Manifestation of a spin-splitting field in a thermally-biased Josephson junction
MManifestation of a spin-splitting field in a thermally-biased Josephson junction.
F. S. Bergeret
1, 2, ∗ and F. Giazotto
3, † Centro de F´ısica de Materiales (CFM-MPC), Centro Mixto CSIC-UPV/EHU,Manuel de Lardizabal 4, E-20018 San Sebasti´an, Spain Donostia International Physics Center (DIPC), Manuel de Lardizabal 5, E-20018 San Sebasti´an, Spain NEST, Instituto Nanoscienze-CNR and Scuola Normale Superiore, I-56127 Pisa, Italy
We investigate the behavior of a Josephson junction consisting of a ferromagnetic insulator-superconductor(FI-S) bilayer tunnel-coupled to a superconducting electrode. We show that the Josephson coupling in the struc-ture is strenghtened by the presence of the spin-splitting field induced in the FI-S bilayer. Such strenghteningmanifests itself as an increase of the critical current I c with the amplitude of the exchange field. Furthermore,the effect can be strongly enhanced if the junction is taken out of equilibrium by a temperature bias. We proposea realistic setup to assess experimentally the magnitude of the induced exchange field, and predict a drasticdeviation of the I c ( T ) curve ( T is the temperature) with respect to equilibrium. PACS numbers: 74.50.+r,74.25.F-
The interplay between superconductivity and ferromag-netism in superconductor-ferromagnet (S-F) hybrids exhibitsa large variety of effects studied along the last years [1, 2].Experimental research mainly focuses on the control of the0 − π transition in S-F-S junctions [3, 4] (S-F-S) and on thecreation, detection and manipulation of triplet correlations inS-F hybrids [5–9]. From a fundamental point of view, thekey phenomenon for the understanding of these effects is the proximity effect in S-F hybrids, and how the interplay betweensuperconducting and magnetic correlations affect their ther-modynamic and transport properties.While most of theoretical and experimental investigationson S-F structures deal mainly with the penetration of the su-perconducting order into the ferromagnetic regions, it is alsowidely known that magnetic correlations can be induced inthe superconductor via the inverse proximity effect [10–13]. Ifthe ferromagnet is an insulator (FI), on the one hand supercon-ducting correlation are weakly suppressed at the FI-S interfaceand a finite exchange field, with an amplitude smaller than thesuperconducting gap ∆ , is induced at the interface. Such ex-change correlations penetrate into the bulk of S over distancesof the order of the coherence length [10]. This results in asplitting of the density of states (DoS) of the superconduc-tor, as observed in a number of experiments [14–17]. Yet, thespin-split DoS of a superconductor may lead to interesting ef-fects such as, for instance, the absolute spin-valve effect [18–20], the magneto-thermal Josephson valve [21, 22], and thelarge enhancement of the Josephson coupling observed in F-S-I-S-F junctions (I stands for a conventional insulator) whenthe magnetic configuration of the F layers is arranged in theantiparallel state [23, 24].In this Letter we show that an enhancement of the Joseph-son effect between two tunnel-coupled superconductors S L and S R can also be achieved if a unique FI is attached to one ofthe S electrodes, for instance, the left lead, as shown schemat-ically in Fig. 1(a). According to the discussion above, thepresence of the FI splits the DoS in the left superconductor. Inprinciple, the presence of the spin-splitting field causes a re-duction of the superconducting gap ( ∆ L ) in the left supercon- FI (a) T L S L - /2 /2 T R S R I t S (b) FIG. 1. (Color online) (a) Scheme of the FI-S-I-S Josephson tunneljunction considered in this paper. T L and T R indicate the temperaturein the left (S L ) and right (S R ) superconductor, respectively, I standsfor a conventional insulator whereas ϕ is the macroscopic quantumphase difference over the junction. t S denotes the thickness of theS L layer. (b) Sketch of a possible experimental setup. Additionalsuperconducting leads tunnel-coupled to S L and S R serve either asheaters (h) or thermometers (th), and allow one to probe the effectof a spin-splitting field through measurement of the junction currentvs voltage characteristics under conditions of a temperature bias, asdiscussed in the text. R t denotes the junction normal-state resistance. ductor, and therefore at first glance one may think that, in turn,the Josephson coupling is suppressed. However, we show thatfor low enough temperatures, the presence of the exchangefield h in one of the two electrodes indeed enhances the crit-ical current ( I c ) with respect to its value at h =
0. This effectis further enhanced by applying a temperature bias across thejunction. Furthermore, by setting S L at T L and S R at T R , thetemperature-dependent I c curves change drastically, showing a r X i v : . [ c ond - m a t . m e s - h a ll ] D ec a sharp step when the energy gaps difference matches the ex-change field. A measurement of I c therefore allows to assessthe magnitude of the induced h . We discuss different realiza-tions, and propose a realistic setup and materials combinationsto demonstrate our predictions.In order to understand the enhancement of the Josephsoncoupling by increasing the exchange field h we provide here asimple physical picture that involves two mechanisms: On theone hand, the Josephson effect in the FI-S-I-S junction of Fig.1(a) is stronger the larger the overlap of the condensates fromthe left an right electrodes is. This overlap is proportional tothe number of Cooper pairs with shared electrons between S L and S R . By increasing the exchange field in the left side ofthe junction, it is energetically more favorable for the elec-trons with spin parallel to the field (spin-up) to be localizedwithin S L , whereas spin-down electrons are preferably local-ized in S R where the exchange field is absent. This means thatthe number of Cooper pairs sharing becomes larger . On theother hand, the Josephson coupling is proportional to the am-plitude of the condensate in each of the electrodes. Thereforeby increasing h or the temperature ( T ) one expects a suppres-sion of the order parameter in the electrodes. The behaviorof the Josephson critical current as a function of h and T istherefore the result of these two competing mechanisms. Inthe structure under consideration the exchange field acts onlyin S L . For low enough temperatures, ∆ L ( h , T ) depends onlyweakly on h . Therefore the first mechanism dominates and I c is enhanced by increasing h [see Fig. 2(a)]. At large enoughtemperatures, ∆ L ( h , T ) is much more sensitive to the exchangefield, and its faster suppression leads to a decrease of I c uponincreasing h . We note that in S R the exchange field is absent,and therefore the suppression of ∆ R is caused only by the in-crease of the temperature. If we now keep S L at low temper-ature and vary only the right electrode temperature ( T R ), thefirst mechanism dominates for any value of T R , and the en-hancement of I c by increasing h can always be observed [ cf. Fig.2(b)]. This is a remarkable effect that we now analyzequantitatively in the following.In order to compute the Josephson current through the junc-tion sketched in Fig. 1(a) we assume that the normal-stateresistance of the tunneling barrier R t is much larger than thenormal-state resistances of the junction electrodes. In such acase, the charge current through the junction can be calculatedfrom the well known expression I = eR t (cid:90) Tr (cid:110) τ [ G L ( E ) , G R ( E )] K (cid:111) dE , (1)where the matrix τ in Eq. (1) is the third Pauli matrix in theparticle/hole space, the Green functions (GFs) G L ( R ) are thebulk GFs matrices for the left and right electrodes, and e isthe electron charge. They are 8 × G L ( R ) = (cid:32) ˇ G RL ( R ) ˇ G KL ( R ) G AL ( R ) (cid:33) . (2) T R / T c I c/ I h / D T / T c T L = 0 . 0 1 T c ( b ) I c/ I h / D ( a ) FIG. 2. (Color online) (a) Junction critical current I c versus exchangefield h calculated for several values of the temperature. Here we set T L = T R = T . (b) Critical current I c versus h calculated for differ-ent values of T R at T L = . T c . ∆ denotes the zero-field, zero-temperature superconducting energy gap with critical temperature T c . We assume that the junction is temperature biased so that theleft (right) electrode is held at a constant and uniform temper-ature T L ( R ) , and ϕ denotes the macroscopic phase differencebetween the superconducting electrodes. In this case the re-tarded (R) and advanced (A) components are 4 × G R ( A ) L ( R ) = ˆ g R ( A ) L ( R ) τ + ˆ f R ( A ) L ( R ) ( i τ cos ( ϕ / ) ± i τ sin ( ϕ / )) , (3)where ˆ g and ˆ f are matrices in spin-space defined by ˆ g RL ( R ) = g R − L ( R ) σ + g R + L ( R ) σ and ˆ f RL ( R ) = f R − L ( R ) σ + f R + L ( R ) σ . In prin-ciple the functions in the left electrode may depend on the spa-tial coordinates. To simplify the problem we assume that thethickness t S of the S L electrode is smaller than the supercon-ducting coherence length and hence the Green’s functions arewell approximated by spatially constant functionsˆ f R ± L ( R ) = ∆ L ( R ) (cid:113) ( E + h + i Γ ) − ∆ L ( R ) ± ∆ L ( R ) (cid:113) ( E − h + i Γ ) − ∆ L ( R ) . (4)ˆ g R ± L ( R ) has a similar form by replacing ∆ L ( R ) in the numera-tors of the previous expressions with E ± h . For the particularsetup of Fig. 1(a), the exchange field in the right electrode isset to zero ( h =
0) and therefore g − L = f − L = f R = f + R .Notice that the gaps ∆ L ( R ) depend on the corresponding tem-perature T L ( R ) and exchange field, and have to be determinedself-consistently. The advanced GFs have the same form afterreplacing i Γ → − i Γ . The latter parameter describes inelasticeffects within the time relaxation approximation [25]. Finally,the Keldysh component of the GF [Eq. (2)] is defined asˇ G KL ( R ) = ( ˇ G RL ( R ) − ˇ G AL ( R ) ) tanh ( E / T L ( R ) ) . (5)By using Eqs. (2-5) we can compute the electric current fromEq. (1). In the absence of a voltage drop across the junction(i.e., V =
0) the charge current equals the Josephson current, I J = I c sin ϕ , where the critical supercurrent is given by theexpression I c = i eR t (cid:90) dE (cid:26)(cid:2) f RR f R + L − f AR f A + L (cid:3) (cid:20) tanh ( E T R ) + tanh ( E T L ) (cid:21) + (cid:2) f RR f A + L − f AR f R + L (cid:3) (cid:20) tanh ( E T R ) − tanh ( E T L ) (cid:21)(cid:27) . (6)The second line of the above expression corresponds to thecontribution from out-of equilibrium conditions due to a tem-perature bias across the junction. It vanishes when both elec-trodes are held at the same temperature and, as we will seebelow, leads to important deviations of I c ( T ) from its equilib-rium behavior.Before analyzing the most general case, we first assumeequilibrium, i.e., T L = T R = T and compute the Josephsoncritical current as a function of the exchange field. This isshown in Fig. 2(a). At low enough temperatures, I c increasesby increasing the exchange field. This is an unexpected resultsince the increase of the exchange field in the left electrodereduces the corresponding self-consistent gap ∆ L , and there-fore at first glance this suppression might lead to a reductionof I c . However, this mechanism competes with the Josephsoncoupling, which, within the simple physical picture given inthe introduction, is enhanced thanks to the fact that the elec-trons of the Cooper pairs with spin projection parallel to thefield h prefer to be localized mainly in S L while those withantiparallel spin are mostly localized in S R .To quantify the effect it is convenient to consider the lim-iting case T → I eqc ( T = ) = ∆ eR t (cid:90) dE (cid:113) E + ∆ Re (cid:113) ( E + ih ) + ∆ , (7)where ∆ is the superconducting gap at T = h =
0. Forsmall values of h (cid:28) ∆ one can expand this expression andfind I eqc ( T = ) ≈ π ∆ eR t (cid:18) + h ∆ (cid:19) , (8) which confirms the enhancement of I c upon increasing h . Inthe opposite limit, i.e., h → ∆ , numerical evaluation of theintegral gives eR t I eqc ( T = , h = ∆ ) ≈ . ∆ which is largerthen the expected value at h =
0, i.e., π ∆ / h → ∆ [23].Therefore, although a larger effect can be achieved in a S-F-I-S-F (or FI-S-I-S-FI junction), for practical purposes the setupof figure Fig. 1(a) with just one single FI is much simpler, andthe measurement of I c enhancement does not require controlof magnetizations direction. Moreover, in our geometry onecan boost the supercurrent enhancement by applying a tem-perature bias across the junction, as we shall discuss in thefollowing.If the temperatures in the superconductors are different( T L (cid:54) = T R ), although each of the electrodes is in local steady-state equilibrium, the junction as a whole is in an out-of-equilibrium condition. In such a situation, also the secondline in Eq. (6) contributes to the amplitude of the critical cur-rent, and leads to new features in the dependence of I c on h ,and on the temperature difference. For instance, one can holdS L at some fixed T L and vary the temperature T R of S R , or viceversa. The critical current can be calculated numerically fromEq. (6). These results are shown in panel (b) of Fig. 2 wherewe set T L = . T c , and T R varies from 0 . T c up to 0 . T c . It isclear that, for large values of the spin-splitting field, I c is largerthan the one for h =
0. It is also remarkable that the effect ismore pronounced the larger is the temperature difference. Fur-thermore, the main enhancement occurs stepwise, and stemsfrom the out-of-equilibrium contribution to I c appearing in Eq.(6). The latter is equivalent to the expression for I J ( V , T ) , theterm proportional to sin ϕ , of a voltage-biased Josephson junc-tion obtained several years ago in Refs.[29, 30]. In our systemthe exchange field plays the role of the voltage bias and, inagreement with Refs. [29, 30], the jump takes place at thevalue of h for which the following condition is satisfied: | ∆ R ( T R ) − ∆ L ( T L , h ) | = h . (9)We stress that while I J ( V , T ) can be accessed experimentallythrough a measurement of the ac Josepshon effect in voltage-biased configuration, the experiment we purpose below re-quires only a rather simple dc measurement at V = in-situ the exchange field present in FI-S layer, and to verifythe I c ( h ) dependence as displayed in Fig. 2. However, thereis a simpler alternative way to proceed and to demonstratethese effects. Toward this end we propose a possible experi-mental setup sketched in Fig. 1(b). The structure can be re-alized through standard lithographic techniques, and consistsof a generic FI-S-I-S Josephson junction where the two S L and S R electrodes are connected to additional superconduct-ing (e.g., made of aluminum) probes through oxide barriers soto realize normal metal-insulator-superconductor (NIS) tun-nel junctions. The NIS junctions are used to heat selectively (a) (b) (c) (d) T R = 0.1 T c I c / I T L / T c T L = T R h / T L = 0.1 T c T R / T c T R = 0.6 T c I c / I T L / T c T L = 0.6 T c T R / T c FIG. 3. (Color online) (a) Critical current I c versus T L calculated forselected values of the exchange field h at T R = . T c . (b) I c versus T R calculated at T L = . T c for the same values of h as in panel (a).(c) The same as in panel (a) calculated for T R = . T c . (d) The sameas in panel (b) calculated for T L = . T c . In all panels the dashedline shows the critical current for T L = T R and in the absence of anexchange field. the S L or S R electrode as well as to perform accurate elec-tron thermometry [31]. Therefore, instead of varying the ex-change field in the Josephson weak-link, one could now holdone of the junction electrodes at a fixed temperature and varythe temperature of the other lead while recording the currentversus voltage characteristics under conditions of a temper-ature bias [32–36]. In this context, the electric current canbe led through the whole structure via suitable outer super-conducting electrodes allowing good electric contact, but pro-viding the required thermal insulation necessary for thermallybiasing the Josephson junction. In addition, the tunnel probesenable to determine independently the energy gaps in the twosuperconducting electrodes through differential conductancemeasurements. Moreover, from the materials side, ferromag-netic insulators such as EuO or EuS [14, 15, 37] combinedwith superconducting aluminum could be suitable candidatesin light of a realistic implementation of the structure. The critical current behavior under thermal-bias conditionsis displayed in Fig. 3 where, in panels (a) and (c), T R is heldat 0 . T c and 0 . T c , respectively, and T L varies. Similarly, inpanels (b) and (d) we keep T L constant at 0 . T c and vary T R . Itclearly appears that the I c ( T ) curves drastically deviates formthose obtained at equilibrium, i.e., for T L = T R [dotted lines inFig. 3)]. If we keep a constant temperature 0 . T c in one of theelectrodes [see Fig. 3(a) and (b)], and change the temperatureof the other it follows that, for low enough temperatures, I c gets larger by increasing the magnitude of the exchange field.This corresponds to the enhancement discussed in Fig. 2.By further increasing the temperature of one of the electrodesleads to a critical current decrease. Notably, I c exhibits a sharpjump at those temperatures such that the condition expressedby Eq. (9) holds. This is a striking effect which can provide,from the experimental side, evidence of the supercurrent en-hancement discussed above. Yet, it can be used as well todetermine the value of the effective exchange field induced inthe superconductor placed in direct contact with the FI layer.It is remarkable that these features can be also observed, al-though reduced in amplitude, in the high-temperature regime[see Figs. 3(c) and (d)]. We emphasize that the effect here dis-cussed is much more pronounced when the left electrode (i.e.,the one with the FI layer) is kept at a low temperature, and onevaries T R . This is simple to understand, since a superconduc-tor with a spin-splitting field is more sensitive to a temperaturevariation: the larger the exchange field the faster one get sup-pression of superconductivity by enhancing the temperature.In conclusion, we have shown that the critical current I c of a FI-S-I-S Josephson junction is drastically modified by thepresence of the exchange field induced in one of the electrodesfrom the contact with a ferromagnetic insulator. In particular,we have demonstrated that the Josephson coupling is strength-ened by the presence of the exchange field and therefore the I c amplitude is enhanced. The enhancement becomes more pro-nounced upon the application of a temperature bias across thejunction. In such a case we predict a change of the I c ( T ) curvewith respect to the equilibrium situation which now shows ajump occurring when the difference of the superconductinggaps equals the amplitude of the exchange field. This behav-ior can be measured through standard techniques as we havediscussed for a realistic experimental setup. Our predictionson Josephson junctions with ferromagnetic insulators are ofgreat relevance since they constitute the building blocks ofrecently proposed nanodevices for spintronics[19, 38] and co-herent spin caloritronics[21, 22].The work of F.S.B was supported by the Spanish Ministryof Economy and Competitiveness under Project FIS2011-28851-C02-02. F.G. acknowledges the Italian Ministry of De-fense through the PNRM project ”Terasuper”, and the MarieCurie Initial Training Action (ITN) Q-NET 264034 for partialfinancial support. F.S.B thanks Prof. Martin Holthaus and hisgroup for their kind hospitality at the Physics Institute of theOldenburg University. ∗ sebastian [email protected] † [email protected][1] F. S. Bergeret, A. F. Volkov, and K. B. Efetov, Rev. Mod. Phys. , 1321 (2005).[2] A. I. Buzdin, Rev. Mod. Phys. , 935 (2005).[3] A. I. Buzdin, L. N. Bulaevskii and S. V. Panyukov, JETP Lett. , 178 (1982).[4] V. V. Ryazanov, V. A. Oboznov, A. Yu. Rusanov, A. V. Vereten-nikov, A. A. Golubov, and J. Aarts, Phys. Rev. Lett. , 2427(2001).[5] F. S. Bergeret, A.F. Volkov, and K.B. Efetov, Phys. Rev. Lett. , 4096 (2001).[6] R. S. Keizer, S. T. B. Goennenwein, T. M. Klapwijk, G. Miao,G. Xiao, and A. Gupta, Nature , 825 (2006).[7] J. W. A. Robinson, J. D. S. Witt, and M. G. Blamire, Science , 59 (2010).[8] T. S. Khaire, M. A. Khasawneh, W. P. Pratt, Jr., and N. O. Birge,Phys. Rev. Lett. , 137002 (2010); C. Klose, T. S. Khaire, Y.Wang, W. P. Pratt, Jr., Norman O. Birge , B. J. McMorran, T.P. Ginley, J. A. Borchers, B. J. Kirby, B. B. Maranville, and J.Unguris, Phys. Rev. Lett. , 127002 (2012).[9] M. S. Anwar, F. Czeschka, M. Hesselberth, M. Porcu, and J.Aarts, Phys. Rev. B , 100501 (2010).[10] T. Tokuyasu, J. A. Sauls, and D. Rainer, Phys. Rev. B , 8823(1988).[11] F. S. Bergeret, A. F. Volkov, and K. B. Efetov, Phys. Rev. B, ,174504 (2004).[12] F. S. Bergeret, A. L. Yeyati, and A. Martin-Rodero, Phys. Rev.B, , 064524 (2005).[13] J. Xia, V. Shelukhin, M. Karpovski, A. Kapitulnik, and A.Palevski, Physical Rev. Lett. , 087004 (2009).[14] X. Hao, J. Moodera, and R. Meservey, Phys. Rev. B , 8235(1990).[15] T. Santos, J. Moodera, K. Raman, E. Negusse, J. Holroyd, J.Dvorak, M. Liberati, Y. Idzerda, and E. Arenholz, Phys. Rev.Lett. , 147201 (2008).[16] G. Catelani, X. S. Wu, and P. W. Adams, Phys. Rev. B ,104515 (2008).[17] Y. M. Xiong, S. Stadler, P. W. Adams, and G. Catelani, Phys.Rev. Lett. , 247001 (2011). [18] R. Meservey and P. M. Tedrow, Phys. Reports , 173 (1994).[19] D. Huertas-Hernando, Yu V. Nazarov, and W. Belzig, Phys.Rev. Lett. , 047003 (2002).[20] F. Giazotto and F. Taddei, Phys. Rev. B , 132501 (2008).[21] F. Giazotto and F. S. Bergeret, App. Phys. Lett. , 132603(2013).[22] F. S. Bergeret and F. Giazotto, Phys. Rev. B , 014515 (2013).[23] F. S. Bergeret, A. F. Volkov, and K. B. Efetov, Phys. Rev. Lett. , 3140 (2001).[24] J. W. A. Robinson, G´abor, B. Hal´asz, A. I. Buzdin, and M. G.Blamire, Phys. Rev. Lett. , 207001 (2010).[25] Throughout the paper we set Γ = − ∆ for the numericalevaluation of I c .[26] Stricktly speaking, in Ref. [23] a F-S-I-S-F structure was con-sidered, where F denotes a metallic (weak) ferromagnet in goodelectric contact with the superconductor.[27] We stress that the enhancement of I c as a function of h in a F-S-I-S-F junctions only takes place if the magnetizations of theF layers are oriented in the antiparallel configuration. In theparallel configuration I c is a monotonically decreasing functionof h .[28] M. Tinkham, Introduction to Superconductivity 2nd Edn. (McGraw-Hill, New York, 1996).[29] A. I. Larkin and Yu. N. Ovchinnikov, Sov. Phys. JETP , 1035(1967).[30] R. E. Harris, Phys. Rev. B , 84 (1974).[31] F. Giazotto, T. T. Heikkil¨a, A. Luukanen, A. M. Savin, and J. P.Pekola, Rev. Mod. Phys. , 217 (2006).[32] S. Tirelli, A. M. Savin, C. P. Garcia, J. P. Pekola, F. Beltram,and F. Giazotto, Phys. Rev. Lett. , 077004 (2008).[33] A. M. Savin, J. P. Pekola, J. T. Flyktman, A. Anthore, and F.Giazotto, Appl. Phys. Lett. , 4179 (2004).[34] A. F. Morpurgo, T. M. Klapwijk, and B. J. van Wees, Appl.Phys. Lett. , 966 (1998).[35] H. Courtois, M. Meschke, J. T. Peltonen, and J. P. Pekola, Phys.Rev. Lett. , 067002 (2008).[36] S. Roddaro, A. Pescaglini, D. Ercolani, L. Sorba, F. Giazotto,and F. Beltram, Nano Res. , 259 (2011).[37] G.-X. Miao, M. M¨uller, and J. S. Moodera, Phys. Rev. Lett. ,076601 (2009).[38] F. Giazotto and F. S. Bergeret, Appl. Phys. Lett.102