UPPLEMENTARY INFORMATION doi: 0.038/nmat78. relaxation time, effective s polarization, and s accumulation in the superconducting state The s-orbit scattering of conducting electrons by impurities in metals is the main origin of the s relaxation at low temperatures. In the presence of non-magnetic impurities in the aluminum layer, the s-orbit coupling arises as follows. The impurity potential V ( r gives rise to an additional electric field E ( / imp V r imp e. Electrons with velocity vp m (p: momentum, m: electron mass subject to the electric field E feel an effective magnetic field B ( pm ( Ec ( emc( p V. imp This leads to a s-orbit coupling interaction VO B B via the impurity potential, where is the electron s s. The matrix element for scattering of the electron from a state k with momentum k and s to a state k takes the form k k ( kk, where O Vso i sovimp is the s-orbit coupling constant which is given by O ( / mc in the -electron model and is enhanced by several orders of magnitude in real metals 44. Thus the s-orbit Hamiltonian in the normal state can be written as: Hso isovimp ( kk ak ak k, k,, (. where a k ( ak is the annihilation (creation operator of an electron with a one-electron energy k in a state with k and. Using the Bogoliubov transformations a u v k k and k k k a v u k k in (., where k k k k ( k is the annihilation (creation operator of a quasi-particle with excitation energy E k k with the superconducting energy gap and u v E k k ( k k are the coherence factors and retaining only the s-flip scattering terms, the s-orbit Hamiltonian in nature materials www.nature.com/naturematerials 00 Macmillan Publishers Limited. All rights reserved.
UPPLEMENTARY INFORMATION doi: 0.038/nmat78 the superconducting state is rewritten as O so imp ( ( [ k k k k k k, ] k k, k, k H i V uu vv. (. k k k The s density in the C layer is given by (/ [ f ( E f ( E ], where f ( Ek k k is the distribution function for occupancy of the state of energy E k and s. Using Fermi s golden rule, the s-relaxation rate ( t so due to s-orbit scattering can be calculated as N(0 N(0 de[ f ( E f ( E ] f (, t (.3 O N N where (8 9 n V N(0 k is the inverse s-flip scattering time in the 4 N imp imp F so normal state, n imp is the impurity concentration, and N(0 is the normal-state density of states at the Fermi level of the C. We made use of the expansion f f( E [ f( E E] with respect to the chemical potential shift for the upand down-s quasiparticles due to s accumulation. We calculate the s density accumulated in the C layer due to by taking into accounts the fact that the s susceptibility ( T N in the presence of s-orbit scattering remains finite ( T N b at low temperatures as shown by Anderson 34, where b (3 N0 is the s-orbit parameter and 0 is the energy gap at zero temperature. Therefore we may have D ( E [ f ( E f ( E ] de b b ( E f( E f( E de, D (.4 whered ( E is the quasi-particle s density of states with the energy gap. Therefore, the s relaxation time ( T ( t so in the superconducting state is nature MATERIAL www.nature.com/naturematerials 00 Macmillan Publishers Limited. All rights reserved.
UPPLEMENTARY INFORMATION doi: 0.038/nmat78 ( T b b f ( N 0 ( T N, ( T ( T N, (.6 0 where N is the s susceptibility in the normal state and ( T N ( b 0 is the s susceptibility in the superconducting state (Yosida function calculated in the absence of s-orbit scattering (b 0. Both f( and 0 ( T N decrease with decreasing temperature below T c and vanish at low temperatures 3. On the other hand, in the presence of the s-orbit scattering, ( T N is finite. ince the superconducting energy gap remains unaffected due to the time-reversal symmetry, f( vanishes at low temperatures. Consequently, the s relaxation time ( T increases rapidly with decreasing temperature in the superconducting state. Fig. 3e shows an enormous increase of ( T at low temperatures well below T c. The s-up and s down tunneling currents through the tunnel barriers in the double tunnel junction with the C layer between two ferromagnets are calculated by using a standard tunnel Hamiltonian and the Bogoliubov transformations. These s-dependent tunneling currents determine the s injection rate (d/dt inj, which is balanced with the s-relaxation rate ( T to yield the s density for the parallel (P and antiparallel (AP states as, P 0, ( ( ( / P D E f E ev f E ev de (.7 AP AP with the effective tunneling polarization, P P/ (, (.8 where t ( T is the ratio of the s relaxation rate / s to the quasi-particle injection rate t ( e N(0 R Ad ( R Ad, where R T A is the product of the T N N T N tunnel barrier resistance R T and area A, d is the C thickness, and N is the resistivity, nature materials www.nature.com/naturematerials 3 00 Macmillan Publishers Limited. All rights reserved.
UPPLEMENTARY INFORMATION doi: 0.038/nmat78 and N is the s diffusion length in the normal state of C. The effect of AP on the superconducting gap AP is described using a modified BC gap equation, which is combined with (.4 and (.7 to numerically solve for AP and self-consistently as functions of the bias voltage V.. Conductance curve broadening by depairing, s-orbit interaction, and quasi-particle life-time effects By taking into account the broadening due to depairing, s-orbit interaction and quasi-particle life-time, the quasi-particle s density of states is written as 8,45 LT u ( E i, V D ( EV, Re LT, (. u ( Ei, V ( V where u are obtained by solving the following coupled equations 46 ; u bu ( u u E BH, (. u u where H is the applied magnetic field, LT is the quasiparticle life-time broadening, is the depairing parameter. ince the Zeeman splitting B H is very small in our system, D ( EV, D ( EV, (i.e. B H is set equal to zero in.. Equation (.4 is rewritten as, with b b 0, 0 D ( E, V f( E f( E de (.3 From the balancing of the s-injection rate and the s-relaxation rate, we have 4 nature MATERIAL www.nature.com/naturematerials 00 Macmillan Publishers Limited. All rights reserved.
UPPLEMENTARY INFORMATION doi: 0.038/nmat78 P D ( E, V f ( E ev f ( E ev de, (.4 where the effective tunneling polarization is given by P N f( E f( E de P [ ( b ] D ( E, V f( E f( E de( b (.5 For the superconducting gap equation, we use the modified BC gap equation D E E = VBC ( E, V tanh tanh de D (.6 kt B kt B where V BC is the BC coupling constant and D is the Debye energy. Equations (.3-(.6 are self-consistently solved with respect to ( V and for a given bias voltage V. The calculated results are used to calculate the tunnel currents, I P and I AP, for the P and AP alignments: IP( V ( E, V f ( E ev f ( E ev/ de, er D (.7 P T P IAP ( V ( E, V f ( E ev f ( E ev/ de, er and the tunneling conductances AP T D (.8 G ( V di ( V dv P P, AP AP G ( V di ( V dv, (.9 and, thereby the tunnel magnetoresistance (TMR. The fitted TMR curve is shown in Fig. d. 3. Hanle effect in F-I-C-I-F In studies using s injection from magnetic electrodes, the Hanle effect is typically carried out with a field perpendicular to the plane of the magnetic electrodes, which are typically much thinner than their lateral dimensions. Under these nature materials www.nature.com/naturematerials 5 00 Macmillan Publishers Limited. All rights reserved.
UPPLEMENTARY INFORMATION doi: 0.038/nmat78 circumstances, fairly large magnetic fields can be applied without upsetting the direction of magnetization of the electrodes due to the large shape anisotropy fields of the electrodes. The application of small magnetic fields perpendicular and out of the plane to the aluminum superconducting layer suppresses superconductivity since the critical field for aluminum is small so the experiment is not possible in this configuration. Although the Hanle effect can be carried out using in-plane magnetic fields perpendicular to the direction of magnetization of the magnetic electrodes, these fields easily upset the magnetic configuration of the electrode making the experiment difficult. Even small magnetic fields should give rise to significant precession of the s of the injected quasi-particles, given the long s life time that we have observed. It is very difficult experimentally to eliminate all residual magnetic fields. These arise not only from the earth s magnetic field which, however, can be screened but, more importantly, from even small amounts of roughness at the surfaces of the magnetic electrodes and from any inhomogeneities in the magnetization of the CoFe electrodes. It is for these reasons that our experiments are carried out in fields of 300 Oe to eliminate the effect of small perturbing extrinsic fields, and high reproducibility of s accumulation has been observed from sample to sample. Our observation of the Hanle effect is shown in Fig. 5a,b in which we measure the dependence of voltage separation of the two conductance peaks in the conductance versus voltage curves (e.g. Fig. a as a function of the in-plane parallel magnetic field H in the presence of small orthogonal in-plane fields H. H is applied using a superconducting solenoid magnet whereas H is applied using two small permanent magnets glued to the sample holder. The spacing between these magnets is varied from run to run to vary H. However, because it is difficult to precisely align H 6 6 nature MATERIAL www.nature.com/naturematerials 00 Macmillan Publishers Limited. All rights reserved.
UPPLEMENTARY INFORMATION doi: 0.038/nmat78 with the sample s magnetic in-plane easy axis we define H and H to be the fields collinear and perpendicular to the sample s easy axis (see inset in Fig. 5a. Important consequences of this misalignment are an asymmetric switching of the s accumulation (i.e. abrupt change for PAP; gradual change for APP and an asymmetry of the Hanle dephasing peak, as discussed later. The magnitude of the switching field of the layer, measured from magnetic hysteresis curves obtained by varying the in-plane collinear field H, is very sensitive to H. The observed dependence of the switching field on H and H is in good agreement with a single domain model of the switching process. Our switching model assumes that the ned layer magnetic moment M is unaffected by the small values of H and H because of the very large exchange bias field at 0.5 K. The energy of the layer can then be written as, E K sin M H K sin M Hcos( u u H M H K sin MHcos( H (3. where K u is the uniaxial anisotropy energy of the layer (and H K, the uniaxial anisotropy field of the layer Ku / M, H ( H H H H is the in-plane applied field, is the angles between M is the magnetic moment of the layer, and ( H M and M ( H and M. The direction of the s of the injected quasiparticles is along the direction of the magnetization of the layer (ned layer for negative (positive bias voltages according to the magnetic layer from which the electrons are injected. Thus, we calculate the fields collinear and perpendicular to the injected s direction i.e. H, H for injection from the layer and H, H for injection from the ned layer as; 7 nature materials www.nature.com/naturematerials 7 00 Macmillan Publishers Limited. All rights reserved.
UPPLEMENTARY INFORMATION doi: 0.038/nmat78 H H cos H sin, (3. H H sin Hcos H H, H H (3.3 When the moment is not collinear to the ned layer moment, the s accumulation is given by; where BC AP ( V ( V, ( P, P D ( E f ( E ev f ( E ev / de (3.4 ( P, accumulation. sin ( P cos ( is the angular dependence of the s In the oblique Hanle effect, the normalized s accumulation can be written as (, F ( exp / cos L sin cos 0 D t t t dt (3.5 0 where 0 = F ( exp / 0 D t t dt and the Larmor frequency ( L g B H / (g is the Lande g-factor, µ B is Bohr magneton, h /, and h is Planck s constant. 0 is the s accumulation when H >> H (or, equivalently, 0 or and FD ( t is the distribution of ss that tunnel out of the Al layer in time t. Eq. (3.5 can be rewritten as (, cos for a long. We define a Hanle dephasing factor to be the complement of the normalized s accumulation, as follows: ( H (, sin ( ( ( ( H ( H. The normalized gap equation then becomes, ( H P AP AP P, ( H ( H P, ( H ( H ( H ( H (3.6 where the three terms on the right side of the equation represent, firstly, the P 8 nature MATERIAL www.nature.com/naturematerials 00 Macmillan Publishers Limited. All rights reserved.
UPPLEMENTARY INFORMATION doi: 0.038/nmat78 configuration gap, secondly, the gap reduction due to s accumulation, and thirdly the suppression of the gap reduction due to the Hanle dephasing contribution. The third term consists of two contributions corresponding to s injection from the layer and s extraction by the ned layer, and vice versa. The s accumulation data in Fig. 5a,b can be well described by equation (3.6 as shown in Fig. 5c,d. First, H K (~ 064 Oe and (~ 05 are found from the abrupt PAP switching field and the APP switching curve for H 0 (Fig. 5a. The resulting fitted curve is shown in Fig. 5c. The same fitting parameters well describe the data for non-zero H as shown in Fig. 5b,d for H Oe. The calculated variations of the layer normalized magnetization for H 0 and Oe are shown in the insets to Fig. 5c,d. With these values of H K and, H is found from the reduction in switching field and (~ 6.55 is fitted from the Hanle dephasing peak position for H 0 (see Fig. 5b,d. Not only the width but the asymmetry of Hanle dephasing peak are well described by the calculation. Both the experiment and model show that the superconducting energy gap, which is suppressed in the AP state by s accumulation, is restored in the presence of small in-plane perpendicular fields due to the Hanle dephasing of the accumulated s density. This effect occurs only for H ~0. Remarkably, when H takes small positive or negative values the Hanle dephasing is not effective so that there can be s accumulation and suppression of the superconducting energy gap. We find clear evidence for such a Hanle dephasing peak in our measurements (see Fig. 5b which can be well accounted for by our model (Fig. 5d. The fact that we observe a Hanle dephasing peak for which the gap is completely restored in small orthogonal in-plane fields near H ~0 shows that the nature materials www.nature.com/naturematerials 9 00 Macmillan Publishers Limited. All rights reserved.
UPPLEMENTARY INFORMATION doi: 0.038/nmat78 ss that are accumulated in the C Al layer in the AP configuration are readily depolarized by small perpendicular in-plane fields. Additional References for upplementary Information 44. 45. 46. Chien, C. L. & Westgate, C. R. (eds., The Hall effect and its applications (Plenum, 980. Fulde, P., High field superconductivity in thin films. Adv. Phys., 667 (973. Alexander, J. A. X. Orlando, T. P. Rainer, D. & Tedrow, P. M., Theory of Fermi-liquid effects in high-field tunneling. Phys. Rev. B 3, 58 (985. 0 0 nature MATERIAL www.nature.com/naturematerials 00 Macmillan Publishers Limited. All rights reserved.