Controllable supercurrent in mesoscopic superconductor-normal metal-ferromagnet crosslike Josephson structures
Tatiana E. Golikova, Michael J. Wolf, Detlef Beckmann, Grigory A. Penzyakov, Igor E. Batov, Irina V. Bobkova, Alexander M. Bobkov, Valery V. Ryazanov
aa r X i v : . [ c ond - m a t . s up r- c on ] A ug Controllable supercurrent in mesoscopic superconductor-normal metal-ferromagnetcrosslike Josephson structures
T. E. Golikova a , M. J. Wolf b , D. Beckmann b , G. A. Penzyakov a ,I. E. Batov a , I. V. Bobkova a,c,d , A. M. Bobkov a , and V. V. Ryazanov a,d a Institute of Solid State Physics RAS, 142432 Chernogolovka, Moscow district, Russia b Institute for Quantum Materials and Technologies,Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany c Moscow Institute of Physics and Technology, Dolgoprudny, 141700 Russia d Faculty of Physics, National Research University Higher School of Economics, Moscow, 101000 Russia
A nonmonotonic dependence of the critical Josephson supercurrent on the injection currentthrough a normal metal/ferromagnet weak link from a single domain ferromagnetic strip has beenobserved experimentally in nanofabricated planar crosslike S-N/F-S Josephson structures. This be-havior is explained by 0- π and π -0 transitions, which can be caused by the suppression and Zeemansplitting of the induced superconductivity due to interaction between N and F layers, and the in-jection of spin-polarized current into the weak link. A model considering both effects has beendeveloped. It shows the qualitative agreement between the experimental results and the theoreticalmodel in terms of spectral supercurrent-carrying density of states of S-N/F-S structure and thespin-dependent double-step nonequilibrium quasiparticle distribution. I. INTRODUCTION
The interplay between spin-singlet superconductiv-ity and ferromagnetism in mesoscopic hybrid structuresleads to variety of novel effects actively investigated inthe last decades. The most remarkable experiments werecarried out using nonlocal technique of detection of thespin accumulation and injection in normal metals and su-perconductors [1–3], coherent effects of crossed Andreevreflection and elastic cotunneling [4–8], Zeeman splittingin superconductors with adjacent ferromagnetic insula-tors [9], charge and spin imbalance [10–14], specific ther-moelectric effects [15–17].An unusual Josephson effect characterized by the in-verse current-phase relation I = − I c sinϕ , the so called π -state of a Josephson junction, was observed in quitemacroscopic trilayered SFS systems with a weak fer-romagnetic interlayer [18, 19]. There the temperatureinduced 0- π transition was demonstrated. Experimen-tally feasible lateral superconductor (S)-normal metal(N)-ferromagnet (F) Josephson junctions manifesting the π -state under certain conditions were studied in detailtheoretically [20]. Implementation of such structures assubmicron-scale π -phase shifters is of particular inter-est for superconducting and quantum electronics [21],which has already been demonstrated with sandwich-type SFS Josephson junctions incorporated in supercon-ducting logical schemes [22] and in the qubit loop [23].The future improvement is associated with the systemsize reduction by means of the fabrication of submicronlateral hybrid structures.The driving out of equilibrium conditions makes itpossible to achieve 0- π transitions even without ferro-magnets as it was predicted [24] and demonstrated [25]for controllable SNS Josephson junctions. The transi-tion into π -state occurred there at a certain value ofthe control voltage across the N-barrier, which corre- sponds to a sign change in the spectrum of supercurrent-carrying states Im[ J ( ǫ )]. The latter is characterized bythe strongly damped oscillations with positive and neg-ative parts which is cut out by the double-step-like elec-tron distribution function or smeared by the thermal dis-tribution function (see Fig.2 in [26]).It is predicted that for similar mesoscopic structureswith ferromagnets [27] (or under an applied magneticfield [28]) the spectral supercurrent is shifted due to thepresence of the exchange or Zeeman field h which leadsto the redistribution of current-carrying states. The re-distribution results in the suppression of the Joseph-son current in equilibrium. The recovery of the super-current by applying a suitable non-equilibrium distribu-tion of quasiparticles was proposed. The recovery waspredicted both for spin-independent [28] and for spin-dependent nonequilibrium quasiparticle distribution [29].The proper spin-dependent quasiparticle distribution isthe most efficient way to recover the supercurrent. More-over, 0 − π transitions driven by the nonequilibrium quasi-particle distribution can be observed [28, 30, 31].In this article, we present the first observation of dou-ble 0- π transition in crosslike S-N/F-S mesoscopic Al-Cu/Fe-Al structures. We claim that this effect is dueto spectral supercurrent modification in the presence ofthe effective exchange field h in the hybrid weak link andcontrollable non-equilibrium distribution of quasiparti-cles. The corresponding calculations for the geometryof our structures and the parameters obtained from themeasurements show qualitative agreement with the ex-periment. II. SAMPLES AND EXPERIMENT
In the investigated structures Al, Cu and Fe were usedas a superconductor (S), a normal metal (N), and a fer- I + - V + - I inj + - Al AlCuFe
200 nm
AlCuFe m m m (a)(b) Figure 1: (a) SEM image of the Al-Cu/Fe-Al junction withadditional contacts to the Fe strip together with the measure-ment scheme. (b) The schematic sketch of the sample withgeometrical dimensions. romagnet (F) correspondingly. Figure 1(a) shows a scan-ning electron microscopy (SEM) image of one of our sam-ples, together with the measurement scheme, Fig. 1(b)demonstrates a schematic sketch of the structure with itsgeometrical dimensions. The submicron-scale crosslikeS-N/F-S junctions were fabricated by means of electronbeam lithography and in situ shadow evaporation. First,a thin (10-15 nm) iron layer was deposited onto an oxi-dized silicon substrate at the first angle to form a ferro-magnetic injector. Then a copper layer of 60 nm (or 30nm) thickness was evaporated at the second angle to cre-ate a complex N/F-weak link in the intersection area of Nand F layers. Finally, a thick (100 nm) layer of aluminumwas deposited at the third angle to form superconduct-ing banks of the junction and electrical contacts to theperpendicular Fe electrode. All these technological stepswere executed without breaking the vacuum, so that allFN and NS interfaces are assumed to be highly trans-parent. The geometrical dimensions are the same forall structures, i.e. the distance between superconductingbanks is 200 nm, the width of iron strip is 160 nm, thewidth of copper strip is 200 nm (Fig. 1 (b)). Since the I c ( n A * ) I c ( n A ) T (K)
Figure 2: (color online) Temperature dependencies of the crit-ical current I c for the Al-Cu-Al (blue circles) and Al-Cu/Fe-Al (black triangles) Josephson junctions together with the fits(solid lines) described in the discussion. resistance of iron film with the specific resistance ρ F =70 µ Ω × cm and thickness d F =10 nm is much larger thanthe resistance of thick copper film ( ρ N = 4.5 µ Ω × cm, d N =60 nm), the main part of current flows through theN layer.The transport measurements were performed in ashielded cryostat at temperatures down to 0.3 K. Low-pass RC filters were incorporated into DC measurementlines directly on the sample holder in order to eliminatethe external noise. The current-voltage characteristics ofcrosslike S-N/F-S junctions were measured by using thestandard four-terminal configuration in presence of theinjection current via the iron electrode as it is shown inFig.1(a). The dimensions and thickness of the ferromag-netic strip have provided the single domain state with thepractically uniform magnetization aligned parallel to thethe strip, as it was demonstrated in our previous work[32].The Josephson supercurrent was observed only forsamples with the thickness of copper layer d N =60 nm.In comparison with a reference S-N-S ( Al-Cu-Al ) Joseph-son junction with the same geometrical dimensions (theonly difference was the absence of a perpendicular ironstrip) the critical current of hybrid S-N/F-S structureswas strongly suppressed. The temperature dependenciesof the critical currents for both structures are shown inFig. 2. The suppression of the Josephson supercurrentin case of S-N/F-S structures is explained by the proxim-ity of the ferromagnetic layer in contact with the copperlayer in the weak link. Although the area of intersectionof N and F layers in the cross-shaped S-N/F-S struc-tures is much less than in the layer-on-layer weak link[32], the F layer still affects considerably the transportsuperconducting coherence properties. The effective spin
Figure 3: (color online) Critical current I c of the crosslikeAl-Cu/Fe-Al junction (A-S1) as a function of the injectioncurrent I inj across the junction at T=0.3 K. Black verticallines are guides to the eye. Inset: The example of current-voltage characteristic of Al-Cu/Fu-Al junction ( I inj = 0) usedfor the critical current determination. polarization is induced into the N layer due to relativelylarge spin diffusion length in Cu ( λ N =1 µ m at 1 K [34])in comparison with the dimensions of Cu strip in theweak link (Fig.1). A mechanism of the supercurrent sup-pression in such systems has already been discussed intheoretical works [27, 28] and it is related to the modifi-cation of equilibrium spectral functions by their shiftingwith the effective exchange energy or Zeeman field h incomparison with the normal metal case.In the main experiments, the current-voltage (I-V)characteristics of the crosslike S-N/F-S junction weremeasured for different values of the injection current I inj across the junction (Fig. 1(a)). From the I-V curves wedetermined the critical current I c for each value of the in-jection current I inj as it is shown in Fig.3. One can see aclear nonmonotonic behavior of the critical current withthe increasing of the injection current. This behaviordoes not depend on the injection current sign. Generally,the I c ( I inj ) dependence demonstrates the critical currentdecrease with the increasing of the injection current andtwo local dips at I inj (1) =0.25 µ A and I inj (2) =1.4 µ A. Wesuppose that the observed behavior is due to transitionsbetween 0- and π - states at node values I inj (1) and I inj (2) .As noted above, it was predicted that the different typesof Josephson junctions with a complex N/F weak linkcan be in the π -state even in equilibrium conditions [20],however, the π -state is also possible in controllable SNSsystems with noneqiulibrium electron distribution due tocurrent injection [25]. In our work, we combine both ap-proaches. Figure 4: FNF configuration for the calculation of the distri-bution function. The direction of the injection current flow isindicated by arrows.
III. DISCUSSIONA. Quasiparticle distribution in the interlayer
We associate the mechanism of the π -state formationwith the redistribution in occupied fractions of positiveand negative supercurrent-carrying density of states (SC-DOS) [25, 27, 28, 30]. For the case under considerationan effective exchange field is induced in the normal layerdue to the proximity with the ferromagnet [20]. It pro-vides more complicated SCDOS in the interlayer thanfor the case of a conventional S/N/S Josephson junction(see detailed discussion below). In its turn this more richstructure of SCDOS makes the Josephson current moresensitive to the quasiparticle redistribution allowing forobservation of the double-transition behavior.At first we describe the nonequilibrium quasiparticledistribution, which is formed in the normal part of theinterlayer due to the current injection from the ferromag-net in our system depicted in Fig. 1. For simplicity, wesuppose that no transverse current enters into the super-conducting banks of the junction (since the low energyprocess taking place at energies much less than supercon-ducting gap of aluminum ∆=180 µ eV is considered, and,apart from this, the S parts are shifted away from the Fone and have no intersection with it). We can considerour crosslike N/F system as F/N/F spin valve in parallelconfiguration (Fig. 4) since the main part of the injectioncurrent flows through the N layer, as it was noted above.The quasiparticle distribution in the F layers can bedescribed in terms of the different electrochemical po-tentials µ ↑ , ↓ [35], while for the N layer such descriptionis inappropriate because the N layer length L is shorterthan all the inelastic relaxation lengths and the Fermi dis-tribution is not formed here. The spin relaxation lengthfor Cu greatly exceeds the N layer length L =200 nm, asit was noted above. Therefore, the spin relaxation termcan also be neglected in the kinetic equation, which canbe written for the distribution function ϕ σ for the spinsubbands separately and takes the form: ∂ x ϕ σ = 0 . (1)This equation should be supplemented by the Kuprianov-Lukichev boundary conditions at x = 0 , L : ∂ x ϕ σ (cid:12)(cid:12)(cid:12) x =0 ,L = ∓ G σ σ N (cid:16) tanh ε − µ L,Rσ ( x = 0 , L )2 T − ϕ σ (cid:17) , (2)where µ L,Rσ ( x = 0 , L ) are electrochemical potentials forleft (right) ferromagnets at the F/N interfaces, respec-tively and G σ is the conductance of F/N interfaces forthe spin subband σ . The solution of Eqs. (1)-(2) takesthe form: ϕ σ = 12 (cid:16) tanh ε − µ Lσ T + tanh ε − µ Rσ T (cid:17) + G σ σ N (1 + G σ L σ N ) (cid:16) tanh ε − µ Rσ T − tanh ε − µ Lσ T (cid:17) ( x − L , (3)where µ Lσ and µ Rσ are taken at the corresponding N/Finterfaces x = 0 , L . The electrochemical potentials in theferromagnets can be found from the appropriate diffusionequation and take the form [35, 36]: µ Lσ ( x ) = A + je ( x − L/ σ F + σCe x/λ F σ σF , (4) µ Rσ ( x ) = − A + je ( x − L/ σ F + σDe − ( x − L ) /λ F σ σF . (5)where λ F is the spin diffusion length in the ferromag-nets, σ σF is the ferromagnet conductivity for a given spinsubband and σ = ± A , C and D are to be found fromthe condition of the continuity of the electric current atthe N/F interfaces for each of the spin subbands sepa-rately. The electric currents j σ in the ferromagnets canbe calculated as j Fσ = ( σ σF /e ) ∂ x µ σ , while in the normalinterlayer it should be calculated according to: j Nσ = σ N e ∞ Z −∞ dε∂ x ϕ σ ( ε ) . (6)Then the condition j Fσ = j Nσ at x = 0 , L gives us con-stants A , C and D = − C , and the electrochemical po-tentials µ Lσ ( x = 0) and µ Rσ ( x = L ) entering Eq. (3) takethe form: µ Lσ ≡ µ σ = jeλ F (1 − κ σ ) P σ κ σ σ σF − κ σ , (7) µ Rσ = − µ Lσ = − µ σ , (8)with κ σ = λ F G σ σ σF (1 + G σ L σ N ) . (9) Figure 5: Characteristic form of the distribution functionplotted according to the first spatially constant term ofEq. (3).
Experimentally relevant parameters allow for disre-garding the incline of the distribution function in theinterlayer, described by the second term in Eq. (3). Thenthe distribution function can be considered as spatiallyconstant and the characteristic example is plotted inFig. 5 as a function of energy. It is seen that the distri-bution functions are different for the both spin subbandsand exhibit the double-step structure, which is typicalfor mesoscopic regions between two leads, where the dis-tribution function is not thermalized and is mainly de-termined by the distributions in the leads.Because of the condition µ Rσ = − µ Lσ the distributionfunction is antisymmetric with respect to ε → − ε . To-gether with the fact that in general µ L,R ↑ = µ L,R ↓ thisleads to the conclusion that in the system under con-sideration two physically different modes of nonequilib-rium quasiparticle distribution are nonzero. In nota-tions of Refs. 37, 38 they are f L = (1 / ϕ ↑ + ϕ ↓ )and f L = (1 / ϕ ↑ − ϕ ↓ ). The first one is the energynonequilibrium mode. In general, it is associated withthe nonequilibrium distribution of quasiparticles over en-ergy levels. For example, excess energy can be pumpedinto the system due to the injection current. The result-ing nonequilibrium distribution can be thermalized (thequasiparticle subsystem is overheated) or not thermal-ized. The latter case is realized in our experiment in theparticular form of the double-step structure of the distri-bution function. f L is the so-called spin-energy mode,which can be associated with, roughly speaking, differentenergies pumped into spin-up and spin-down quasiparti-cles. If the spin-up and spin-down quasiparticle distribu-tions are thermalized separately, it shows up as differenteffective temperatures for spin-up and spin-down quasi-particles. In our case of non-thermalized quasiparticledistributions it manifests itself by different widths of thedouble-step structure for spin-up and spin-down subsys-tems, see Fig. 5. The other two possible nonequilibriummodes (the charge imbalance and spin imbalance) are notexcited in the system. B. Critical current of the junction underqusiparticle injection
Below we describe the calculations of a supercurrent inthe S-N/F-S contact. We consider S as an ordinary super-conductor in equilibrium with the superconducting gap∆, complex N/F weak link as a normal metal with non-equilibrium distribution and Zeeman splitting h and thedepairing parameter Γ. The latter parameter accountsfor the leakage of the superconducting correlations intothe ferromagnet and depairing there [39]. d is the lengthof the N/F area (the distance between superconductingbanks), G SF is the specific conductance of the S-N/Fboundaries, R NF is the normal state resistance of theN/F area, σ NF is the conductivity and D NF is the dif-fusion coefficient. The calculation is performed in theframework of Usadel equations for Green’s functions inthe Keldysh technique. We linearize the Usadel equationin the interlayer region assuming that the S-N/F inter-faces are low transparent. The assumption works quitewell as it is indicated by further numerical estimates ofthe interface transparency. The critical current can beexpressed via the SCDOS and the distribution functionas follows: j c = d eR NF Z ∞−∞ dε X σ ( ϕ σ ( ε ) + ˜ ϕ σ ( ε )) Im [ J ε,σ ] , (10)where Im[ J ε,σ ] is the SCDOS for a given spin subbandwith J ε,σ = sinh ( λ σ d )( G SF /σ NF ) ( g RS − λ σ (cid:16) sinh ( λ σ d ) + G SF g RS σ NF λ σ cosh ( λ σ d ) (cid:17) . (11) g RS = − i ( ε + iδ ) / p ∆ − ( ε + iδ ) is the retarded Green’sfunction and λ σ = p − i ( ε + σh + i Γ) /D NF .In general, ˜ ϕ σ ( ε ) = − ϕ − σ ( − ε ). In our case, due tothe antisymmetry of the distribution function this leadsto ϕ σ ( ε ) + ˜ ϕ σ ( ε ) = ϕ ↑ ( ε ) + ϕ ↓ ( ε ), i.e. the effective quasi-particle distribution, which ”occupies” the SCDOS inEq. (10), does not depend on spin. Therefore, only theenergy nonequilibrium quasiparticle mode f L is relevantfor the current situation.The SCDOS for our N/F interlayer is demonstratedin Fig. 6(b). The plot corresponds to the parameters˜ G = G SF ξ /σ NF = 0 . d = 1 . ξ , h = 1 .
41∆ andΓ = 0 . ξ = p D NF / ∆ ≈
190 nm. Theseparameters are found by fitting the experimental datapresented in Fig. 2. d is an effective length of the normalinterlayer, it does not exactly coincides with the actuallength of the Cu region because the Cu regions under-neath the Al leads and Cu regions between the Al leadsand the Fe strip are partially proximitized by the Al su-perconductivity.The SCDOS for the N/F interlayer should be com-pared to the SCDOS for a usual N interlayer correspond-ing to h = 0, Γ = 0 [Fig. 6(a)]. It is seen that the (a) (b) Figure 6: SCDOS J ε = Im J ε, ↑ + Im J ε, ↓ as a function ofquasiparticle energy. (a) SCDOS for a normal interlayer with h = Γ = 0; (b) SCDOS for the N/F interlayer with h = 1 . . d = 1 . ξ and G SF ξ /σ N = 0 .
13 for theboth panels.
SCDOS is a sign changing function. This is the reasonproviding the possibility for 0- π transitions driven by anexternal parameter controlling the quasiparticle distri-bution. For the case of only one zero-crossing point at ε >
0, as it takes place for the SCDOS in the N interlayer[Fig. 6(a)], no more than one 0- π transition driven by in-jection, which results in the double-step or overheatedquasiparticle distribution, is possible. This situation hasbeen realized experimentally [25]. On the contrary, theSCDOS for our system manifests two zero-crossing pointsat ε > π transitions driven by injection.The evolution of the distribution modes f L and f L upon increasing of the injection current is presented inFigs. 7(a,b), respectively. As it was discussed above forthe problem under consideration, the critical current isonly determined by f L . At zero temperature it manifestsa four-step structure as a function of energy. The stepsoccur at ε = ± µ ↑ , ± µ ↓ . According to Eq. (5), the elec-trochemical potentials are proportional to the injectioncurrent j . Therefore, µ ↑ , ↓ = α ↑ , ↓ J inj , where taking theparameters of our F/N/F structure ( σ ↑ F − σ ↓ F ) /σ F = 0 . G NF = G ↑ + G ↓ = 0 . G SF , ( G ↑ − G ↓ ) /G NF =0 . λ F =8.5 nm [42] we obtain α ↑ = 1 . /µA =279 eV /A and α ↓ = 1 . /µA = 281 eV /A . It is seenthat µ ↑ and µ ↓ are very close for the particular parame- (a) (b) Figure 7: Quasiparticle distribution: nonequilibrium modes(a) f L = ( ϕ ↑ + ϕ ↓ ) / f L, = ( ϕ ↑ − ϕ ↓ ) / σ ↑ F − σ ↓ F ) /σ F = 0 . G NF = G ↑ + G ↓ = 0 . G SF , ( G ↑ − G ↓ ) /G NF = 0 . T = 0 . K , β = 0 . K/ ( µA ) . ters of the structure. As a result, the spin-energy mode f L is rather small here, as it is demonstrated in Fig. 7(b)and the four-step structure is very close to the double-step one even at T = 0. Due to nonzero temperature thisstep structure is smeared. The temperature smearing isfurther increased because of the Joule heating by the in-jection current, which is modelled by T → T + βJ inj with β = 0 . K/ ( µA ) .As it is seen from Fig. 7(a), upon increasing the in-jection current the quasiparticles are redistributed tohigher energies leaving the low-energy states. This leadsto gradual turning off the low-energy parts of the SC-DOS. Therefore, the main part of the SCDOS contribut-ing to the critical current changes sign upon increasingthe injection current and the maximal number of 0- π -transitions is given by the number of zero-crossings inthe SCDOS, as it was already mentioned above.The critical current calculated in the framework of ourtheoretical model as a function of the injection currentis presented in Fig. 8. It manifests two 0- π -transitionsin qualitative agreement with the experimental results.However, the quantitative agreement is lacking because ofthe complicated geometry of the experimental structure.To obtain better approximation one should take into ac-count that the real junction is not a simple S-N/F-S junc-tion but contains regions of proximity superconductivity Figure 8: Critical current of the S-N/F-S Josephson junctioncalculated as a function of the injection current. The parame-ters of the junction correspond to SCDOS shown in Fig. 6(b)and distribution function presented in Fig. 7. underneath the superconducting leads, which are verysensitive to the nonequilibrium quasiparticle distributionin the interlayer.
IV. CONCLUSION
To conclude, we have demonstrated experimentally thedouble 0- π transition in the crosslike S-N/F-S Josephsonjunctions driven in non-equilibrium by applying an in-jection current across the complex weak link. We ascribethis effect to the appearance of two zero-crossing pointsin the supercurrent-carrying density of states (SCDOS)caused by the Zeeman splitting of the superconductingcorrelations in the N/F interlayer. The model taking intoaccount SCDOS of a S-N/F-S structure and spin injec-tion into the N layer is developed to describe the observedeffect. It has been concluded that the non-equilibrium inour case is not thermal, mainly described by the energymode with a small amount of the spin-energy nonequilib-rium mode without a contamination of charge and spinimbalance. Thus, we have shown that the Zeeman split-ting due to N/F proximity has provided the origin of twonodes of the alternating SCDOS, which is practically im-possible to realize in simple SNS structures without theF-sublayers used in the pioneering works [25, 26]. In ourparticular case, the spin-energy mode turned out to benegligible. To demonstrate the spin-energy mode mani-festations, it is necessary that the distribution functionhas a spin splitting (i.e. the difference between µ ↑ and µ ↓ curves in Fig.5) significantly exceeding its temperaturesmearing. 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