Spin-orbit coupling suppression and singlet-state blocking of spin-triplet Cooper pairs

Spin-information can be transferred between ferromagnets through a superconducting spacer via spin-polarized quasiparticles or triplet Cooper pairs. Below the critical temperature of an s-wave superconductor, an energy gap opens in the density of states below which the electrons pair up with antiparallel spins ("singlet pairs") meaning singlet supercurrents do not carry a net spin. However, in this state the spin-relaxation time for spin-polarized quasiparticle (i.e. non-superconducting carrier) currents injected from a ferromagnet into a superconductor at the energy gap edge, is enhanced by over 6 orders of magnitude over the normal state. Spin can also be carried directly in the superconducting state through the conversion of singlet pairs into spin-polarized triplet pairs at magnetically inhomogeneous superconductor/ferromagnet interfaces. Although the decay envelope of triplet pairs within ferromagnetic materials is well studied, little is known about their decay in non-magnetic metals and superconductors, and in particular in presence of spin-orbit coupling (SOC). Here we report devices in which triplet supercurrents are created and are injected into the s-wave superconductor Nb with strong SOC. In the normal state of Nb, triplet pairs decay over a distance of 5 nm, which is an order of magnitude smaller than the decay of zero spin singlet pairs due to the SOC interacting with the spin associated with a triplet supercurrent. In the superconducting state of Nb, triplet supercurrents are blocked by the lack of available equilibrium states in the singlet superconducting gap. The results offer new insight into the dynamics between s-wave singlet and triplet states.

A significant difference in the decay lengths is also expected for triplet and singlet pairs within an swave S. An attraction between electrons with opposite spin projections inside the s-wave superconductor supports the transfer of singlet Cooper pairs through the S layer without any damping. At the same time, the triplet Cooper pairs penetrating the superconductor experience the spatial decay of their wave function due to the absence of the pairing between electrons with equal spin projections.
Here we investigate triplet coherence in a metal (Nb) with strong SOC in both the normal and superconducting states, by fabricating four series of S/FL/S'/FR/S devices: (1) "triplet control devices" Nb(300)/Cr(1)/Fe(dFe)/Cr(1)/Nb(300) (thicknesses in nm) without a central layer of Nb' and varying total thickness of Fe (3 to 15 nm) to confirm singlet-to-triplet pair conversion at Fe/Cr spin mixer interfaces; (2) "singlet devices" Nb(300)/Cr(1)/Fe(2)/Nb'(dNb')/Fe(2)/Cr(1)/Nb(300) in which the total Fe thickness is low enough such that a residual singlet supercurrent is measurable; and two series of "triplet devices" with (4) Nb(300)/Cr(1)/Fe(7.5)/Nb'(dNb')/Fe(2.0)/Cr(1)/Nb(300) layers with a total Fe thickness exceeding the maximum thickness for which a singlet supercurrent is observed in Nb/Fe/Nb devices (5.5 nm) 13 . Each set of devices were prepared in a single deposition run. In device series (2)- (4) there are no intentional spin-mixing and spin-rotation interfaces between the Fe layers and the central Nb' layer and hence a triplet pair wavefunction should not be generated in Nb' in the superconducting state. We note that the focus of this study is not the triplet pair correlation in a source S 25,26 created by the pair conversion but the proximityinduced triplet correlation in S' located away from a source S.
Current-perpendicular-to-plane S/FL/S'/FR/S Josephson devices are fabricated using a focused ion beam microscope technique that is described in detail elsewhere 35 . Due to variations in the cross-sectional areas of the devices, the Josephson critical current (Ic) is multiplied by the device normal state resistance Rn (measured at high current bias) to give the characteristic voltage (IcRn). The IcRn of all devices is systematically investigated as a function of dNb' in the 0 to 40 nm range.
We first discuss the triplet control devices. In Fig. 1a we compare the IcRn for these devices with Nb/Fe/Nb devices (blue curve) previously measured by our group 13 versus Fe layer thickness (dFe) at 1.6 K.
The Nb/Fe/Nb devices do not have (intentional) spin-mixer interfaces and so transport is spin-singlet. For dFe < 5 nm, supercurrents are detectable in both types of devices, but for dFe > 5 nm supercurrents are only detectable in the triplet control devices confirming spin-mixing and spin-rotation at the Fe/Cr interfaces.
By applying a magnetic field (H) parallel to the interfaces, the Ic of the triplet control devices is modulated (inset of Fig. 1b). Ic (H) is hysteretic and the maximum values of Ic are obtained at non-zero applied field (μ0H = δ) due to the barrier magnetization. In Fig. 1b we have plotted δ at 1.6 K (left-axis) versus dFe, which shows a linear increase in δ with dFe, consistent with the linear rise in the magnetic moment (ms) per unit area with dFe for the unpatterned Nb/Cr/Fe/Cr/Nb films measured using a vibrating sample magnetometer at 300 K (right-axis). Both δ and ms per unit area are proportional to dFe, suggesting that the Fe layers are homogeneously magnetized at magnetic saturation in both the unpatterned films and devices.  Nb(300)/Cr(1)/Fe(dFe)/Cr(1)/Nb(300) triplet control devices (red diamonds) at 1.6 K along with the known dFedecay of IcRn for Nb(300)/Fe(dFe)/Nb(300) (the blue curve) singlet devices with a coherence length of 1.0 nm 13 .
The (red) curve is a least square fit giving a triplet coherence length of ξF triplet = 5.3 ± 1.9 nm. b, In-plane magnetic hysteresis (δ; red diamonds, left axis) estimated from the Nb(300)/Cr(1)/Fe(dFe)/Cr(1)/Nb(300) triplet control devices at 1.6 K where δ is the maximum field shift in Ic (H). The right axis shows the magnetic moment at magnetic saturation per unit area (ms/m 2 ) determined from unpatterned Nb(300)/Cr(1)/Fe(dFe)/Cr(1)/Nb(300) films (blue triangles). The (blue) curve is a least-squares regression line fit to ms/m 2 versus dFe with a volume magnetization of 618 emu cm ─3 and a magnetically dead layer at each In Fig. 2a we have plotted IcRn versus dNb' for the singlet devices which show two Nb'-thickness regimes: for dNb' < 30 nm, IcRn slowly decreases with increasing dNb' and rises beyond 30 nm, indicating the onset of superconductivity in Nb' which leads to two Josephson devices operating in series with the effective barrier thickness reduced as illustrated in Figs. 2b and 2c. Since the potential injection of spin-polarized quasiparticles suppresses the onset superconductivity of Nb', it is difficult to distinguish the critical current of Nb' and the Josephson critical current of the two devices. However, the formation of the two Josephson devices in series is confirmed by a second harmonic Fraunhofer pattern which results from the overlap of the Andreev bound states in Nb' [36][37][38] . In Fig. 2d, we have plotted the positive field direction in Ic (H) for two representative devices for two different values of dNb' (20 and 30 nm). Ic is modulated with magnetic flux (Φ) according to sinc (nΦ/Φ0), where Φ0 is a flux quantum, but the periodicity (1/n) is halved (n = 2) for the 30 nm device, consistent with a second harmonic current-phase relationship. In Fig. 2e (left-axis), we have plotted n versus dNb', which shows n = 1 behaviour for all thicknesses except for the 30 nm device (which matches the singlet coherence length). The dNb' = 40 nm device shows n = 1 behaviour (i.e. the first harmonic), consistent with weakly overlapped Andreev bound states 36,37 .
Typical R (T) curves for dNb' < 15 nm are shown in Fig. 3c. The 300 nm-thick top and bottom Nb layers become superconducting below 9 K, showing a drop in R with the resistance continuously decreasing with decreasing temperature as superconductivity gradually proximitizes the Cr/Fe/Nb/Fe/Cr barrier. The barriers are completely proximitized (R = 0) below 4 K. The decay in IcRn versus dNb' is exponential [IcRn = exp(─ξN triplet /dNb')] with a triplet coherence length of ≈ 3.2 -5.7 nm, which is an order of magnitude smaller than the singlet coherence length in Nb' estimated from Fig. 2a. The strong pair breaking effect on triplet pairs is likely due to strong SOC in normal state Nb [41][42][43] , which suppresses the triplet pairing coherence due to scattering of the spin associated with the triplet supercurrent 6,31 . We note that, for all temperatures, we do not observe magnetoresistance from the Fe/Nb'/Fe barriers in these devices, suggesting a short spindiffusion length in thin Nb' layers ( < 10 nm) in these particular devices due to SOC [41][42][43] Table 1 together with the mean free path and the spin diffusion lengths. In F, the coherence length of triplet pairs is long-ranged and close to the spin-diffusion length, while singlet pairs affected by the exchange field are short-ranged (Fig. 1a). In N with strong SOC, the coherence length of the triplet pairs is short-ranged (Fig. 3a) due to the short spin-diffusion length (see Supplementary Materials) while singlet pairs are unaffected by SOC and are long-ranged (Fig. 2a).
In S', singlet pairs couple with the singlet wavefunction of S' and creates two-series junction behaviour and hence singlet supercurrents do not show a decay (Fig. 2a). Triplet pairs however are not able to couple with the singlet wavefunction of S' and decay within the order of the singlet coherence length (30 nm; Fig. 3a). Triplet pairs which are not coupled with the singlet wavefunction can decay through two possible mechanisms in superconducting Nb'. One is SOC which can exist also below Tc and results in a short spin diffusion length. The spin-diffusion length of quasiparticles in the superconducting state is theoretically comparable to that in the normal state 45 . However, this is not necessarily the case for triplet pairs which are unaffected by the energy dispersion of quasiparticles and hence it is unreasonable to simply assume that the triplet blocking is only due to SOC. Another possibility is a competition between the singlet and the triplet pairing states 46 resulting from the fact that they have an opposite influence on the electron density of states at the Fermi energy, i.e. the singlet pairing decreases it, while the triplet correlations lead to its increase.
To show the absence of phase-coupling between triplet pairs and the singlet wavefunction and model the decay of triplet supercurrents in a singlet superconductor, we calculate the critical current density in a SL/FL/S'/FR/SR device in which SL/FL and FR/SR spin-mixing/rotation interfaces are atomically thin. The central S' layer has a superconducting gap of Δ0 which is smaller than that of SL and SR (Δ1). The magnetic exchange fields of FL and FR layers (spin-rotation axis) are parallel to each other and strong enough to block the transport of minority spin triplet pairs. By solving the Gor'kov equations (see Supplementary Materials for details), we obtain the critical current density which appears to be completely triplet in our case: where hL (hR) is the magnetic exchange field in FL (FR) and θL (θR) is the magnetization angle between the magnetic exchange field at the SL/FL (FR/SR) interface and FL (FR). We note that equation (1)

Competing interests
The authors declare no competing interests.

Methods
Film growth. Unpatterned films were fabricated on 5 mm × 5 mm quartz substrates by DC magnetron sputtering in an ultrahigh-vacuum chamber with a base pressure better than 10 -6 Pa. The sputtering targets were pre-sputtered for approximately 20 minutes to clean the surfaces and the films were grown using an Ar pressure of 1.5 Pa. Multiple quartz substrates were placed on a rotating circular table that passed in series under stationary magnetrons so that multiple samples with different layer thicknesses could be grown in the same deposition run. The thickness of each layer was controlled by adjusting the angular speed of the rotating table at which the substrates moved under the respective targets and the sputtering power.
Device fabrication. Standard optical lithography and Ar-ion milling define 4-μm-wide tracks, which were narrowed down by focused-ion-beam milling to make current-perpendicular-to-plane devices. Further details on the device fabrication process are described elsewhere 35 . A typical device dimension is 500 nm × 500 nm.
Transport measurements. A pulse tube cryogen-free system (Cryogenic Ltd) was used to cool the devices down to 1.

Theory of the suppression of spin-triplet Josephson currents in a singlet superconductor
We consider a S1/F1/S'/F2/S2 Josephson junction (see Fig. S2) consisting of atomically thin superconductors (S1 and S2), ferromagnets (F1 and F2) and a central superconductor (S'). The neighbouring layers are coupled by the transfer integrals ti, (i = 1, 2, 3, 4) of the tight-binding model. The critical temperature of the superconducting leads S1 and S2 (Tc1 = Tc2) is higher than that of the central superconductor S' (Tc0). Thus, the central layer S' can be both in the normal and in the superconducting states at T < Tc1. We assume that T ≈ Tc0 < Tc1, ti << Tc0 and the interlayer tunneling conserves the momentum. Also, we assume that the S1/F1 and F2/S2 interfaces are magnetized. The misalignment angle θi (i = 1, 2) between the exchange field hi = hi(cos θi z + sin θi x) at the Si/Fi interface and the spin-rotation axis z in the Fi layer gives rise to the emergence of the spin-triplet superconducting correlations. We assume 100 % spin polarization of F1 and F2 layers and therefore the transport of minority spin triplet pairs is blocked. Figure S2: S1/F1/S'/F2/S2 Josephson junction consisting of atomically thin layers.
The superconducting gaps in S', S1 and S2 layers are Δ0, Δ1 = |Δ1|e -iφ/2 and Δ2 =|Δ2|e iφ/2 with |Δ1| = |Δ2| (the phase difference across the junction equals to φ). The energy spectrum in the superconductors is ξ (p), while in the ferromagnets it is spin-dependent: ξ↑ = ξ (p) and ξ↓ = +∞. We denote the electron annihilation operators in S1, F1, S', F2, S2 layers as  , , , and  . The Hamiltonian in the system under consideration is where the coefficient (i = 1,2) describes the modified kinetic energy due to the magnetized interfaces and is the spin-dependent kinetic energy of the ferromagnets (assumed identical for F1 and F2), given by The second term in (S1), describes the superconductivity in the three superconductors as creation of a Cooper pair in which one electron is located in S′ and the other electron is in F1, F2, S1 and S2, respectively.
Rewriting the commutation relations in terms of the Green's functions and applying the Fourier transform such that i∂Ψ/∂τ = −iωΨ, we find the following closed set of matrices Gor'kov equations in frequency representation: for S': for F1: for F2: for S1: where is the 2 × 2 identity matrix and σy is the second Pauli matrix.
The definition of the current through the junction is imposed by the tunnelling Hamiltonian model. In this description, the charge current corresponds to the number of particles travelling from one layer to another, as described by the tunnelling Green's function . Since the charge current flows across all the layers, it can be obtained from the Green's function at an arbitrary layer. The model presented here focuses on the Green's functions in the central S' layer, such that we express the current in terms of . The current consists of the sum of spin-up and spin-down currents, however, since we consider fully polarized ferromagnets, we only need to take the component into account.
Hence, the Josephson current density ( ) across the junction is expressed via the Fourier component where ν0 and Tτ are the electron density of states at the Fermi level and the time-ordered product for the imaginary time τ, respectively. By solving the system above, we find the exact expression for E ψ . We obtain ↓↓ ( ; ) = 0 , which corresponds to the absence of the spin-down state, resulting from fully spin-polarized ferromagnets. By expanding ↑↑ up to seventh order over ti << T,(i = 1, 2, 3, 4) (assuming ℎ , ℎ ≪ ≈ and |Δ | ≪ ) we find: such that the Josephson current density (S10) becomes = 8 | | (ℎ sin )(ℎ sin )( − | | )sin . (S11) Note that the temperature dependence of the critical current in the absence of the singlet superconductivity (|Δ0| = 0) may be non-monotonous, similar to the results obtained by Eschrig and Löfwander 1,2 for the triplet supercurrents in a half-metallic Josephson junction with spin-active interfaces.
At low temperatures, T << Tc2 and |Δ1| >> T, the coefficients a and b reduce to where is the Riemann zeta function.