bogoli-ubov transformation. This gives the following antiferromag-netic Hamiltonian: 𝐻 AFI = ∑︁ 𝒌 ∈ (cid:94) (cid:16) 𝜔 𝛼 𝒌 𝛼 † 𝒌 𝛼 𝒌 + 𝜔 𝛽 𝒌 𝛽 † 𝒌 𝛽 𝒌 (cid:17) + √︁ 𝑁 𝐴 𝑆 ( 𝑢 + 𝑣 ) 𝛾 (cid:104) ℎ − (cid:16) 𝛼 + 𝛽 † (cid:17) + ℎ + (cid:16) 𝛼 † + 𝛽 (cid:17)(cid:105) (6)where (cid:94) is the first magnetic Brillouin zone, which is the 1BZcorresponding to the 𝐴 sublattice, 𝑁 𝐴 is the number of lattice points in the 𝐴 sublattice, 𝑆 is the spin at each lattice point and 𝑢 𝒌 = 𝐽𝑧 + 𝐾 √︃ ( 𝐽𝑧 + 𝐾 ) − ( 𝐽𝛾 𝒌 ) , (7a) 𝑣 𝒌 = − 𝐽𝛾 𝒌 √︃ ( 𝐽𝑧 + 𝐾 ) − ( 𝐽𝛾 𝒌 ) , (7b) 𝜔 𝛼 𝒌 = 𝑆 √︃ ( 𝐽𝑧 + 𝐾 ) − ( 𝐽𝛾 𝒌 ) + 𝛾ℎ 𝑧 , (7c) 𝜔 𝛽 𝒌 = 𝑆 √︃ ( 𝐽𝑧 + 𝐾 ) − ( 𝐽𝛾 𝒌 ) − 𝛾ℎ 𝑧 . (7d)Here, ℎ 𝑧 is the 𝑧 -component of the external magnetic field,which is the same as the magnetization direction in the an-tiferromagnet and the direction of the easy-axis anisotropy.Moreover, ℎ ± = ℎ 𝑥 ± 𝑖ℎ 𝑦 and 𝛾 𝒌 = ∑︁ (cid:104) 𝜹 (cid:105) cos ( 𝒌 · 𝜹 ) = 𝛾 − 𝒌 , (8)where the sum goes over the nearest neighbour displacementvectors 𝜹 , and 𝑧 is the number of nearest neighbours.To write 𝐻 int in terms of Fourier components requires usto connect the reciprocal space in the superconductor withthe reduced Brillouin zone of the magnetic lattice in the AFI.This gives rise to so-called Umklapp scattering, where thewavevector falls outside the 1BZ in the AFI [23]. Whetherthis effect is present depends on the interface. Dependingon how the interface slices the biparte lattice of the AFI, theinterface can have a different number of atoms belonging tothe 𝐴 and 𝐵 lattices. If the interface has an equal number ofatoms from each sublattice and the coupling strengths 𝐽 𝐴 and 𝐽 𝐵 are equal, we call it a compensated interface. Otherwise,it is uncompensated. We let 𝒙 = to be the location of alattice point belonging to the 𝐴 sublattice and 𝒙 be such thatall lattice points at the interface can be written 𝒙 + ˜ 𝒙 𝑖 , where 𝒙 · ˜ 𝒙 𝑖 = 𝛿 𝐴 𝒒 (cid:107) , 𝒌 (cid:107) = 𝒒 · ˜ 𝒙 𝑖 − 𝒌 · ˜ 𝒙 𝑖 = 𝜋𝑛 + 𝑑 for all vectors ˜ 𝒙 𝑖 such that 𝒙 + ˜ 𝒙 𝑖 is in the 𝐴 -sublatticeat the interface and for some integer 𝑛 and a constant 𝑑 that is independent of ˜ 𝒙 𝑖 . Similarly, 𝛿 𝐵 𝒒 (cid:107) , 𝒌 (cid:107) = 𝒒 · ˜ 𝒙 𝑖 − 𝒌 · ˜ 𝒙 𝑖 = 𝜋𝑛 + 𝑑 for all lattice vectors 𝒙 + ˜ 𝒙 𝑖 inthe 𝐵 sublattice at the interface and for some integer 𝑛 anda constant 𝑑 that is independent of ˜ 𝒙 𝑖 . We can determine 𝑑 by noting that both ˜ 𝒙 𝑖 and 2 ˜ 𝒙 𝑖 is in the 𝐴 sublattice, so2 𝑑 = 𝑑 + 𝜋𝑛 = ⇒ 𝑑 = 𝜋𝑚 for some integer 𝑚 . Hence,we can set 𝑑 =
0. Similarly, if ˜ 𝒙 𝑖 is in the 𝐵 sublattice, then2 ˜ 𝒙 𝑖 is in the 𝐴 sublattice, so 4 𝜋𝑛 + 𝑑 = 𝜋𝑚 = ⇒ 𝑑 = 𝑙𝜋 for some integer 𝑙 . The 𝒌 -vectors that result in 𝑙 being anodd-number give rise to the Umklapp scattering channel. Wecan drop the superscripts because 𝛿 𝐴 𝒒 (cid:107) , 𝒌 (cid:107) = ⇐⇒ 𝛿 𝐵 𝒒 (cid:107) , 𝒌 (cid:107) = 𝐵 sublattice is midwaybetween two lattice points in the 𝐴 sublattice and vica versa.Finally, if the number of lattice points at the interface is equalon the superconductor and the antiferromagnet, then half of thepossible 𝒌 -vectors in the superconductor will give 𝑙 = 𝑙 =
1. There is a vector 𝑮 connectingthe region in (cid:3) with 𝑙 = 𝑙 = (cid:3) , is therefore also cubical. Meanwhile, thesublattice in the AFI is face-centered cubic, so (cid:94) is the truncatedoctahedron inscribed in (cid:3) . A wavevector in the corner of (cid:3) willbe in the center of the second Brillouin zone in the AFI. If welet 𝑮 be the vector in a corner of (cid:3) , then exp ( 𝑖 𝑮 · 𝒙 𝑖 ) is 1 when 𝒙 𝑖 is in the 𝐴 sublattice and − 𝒙 𝑖 is in the 𝐵 sublattice.Thus 𝑮 is the vector that connects the region of 𝒌 -vectors in (cid:3) with regular scattering and those with Umklapp-scattering.Using this notation, 𝐻 int can be written 𝐻 int = ∑︁ 𝒌 ∈ (cid:3) ∑︁ 𝒒 ∈ (cid:94) (cid:104) 𝑇 𝛼 𝒒𝒌 𝛼 𝒒 𝑠 − 𝒌 + 𝑇 𝛽 † 𝒒𝒌 𝛽 † 𝒒 𝑠 − 𝒌 + h.c. (cid:105) + 𝐻 𝑍 int , (9)where 𝐻 𝑍 int = − √︁ 𝑆𝑁 𝐴 ∑︁ 𝒌 ∈ (cid:3) 𝛿 𝒌 (cid:107) , (cid:16) ¯ 𝐽 𝐴 − (− ) 𝑙 ¯ 𝐽 𝐵 (cid:17) 𝑠 𝑧 𝒌 (10)is the Zeeman energy and 𝑇 𝛼 𝒒𝒌 = − e 𝑖 𝒙 ·( 𝒌 + 𝒒 ) (cid:104) ¯ 𝐽 𝐴 𝑢 𝒒 + (− ) 𝑙 ¯ 𝐽 𝐵 𝑣 𝒒 (cid:105) 𝛿 𝒌 (cid:107) , − 𝒒 (cid:107) , (11a) 𝑇 𝛽 † 𝒒𝒌 = − e 𝑖 𝒙 ·( 𝒌 − 𝒒 ) (cid:104) ¯ 𝐽 𝐴 𝑣 𝒒 + (− ) 𝑙 ¯ 𝐽 𝐵 𝑢 𝒒 (cid:105) 𝛿 𝒌 (cid:107) , 𝒒 (cid:107) . (11b)Additionally, ¯ 𝐽 𝐴 = 𝐽 𝐴 √ 𝑆𝑁 (cid:107) 𝐴 𝑁 𝑆 √ 𝑁 𝐴 , (12a)¯ 𝐽 𝐵 = 𝐽 𝐵 √ 𝑆𝑁 (cid:107) 𝐵 𝑁 𝑆 √ 𝑁 𝐴 , (12b)where 𝑁 𝑆 is the number of lattice points in the superconductorand 𝑁 (cid:107) 𝐴 ( 𝑁 (cid:107) 𝐵 ) is the number of lattice points belonging to the 𝐴 ( 𝐵 ) sublattice at the interface, and 𝑠 𝑧 𝒌 = ∑︁ 𝒒 ∈ (cid:3) (cid:16) 𝑐 † 𝒒 ↑ 𝑐 𝒒 + 𝒌 ↑ − 𝑐 † 𝒒 ↓ 𝑐 𝒒 + 𝒌 ↓ (cid:17) , (13a) 𝑠 − 𝒌 = ∑︁ 𝒒 ∈ (cid:3) 𝑐 † 𝒒 ↓ 𝑐 𝒒 + 𝒌 ↑ . (13b) III. GREEN’S FUNCTIONS
In order to calculate the spin current we will make use ofGreen’s functions corresponding to three different types ofoperators. Let 𝜓 be either 𝛼 , 𝛽 † or 𝑠 + , then the lesser, retardedand advanced Green’s functions are 𝐺 <𝜓 ( 𝑡 , 𝑡 , 𝒌 ) = − 𝑖 (cid:68) 𝜓 † 𝒌 ( 𝑡 ) 𝜓 𝒌 ( 𝑡 ) (cid:69) , (14a) 𝐺 𝑅𝜓 ( 𝑡 , 𝑡 , 𝒌 ) = − 𝑖𝜃 ( 𝑡 − 𝑡 ) (cid:68) (cid:104) 𝜓 𝒌 ( 𝑡 ) , 𝜓 † 𝒌 ( 𝑡 ) (cid:105) (cid:69) , (14b) 𝐺 𝐴𝜓 ( 𝑡 , 𝑡 , 𝒌 ) = 𝑖𝜃 ( 𝑡 − 𝑡 ) (cid:68) (cid:104) 𝜓 𝒌 ( 𝑡 ) , 𝜓 † 𝒌 ( 𝑡 ) (cid:105) (cid:69) , (14c) respectively. The subscript 0 means that the expectation valuesare taken in the absence on 𝐻 int . This is done because we willtreat 𝐻 int as a perturbation in the interaction picture. We willalso define the distribution function 𝑓 𝜓 ( 𝜀, 𝒌 ) (cid:66) 𝐺 <𝜓 ( 𝜀, 𝒌 ) 𝑖 Im 𝐺 𝑅𝜓 ( 𝜀, 𝒌 ) , (15)where the Green’s functions in Eq. (14) are Fourier transformedwith respect to the relative time 𝑡 − 𝑡 . In thermal equilibrium, 𝑓 𝜓 ( 𝜀, 𝒌 ) is equal to the Bose-Einstein distribution function 𝑛 𝐵 ( 𝑇, 𝜀 ) .First consider the effect of spin pumping. We add spinpumping in the AFI by letting ℎ ± ( 𝑡 ) = ℎ e ∓ 𝑖 Ω 𝑡 . The readeris referred to Appendix A for the detailed calculation, whichshows that the retarded Green’s functions are unaffected tosecond order in ℎ . Since the unperturbed Hamiltonian isdiagonal in 𝛼 and 𝛽 , this means that the retarded Green’sfunctions for 𝛼 and 𝛽 † are 𝐺 𝑅𝛼 ( 𝜀, 𝒌 ) = 𝜀 − 𝜔 𝛼 𝒌 + 𝑖𝜂 𝛼 , (16a) 𝐺 𝑅𝛽 † ( 𝜀, 𝒌 ) = − 𝐺 𝐴𝛽 (− 𝜀, 𝒌 ) = 𝜀 + 𝜔 𝛽 𝒌 + 𝑖𝜂 𝛽 , (16b)where 𝜂 𝛼 and 𝜂 𝛽 are the lifetimes of the 𝛼 and 𝛽 magnons. Thedistribution functions are modified by the oscillating magneticfield, and to second order in ℎ 𝑓 𝜈 ( 𝜀, 𝒌 ) = 𝑛 𝐵 ( 𝜀, 𝑇 )+ 𝜋𝑁 𝐴 𝑆 [( 𝑢 + 𝑣 ) 𝛾ℎ ] 𝜂 𝜈 𝛿 𝒌 , 𝛿 ( 𝜀 − Ω ) , (17)where 𝜈 ∈ { 𝛼, 𝛽 † } .The dynamic spin susceptibility 𝐺 𝑅𝑠 + is more complicated,but can be calculated from the imaginary time Green’s functionby use of analytical continuation and Matsubara summationtechniques. This is shown in Appendix B, and the result is 𝐺 𝑅𝑠 + ( 𝜀, 𝒌 ) = − ∑︁ 𝒒 ∑︁ 𝜔 = ± 𝐸 ∑︁ ˜ 𝜔 = ± ˜ 𝐸 (cid:18) + 𝜉 ˜ 𝜉 + Δ 𝜔 ˜ 𝜔 (cid:19) × 𝑛 𝐹 ( ˜ 𝜔, 𝑇 ) − 𝑛 𝐹 ( 𝜔, 𝑇 ) 𝜀 + 𝑖𝜂 SC − ( ˜ 𝜔 − 𝜔 ) , (18)where 𝜉 = 𝜉 𝒒 , ˜ 𝜉 = 𝜉 𝒒 + 𝒌 , 𝐸 = √︁ 𝜉 + Δ and ˜ 𝐸 = √︁ ˜ 𝜉 + Δ , 𝑛 𝐹 is the Fermi-Dirac distribution function. Since the spin-pumping in the AFI does not affect the Hamiltonian in the super-conductor, the distribution function is 𝑓 𝑠 + ( 𝜀, 𝒌 ) = 𝑛 𝐵 ( 𝜀, 𝑇 ) . IV. SPIN CURRENT
To find the spin current we follow Kato et al. [15] and usethat 𝐼 𝑠 = − 𝜕𝜕𝑡 (cid:10) 𝑠 𝑧 (cid:11) = − 𝑖 (cid:10) (cid:2) 𝐻, 𝑠 𝑧 (cid:3) (cid:11) . (19)From the fact that 𝑠 𝑧 commutes with 𝐻 SC + 𝐻 AFI , (cid:2) 𝑠 − 𝒒 , 𝑠 𝑧 (cid:3) = 𝑠 − 𝒒 ,and [ 𝐴 † , 𝐵 ] = −[ 𝐴, 𝐵 † ] † , we find that [ 𝐻, 𝑠 tot ] = ∑︁ 𝒌 ∈ (cid:3) ∑︁ 𝒒 ∈ (cid:94) (cid:104) 𝑇 𝛼 𝒒𝒌 𝛼 𝒒 𝑠 − 𝒌 + 𝑇 𝛽 † 𝒒𝒌 𝛽 † 𝒒 𝑠 − 𝒌 − h.c. (cid:105) . (20)Thus, the spin current is 𝐼 𝑠 ( 𝑡 ) = ∑︁ 𝒌 ∈ (cid:3) ∑︁ 𝒒 ∈ (cid:94) ∑︁ 𝜈 ∈{ 𝛼, 𝛽 † } Im (cid:68) 𝑇 𝜈 𝒒𝒌 𝑠 − 𝒌 ( 𝑡 ) 𝜈 𝒒 ( 𝑡 ) (cid:69) . (21)We evaluate this expectation value in the interaction pictureand treating the interfacial exchange interaction as a pertur-bation using the Keldysh formalism. First, let 𝐺 𝜓 with nosuperscript denote contour-ordered Green’s functions, 𝐺 𝜓 ( 𝜏 , 𝜏 , 𝒌 ) = − 𝑖 (cid:68) T 𝑐 𝜓 𝒌 ( 𝜏 ) 𝜓 † 𝒌 ( 𝜏 ) (cid:69) , (22)where T 𝑐 means that 𝜓 𝒌 and 𝜓 † 𝒌 are ordered with respect to 𝜏 and 𝜏 along the complex Keldysh contour, C . Next, we define 𝐶 ( 𝜏 , 𝜏 ) (cid:66) (cid:68) T 𝑐 𝑇 𝜈 𝒒𝒌 𝜈 𝒒 ( 𝜏 ) 𝑠 − 𝒌 ( 𝜏 ) (cid:69) , (23)where 𝜈 is either 𝛼 or 𝛽 † .Going to the interaction picture with 𝐻 int as the interaction,we get 𝐶 ( 𝜏 , 𝜏 ) = (cid:68) T 𝑐 𝑇 𝜈 𝒒𝒌 𝜈 𝒒 ( 𝜏 ) 𝑠 − 𝒌 ( 𝜏 ) e − 𝑖 ∫ C d 𝜏𝐻 int ( 𝜏 ) (cid:69) ≈ (cid:28) T 𝑐 ∫ C d 𝜏 (cid:12)(cid:12)(cid:12) 𝑇 𝜈 𝒒𝒌 (cid:12)(cid:12)(cid:12) 𝜈 𝒒 ( 𝜏 ) 𝜈 † 𝒒 ( 𝜏 ) 𝑠 +− 𝒌 ( 𝜏 ) 𝑠 − 𝒌 ( 𝜏 ) (cid:29) = 𝑖 (cid:12)(cid:12)(cid:12) 𝑇 𝜈 𝒒𝒌 (cid:12)(cid:12)(cid:12) (cid:2) 𝐺 𝜈 ( 𝒒 ) • 𝐺 𝑠 + ( 𝒌 ) (cid:3) ( 𝜏 , 𝜏 ) , (24)where we have used the bullet product • to denote integration ofthe internal complex time parameter along the Keldysh contour.In the second equality it was used that − 𝑖 (cid:10) T 𝑐 𝑠 +− 𝒌 (cid:48) ( 𝜏 ) 𝑠 − 𝒌 ( 𝜏 ) (cid:11) = 𝛿 𝒌 , 𝒌 (cid:48) 𝐺 𝑠 + ( 𝜏, 𝜏 , 𝒌 ) , (25)as can be confirmed by using Wick’s theorem. Next, if wechoose 𝜏 to be placed later in the contour we have 𝐶 ( 𝜏 , 𝜏 ) = 𝐶 < ( 𝜏 , 𝜏 ) = (cid:68) 𝑇 𝜈 𝒒𝒌 𝑠 − 𝒌 ( 𝜏 ) 𝜈 𝒒 ( 𝜏 ) (cid:69) . (26)From the Langreth rules we have 𝐶 < ( 𝑡, 𝑡 ) = (cid:2) 𝐺 𝑅𝜈 ( 𝒒 ) ◦ 𝐺 <𝑠 + ( 𝒌 ) + 𝐺 <𝜈 ( 𝒒 ) ◦ 𝐺 𝐴𝑠 + ( 𝒌 ) (cid:3) ( 𝑡, 𝑡 ) , (27)where the circle product ◦ means integration over the internalreal time coordinate. The circle products are the same asnormal convolution products, since 𝐺 𝑅𝜓 ( 𝑡 , 𝑡 ) and 𝐺 <𝜓 ( 𝑡 , 𝑡 ) only depend on time through the relative time 𝑡 − 𝑡 . Thus,by writing Eq. (27) in terms of Fourier transformed Green’s functions, the circle products become normal products, so, byinserting it into Eq. (21), 𝐼 𝑠 = ∫ d 𝜀 𝜋 ∑︁ 𝒌 ∈ (cid:3) ∑︁ 𝒒 ∈ (cid:94) ∑︁ 𝜈 ∈{ 𝛼, 𝛽 † } (cid:12)(cid:12)(cid:12) 𝑇 𝜈 𝒒𝒌 (cid:12)(cid:12)(cid:12) Im 𝐺 𝑅𝜈 ( 𝜀, 𝒒 )× Im 𝐺 𝑅𝑠 + ( 𝜀, 𝒌 ) (cid:104) 𝑓 𝜈 ( 𝜀, 𝒒 ) − 𝑓 𝑠 + ( 𝜀, 𝒌 ) (cid:105) , (28)where we used that 𝐺 𝐴𝜓 ( 𝜀 ) = [ 𝐺 𝑅𝜓 ( 𝜀 )] ∗ .Inserting Eqs. (11), (16) and (17) into Eq. (28) and usingEq. (7) gives 𝐼 𝑠 = 𝐼 𝑟 + 𝐼 𝑈 , (29)where 𝐼 𝑟 = − ¯ 𝐽 𝐴 𝛾 ℎ (cid:169)(cid:173)(cid:173)(cid:171) (cid:16) Ω − 𝜔 𝛼 (cid:17) + ( 𝜂 𝛼 ) (cid:20) 𝑈 𝐾 + ( − 𝑐 ) + 𝑈 𝐾 (cid:21) + (cid:16) Ω + 𝜔 𝛽 (cid:17) + (cid:0) 𝜂 𝛽 (cid:1) (cid:20) 𝑐𝑈 𝐾 + ( 𝑐 − ) + 𝑈 𝐾 (cid:21) (cid:170)(cid:174)(cid:174)(cid:172) × ∑︁ 𝒌 ∈ (cid:3) ,𝑙 = Im 𝐺 𝑅𝑠 + ( Ω , 𝒌 ) 𝛿 𝒌 (cid:107) , (30)and 𝐼 𝑈 = − ¯ 𝐽 𝐴 𝛾 ℎ (cid:169)(cid:173)(cid:173)(cid:171) (cid:16) Ω − 𝜔 𝛼 (cid:17) + ( 𝜂 𝛼 ) (cid:20) 𝑈 𝐾 + ( + 𝑐 ) + 𝑈 𝐾 (cid:21) + (cid:16) Ω + 𝜔 𝛽 (cid:17) + (cid:0) 𝜂 𝛽 (cid:1) (cid:20) 𝑐𝑈 𝐾 + ( 𝑐 + ) + 𝑈 𝐾 (cid:21) (cid:170)(cid:174)(cid:174)(cid:172) × ∑︁ 𝒌 ∈ (cid:3) ,𝑙 = Im 𝐺 𝑅𝑠 + ( Ω , 𝒌 + 𝑮 ) 𝛿 𝒌 (cid:107) , . (31)Here, 𝑈 𝐾 = 𝐾 /( 𝐽𝑧 ) and 𝑐 = ¯ 𝐽 𝐵 / ¯ 𝐽 𝐴 is the interface asym-metry parameter that gives the degree to which the interfaceis compensated. The sums are restricted to include only the 𝒌 -vectors that satisfy 𝛿 𝒌 (cid:107) , = 𝑙 = 𝑮 is the vectorthat connects these to the 𝒌 -vectors with 𝑙 =
1. When both theSC and AFI are cubical with a lattice parameter 𝑎 and a com-pensated interface, then 𝑮 = 𝜋 ( 𝒆 𝑥 + 𝒆 𝑦 + 𝒆 𝑧 )/ 𝑎 . In order forthe Umklapp scattering to produce a nonzero 𝐼 𝑈 , it is necessarythat there exists 𝒌 , 𝒒 ∈ (cid:3) such that both 𝒒 and 𝒒 + 𝒌 + 𝑮 areclose to the Fermi surface and 𝛿 𝒌 (cid:107) , =
1. In a cubical latticethe minimal value of 𝒌 + 𝑮 is √ 𝜋 / 𝑎 , so the maximal diameterof the Fermi surface must be at least √ 𝜋 / 𝑎 . The Umklappcurrent is also zero if the interface is fully uncompensated.In this case there is no Umklapp scattering and the current issimply 𝐼 𝑠 = 𝐼 𝑟 with 𝑐 = V. NUMERICAL RESULTS
Next we show numerical results for a cubical lattice withlattice constant 𝑎 such that 𝜉 𝒌 = − 𝑡 ∑︁ 𝑖 ∈{ 𝑥,𝑦,𝑧 } cos ( 𝑎𝑘 𝑖 ) − 𝜇, (32)where 𝑡 is the hopping parameter. In Fig. 2 we show the spincurrent into the superconductor, 𝐼 SC 𝑠 , normalized by the normalstate value, 𝐼 NM 𝑠 for different temperatures 𝑇 and precessionfrequencies Ω . In this case we have used 𝜇 = − 𝑡 , which meansthat 𝐼 𝑈 =
0. However, we find that both 𝐼 𝑟 and 𝐼 𝑈 scale inthe same way as functions of Ω and 𝑇 also for other valuesof 𝜇 . In Fig. 2 we have also used 𝑈 𝑘 = .
01, which is closeto the reported value for MnF [24, 25], 𝑡 = Δ , 𝑐 = . 𝜂 𝛼 = 𝜂 𝛽 = Δ × − and 𝜔 𝛼 = 𝜔 𝛽 = Δ . This correspondsto a resonance frequency of 1 THz when Δ = 𝐼 SC 𝑠 / 𝐼 NM 𝑠 as a function of Ω / Δ forvarious 𝑇 . It can be seen that at zero temperature the spincurrent in the superconducting case is zero for Ω < Δ . For 𝑇 > 𝐼 SC 𝑠 / 𝐼 NM 𝑠 initially decreases as Ω increases andreaches a minimum at Ω = Δ ( 𝑇 ) .This can be understood physically in the following way. Thespin-current is generated by spin-flip scatterings which exciteparticles by energy Ω and flip their spin. This can be seen fromEqs. (9), (18), (30) and (31) when 𝜂 SC (cid:28)
1. In this case the sumin Eq. (18) only contribute to imaginary part of 𝐺 𝑅𝑠 + ( Ω , 𝒌 ) when˜ 𝜔 − 𝜔 = Ω , and only when 𝑛 𝐹 ( 𝜔, 𝑇 ) − 𝑛 𝐹 ( ˜ 𝜔, 𝑇 ) ≠
0. In thenormal metal case there is a number of electrons proportional to Ω around the Fermi surface which can be excited to an availablestate. Hence, the dynamic spin-susceptibility is proportional to Ω .In a superconductor the spin-flip scatterings can happenby breaking a Cooper pair or exciting a quasiparticle fromabove the gap to a higher energy. When Ω < Δ ( 𝑇 ) only thelatter is possible. Thus, in order to get a nonzero spin-currentwhen Ω < Δ ( 𝑇 ) the temperature must be large enough forquasiparticle states above the gap to be occupied. This is why,in Fig. 3, the current is identically zero in the superconductorwhen 𝑇 = Ω < Δ . On the other hand, when thetemperature is close to the critical current there can be manyavailable quasiparticles available because the density of statesis peaked around the gap. This peak in the density of states iswhy the spin-current in a superconductor can be larger than thespin-current in a normal metal, but only when the temperatureis close to the critical temperature. It is also only larger when Ω (cid:28) Δ ( 𝑇 ) , which is because the lack of states below the gapin the superconductor means that the spin susceptibility cannot increase as fast as in the normal state when Ω increases. Inthe normal state there is a range of energies ∝ Ω around theFermi surface that can be excited to an available state, but in thesuperconducting state the number of states that can be excitedis limited by the number of quasiparticles present. Increasing Ω therefore decreases the ratio 𝐼 SC 𝑠 / 𝐼 NM 𝑠 when Ω < Δ ( 𝑇 ) , ascan be seen in Figs. 2 and 3. At Ω = Δ ( 𝑇 ) the breaking ofCooper pairs becomes possible as a spin-transfer mechanism,which is why 𝐼 SC 𝑠 / 𝐼 NM 𝑠 starts to increase. This can be seen mostclearly in Fig. 3.Figure 4 shows the ratio between regular spin-current 𝐼 𝑟 andthe Umklapp contribution 𝐼 𝑈 for 𝜇 = − . 𝑡 . The result is shownfor axis anisotropy 𝑈 𝑘 = .
01, which correspond to MnF [24,25], and 𝑈 𝑘 = .
37, corresponding to FeF [26]. In both casesthe regular current dominates when the interface asymmetryparameter 𝑐 is small, meaning that the superconductor iscoupled more strongly to one of the sublattices in the AFI. TheUmklapp contribution becomes more important as 𝑐 increasesand when 𝑈 𝑘 is small the Umklapp contribution eventuallybecomes larger than the contribution from the regular scatteringchannel. This is consistent with the work by Kamra and Belzigshowing that the in the absence of easy-axis anisotropy thecross-sublattice contribution quench the spin-current from theregular scattering channel [19]. However, here we see that ifwe include the Umklapp scattering the spin-current will not goall the way to zero, even in the absence of easy-axis anisotropy.Mathematically, this can be seen from Eqs. (30) and (31): when 𝑈 𝑘 = 𝐼 𝑟 ∝ ( − 𝑐 ) and 𝐼 𝑈 ∝ ( + 𝑐 ) . However,when 𝑈 𝑘 = .
37 the regular contribution remains dominant forall values of 𝑐 . VI. EXPERIMENTAL DETECTION
Although the spin-current can be enhanced in SC/AFI bi-layers as compared to NM/AFI bilayers, it can be difficult toobserve this enhancement experimentally. This is because thespin-current is strongly peaked around the antiferromagneticresonance frequencies 𝜔 𝛼 / 𝛽 . In antiferromagnets this reso-nance frequency is on the order of 1 THz, which is much largerthan in ferromagnets [1]. This is an advantage for spintronics asit allows for ultrafast information processing, but in the contextof this paper it means that observation of the enhancementproduced by the superconducting order is hard to experimen-tally verify. A resonance frequency of 1 THz means that thespin-current is most easily observed at Ω / Δ ≈
4, assumingthat Δ = Ω / Δ < . Δ . This is also below 1 THz, but not out of reach. Theresonance frequency of MnF , which was used in the detectionof spin-pumping by Vaidya et al. , was reported to be around250 GHz [20]. This corresponds to Ω ≈ ≈ Δ , whichmakes the low-temperature suppression shown in Figs. 2 and 3detectable.One way to potentially detect the spin-current enhancementat low frequencies is to apply a constant magnetic field alongthe 𝑧 -axis. This was also done by Vaidya et al. , who reducedthe frequency of MnF to 120 GHz by applying a magneticfield of 4 . a) . . . . . . . . . . T/T c Ω / ∆ . . . . . I SC s /I NM s b) . . . . . . . . . . T/T c Ω / ∆ . . . . . . I SC /I FIG. 2: The spin-current into a superconductor with gap given byEq. (5), 𝐼 SC 𝑠 , for different precession frequencies Ω and temperatures 𝑇 and normalized by the normal state spin-current, 𝐼 NM 𝑠 , found bysetting Δ = 𝐼 = 𝛾 ℎ ¯ 𝐽 𝐴 𝑁 𝑆 𝑁 ⊥ 𝑆 / (cid:2) ( 𝜋 ) Δ (cid:3) in b). 𝑁 ⊥ 𝑆 is the number of lattice points in the superconductor in thedirection transverse to the interface, Δ is the superconducting gap at 𝑇 = 𝑇 𝑐 is the critical temperature. . . . . . .
52 2∆(0 . T c )2∆(0 . T c )2∆(0 . T c )2∆(0)Ω / ∆ I S C s / I N M s T /T c = 0 . T /T c = 0 . T /T c = 0 . T /T c = 0 . FIG. 3: The spin-current into a superconductor, 𝐼 SC 𝑠 normalized bythe normal state spin-current, 𝐼 NM 𝑠 , found by setting the gap Δ =
0, asa function of the spin-pumping precession frequency Ω . Here, Δ ( 𝑇 ) is the energy gap that solves Eq. (5) and 𝑇 𝑐 is the critical temperature. . . . . − c I r / I U MnF , U k = 0 . FeF , U k = 0 . FIG. 4: The ratio of the spin-current contribution from the regularscattering channel, 𝐼 𝑟 , and the Umklapp scattering channel, 𝐼 𝑈 , asa function of the interface asymmetry parameter 𝑐 for Ω / Δ = . 𝑇 / 𝑇 𝑐 = . 𝜇 = − . 𝑡 . The results are shown for easy axis anisotropyvalues 𝑈 𝑘 = .
01 and 𝑈 𝑘 = .
37, where the former is found in MnF and the latter is found in FeF [24–26]. resonance frequencies are 𝜔 res = 𝜔 √︁ 𝑈 𝑘 ( + 𝑈 𝑘 ) ± 𝛾ℎ 𝑧 . (33)where 𝜔 = 𝐽𝑧𝑆 . Thus, by applying a magnetic field of 𝜔 √︁ 𝑈 𝑘 ( + 𝑈 𝑘 )/ 𝛾 , the resonance frequency can be pushedwell below Δ , making the enhancement in spin-current dueto superconductivity detectable. This is illustrated in Fig. 5.At 𝛾ℎ 𝑧 = Ω = Δ , where the peak in 𝐼 SC 𝑠 isonly slightly smaller than the peak in 𝐼 NM 𝑠 , in accordance withFig. 3. However, when 𝛾ℎ 𝑧 = − . Δ , the peak is shifted to Ω = . Δ and the peak in the superconducting case is taller.How large the applied magnetic field is required to be dependon the gyromagnetic ratio 𝛾 as well as 𝜔 and 𝑈 𝑘 , but it will ingeneral be several tesla. Experimental ingenuity is thereforerequired in order to make sure that the superconductivity isnot completely suppressed by the magnetic field. This couldfor instance be done by shielding the superconductors or usingsuperconductors that can withstand large magnetic field from acertain direction, such as Ising superconductors. VII. CONCLUSION
We have derived an expression for the spin-current in SC/AFIbilayers undergoing spin-pumping, valid for both compensatedand uncompensated interfaces and taking into considerationboth the regular scattering channel and the Umklapp scatteringchannel. We found that for temperature 𝑇 well below thecritical temperature 𝑇 𝑐 , the spin-current is strongly suppressedas long as the precession frequency of the applied magneticfield is less than 2 Δ ( 𝑇 ) . This is because the energy gap in thesuperconductor inhibits spin-flip scatterings below the gap andthere are few quasiparticles present that can be scattered tohigher energies. However, at temperatures close to 𝑇 𝑐 thereare quasiparticles present and because of their large density of . . γh z = 0 γh z = − . Ω / ∆ I s / m a x (cid:0) I N M s (cid:1) I SC s I NM s FIG. 5: The superconductor spin-current 𝐼 SC 𝑠 and normal metal spin-current 𝐼 NM 𝑠 normalized by the maximal value of 𝐼 NM 𝑠 as a function ofthe precession frequency Ω for two different values of constant externalmagnetic field ℎ 𝑧 . Here, 𝛾 is the gyromagnetic ratio, 𝑇 / 𝑇 𝑐 = . 𝜂 𝛼 / 𝛽 / Δ = . 𝑐 = . 𝑈 𝑘 = . 𝑇 𝑐 is the critical temperatureand Δ is the superconducting gap at zero temperature. states close to the gap, the spin-current can be more than twiceas large as for NM/AFI bilayers when the precession frequencyis significantly less than the gap. The spin-current contributionfrom the Umklapp channel is typically much smaller than thecontribution from the regular scattering channel, but it can besignificant if the Fermi surface is large, the easy axis anisotropyis small and the interface is compensated.The relevant precession frequencies where the spin-current inSC/AFI is enhanced compared to NM/AFI is much lower thanthe typical resonance frequencies of antiferromagnets, whichmakes the detection of this effect experimentally challenging. Apossible solution lies in the shifting of the resonance frequencyby a static magnetic field. Acknowledgments
This work was supported by the Research Council of Norwaythrough grant 240806, and its Centres of Excellence fundingscheme grant 262633 “
QuSpin ”. J. L. also acknowledge fundingfrom the NV-faculty at the Norwegian University of Scienceand Technology.
Appendix A: AFI Green’s functions
In this section we calculate the correction to the magnonGreen’s functions due to the precessing external magnetic field.The Hamiltonian for the antiferromagnetic insulator is givenby Eq. (3), and we treat 𝑉 (cid:66) √︁ 𝑁 𝐴 𝑆 ( 𝑢 + 𝑣 ) 𝛾 (cid:104) ℎ − (cid:16) 𝛼 + 𝛽 † (cid:17) + ℎ + (cid:16) 𝛼 † + 𝛽 (cid:17)(cid:105) (A1) as a perturbation. In order to calculate 𝐺 𝑅𝜈 and 𝐺 <𝜈 , where 𝜈 ∈ { 𝛼, 𝛽 † } , we will first calculate the contour-ordered Green’sfunction. This, in turn, is done by adding an infinitesimalimaginary part to the otherwise real time coordinates andintegrating over the complex Keldysh contour.To second order in 𝑉 , the contour-ordered Green’s functionis 𝐺 𝜈 ( 𝜏 , 𝜏 , 𝒌 ) = − 𝑖 (cid:68) T 𝑐 𝜈 𝒌 ( 𝜏 ) 𝜈 † 𝒌 ( 𝜏 ) e − 𝑖 ∫ C d 𝜏𝑉 ( 𝜏 ) (cid:69) = − 𝑖 (cid:68) T 𝑐 𝜈 𝒌 ( 𝜏 ) 𝜈 † 𝒌 ( 𝜏 ) (cid:69) − (cid:28) T 𝑐 ∫ C d 𝜏 𝜈 𝒌 ( 𝜏 ) 𝜈 † 𝒌 ( 𝜏 ) 𝑉 ( 𝜏 ) (cid:29) + 𝑖 (cid:28) T 𝑐 ∫ C d 𝜏 (cid:48) d 𝜏 𝜈 𝒌 ( 𝜏 ) 𝜈 † 𝒌 ( 𝜏 ) 𝑉 ( 𝜏 ) 𝑉 ( 𝜏 (cid:48) ) (cid:29) + O ( 𝑉 ) , (A2)where T 𝑐 means ordering along the Keldysh contour C and thesubscript 0 means that the expectation values are evaluated in theabsence of 𝑉 . The first order term is odd in magnon operatorsand is therefore zero. Inserting Eq. (A1), the correction to theequilibrium Green’s function is Δ 𝐺 𝜈 ( 𝑡 , 𝑡 , 𝒌 ) (cid:66) 𝐺 𝜈 ( 𝑡 , 𝑡 , 𝒌 ) − 𝐺 𝜈 ( 𝑡 , 𝑡 , 𝒌 ) = 𝑖𝜆 (cid:28) T 𝑐 ∫ C d 𝜏 (cid:48) d 𝜏 𝜈 𝒌 ( 𝑡 ) 𝜈 † ( 𝜏 ) ℎ + ( 𝜏 ) ℎ − ( 𝜏 (cid:48) ) 𝜈 ( 𝜏 (cid:48) ) 𝜈 † 𝒌 ( 𝑡 ) (cid:29) , (A3)where 𝜆 = √︁ 𝑁 𝐴 𝑆 ( 𝑢 + 𝑣 ) 𝛾. (A4)We can use Wick’s theorem to evaluate the rewrite this as (cid:28) T 𝑐 ∫ C d 𝜏 (cid:48) d 𝜏 𝜈 𝒌 ( 𝑡 ) 𝜈 † ( 𝜏 ) ℎ + ( 𝜏 ) ℎ − ( 𝜏 (cid:48) ) 𝜈 ( 𝜏 (cid:48) ) 𝜈 † 𝒌 ( 𝑡 ) (cid:29) = ∫ 𝐶 d 𝜏 (cid:48) d 𝜏 ℎ + ( 𝜏 ) ℎ − ( 𝜏 (cid:48) ) (cid:34)(cid:68) T 𝑐 𝜈 𝒌 ( 𝑡 ) 𝜈 † ( 𝜏 ) (cid:69) (cid:68) T 𝑐 𝜈 ( 𝜏 (cid:48) ) 𝜈 † 𝒌 ( 𝑡 ) (cid:69) + (cid:68) T 𝑐 𝜈 𝒌 ( 𝑡 ) 𝜈 † 𝒌 ( 𝑡 ) (cid:69) (cid:68) T 𝑐 𝜈 ( 𝜏 (cid:48) ) 𝜈 † ( 𝜏 ) (cid:69) (cid:35) . (A5)The second term is zero, which we show in the following. First,define Σ ( 𝜏 , 𝜏 ) = ℎ + ( 𝑡 ) ℎ − ( 𝑡 ) = (cid:10) T 𝑐 ℎ + ( 𝜏 ) ℎ − ( 𝜏 ) (cid:11) . (A6)Then, ∫ C d 𝜏 (cid:48) d 𝜏 ℎ + ( 𝜏 ) ℎ − ( 𝜏 (cid:48) ) (cid:68) T 𝑐 𝜈 𝒌 ( 𝜏 ) 𝜈 † 𝒌 ( 𝜏 ) (cid:69) (cid:68) T 𝑐 𝜈 ( 𝜏 (cid:48) ) 𝜈 † ( 𝜏 ) (cid:69) = − 𝐺 𝜈 ( 𝜏 , 𝜏 , 𝒌 ) ∫ C d 𝜏 (cid:2) Σ • 𝐺 𝜈 (cid:3) 𝒌 = ( 𝜏, 𝜏 ) = − 𝐺 𝜈 ( 𝜏 , 𝜏 , 𝒌 ) (cid:18)∫ ∞−∞ d 𝑡 + ∫ −∞∞ d 𝑡 (cid:19) (cid:2) Σ • 𝐺 𝜈 (cid:3) 𝒌 = ( 𝑡, 𝑡 ) = , (A7)where it was used that C goes from −∞ − 𝑖𝛿 to ∞ − 𝑖𝛿 and thenfrom ∞ + 𝑖𝛿 to −∞ + 𝑖𝛿 with 𝛿 ∈ R being an infinitesimal. Thebullet product is ( 𝐴 • 𝐵 )( 𝜏 , 𝜏 ) = ∫ C d 𝜏 𝐴 ( 𝜏 , 𝜏 ) 𝐵 ( 𝜏, 𝜏 ) . (A8)Hence, we are left with Δ 𝐺 𝜈 ( 𝜏 , 𝜏 , 𝒌 ) = − 𝑖𝛿 𝒌 , 𝜆 (cid:16) 𝐺 𝜈 • Σ • 𝐺 𝜈 (cid:17) ( 𝜏 , 𝜏 ) . (A9)To get the real-time Green’s functions we can use the Langrethrules. If 𝐶 ( 𝜏 , 𝜏 ) = ( 𝐴 • 𝐵 )( 𝜏 , 𝜏 ) , (A10a) 𝐷 ( 𝜏 , 𝜏 ) = ( 𝐴 • 𝐵 • 𝐶 )( 𝜏 , 𝜏 ) , (A10b)where 𝐴 and 𝐵 are contour-ordered functions, then the cor-responding advanced, retarded and lesser Green’s functionssatisfy [27] 𝐶 < = 𝐴 𝑅 ◦ 𝐵 < + 𝐴 < ◦ 𝐵 𝐴 , (A11a) 𝐶 𝑅 / 𝐴 = 𝐴 𝑅 / 𝐴 ◦ 𝐵 𝑅 / 𝐴 , (A11b) 𝐷 < = 𝐴 𝑅 ◦ 𝐵 𝑅 ◦ 𝐶 < + 𝐴 𝑅 ◦ 𝐵 < ◦ 𝐶 𝐴 + 𝐴 < ◦ 𝐵 𝐴 ◦ 𝐶 𝐴 , (A11c) 𝐷 𝑅 / 𝐴 = 𝐴 𝑅 / 𝐴 ◦ 𝐵 𝑅 / 𝐴 ◦ 𝐶 𝑅 / 𝐴 , (A11d)where the circle product is ( 𝐴 ◦ 𝐵 )( 𝑡 , 𝑡 ) = ∫ ∞−∞ d 𝑡 𝐴 ( 𝑡 , 𝑡 ) 𝐵 ( 𝑡, 𝑡 ) . (A12)Using Eq. (A11c) as well as Σ < = Σ and Σ 𝑅 = Σ 𝐴 = Δ 𝐺 𝑅 / 𝐴𝜈 = Δ 𝐺 <𝜈 ( 𝑡 , 𝑡 , 𝒌 ) = − 𝑖𝛿 𝒌 , 𝜆 (cid:16) 𝐺 𝑅𝜈 ◦ Σ ◦ 𝐺 𝐴𝜈 (cid:17) ( 𝑡 , 𝑡 ) . (A13)Next, if we let ℎ 𝑥 ( 𝑡 ) = ℎ cos ( Ω 𝑡 ) and ℎ 𝑦 ( 𝑡 ) = − ℎ sin ( Ω 𝑡 ) we get ℎ ± ( 𝑡 ) = ℎ exp (∓ 𝑖 Ω 𝑡 ) , so Σ ( 𝑡 , 𝑡 ) = ℎ e − 𝑖 Ω ( 𝑡 − 𝑡 ) . (A14)The circle products in Eq. (A13) reduce to normal convolu-tions because 𝐺 𝜈 and Σ only depend on the relative time. Thus,they further reduce to ordinary products in energy-space. TheFourier transform of Σ is Σ ( 𝜀 ) = ∫ ∞−∞ d ( 𝑡 − 𝑡 ) Σ ( 𝑡 , 𝑡 ) e 𝑖𝜀 ( 𝑡 − 𝑡 ) = 𝜋ℎ 𝛿 ( 𝜀 − Ω ) . (A15)We also have that [27] 𝐺 𝐴𝜈 ( 𝜀, 𝒌 ) = (cid:2) 𝐺 𝑅𝜈 ( 𝜀, 𝒌 ) (cid:3) ∗ , (A16)so, to second order in ℎ , Δ 𝐺 <𝜈 ( 𝜀, 𝒌 ) = − 𝑖𝜋ℎ 𝜆 (cid:12)(cid:12) 𝐺 𝑅𝜈 ( 𝜀, 𝒌 ) (cid:12)(cid:12) 𝛿 𝒌 , 𝛿 ( 𝜀 − Ω ) . (A17)Inserting this into the definition of the distribution function andusing Eq. (16) finally gives us Eq. (17). Appendix B: BCS dynamic spin susceptibility
To calculate Im 𝐺 𝑅𝑠 + ( 𝜀, 𝒌 ) we will use the imaginary timeGreen’s function [28]¯ 𝐺 𝑠 + ( 𝜏 , 𝜏 , 𝒌 ) = −(cid:104) T 𝜏 𝑠 +− 𝒌 ( 𝜏 ) 𝑠 − 𝒌 ( 𝜏 )(cid:105) , (B1)where T 𝜏 means time-ordering in 𝜏 , together with the connec-tion through analytical continuation, 𝐺 𝑅𝑠 + ( 𝜀, 𝒌 ) = ¯ 𝐺 𝑠 + ( 𝜀 + 𝑖𝜂 SC , 𝒌 ) , (B2)where¯ 𝐺 𝑠 + ( 𝑖𝜔 𝑛 , 𝒌 ) = ∫ 𝛽 d ( 𝜏 − 𝜏 ) ¯ 𝐺 𝑠 + ( 𝜏 , 𝜏 , 𝒌 ) e 𝑖𝜔 𝑛 ( 𝜏 − 𝜏 ) (B3)and 𝜔 𝑛 = 𝑛𝜋𝛽 (B4)are bosonic Matsubara frequencies. The inverse temperature is 𝛽 = / 𝑇 .We will also make use of the Nambu spinors 𝜙 † 𝒌 = (cid:16) 𝑐 † 𝒌 , ↑ 𝑐 − 𝒌 , ↓ (cid:17) . (B5)With these spinors we can write 𝑠 − 𝒌 = ∑︁ 𝒒 𝜙 − 𝒒 , 𝜙 𝒒 + 𝒌 , , 𝑠 +− 𝒌 = ∑︁ 𝒒 𝜙 † 𝒒 + 𝒌 , 𝜙 †− 𝒒 , . (B6)Thus,¯ 𝐺 𝑠 + ( 𝜏 , 𝜏 , 𝒌 ) = − ∑︁ 𝒒𝒒 (cid:48) (cid:68) T 𝜏 𝜙 † 𝒒 + 𝒌 , ( 𝜏 ) 𝜙 †− 𝒒 , ( 𝜏 ) 𝜙 − 𝒒 (cid:48) , ( 𝜏 ) 𝜙 𝒒 (cid:48) + 𝒌 , ( 𝜏 ) (cid:69) = ∑︁ 𝒒𝒒 (cid:48) (cid:32)(cid:68) T 𝜏 𝜙 † 𝒒 + 𝒌 , ( 𝜏 ) 𝜙 − 𝒒 (cid:48) , ( 𝜏 ) (cid:69) (cid:68) T 𝜏 𝜙 †− 𝒒 , ( 𝜏 ) 𝜙 𝒒 (cid:48) + 𝒌 , ( 𝜏 ) (cid:69) − (cid:68) T 𝜏 𝜙 † 𝒒 + 𝒌 , ( 𝜏 ) 𝜙 𝒒 (cid:48) + 𝒌 , ( 𝜏 ) (cid:69) (cid:68) T 𝜏 𝜙 †− 𝒒 , ( 𝜏 ) 𝜙 − 𝒒 (cid:48) , ( 𝜏 ) (cid:69)(cid:33) = ∑︁ 𝒒 (cid:104) G , ( 𝜏 , 𝜏 , 𝒒 + 𝒌 ) G , ( 𝜏 , 𝜏 , − 𝒒 )− G , ( 𝜏 , 𝜏 , 𝒒 + 𝒌 ) G , ( 𝜏 , 𝜏 , − 𝒒 ) (cid:105) , (B7)where G ( 𝜏 , 𝜏 , 𝒌 ) = − (cid:68) T 𝜏 𝜙 𝒌 ( 𝜏 ) 𝜙 † 𝒌 ( 𝜏 ) (cid:69) = 𝛽 ∑︁ 𝑛 ( 𝑖𝜈 𝑛 ) − 𝜉 𝒌 − | Δ | (cid:18) 𝑖𝜈 𝑛 + 𝜉 𝒌 − Δ − Δ ∗ 𝑖𝜈 𝑛 − 𝜉 𝒌 (cid:19) e − 𝑖𝜈 𝑛 ( 𝜏 − 𝜏 ) , (B8)is the BCS single-particle Green’s function. Here, 𝜈 𝑛 = ( 𝑛 + ) 𝜋 / 𝛽 are fermionic Matsubara frequencies. Inserting this intoEq. (B3), we get¯ 𝐺 𝑠 + ( 𝑖𝜔 𝑛 , 𝒌 ) = 𝑇 ∑︁ 𝒒 ,𝑚 (cid:104) G , (− 𝑖𝜈 𝑚 − 𝑖𝜔 𝑛 , 𝒒 + 𝒌 ) G , ( 𝑖𝜈 𝑚 , − 𝒒 )− G , (− 𝑖𝜈 𝑚 − 𝑖𝜔 𝑛 , 𝒒 + 𝒌 ) G , ( 𝑖𝜈 𝑚 , − 𝒒 ) (cid:105) = 𝑇 ∑︁ 𝒒 ,𝑚 (cid:104) G , ( 𝑖𝜈 𝑚 + 𝑖𝜔 𝑛 , 𝒒 + 𝒌 ) G , ( 𝑖𝜈 𝑚 , − 𝒒 )+ G , ( 𝑖𝜈 𝑚 + 𝑖𝜔 𝑛 , 𝒒 + 𝒌 ) G , ( 𝑖𝜈 𝑚 , − 𝒒 ) (cid:105) = 𝛽 ∑︁ 𝒒 ,𝑚 Tr [ G ( 𝑖𝜈 𝑚 + 𝑖𝜔 𝑛 , 𝒒 + 𝒌 ) G ( 𝑖𝜈 𝑚 , 𝒒 )] . (B9)In the last equality we have used that G ( 𝑖𝜈 𝑚 , − 𝒒 ) = G ( 𝑖𝜈 𝑚 , 𝒒 ) , G , ( 𝑖𝜈 𝑛 , 𝒌 ) G , ( 𝑖𝜈 𝑚 , 𝒒 ) = G , ( 𝑖𝜈 𝑛 , 𝒌 ) G , ( 𝑖𝜈 𝑚 , 𝒒 ) and ∑︁ 𝒒 ,𝑚 G , ( 𝑖𝜈 𝑚 + 𝑖𝜔 𝑛 , 𝒒 + 𝒌 ) G , ( 𝑖𝜈 𝑚 , 𝒒 ) = ∑︁ 𝒒 (cid:48) ,𝑘 G , (− 𝑖𝜈 𝑘 , 𝒒 (cid:48) ) G , (− 𝑖𝜈 𝑘 − 𝑖𝜔 𝑛 , 𝒒 (cid:48) + 𝒌 ) ∑︁ 𝒒 (cid:48) ,𝑘 G , ( 𝑖𝜈 𝑘 , 𝒒 (cid:48) ) G , ( 𝑖𝜈 𝑘 + 𝑖𝜔 𝑛 , 𝒒 (cid:48) + 𝒌 ) , (B10)where, in the first equality, we introduced 𝒒 (cid:48) = − 𝒒 − 𝒌 and 𝑖𝜈 𝑘 = − 𝑖𝜈 𝑚 − 𝑖𝜔 𝑛 .Next, we can use the spectral form, G ( 𝑖𝜈 𝑚 , 𝒒 ) = ∫ ∞−∞ d 𝜔 (− 𝜋 ) Im G ( 𝜔 + 𝑖𝜂 SC , 𝒒 ) 𝑖𝜈 𝑚 − 𝜔 , (B11)and the Matsubara sum identity1 𝛽 ∑︁ 𝑚 𝑖𝜈 𝑚 + 𝑖𝜔 𝑛 − ˜ 𝜔 × 𝑖𝜈 𝑚 − 𝜔 = 𝑛 𝐹 ( 𝜔, 𝑇 ) − 𝑛 𝐹 ( ˜ 𝜔, 𝑇 ) 𝑖𝜔 𝑛 − ( ˜ 𝜔 − 𝜔 ) , (B12) where we have used that 𝜈 𝑚 are fermionic Matsubara frequen-cies, giving rise to the Fermi-Dirac distribution function 𝑛 𝐹 .We have also used that 𝑛 𝐹 ( 𝜔 − 𝑖𝜔 𝑛 ) = 𝑛 𝐹 ( 𝜔 ) since 𝜔 𝑛 is abosonic Matsubara frequency. Additionally, Eq. (B8) gives,assuming Δ real,Im G ( 𝜔 + 𝑖𝜂 SC , 𝒌 ) = − 𝜋 √︃ 𝜉 𝒌 + | Δ | (cid:18) 𝜔 + 𝜉 𝒌 − Δ − Δ 𝜔 − 𝜉 𝒌 (cid:19) × (cid:20) 𝛿 (cid:18) 𝜔 − √︃ 𝜉 𝒌 + | Δ | (cid:19) − 𝛿 (cid:18) 𝜔 + √︃ 𝜉 𝒌 + | Δ | (cid:19)(cid:21) (B13)in the limit 𝜂 SC → + . Hence, if we define 𝐸 𝒌 (cid:66) √︃ 𝜉 𝒌 + | Δ | ,lim 𝜂 SC → + Tr (cid:2) Im G ( ˜ 𝜔 + 𝑖𝜂 SC , 𝒒 + 𝒌 ) Im G ( 𝜔 + 𝑖𝜂 SC , 𝒒 ) (cid:3) = 𝜋 𝜔 ˜ 𝜔 + 𝜉 ˜ 𝜉 + Δ 𝐸 ˜ 𝐸 [ 𝛿 ( 𝜔 − 𝐸 ) − 𝛿 ( 𝜔 + 𝐸 )]× (cid:2) 𝛿 (cid:0) ˜ 𝜔 − ˜ 𝐸 (cid:1) − 𝛿 (cid:0) ˜ 𝜔 + ˜ 𝐸 (cid:1)(cid:3) , (B14)where 𝜉 = 𝜉 𝒒 , ˜ 𝜉 = 𝜉 𝒒 + 𝒌 , 𝐸 = 𝐸 𝒒 and ˜ 𝐸 = 𝐸 𝒒 + 𝒌 . Inserting thisinto Eq. (B9) gives¯ 𝐺 𝑠 + ( 𝑖𝜔 𝑛 , 𝒌 ) = − ∑︁ 𝒒 ∑︁ 𝜔 = ± 𝐸 ∑︁ ˜ 𝜔 = ± ˜ 𝐸 𝜔 ˜ 𝜔 + 𝜉 ˜ 𝜉 + Δ 𝜔 ˜ 𝜔 × 𝑛 𝐹 ( ˜ 𝜔, 𝑇 SC ) − 𝑛 𝐹 ( 𝜔, 𝑇 SC ) 𝑖𝜔 𝑛 − ( ˜ 𝜔 − 𝜔 ) . (B15)From Eq. (B2) we then finally have Eq. (18). [1] V. Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono, andY. Tserkovnyak, Rev. Mod. Phys. , 015005 (2018).[2] F. S. Bergeret, A. F. Volkov, and K. B. Efetov, Rev. Mod. Phys. , 1321 (2005).[3] A. I. Buzdin, Rev. Mod. Phys. , 935 (2005).[4] J. Linder and J. Robinson, Nature Phys. , 307 (2015).[5] M. Eschrig, Rep. Prog. Phys. , 104501 (2015).[6] J. Linder and A. V. Balatsky, Rev. Mod. Phys. , 045005 (2019).[7] S. A. Kivelson and D. S. Rokhsar, Phys. Rev. B , 11693 (1990).[8] H. L. Zhao and S. Hershfield, Phys. Rev. B , 3632 (1995).[9] C. H. L. Quay, D. Chevallier, C. Bena, and M. Aprili, NaturePhys. , 84 (2013).[10] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. Lett. , 117601 (2002).[11] K.-R. Jeon, C. Ciccarelli, H. Kurebayashi, J. Wunderlich, L. F.Cohen, S. Komori, J. W. A. Robinson, and M. G. Blamire, Phys.Rev. Applied , 014029 (2018). [12] K.-R. Jeon, C. Ciccarelli, A. J. Ferguson, H. Kurebayashi, L. F.Cohen, X. Montiel, M. Eschrig, J. W. A. Robinson, and M. G.Blamire, Nature Mater. , 499 (2018).[13] Y. Yao, Q. Song, Y. Takamura, J. P. Cascales, W. Yuan, Y. Ma,Y. Yun, X. C. Xie, J. S. Moodera, and W. Han, Phys. Rev. B ,224414 (2018).[14] M. Inoue, M. Ichioka, and H. Adachi, Phys. Rev. B , 024414(2017).[15] T. Kato, Y. Ohnuma, M. Matsuo, J. Rech, T. Jonckheere, andT. Martin, Phys. Rev. B , 144411 (2019).[16] M. A. Silaev, Phys. Rev. B , 144521 (2020).[17] M. A. Silaev, Phys. Rev. B , 180502(R) (2020).[18] R. Cheng, J. Xiao, Q. Niu, and A. Brataas, Phys. Rev. Lett. ,057601 (2014).[19] A. Kamra and W. Belzig, Phys. Rev. Lett. , 197201 (2017).[20] P. Vaidya, S. A. Morley, J. van Tol, Y. Liu, R. Cheng, A. Brataas,D. Lederman, and E. del Barco, Science , 160 (2020), https://science.sciencemag.org/content/368/6487/160.full.pdf .[21] S. Takei, B. I. Halperin, A. Yacoby, and Y. Tserkovnyak, Phys.Rev. B , 094408 (2014).[22] E. L. Fjærbu, N. Rohling, and A. Brataas, Phys. Rev. B ,144408 (2017).[23] E. L. Fjærbu, N. Rohling, and A. Brataas, Phys. Rev. B ,125432 (2019).[24] M. Hagiwara, K. Katsumata, I. Yamada, and H. Suzuki, J. Phys.:Condens. Matter , 7349 (1996).[25] F. M. Johnson and A. H. Nethercot, Phys. Rev. , 705 (1959).[26] R. C. Ohlmann and M. Tinkham, Phys. Rev. , 425 (1961).[27] J. Rammer, Quantum field theory of non-equilibrium states ,Vol. 22 (Cambridge University Press Cambridge, 2007).[28] H. Bruus and K. Flensberg,