Phase transitions in superconductor/ferromagnet bilayer driven by spontaneous supercurrents
Zh. Devizorova, A.V. Putilov, I. Chaykin, S. Mironov, A.I. Buzdin
aa r X i v : . [ c ond - m a t . s up r- c on ] F e b Phase transitions in superconductor/ferromagnet bilayer driven by spontaneoussupercurrents
Zh. Devizorova, A. V. Putilov, I. Chaykin,
1, 3
S. Mironov, and A.I. Buzdin
4, 5 Moscow Institute of Physics and Technology, 141700 Dolgoprudny, Russia Institute for Physics of Microstructures, Russian Academy of Sciences, 603950 Nizhny Novgorod, GSP-105, Russia Kotelnikov Institute of Radioengineering and Electronics of Russian Academy of Sciences, Moscow, 125009, Russia University Bordeaux, LOMA UMR-CNRS 5798, F-33405 Talence Cedex, France World-Class Research Center “Digital biodesign and personalized healthcare”,Sechenov First Moscow State Medical University, Moscow 119991, Russia
We investigate superconducting phase transition in superconductor(S)/ferromagnet(F) bilayerwith Rasba spin-orbit interaction at S/F interface. This spin-orbit coupling produces spontaneoussupercurrents flowing inside the atomic-thickness region near the interface, which are compensatedby the screening Meissner currents [S. Mironov and A. Buzdin, Phys. Rev. Lett , 077001 (2017)].In the case of thin superconducting film the emergence of the spontaneous surface currents causesthe increase of the superconducting critical temperature and we calculate the actual value of thecritical temperature shift. We also show that in the case of type-I superconducting film this phasetransition can be of the first order. In the external magnetic field the critical temperature dependson the relative orientation of the external magnetic field and the exchange field in the ferromagnet.Also we predict the in-plane anisotropy of the critical current which may open an alternative wayfor the experimental observation of the spontaneous supercurrents generated by the SOC.
I. INTRODUCTION
The superconducting states carrying spontaneous cur-rent in the systems with broken time reversal symme-try were the subject of interest for more than twentyyears . Such spontaneous supercurrents were pre-dicted for d -wave or chiral p -wave superconduc-tors, for a mesoscopic normal metal film in contact witha superconductor and at the interface between a su-perconductor and a ferromagnet . These currents aretypically carried by Andreev edges states and appearat temperature T well below the superconducting criticaltemperature T c .Recently the spontaneous supercurrents were predictedto appear at the interface between s-wave superconduc-tor (S) and a ferromagnetic (F) insulator . Contraryto the spontaneous supercurrents carried by Andreevbound states , these currents appears at the super-conducting transition, i.e. at T = T c . The crucial con-dition for the emergence of these currents is the pres-ence of Rashba spin-orbit coupling (SOC) at the S/Finterface . Indeed, this SOC produces the additionalterm ∝ ( σ × p ) · n in the effective Hamiltonian of a con-ducting electron ( n is the unit vector perpendicular to theS/F interface). As a result, spin and momentum appearto be coupled, which produces the nontrivial “helicity”of the electronic energy bands. Since the exchange fieldmakes the spin-up state energetically more favourablethan the spin-down one, one may expect the emergenceof the electric current. Note that the helical states also play an important role in the emergence of Majoranamodes , the formation of Josephson ϕ junctions withspontaneous nonzero phase difference across the junctionin the ground state , and the appearance of Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) like states with finiteCooper-pair momentum . Since spontaneous supercurrents flowing at the S/F in-terface with SOC appear at T = T c , these currents canaffect the parameters of the phase transition (the super-conducting critical temperature, the phase diagram inthe external magnetic field, the critical current, etc). Al-though the spontaneous supercurrents result in the localenhancement of the superconductivity near the S/F in-terface, it was shown that for the large thickness of thesuperconductor they do not affect the superconductingtransition temperature and, thus, with the increase intemperature the superconductivity becomes destroyed inthe whole bulk of the sample. This situation is in contrastwith the well-known phenomena arising in the supercon-ductors containing twinning planes which locally increasethe critical temperature and favor the emergence of thelocalized superconducting states above the bulk criticaltemperature (see, e.g., Ref. 26 for review).In the present paper we study the effect of the sponta-neous supercurrents on the superconducting phase tran-sition in S/F bilayer with finite thickness of the super-conducting layer and SOC of the Rashba type at the S/Finterface. We show that in the case of thin superconduct-ing film these currents causes the increase of the super-conducting critical temperature T c and we calculate thecorresponding critical temperature shift. Surprisingly, intype-I superconductors the superconducting phase tran-sition is of the first order even in the absence of externalmagnetic field. At the same time, in external magneticfield the critical temperature strongly depends on therelative orientation between external magnetic field andthe exchange field in the ferromagnet. Note that in thecase of positive (negative) SOC parameter T c is signifi-cantly higher (lower) for the parallel magnetic configu-ration in comparison with the antiparallel one. At thesame time, in the case of type-II superconducting layerthe emergence of the spontaneous supercurrents also in-creases the superconducting critical temperature in theabsence of external magnetic field and makes in sensitiveto the relative orientation of external magnetic field andthe exchange field, if the sample placed into the field, butthe phase transition is of the second order. All describedphenomena can serve the hallmarks of the spontaneoussupercurrents and can be used for the experimental de-tection of these currents.The paper is organized as follows. In section II we in-troduce the model and study the phase transition in theS/F bilayer with type-I superconductor. In section IIIwe calculate the dependence of the critical temperatureon the external magnetic field. In section IV we con-sider type-II superconductor. In section V we analyzethe anisotropy of the critical current. In section VI wesummarize our results. II. FIRST ORDER PHASE TRANSITION INS/F BILAYER WITH THIN TYPE-ISUPERCONDUCTOR
We consider thin superconducting (S) film of the thick-ness L placed in contact with a ferromagnetic insulator(F), see Fig.1. We assume the presence of Rashba spin-orbit coupling (SOC) at the S/F interface. The free en-ergy functional of the system under consideration is givenby the following expression : F = Z Z Z dV (cid:26) α | ψ | + β | ψ | + 14 m | ˆ D ψ | +(rotA) π + [ n × h ] ǫ ( r ) (cid:16) ψ ∗ ˆ D ψ + ψ ˆ D † ψ ∗ (cid:17)(cid:27) . (1)Here α = a ( T − T c ) and β are the standard GL coeffi-cients, T c is the critical temperature of bulk supercon-ductor, ˆ D = − i ~ ∇ + ec A , A is the vector-potential, n is the unit vector perpendicular to the S/F interface anddirected from S to F layer, h is the exchange field in theferromagnet, ψ is the order parameter of the supercon-ductor and ǫ is the spin-orbit constant, which is nonzeroonly in the atomically thin area of the width l so nearthe S/F interface. Without loss of generality, let us as-sume that the exchange field is directed along the z -axis,i.e. h = h e z , the x -axis coincides with n and the S/Finterface is located at x = L .In such system spontaneous supercurrents flow nearthe S/F interface . These currents cause the increase ofthe superconducting order parameter in the area of thewidth ∼ ξ (superconducting coherence length) near S/Finterface. If the thickness L of the superconducting filmis much smaller than ξ , i.e. L ≪ ξ , one can expect theincrease of the superconducting critical temperature T c .Here we find the actual temperature shift for type-I su-perconductor, i.e. for the case λ ≪ L ≪ ξ , where λ is theLondon penetration depth. Also we show, that the phasetransition is of the first order. For this purpose, let uscalculate the free energy for such a system. Due to the condition L ≪ ξ we may take ψ ( x ) ≈ const . Let us intro-duce the dimensionless order parameter ϕ = p β/ | α | ψ ,where α = − aT c .Since the spontaneous supercurrents are fully com-pensated by the Meissner currents, the magnetic field B = (0 , , B z ) is absent outside the superconductor andhas the following form: B z ( x ) = B exp (cid:18) x − Lλ (cid:19) , < x < L, , x > L, x ≤ . (2)Here λ = λ /ϕ , where λ = mc β/ (8 πe | α | ) is thezero-temperature London penetration depth for the bulksuperconductor. Note that inside the superconductingslab we neglect the second exponent [see the first line inEq.(2)] since we assume λ ≪ L . The resulting magneticfield Eq.(2) is continuous at x = 0 and experiences a jumpby the value B at x = L due to the surface spontaneoussupercurrents flowing along S/F interface. To find theactual value of the jump we substitute the magnetic fieldto the free energy and minimize it with respect to B .We obtain: A = ∆ Hϕ . (3)Here ∆ H = 4 √ H cm k so l so ( ξ /λ ) is the jump of mag-netic field at S/F interface due to spontaneous surfacesupercurrents, where H cm = p πα /β is the thermo-dynamic critical magnetic field, k so = mhǫ/ ~ and ξ = p ~ / m | α | .The resulting free energy reads as: F = V (cid:18) α | α | β ϕ + | α | β ϕ (cid:19) − S ∆ H π λ ϕ , (4) xz FS h L<< x j s j M y (cid:3) const SOC
Figure 1. The sketch of a superconducting (S) film placed incontact with thin ferromagnetic (F) layer. The spin-orbit cou-pling at the S/F inteface produces spontaneous supercurrent j s causing the increase of the superconducting order parame-ter inside the superconductor. where V is the volume of the superconducting slab and S is the surface area of S/F boundary.We find the value of the critical temperature T c andthe order parameter ϕ cr at T = T c by minimizing F with respect to ϕ and using the condition F = 0, whichis fulfilled at the critical temperature. We find: ϕ cr = 12 λ L (cid:18) ∆ HH cm (cid:19) , (5) T c T c = 1 + 18 (cid:18) λ L (cid:19) (cid:18) ∆ HH cm (cid:19) . (6)One see that the critical temperature, indeed, increasesdue to the spin-orbit interaction at the S/F interface.Moreover, since the order parameter does not equal tozero at the transition temperature, in the structure underconsideration the phase transition is of the first order.Note that the critical temperature Eq.(6) is not divergentwhen the slab thickness L tends to zero, since we considerthe situation λ ≪ L .The obtained result is valid only if the assumption λ ≪ L is fulfilled at T = T c . To check this, we find the Londonpenetration depth at the critical temperature: λ cr = 2 L (cid:0) ∆ HH cm (cid:1) . (7)Since ∆ H/H cm ∼ k so l so ( ξ /λ ) and ( ξ /λ ) ≫ λ cr ≪ L is valid if k so l so is not very small.Let us estimate ∆ T c = T c − T c . Since ǫ = v so /E F ,where v so is the spin-orbit velocity and E F is the Fermienergy, the jump of the magnetic field at x = L dueto the spontaneous supercurrents can be estimated as∆ H ≈ √ H cm ( h/E F )( v so /v F )( ξ /λ ), where v F is theFermi velocity. It is reasonable to take h/E F ∼ . v so /v F ∼ .
1. If we also assume ξ /λ ∼
50, then we find∆ H ≈ H cm . Taking λ /L ∼ . T c /T c ≈ .
1. Since ∆ T ≪ T c the Ginzburg-Landau approach isapplicable.Note that the first-order phase transition in S/F bilayerwith Rashba-type spin-orbit interaction and λ ≪ L ≪ ξ can serve as a hallmark of the spontaneous supercurrentsflowing at S/F interface .The suitable system for the observation of the dis-cussed effects may be based on thin epitaxially grownlayers of extreme type-I superconductors. For exam-ple, recently it was demonstrated the epitaxial growthof high-quality single-crystalline aluminum films . III. PHASE TRANSITION IN EXTERNALMAGNETIC FIELD
In the external magnetic field H the critical tempera-ture of superconducting phase transition depends on therelative orientation of H and the exchange field h . For simplicity, let us restrict ourselves to the case H = H e z ,where H can be both positive and negative. To find ac-tual T c ( H ) dependence let us write down the Gibbs freeenergy of the system: G = Z Z Z dV (cid:26) α | ψ | + β | ψ | + 14 m | ˆ D ψ | +( B − H ) π + [ n × h ] ǫ ( r ) (cid:16) ψ ∗ ˆ D ψ + ψ ˆ D † ψ ∗ (cid:17)(cid:27) . (8)As before, we consider the case L ≪ ξ and assume ψ ≈ const. Since we also assume λ ≪ L , the magneticfield B = B e z = rot A inside the superconducting slabreads: B = H exp (cid:16) − xλ (cid:17) + (cid:0) H + ∆ Hϕ (cid:1) exp (cid:18) x − Lλ (cid:19) . (9)Calculating the resulting Gibbs free energy we obtain: GV = α | α | β ϕ + | α | β ϕ + H π − H ∆ H π λ L ϕ − ∆ H π λ L ϕ (10)Note that the surface contribution to the free energycaused by spontaneous supercurrents results in the in-creases (decreases) of the free energy if H ↑↑ h ( H ↑↓ h ).The critical temperature and the order parameter atthe critical point ϕ cr can be found from the system ofequations G = 0 and ∂ ϕ G = 0, which are fulfilled at T = T c . From the second equation we find the criticaltemperature as a function of ϕ cr : T c T c = 1 − ϕ cr + 34 (cid:18) ∆ HH cm (cid:19) λ L ϕ cr ++ 12 ϕ H H cm ∆ HH cm λ L . (11)At the same time, the order parameter obeys the equa-tion ϕ cr − (cid:18) ∆ HH cm (cid:19) λ L ϕ cr − (cid:18) H H cm (cid:19) ++ H H cm ∆ HH cm λ L ϕ cr = 0 . (12)Solving the equations we indeed find that the criti-cal temperature of the superconducting phase transitionstrongly depends on the relative orientation of the ex-ternal magnetic field and the exchange field: for posi-tive (negative) SOC parameter ǫ the T c is higher for theparallel (anti-parallel) orientation compared to the anti-parallel (parallel) one (see Fig. 2). This findings provides H h ↑↑ È H È÷ H cm T c ÷ T c H h ↑↓ e < H h ↑↑ e > H h ↑↓ Figure 2. The dependence of the superconducting criticaltemperature T c of the S/F bilayer with λ = 0 . L and ∆ H =5 H cm on the external magnetic field H = H e z . Here ǫ isthe spin-orbit coupling constant. a tool for the experimental observation of the predictedeffect, although the correction to T c due to SOC is small.Changing the direction of the external magnetic field,one can observe the variation of superconducting criticaltemperature. Note that the dependence of the criticaltemperature on the magnetic configuration is also canbe used for the experimental detection of the sign of thespin-orbit parameter. Indeed, the critical temperature ishigher for the parallel orientation between the externalmagnetic field and the exchange field in comparison withthe antiparallel one only if the spin orbit constant is pos-itive, while for the negative SOC parameter the situationis reversed.Since λ ≪ L we can find approximate analytical ex-pressions for T ↑↑ c and T ↑↓ c expanding the results obtainedfrom Eq.(11) and Eq.(12) over λ /L . To have good agree-ment between the exact results and the approximate one,we should expand these expressions up to the third or-der over λ /L . If ∆ H >
H <
0) for the parallel(anti-parallel) magnetic configuration we obtain: T ↑↑ ( ↑↓ ) c T c = 1 − | H | H cm + 12 s | H | H cm ∆ HH cm (cid:18) HH cm (cid:19) λ L ++ 132 (cid:18) ∆ HH cm (cid:19) "(cid:18) ∆ HH cm (cid:19) − (cid:18) λ L (cid:19) . (13)At the same time, if ∆ H >
H <
0) for the an-tiparallel (parallel) magnetic configuration we find: T ↑↓ ( ↑↑ ) c T c = 1 − | H | H cm − s | H | H cm ∆ HH cm λ L ++ 1 √ (cid:18) ∆ HH cm (cid:19) . H H cm + √ ! (cid:18) λ L (cid:19) . (14)Note that the above expressions are invalid, when H tends to zero. Assuming H ≪ ∆ H from Eq.(11) andEq.(12) we find: T ↑↑ ( ↑↓ ) c T c = 1 + 18 (cid:18) ∆ HH cm (cid:19) (cid:18) λ L (cid:19) ± | H | ∆ H . (15)Note that the dependence of the critical temperatureon the in plane magnetic field orientation is another hall-mark of spontaneous supercurrents flowing at the S/Finterface (in addition to the appearance of the straymagnetic field near S/F interface and the anisotropy ofthe upper critical field predicted in Ref.[12]). This factcan be used for the experimental detection of the spon-taneous supercurrents. IV. PHASE TRANSITION IN THE CASE OFTHIN TYPE-II SUPERCONDUCTING LAYER
In this section we analyze the peculiarities of the su-perconducting phase transition in the S/F bilayer (seeFig. 1) for the case when the superconducting layer isof the II type. We show that in this case the criticaltemperature also increases in the absence of the exter-nal magnetic field H and depends on the relative ori-entation between the external magnetic field and the ex-change field. Let us consider the temperatures close tothe superconducting transition temperature and the ex-ternal magnetic field directed along the z axis so that H = (0 , , H ). At the point of the phase transition thesuperconducting order parameter is small which allow usto neglect the term ∝ ψ | ψ | and the term associated withthe spontaneous supercurrents in the Ginzburg-Landau(GL) equation. Choosing the gauge of the vector poten-tial in the form A = (0 , H x,
0) and searching the solu-tion in the form ψ = e ik y y ψ ( x ) we obtain the followingGL equation: αψ ( x ) − ~ m ∂ xx ψ ( x ) + ( ~ k y + 2 eH x/c )4 m ψ ( x ) = 0 , (16)with the boundary conditions ∂ x ψ (0) = 0 , ∂ x ψ ( L ) = 8 mhǫl so ~ (cid:18) ~ k y + 2 eH Lc (cid:19) ψ ( L ) . (17)In the absence of the external magnetic field the or-der parameter reads as ψ ( x ) = A cosh qx , where q = (cid:0) mα/ ~ + k y (cid:1) . Calculating the free energy, we find: FS = A L α + ~ k y m ! (cid:18) sinh 2 ql qL + 1 (cid:19) + A ~ q m ×× L (cid:18) sinh 2 ql qL − (cid:19) − A hǫl so ~ k y cosh qL. (18)Assuming qL ≪ k y , we find the optimal modulation vector k y =4 k so l so /L . Since at the critical point F=0, we obtain theincrease of the superconducting critical temperature: T c − T c T c = 16 k so l so (cid:18) ξ L (cid:19) . (19)Note that the effect is absent in the limit L → ∞ , i.e. forthick superconducting slab.When the sample is placed in the external magneticfield it is convenient to choose the origin of the x -axisat the center of S film, so the S/F boundary is locatedat x = L/ x = − L/
2. Fol-lowing the procedure described in Ref. [28] we introducethe dimensionless coordinate X = 2 x/L , the modulationvector K y = k y L/
2, the parameters ˜ H = eH L / (2 ~ c ), ǫ = − mαL / ~ and rewrite the GL equation in the fol-lowing form: ∂ XX ψ ( X ) + ( K y + ˜ H X ) ψ ( X ) = ǫ ψ ( X ) . (20)At the same time, the boundary conditions read ∂ X ψ (1) = s ( K y + ˜ H ) ψ (1) , ∂ X ψ ( −
1) = 0 , (21)where s = 8 k so l so .Next it is useful to introduce the new variable t = q | ˜ H | ( X + K y /h ). The resulting GL equation andthe boundary conditions are the following: − ∂ tt ψ ( t ) + 14 t ψ ( t ) = ǫ | ˜ H | ψ ( t ) , (22) ∂ t ψ |√ | ˜ H | ( − K y / ˜ H ) = 0 , (23) (cid:18) ∂ t ψψ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) √ | ˜ H | (1+ K y / ˜ H ) = s ( K y + ˜ H ) q | ˜ H | . (24)The solution of Eq.(22) has the form of the linearcombination of the Weber functions ψ ( t ) = A ν D ν ( t ) + B ν D ν ( − t ), where 2 ν + 1 = ǫ / | ˜ H | . Substituting it intothe boundary conditions we obtain the following equa-tion, which implicitly defines the function T c ( H ): D ′ ν ( α + ) D ′ ν ( − α − ) − D ′ ν ( − α + ) D ′ ν ( α − ) = s ( K y + ˜ H ) q | ˜ H |× (cid:20) D ν ( α + ) D ′ ν ( − α − ) − D ν ( − α + ) D ′ ν ( α − ) (cid:21) , (25)where α ± = q | ˜ H | ( ± K y / ˜ H ).The maximal value of T c at fixed magnetic field corre-sponds to the minimal value of ν , which satisfies Eq.(25).At fixed K y and ˜ H this equation has infinite but dis-crete number of solutions for ν , and we find the minimalone. Then we minimize it with respect to K y and findthe minimal value ν for fixed ˜ H . As a result, we ob-tain the dependence ǫ ( ˜ H ) (i.e. T c ( H )) in the form (cid:16) H h ↑↑ H h ↑↓ h= | | H e ~ Figure 3. The dependence of the critical temperature T c on the external magnetic field H = (0 , , H ). Here ǫ =(1 − T c /T c ) L / (4 ξ ) and ˜ H = eH L / (2 ~ c ). ǫ = (2 ν + 1) | ˜ H | , see. Fig.3. The critical tempera-ture is higher for the parallel orientation of the externalmagnetic field and the exchange field in comparison withthe antiparallel one. Here we assume that the spin-orbitcoupling parameter ǫ is positive.As we can see from the inset in Fig.3 the weak parallelmagnetic field leads to the initial increase of the criticaltemperature, which is replaced by the usual decrease athigher magnetic field. Such peculiar behavior remind theexperimentally observed in Ref.[29] increase of the criticaltemperature in thin Pb films. We may speculate that thelocal SOC could be generated at Pb/substrate interface,while the role of the exchange field is played by a Zeemanfield. V. CRITICAL CURRENT
In this section we show that the spin-orbit cou-pling makes the in-plane critical current of S/F bilayeranisotropic. Although the total spontaneous supercon-ducting current generated by the SOC is zero, it is non-uniformly distributed across the layers. As a result, forthe fixed direction of the exchange field the local currentdensity (and, thus, the local damping of the supercon-ducting order parameter) at a certain point of the S filmbecomes dependent on the angle θ between the externalcurrent and the spontaneous current flowing along theS/F interface. Consequently, the maximal current whichdoes not destroy the superconducting state (critical cur-rent) also becomes dependent on θ (diode-like effect).To calculate the critical current of the S/F bilayer weagain consider the system sketched in Fig. 1. First, weconsider the situation when the external transport cur-rent of the linear density J is directed along the y axis.Since the magnetic field produced by both the current J and the spontaneous surface current due to SOC is di-rected along the z axis and depends only on the x axis wemay choose the vector potential in the form A = A ( x )ˆ y .Also we choose the order parameter ϕ to be real. Thenthe density of the free energy accounting the non-uniformprofile of the order parameter ϕ and the vector potential A across the structure can be written in the form FV = H cm π (cid:0) − τ ϕ + ϕ + ξ ϕ ′ (cid:1) ++ A ′ π + A ϕ πλ − ∆ HAϕ π δ ( x − L ) , (26)where τ = 1 − T /T c , ϕ ′ ≡ ∂ϕ/∂x and A ′ ≡ ∂A/∂x .Varying the free energy with respect to ϕ ( x ) and A ( x )inside the S layer we drive to the standard set of theGinzburg-Landau equations − ξ ϕ ′′ − τ ϕ + ϕ + A ϕ/ (2 H cm λ ) = 0 , (27) − λ A ′′ + ϕ A = 0 , (28)supplemented by the boundary conditions accounting thesurface energy contribution due to the spin-orbit coupling[the last term with δ –function in Eq. (26)] and the mag-netic field ± (2 π/c ) J generated by the external transportcurrent J at the outer boundaries x = 0 and x = L ofthe superconducting film: ϕ ′ (0) = 0 , ϕ ′ ( L ) = ∆ HA ( L ) ϕ ( L ) /H cm , (29) A ′ (0) = − (2 π/c ) J, A ′ ( L ) = (2 π/c ) J + ∆ Hϕ ( L ) . (30)The accurate solution of Eqs. (27)-(28) requires focus-ing on two features responsible for the anisotropy of thecritical current. The first one is the non-uniform dis-tribution of the screening Meissner current across the Sfilm. The second one is the damping of the order pa-rameter at the S/F interface by the transport currentand the subsequent renormalization of the spontaneoussurface current (and the screening Meissner one). Thus,although in the case of thin S layer the terms containingspatial derivatives of the order parameter and the vectorpotential cannot be neglected.In order to find the analytical solution of the Ginzdurg-Landau equations we make several assumptions simpli-fying the calculations. First, we restrict ourselves tothe most interesting case of the type-II superconductorand assume that the thickness L of the S film is muchsmaller than the superconducting coherence length sothat L ≪ ξ ≪ λ . Second, we consider the limit of smallspin-orbit coupling assuming the dimensionless param-eter µ = ∆ Hλ / ( H cm L ) to be small ( µ ≪ A ( x ) and ϕ ( x ) over x keeping the terms up to ( L/ξ ) in order toaccount the non-uniform distribution of the supercon-ducting current across the S film: A = A + A x + A x + A x ,ϕ = ϕ + ϕ x + ϕ x + ϕ x . Also in the resulting perturbation theory it is enoughto consider the terms proportional to µ and neglect thehigher order contributions.Substituting the expansion for A and ϕ into the equa-tions (27)–(28) and the boundary conditions (29)–(30), we find the dependence of the external current J on thedimensionless vector potential a = A / ( H cm λ ): J = cH cm L/ (8 πλ ) (cid:0) τ a − τ µ + 3 µa − a (cid:1) ×× (cid:2) − L (2 τ + 4 a µ − a ) / (8 λ ) (cid:3) (31)Then the critical current J c of the S/F bilayer can beobtained as the maximum of the dependence J ( a ) for a >
0. This maximum corresponds to a = µ and canbe written in the form J ± c = √ LcH cm τ / π √ λ (cid:18) ± ∆ H L √ τ √ H cm λ (cid:19) (32)Here the sign + ( − ) corresponds to the current flowingparallel (anti-parallel) to the y axis.The expression (32) clearly shows the anisotropy ofthe critical current which differs for the two opposite di-rections of the current flow. The difference between thecritical currents J ± c is proportional to the spontaneousmagnetic field ∆ H generated at the S/F interface dueto the SOC. Note that the critical current is higher ifthe exchange field in the F layer is parallel to the mag-netic field generated by the external current at x = L and lower in the opposite case. The equation (32) canbe straightforwardly generalized for the case of arbitrarydirection of the external current in the plane of the S/Fstructure. In this case the ± sign in the brackets shouldbe replaced with cos θ where θ is the angle between thedirection of the current and the y axis.The predicted diode effect provides the alternative wayfor the experimental observation of the spontaneous cur-rents generated by the SOC. To protect the S/F bilayerfrom the distraction caused by the heating effects onemay use the pulse currents . VI. CONCLUSION
To sum up, we develop the theory of superconductingphase transition in superconductor/ferromagnet bilayerwith Rasba-type spin-orbit interaction at the S/F inter-face and L ≪ ξ (see Fig.1) using the Ginzburg-Landauapproach. In the case of λ ≪ L the phase transitionis of the first order one even in the absence of exter-nal magnetic field. Moreover, its critical temperatureis higher, than in bulk superconductor. In the exter-nal magnetic field H the critical temperature dependson mutual orientation of H and the exchange field in-side the ferromagnet and the sign of spin-orbit couplingparameter: for positive (negative) SOC parameter it ishigher (lower) for the parallel orientation in comparisonwith antiparallel one (see Fig.2). Both these results aremanifestations of the spontaneous supercurrents flowingat the S/F interface and can serve hallmarks of thesecurrents. Moreover, the dependence of the critical tem-perature on the magnetic configuration can be used forthe experimental detection of the sign of SOC parameter.In the case of type-II superconducting layer the phasetransition is the second order one, its critical tempera-ture also increases in the absence of external magneticfield and depends on the relative orientation between ex-ternal magnetic field and the exchange field, if the formeris present. We also show that the critical current of theS/F bilayer reveals anisotropy in the plane of the layers.The resulting diode-like effect may provide an alternativeway for the experimental observation of the spontaneoussuperconducting currents generated by the SOC at theS/F interface. ACKNOWLEDGMENTS
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