Superconducting proximity effect in a diffusive ferromagnet with spin-active interfaces

Superconducting proximity effect in a diffusive ferromagnet with spin-active interfaces

A. Cottet and W. Belzig

Department of Physics and Astronomy, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland 共Received 27 September 2005; published 21 November 2005兲

We reconsider the problem of the superconducting proximity effect in a diffusive ferromagnet bounded by tunneling interfaces, using spin-dependent boundary conditions. This introduces for each interface a phase- shifting conductanceG_␾which results from the spin dependence of the phase shifts acquired by electrons upon scattering on the interface. We show thatG_␾strongly affects the density of states and supercurrents predicted for superconducting/ferromagnetic hybrid circuits. We show the relevance of this effect by identifying clear signatures ofG_␾in the data of T. Kontoset al.关Phys. Rev. Lett. 86, 304共2001兲, 89, 137007共2002兲兴. DOI:10.1103/PhysRevB.72.180503 PACS number共s兲: 74.50.⫹r, 74.20.⫺z, 73.23.⫺b Superconducting/ferromagnetic 共S/F兲 hybrid structures

raise the fundamental question of what happens when two phases with different broken symmetries interact. When aF metal with uniform magnetization is connected to a BCS superconductor, the singlet electronic correlations character- istic of theSphase propagate intoFvia Andreev reflections which couple electrons and holes with opposite spins and excitation energies. In the diffusive case, this propagation occurs on a scale limited by the ferromagnetic exchange field. The decay of the correlations inF is accompanied by oscillations of the superconducting order parameter because the exchange field induces an energy shift between the cor- related electrons and holes.^1,2This has been observed experi- mentally through oscillations of the density of states共DOS兲 in F,³ or of the critical current I₀ through S/F/S struc- tures,^4–7with the thickness ofFor the temperature. Remark- ably, the oscillations of I₀ have allowed one to obtain ␲ junctions, i.e., Josephson junctions withI₀⬍0,⁸which could find applications in the field of superconducting circuits.⁹

The interface between a ferromagnet and a nonmagnetic material can scatter electrons with spin that is parallel or antiparallel to the magnetization of the ferromagnet with dif- ferent phase shifts. The spin dependence of the interfacial phase shifts共SDIPS兲is a general concept in the field of spin- dependent transport. The SDIPS implies that spins noncol- linear to the magnetization precess during the scattering by the interface. This so-called spin mixing is expected to dras- tically affect the behavior ofF/normal metal systems¹⁰when several F electrodes with noncollinear magnetization are used. The same phenomenon is predicted to occur in F/coulomb blockade island,¹¹andF/Luttinger liquid¹²hybrid circuits. InS/Fhybrid systems,^13–15the SDIPS is even pre- dicted to affect the system in collinear configurations, due to the coupling of electrons and holes with opposite spins by the Andreev reflections. However, few experimental signa- tures of the SDIPS have been identified up to now共e.g., Ref.

13 proposes for the data of Ref. 16 an interpretation based on the SDIPS兲.

In this Rapid Communication, we reconsider the problem of the superconducting proximity effect in a diffusiveF. Up to now the tunnelS/F contacts used to produce this effect were described 共see, e.g., Ref. 2兲 with spin-independent boundary conditions 共BC兲 derived in Ref. 17 for the spin-

degenerate case. Instead of that, we use spin-dependent BC based on Ref. 15. These BC introduce a phase-shifting con- ductanceG_␾which takes into account the SDIPS. We show that G_␾ strongly affects the phase and the amplitude of the oscillations of the DOS orI₀ with the thickness of F. Our approach thus provides a framework for future work onS/F diffusive circuits with tunneling interfaces. We show its rel- evance by a comparison with the data of Refs. 3 and 5 which shows that strong experimental manifestations of the SDIPS have already been observed through the superconducting proximity effect.

We consider aS/Fhybrid circuit with a singleFelectrode homogeneously magnetized in direction zជ. In the diffusive limit, the electrons in a superconducting or ferromagnetic electrode␣ can be described with quasiclassical and diffu- sive Green’s functions Gˇ

␣ in the Keldysh丢Nambu丢spin space 共we use the notations of Ref. 15兲. The BC at a S/F interface can be calculated by assuming that the interface potential locally dominates the Hamiltonian, i.e., at a short distance it causes only ordinary scattering共with no particle- hole mixing兲. We characterize this scattering with transmis- sion and reflection amplitudes t_n,␴^S共F兲 and r_n,␴^S共F兲 for electrons coming from theS共F兲side in channelnwith a spin␴^parallel 共␴⁼↑兲or antiparallel共␴⁼↓兲tozជ. In practice, the planarS/F contacts used to induce the superconducting proximity effect in a diffusive ferromagnet are likely to be in the tunnel limit,^18,19due, e.g., to a mismatch of band structure between SandF, thus we assumeTn=兺_␴兩t_n,␴^S 兩²Ⰶ1. We also consider that the system is weakly polarized. Following Ref. 15 and 20, the BC at the right-hand sideF of aS/Finterface is

2gFGˇ

F

⳵^GˇF

⳵^{x =}

冋

^{GtGˇS+iG␾␴z␶ˇ3+G2MRDˇ+,GˇF}

册

+关iG_␹Gˇ

SDˇ

−+iG_␰Dˇ

−Gˇ

F,Gˇ

F兴 共1兲

withDˇ

±=␴z␶^ˇ3Gˇ

S±Gˇ

S␴z␶^ˇ3. Here,␴zand␶^ˇ3are Pauli matrices in spin and Nambu space, respectively. The conductivity ofF times the area of the junction, notedg_F, is assumed to be spin independent. The coefficient G_t=G_Q兺nT_n is the tunneling conductance,G_MR=G_Q兺n共兩t_n,↑^S 兩²−兩t_n,↓^S 兩²兲is the magnetoresis- tance term which leads to a spin polarization of the current, andG_␾= 2GQ兺n共␳n

F− 4关␶n

S/Tn兴兲 is the phase-shifting conduc- PHYSICAL REVIEW B72, 180503共R兲 共2005兲

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tance, with ␶n

S= Im关t_n,↑^S t_n,↓^S*兴, ␳n

F= Im关r_n,↑^F r_n,↓^F*兴 and G_Q=e²/h.

These three terms already appeared in Ref. 15 for studying normal electrodes in contact withS and F reservoirs 共with no proximity effect in F兲. The extra terms in G_␰

= −G_Q兺n␶n

S and G_␹=G_Q兺nT_n共␳n F+␶n

S兲/ 4 occur because there are superconducting correlations at both sides of the inter- face. Note thatG_␾,G_␹, andG_␰can be finite only if the phase shifts acquired by the electrons upon reflection or transmis- sion at the interface are spin dependent. The exact values of these conductance coefficients depend on the microscopic structure of the interface. However, we can estimate their relative orders of magnitude in a rectangular potential barrier model by describing the ferromagnetism of F with an ex- change fieldEexthat is much smaller than the spin-averaged Fermi energyE_F of F. This gives expressions ofG_MR, G_␾, G_␹, and G_␰ linear with Eex/EF. The tunnel limit can be reached by considering a strong mismatch between the Fermi wave vectors in S and F 共case 1兲 or a high enough barrier 共case 2兲. In both limits we find兩G_MR兩,兩G_␹兩,兩G_␰兩ⰆG_t, which allows us to neglect these terms in the following. In case 1, we find兩G_␾兩ⰆGtwhereas in case 2,兩G_␾兩 can be larger than Gt. Thus we will study the consequences of the spin- dependent BC for an arbitrary value of兩G_␾兩/Gt. In addition, in case 1 we findG_␾⬍0 but in case 2, the sign of G_␾ de- pends on the details of the barrier, thus we will consider both signs forG_␾.

In equilibrium, we can use normal and anomalous quasi- classical Matsubara Green’s functions parametrized, respec- tively, as cos共⌳_␴兲 and sin共⌳_␴兲exp共i␸_␴兲 to describe the nor- mal excitations and the condensate of pairs共see, e.g., Ref.

21兲. The spatial variations of the superconducting correla- tions inF are described by the Usadel equations⳵^Q␴/⳵^{x= 0} and

⳵²⌳_␴/⳵^x2=k_␴²sgn共␻n兲sin共⌳_␴兲/␰F

2+Q_␴²cos共⌳_␴兲/sin³共⌳_␴兲, with ␰F=共បD/Eex兲^1/2, ␻n=共2n+ 1兲␲^kBT. Here, Q_␴

= sin²共⌳_␴兲⳵␸_␴^/⳵^x is the spectral current 共constant with x兲 and D the diffusion coefficient. We introduced k_␴

=兵2关i␴^sgn共␻n兲+兩␻n兩/E_ex兴其^1/2for later use.²NeglectingG_MR, G_␹, andG_␰in共1兲yields

gF

⳵⌳_␴

⳵^{x =iG␾}␴sin共⌳_␴兲+Gt关cos共⌳S兲sin共⌳_␴兲

− sin共⌳S兲cos共⌳_␴兲cos共␸␴−␸S兲兴, 共2兲

gF

⳵␸␴

⳵^{x sin共⌳␴兲=Gtsin共⌳S兲sin共}␸␴−␸S兲. 共3兲 In Eqs. 共2兲 and 共3兲, we used rigid BC for S, i.e., ⌳_␴=⌳S

= arctan关⌬/␻n兴, with⌬ the gap ofS.

In the following, we consider the limit of a weak proxim- ity effect in F, i.e., ⌳_␴=␪_␴ for ␻n⬎0 and ⌳_␴=␲⁻␪_␴ for

␻n⬍0 with 兩␪_␴共x兲兩Ⰶ1. We first study geometries with Q_␴

= 0, i.e., no supercurrent flows through the device. In this case, the proximity effect in F can be probed through measurements of the density of states N共␧兲=N₀兵1

−兺_␴Re关␪_␴²共x兲兴/ 4其 关with ␻n= −i␧+ 0⁺ and sgn共␻n兲= 1兴. The simplest case of a singleS/Finterface withFatx⬎0 yields

␪_␴^SF共x兲= ␥tsin共⌳S兲

␥t兩cos共⌳S兲兩+i␥_␾␴^sgn共␻n兲+k_␴e^{−k␴x/␰F} 共4兲 with ␥t共␾兲=Gt共␾兲␰F/gF. In the limit ⌬ⰆEex where k_␴= 1 +i␴sgn共␻n兲, the weak proximity effect hypothesis leading to 共4兲is valid for any values of␥␾and␧if␥tⰆ1. Sincek_␴has finite real and imaginary parts,␪_␴^SF共x兲shows the well-known exponentially damped sinusoidal oscillations withd. The re- markable point in共4兲is that␥␾ shifts these oscillations and modifies their amplitude 关see Fig. 1 which shows the DOS following from共4兲兴. We also study theS/F/Igeometry, with F at x苸关0 ,d兴 and the insulating layer I at x⬎d, for later comparison with the experimental data of Ref. 3. Using共2兲 for theS/F interface and⳵␪_␴/⳵^{x= 0 for F/I yields}

␪_␴^SFI共x兲=␪_␴^dcosh

冉

^共x−d兲k␰^␴F

冊 冋

^cosh

^冉

^k␴␰^dF

冊 册

^{−1 共5兲}

with

␪_␴^d= ␥tsin共⌳S兲

关␥t兩cos共⌳S兲兩+i␥_␾␴^sgn共␻n兲+k_␴tanh共k_␴d/␰F兲兴. In the limit⌬ⰆE_exandd艌␰F, the␪linearization leading to 共5兲is again valid for any␥_␾and␧if␥tⰆ1. From Fig. 1,␥_␾ has qualitatively the same effect on ␪_␴^SFI共x兲 as on ␪_␴^SF共x兲.

More quantitatively, for dⰇ␰F one has ␪_␴^SFI共x=d兲/␪_␴^SF共x

=d兲= 2共Ref. 22兲and for lower values ofd, this ratio depends ond.

Another way to probe the superconducting proximity ef- fect in F is to measure the supercurrent through a S/F/S Josephson junction. We consider a junction with F at x 苸关0 ,d兴 and a right 共left兲 superconducting reservoir, called R共L兲 at a constant phase 共−兲␸S/ 2. A supercurrent I_S

=␲^gFkBT兺n苸Z,␴=±1Q_␴共␻n兲/ 2eflows through this device.²We focus on the asymmetric limit␥t

RⰆ␥t

L, which corresponds to the experiment of Ref. 5, and assume␥_␾^{R= 0.23 We allowL} andR to have different superconducting gaps⌬^R共L兲, so that

⌳_␴=⌳_S^R共L兲 inR共L兲. Solving this problem perturbatively with respect to theS/F/Icase yields

FIG. 1. Zero energy density of states atx=din aS/F/Istruc- ture, in terms of兵关N共0兲/N₀兴− 1其/␥t

2 as a function ofd/␰F, for␥t

Ⰶ1 and different values of ␥_␾ 共full lines兲. The dotted lines show 4[关N共0兲/N₀兴− 1]/␥t

2atx=din a semi-infiniteS/Fstructure with the same values of␥tand␥_␾. The inset shows the DOS atx=d as a function of␥␾for theS/F/Istructure.

A. COTTET AND W. BELZIG PHYSICAL REVIEW B72, 180503共R兲 共2005兲

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180503-2

Q_␴共␻n兲=␪_␴^d␥t Rsin共⌳S

R兲sin共␸S兲

冋

^␰Fcosh

^冉

^k␴␰^dF

冊 册

^{−1, 共6兲}

where ␪_␴^d corresponds to the expression given above with

⌳S=⌳S

L and ␥t共␾兲=␥tL共␾兲

. The supercurrent has the form IS

=I₀sin共␸S兲 because most of the phase drop occurs at R. In the limit⌬^L=⌬^R=⌬ⰆEex,␥t

LⰆ1 andd/␰FⰇ1,共6兲yields

eI₀

␥t

LG_t^R⌬=␲^tanh

冉

2k^⌬BT

冊 冤

^{关sin1 +}

^冉

^{␰共d1 +F+␥␭共␾L␥兲2␾L兴兲1/2}

^冊

^e−d/␰F

冥

^共7兲

with ␭共␥_␾^L兲= arg关i−共1 +␥_␾^L兲兴. It is already known that the state of the junction depends ond. Equation 共7兲 shows that

␥_␾^L shifts the oscillations of theI₀共d兲curve. Thus, for a given value ofd, the state of the junction can be 0 as well as␲^, depending on␥_␾^L. Figure 2 shows that this effect still occurs when one goes beyond the large d/␰F approximation. Note that in the limit⌬ⰆEex and␥t

LⰆ1 used to obtain 共7兲, it is not possible to find a temperature crossover for the sign ofI₀ as observed in Refs. 4 and 6. However, we expect to find such a temperature crossover with a 0 /␲ ^or ␲/ 0 transition, depending on the value of␥_␾^L, if the energy dependence ofk_␴ is taken into account.²⁵

To show the relevance of our approach, we compare our predictions with the measurements of Refs. 3 and 5. We first consider the兩I0兩measured in an asymmetricS/F/Sjunction, i.e., Nb/ Pd_1−xNi_x/ Alox/ Al/ Nb with x⬃0.1 and ␥t

L/␥t R

⬃10⁵. We assume that the contacts have TnⰆ1, which al- lows to use Eqs.共2兲and共3兲. We will use the experimentally determined values ⌬^Al/Nb= 0.6 meV and ⌬^Nb= 1.35 meV ⰆEex, which impliesk_␴⬃1 +i␴^{, andT}= 1.5 K. Samples with different thicknessesd of PdNi were measured共see Fig. 3兲. Interpreting these data requires a careful analysis of the in- fluence of d on the different parameters. We have gF

= 2e²N₀DAand␰F=

冑

_បD/E_ex, withD=v_Fl/ 3 andAthe con- ductors cross section. Curie temperature measurements show that the exchange field Eex increases linearly with d.²⁶ In addition, we first assume that the mean free pathlis constant withd, as confirmed by resistivity measurements for d⬎d₀

= 80 Å. This allows us to parametrize the problem with ␥t L

=a_t⁰␰F

0/

冑

d₀d and␰F=␰F

0

冑

d₀/dwhere␰F

0 anda_t⁰=G_t^Ld₀/gF are constant withd. We also assume that G_␾^L is proportional to E_ex as found above in the rectangular barrier model for E_ex ⰆEF, so that we take␥_␾^L=␥_␾⁰

冑

d/d₀with␥_␾⁰ constant withd.

We neglect ␥_␾^R due to the existence of a strong insulating barrier atR.²⁴ The absolute amplitude ofEexwas not deter- mined exactly, so that␰F0 can be considered as a fitting pa- rameter as well asa_t⁰and␥_␾⁰. This makes in total three fitting parameters but we expect to find fora_t⁰ a value close to the value 0.2 found from minigap measurements in Nb/ Pd.²⁵We have calculated兩I₀兩by summing共6兲on energy and spin. It is not possible to account for the data with␥_␾⁰= 0. On the con- trary, a good agreement with the experiment is obtained by using a_t⁰= 0.4, ␰F

0= 36 Å, and ␥_␾^{0= −1.3} 共full lines in Fig.

3兲.^26,27We have checked that this choice of parameters ful- fills the hypothesis兩␪␴共x兲兩Ⰶ1 made in our calculations. Re- markably, ford⬃d₀in Fig. 3, the theory for␥_␾⁰= −1.3 gives I₀⬍0 in agreement with subsequent experiments,^27,28 whereas it givesI₀⬎0 for␥_␾⁰= 0 if one keeps the same orders of magnitude for a_t⁰ and ␰F

0. For d⬍d₀, l is linear with d, which we have taken into account by using␰F=␰F0,␥_␾^L=␥_␾^0, and␥t

L=a_t⁰␰F

0/d, with the same values of a_t⁰,␥_␾⁰, and ␰F 0 as previously given 共dashed lines in Fig. 3兲. This approach gives a surprising agreement with the data, which seems to indicate that the Usadel description still works for d⬍d₀ although l is linear with d.²⁹ Kontos et al. have also per- formed DOS measurements in Nb/ Pd_1−xNix/ Alox/ Al,³prior to theI₀measurements. We have assumed again thatEexwas linear withd in these measurements, to try to interpret the N共0兲=f共d兲 curve with the same fitting procedure as for I₀. We have generalized Eq.共5兲to second order in ␪_␴ because the values ofd/␰Fare slightly lower than for theI₀measure- FIG. 2. Critical currentI₀of an asymetricS/F/Sjunction as a

function of d/␰F, calculated from Eq. 共6兲 for ␥_t^RⰆ␥_t^LⰆ1, ⌬^L共R兲

=⌬ⰆE_exandk_BT/⌬= 0.15共full lines兲. The dashed lines show the larged/␰Fapproximation of Eq.共7兲. The inset is a phase diagram indicating the equilibrium state of the junction共0 or␲兲depending on␥_␾andd/␰F.

FIG. 3. Critical current measured as in Ref. 5 through Nb/ Pd_1−xNi_x/ Alox/ Al/ Nb junctions as a function of the thickness dof Pd_1−xNi_x共symbols兲. The lines are theoretical curves calculated from Eq.共6兲ford⬎d₀共full lines兲andd⬍d₀共dashed lines兲with the fitting parametersa_t⁰= 0.4,␰F

0= 36 Å and the experimentally deter- mined parameters⌬^Nb= 1.35 meV, ⌬^Al/Nb= 0.6 meV,d₀= 80 Å and T= 1.5 K. The data are well fitted with␥_␾⁰= −1.3. We also show the theory for ␥_␾⁰= 0. Inset: DOS measured by Ref. 3 in Nb/

Pd_1−xNi_x/ Alox/ Al, as a function of d. The full and dotted lines show the DOS atx=dcalculated from the second-order generaliza- tion of共5兲 共see text兲, ford⬎d₀and d⬍d₀, respectively. We used

␰F

0= 50 Å and ␥_␾⁰= −1.6 or ␥_␾⁰= 0, all the other parameters being unchanged.

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180503-3

ments. Again it is impossible to interpret the data with ␥_␾⁰

= 0. We obtain a satisfactory fit by choosing␰F

0= 50 Å and

␥_␾⁰= −1.6, all the other parameters used being the same as in the previous case. Finding a␰F

0 higher than for theI₀data is in agreement with the fact that the samples used for measur- ing the DOS were realized with a lower concentrationxof Ni.

In summary, we have studied the effect of spin-dependent boundary conditions on the superconducting proximity effect in a diffusive ferromagnet bounded by tunneling interfaces.

We have shown that the phase-shifting conductancesG_␾, de- scribing the spin activity of the interfaces in this context,

strongly affect the behavior of the system and allow a con- sistent microscopic explanation of the DOS and supercurrent data of Refs. 3 and 5. This suggests that such effects will have to be considered in any future work on S/F hybrid circuits. In the context of spintronics, this approach might also provide a way to characterize spin-active interfaces.

We thank T. Kontos for raising the question which led us to perform this study and for providing us with the experi- mental data. We thank C. Bruder, T. T. Heikkilä, and D.

Huertas-Hernando for discussions. This work was financed by the Swiss NSF and the NCCR Nanoscience.

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23With a high 共rectangular兲 barrier, one has G_tⰆ兩G_␾兩, but if the barrier is too high, one can check that I₀ is unaffected byG_␾^R becauseG_␾^RⰆG_t^L,G_␾^L.

24This will be investigated elsewhere.

25T. Kontos, Ph.D. thesis, Université Paris-Sud, Orsay, France, 2002.

26Another interpretation of Ref. 5 was proposed in Ref. 27 for␥_␾

= 0, but with values of ⌬L⌬R and T in disagreement with the experimentally determined parameters and continuous BC ⌳_␴

=⌳SLforx= 0.

27A. Buzdin and I. Baladié, Phys. Rev. B 67, 184519共2003兲.

28A. Bauer, J. Bentner, M. Aprili, M. L. Della-Rocca, M. Reinwald, W. Wegscheider, and C. Strunk, Phys. Rev. Lett. 92, 217001 共2004兲.

29The ballistic approach of M. Zareyan, W. Belzig, and Yu. V. Naz- arov, Phys. Rev. Lett. 86, 308共2001兲; Phys. Rev. B 65, 184505 共2002兲could not account for the fullddependence of Ref. 5.

A. COTTET AND W. BELZIG PHYSICAL REVIEW B72, 180503共R兲 共2005兲

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