# Double-reversal thickness dependence of critical current in superconductor-ferromagnet-superconductor Josephson junctions.

*August 1, 2021*

###### Abstract

We report the first experimental observation of the two-node thickness dependence of the critical current in Josephson junctions with a ferromagnetic interlayer. Vanishings of the critical current correspond to transitions into -state and back into conventional -state. The experimental data allow to extract the superconducting order parameter oscillation period and the pair decay length in the ferromagnet. We develope a theoretical approach based on Usadel equations, which takes into account the spin-flip scattering. Results of numerical calculations are in good agreement with the experimental data.

One of the exciting topics in studying the coexistence of superconductivity () and ferromagnetism () is proximity-induced sign-reversal superconductivity in a ferromagnet close to an -interface Buz82 ; Buz92 . The superconducting order parameter does not simply decay into the ferromagnet but also oscillates. An undoubted evidence of sign-reversal spatial oscillations of the superconducting order parameter in a ferromagnet was the observation of the -state in Josephson junctions PRL ; Aprili ; Sellier ; JLTP . ’-junctions’ Bul77 are weakly coupled superconducting structures which demonstrate -shift of the macroscopic phase difference in the ground state. The relation between the superconducting current and the phase difference in a Josephson junction is described by a -periodic function. In the simplest case of a tunnel barrier or a barrier, made of dirty normal metal, one finds . The Josephson -junction has an anomalous current-phase relation , i.e. it is characterized (nominally) by the negative critical current Bul77 . Spatial oscillations of the superconducting order parameter in a ferromagnet close to an -interface was predicted in Ref. Buz92 . A physical origin of the oscillations is the exchange splitting of the spin-up and spin-down electron subbands in a ferromagnet. It was discussed in Refs. PRL ; Aprili ; Sellier ; JLTP that in order to observe manifestations of the transition into the -state one should fabricate an sandwich with the -layer thickness close to integer numbers of half-periods of the order parameter spatial oscillations /2. The period is , where the oscillation (or ”imaginary”) length can be extracted from the complex coherence length in a ferromagnet:

(1) |

The latter approximation corresponds to the case , which is valid for experiments discussed below.

Detailed experimental studies of the critical current thickness dependence for Josephson junctions has been started by us in Ref. JLTP . A very large decay of the critical current and its sharp reentrant behavior for thicknesses close to 23 nm have been observed. An analysis of the experimental data and their comparison with the modern model described below have shown that the observed deep minimum is probably the reverse transition from the - into the -state at the -layer thickness close to full oscillation period while the first node of the dependence has to be at the thickness of about 10 nm. Thus, the presented work is devoted to finding of the two-node behavior of the junction critical current as well as to discussion of mechanisms of the strong order parameter decay in a ferromagnetic interlayer.

In fact a nonmonotonic dependence close to -transition was observed for the first time in Ref. PRL and has been presented there as a number of curves for different thicknesses . Later Kontos et al Aprili for and then Sellier et al Sellier for junctions measured detailed reentrant curves for -interlayer thicknesses close to -transition. In this work we have investigated the thickness dependence of the junction critical current density in a wide thickness range for sandwiches fabricated as described in Ref. JLTP . All junctions had their lateral sizes smaller than the Josephson length and uniform current distribution. To do this the junctions with F-layer thicknesses of less than 17 nm were made with the contact area and all the rest had the area . Weakly-ferromagnetic -interlayers had the Curie temperature of about 60 K. In the thickness interval we had about 6 orders of the critical current density change with vanishings at two values as it is presented in Fig. 1.

One can see that undoubtly the curve demonstrates both direct -transition and reverse transition from - to -state. In transition points the critical current is equal to zero and then should formally change its sign. Since in real experiments we could measure only magnitude of the critical current, the dependence between two sharp cusps is the negative (corresponding to the -state) branch of the curve which is reflected into the positive region. Due to slight temperature dependence of the order parameter oscillation period in our weak ferromagnet (described by (1)) we could pass through the transition points using samples with critical -layer thicknesses 11 nm and 22 nm by means of temperature decrease. Temperature - and -transitions are presented in middle panels of Fig. 2. Upper and lower panels show the critical current temperature behavior for samples with close F-layer thicknesses. One can see that we lost a possibility to detect temperature transitions changing the thickness only by . This implies that the temperature decrease from down to is accompanied by the decrease of in the spatial oscillation period and by the decrease of about in the oscillation length. In this temperature range the change of is about as it has been estimated from curves at different temperatures. At the same time simple evaluations of (obtained from the slope of the envelope) and (estimated from the interval between two minima) show a large difference between these two lengths (1.3 nm and 3.5 nm, correspondingly) that can not be explained by the thermal contribution described by (1).

So, to carry out a theoretical analysis of the results obtained, we need to specify the nature of additional depairing processes that increase and decrease . As the layer is an alloy, a role of the magnetic scattering may be quite important Sellier ; JLTP . Magnetic inhomogeneity is related above all to -rich clusters Mills ; Houghton arising in ferromagnet for close to 0.5. In the region of these concentrations when the Curie temperature is small, we may expect that the inverse spin-flip scattering time could be of the order of the average exchange field or even larger. This circumstance strongly modifies the proximity effect in the systems. A role of spin-orbit scattering should be neglected for the alloy since it is substantial only in ferromagnets with large atomic number . To take into account the exchange field and the magnetic scattering in the framework of Usadel equations it is necessary just to substitute Matsubara frequencies by Buzdin85 , where is the normal Green’s function. Note that this procedure assumes a presence of the relatively strong uniaxial magnetic anisotropy which prevents mixing of spin-up and spin-down Green functions Buz2003 .

To have some idea about the influence of the magnetic scattering on the proximity effect we may start with the linearized Usadel equation usad70 for the anomalous Green’s function in a ferromagnet

(2) |

The exponentially decaying solution has the form

(3) |

with

Here and . The anomalous Green’s function at gives us an idea about the spatial variation of the Cooper pair wave function. In the limit of the vanishing magnetic scattering and the decaying and the oscillating lengths are practically the same. However, if the spin-flip scattering time becomes relatively small , the decaying length could be substantially smaller than the oscillating length. This results in much stronger decrease of the critical current in junctions with increase of the layer thickness.

We have seen it experimentally JLTP that the form of dependence varies a little with temperature, so a good idea about this dependence may be already obtained from the temperature region near . Using in the form

(4) |

we can obtain (see Ref. Buz2003 ) for the case of good -interface transparency and the following expression for the critical current:

(5) |

where are taken in the limit of .

Now we shall address the question of the exact thickness and temperature dependence of the critical current in junctions. To deal with the complete set of the Usadel equations it is convenient to apply the usual parametrization of Green’s functions : and . Then for the Usadel equation is written as

(6) |

If the temperature variation of the exchange field is negligible at the most direct way how the temperature could interfere is through the Matsubara frequencies. The presence of the magnetic scattering provides another mechanism of the critical current temperature dependence - through the normal Green’s function The important range of the Matsubara frequencies variation is of the order of for superconductivity. Then in the case of the relatively strong magnetic scattering the second mechanism of the temperature dependence will be predominant.

In the limit of relatively large -layer thicknesses and rigid boundary conditions we may obtain an analytical solution of Eq.(6) and the expression for the critical current density reads

(7) |

with the function

and , , where , and

In the limit and Eq. (7) coincides with that obtained previously in Ref. Buzdin91 . The theoretical fit of our experimental results which is based on Eq. (7) is presented in Fig. 1 by the solid line and in Fig. 2 by dashed lines. Besides the dashed line in Fig. 1 shows calculations made using Eq. (5). One can see a good agreement obtained with the following parameters: , , . The fitting also yields considerable value of ’dead’ layers : , which do not take part in creating of the ’sign-reversal’ superconductivity. The dead layer may arise due to not full correspondence of the theoretical approach and the real system. On the other hand, other experiments Courtois also demonstrate the existence of large enough (2-3 nm) nonmagnetic layers at -interfaces.

A final remark concerns the real transparency of -interfaces in our sandwiches. In modern theories an interface transparency is characterized by a parameter , where is interface resistance per unit area, is the junction area, is -layer resistivity and . To estimate and we have carried out detailed measurements of -junctions -characteristics. The upper inset in Fig. 3 shows that -characteristics are described by the expression .

The linear approximation presented in Fig. 3 has given for junctions with the area of and . It allows to estimate following ferromagnet parameters: the electron mean free path 1 nm, the diffusion coefficient and the characteristic spatial scale . Values obtained determine the good enough transparency parameter that confirms the validity of the approximation used.

An additional breakthrough of the work is fabrication of -junctions with large enough critical current density. Solving of this problem has enabled detailed experimental investigations of -transition peculiarities, reliable detections of second harmonic in the current-phase relation Sellier2004 and the -coexistence Radovich . High magnitude of the critical current also allows to use -junctions as stationary phase -shifters in novel modifications of the digital and quantum logic appl . In proposed logic circuits -junctions are connected together with ordinary tunnel junctions and should not introduce themselves any noticeable phase shift during dynamical switchings in the rest of the circuit. This is possible only if the -junction critical current is much larger then critical currents of other junctions. The -junctions are based on the standard niobium thin film technology so they can be incorporated directly into existing architectures of the superconducting electronics.

Thus, both and reverse transitions have been detected in junctions for the first time. The double-reversal thickness dependence of the critical current is a most striking evidence of the superconducting order parameter spatial oscillations in a ferromagnet close to -interface. We have also observed that the oscillation length in the ferromagnetic alloy is considerably larger than the the pair decay length. We have presented a theoretical description of an extra mechanism of the order parameter decay in , mainly related to the strong spin-flip scattering on magnetic inhomogeneity.

We are grateful to A.Bobkov, I. Bobkova, Y. Fominov, A. Golubov, M. Kupriyanov and A. Rusanov for helpful discussions and to N. S. Stepakov for assistance during experiments. This work was supported by Russian Foundation for Basic Research, Programs of Russian Academy of Sciences, INTAS(grant no. 01-0809) and partially by ESF ”Pi-shift” Programme and French ECO-NET 2005 Programme”.

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