Little-Parks Oscillations in Hybrid Ferromagnet-Superconductor Systems
On the basis of of linearized Usadel equations we consider superconductivity nucleation in multiply connected mesoscopic superconductor/ferromagnet hybrids such as thin-walled superconducting cylinders placed in electrical contact with a ferromagnetic metal. We study the interplay between the oscillations of due to the Little–Parks effect and the oscillations due to the exchange field. We demonstrate that the exchange field provokes the switching between the superconducting states with different vorticities and this may result in the increase the critical temperature of the superconducting transition in the magnetic field. Moreover we analyse the influence of the S/F transparency on the realisation of the states with higher vorticities.
pacs:74.25.Dw, 74.45.+c, 74.78.Na
Little-Parks effect, Little-Parks-PR64 () i.e., the oscillations of the critical temperature of multiply-connected superconducting samples in an applied magnetic field , is one of the striking phenomena demonstrating coherent nature of the superconducting state. Nevertheless, such oscillatory phase transition line was shown to be inherent to simply connected mesoscopic samples with the lateral size of the order of several coherence lengths Buisson-PL90 (); Bruyndoncx-PRB99 (); Jadallah-PRL99 () and to hybrid ferromagnet (F)-superconductor (S) systems with magnetic dots or domains Lange-PRL03 (); Aladyshkin-JCM03 (); Aladyshkin-PRB03 (); Schildermans-PRB08 (), that create a ”magnetic template” for nucleation of the superconducting order parameter. These oscillations reflect the switching between vortex states characterized by different winding numbers and are associated with the orbital effect Ginzburg-JETP56 ().
Another mechanism of switching between the superconducting states with different vorticities in multiply connected hybrid S/F structures such as thin-walled superconducting shell placed in electrical contact with a ferromagnetic cylinder was suggested recently in Samokhvalov-PRB07 (). This mechanism is caused by the exchange interaction and associated with the damped-oscillatory behavior of the Cooper pair wave function in a ferromagnet Buzdin-RMP05 (). It was shown that under certain conditions the exchange interaction can stimulate the superconducting states with a nonzero vorticity in the absence of external magnetic field. The interplay between the exchange and orbital effects may result in a subsequent switching between the states with different vorticities, as the F core radius increases. An obvious consequence of these transitions between the states with different should be a nonmonotonic dependence of the critical temperature on the F core radius and exchange field. Similar oscillating behavior of on a ferromagnetic layer thickness has been predicted for layered S/F structures Buzdin-JETPL90 (); Radovic-PRB91 (). Note, that the unusual ground states with spontaneously formed vortices in multiply-connected S/F hybrids resemble the behavior of Josephson -junction Buzdin-JETPL82 (); Buzdin-JETPL91 (); Ryazanov-PRL01 () with a step-like change in the F layer thickness Bulaevsky-SSC78 (); Goldobin-PRL06 (); Frolov-PRB06 ().
It is the purpose of this paper to study the influence of an external magnetic field on the proximity induced switching between the vortex states. We focus on the behavior of critical temperatures for superconducting states with different vorticities and, thus, we study the Little-Parks effect affected by the exchange interaction.
The paper is organized as follows. In Sec. II we briefly discuss the basic equations. In Sec. III we study the switching between different vortex states for two model S/F systems placed in the external magnetic field. The first system consists of a thin-walled superconducting cylindrical shell surrounding a cylinder of a ferromagnetic metal. As a second example we consider a cylindrical cavity in a bulk ferromagnet covered by a thin layer of superconducting material. For both cases we assume that there is a good electrical contact between the F and S regions, to assure a rather strong proximity effect. We summarize our results in Sec. IV.
The calculations of the second-order superconducting phase transition temperature are based on the linearized Usadel equations Usadel-PRL70 () for the averaged anomalous Green’s functions and for the F and S regions, respectively (see Ref. Buzdin-RMP05 () for details). The superconducting critical temperature and exchange field is assumed to satisfy the dirty-limit conditions and , where is the elastic electron–scattering time.
In the F (S) region the linearized Usadel equations take the form
Here and are the diffusion constants in the ferromagnet and superconductor, respectively, is a Matsubara frequency at the temperature , and is the flux quantum. Hereafter we consider only a step-like exchange field profile and, thus, neglect both a reduction of the magnetization in the ferromagnet and magnetization leakage into the superconductor. Such assumption is justified above due to the local nature of the exchange interaction. Calculating the value itself we can assume the superconducting order parameter to be vanishingly small and, thus, it is natural to neglect the spreading of the magnetic moment into the superconductor (see, e.g.,Krivoruchko-PRB02 (); Bergeret-PRB04 ()).
The superconducting critical temperature is determined from the self-consistency condition for the gap function:
and at the interface between the F and S metals: Kuprianov-JETP88 ()
Here is the superconducting (normal-metal) coherence length, and are the normal-state conductivities of the F and S metals, is related to the S/F boundary resistance per unit area through , and denotes a derivative taken in the direction to the outer normal to the S surfaces, i.e., the vector is directed from the S to the F metal at the S/F interface. For the sake of simplicity we assume . According to the equations (1),(2),(3) there is a symmetry , so that we can treat only positive values.
We consider here a generic example of hybrid S/F systems with a cylindrical symmetry: a thin-walled superconducting hollow cylinder placed in an electrical contact with a ferromagnet (see Fig. 1).
The ferromagnetic material is assumed to fill either the internal (Fig. 1a) or external (Fig. 1b) region of the system. We choose cylindrical coordinates () as it is shown in Fig. 1, take the gauge , and look for a homogeneous along solution of the equations (1),(2),(3) characterized by certain angular momentum :
The vorticity parameter just coincides with the angular momentum of the Cooper pair wave function.
Certainly, the magnetization inside the region makes contribution to the vector potential :
and can modify the conditions of superconductivity nucleation in the S shell. Choosing the magnetization direction along the axis , we can roughly estimate the relevant change in the total magnetic flux as follows: . In principle, this change in the magnetic flux modifies the period of the standard Little-Parks oscillations and breaks the symmetry of the dependence with respect to the external magnetic field inversion: . However, for typical parameters , , and of order of several lengths we get , and the asymmetry of the Little-Parks curve may be neglected. The above effect of the magnetization can be also weakened provided we decrease the size of the S/F system along the axis, going over to the case of a thin S/F disk when the field is suppressed due to the demagnetization factor. Note that choosing the magnetization direction in the plane perpendicular to the cylinder axis we can get rid of the above magnetic flux correction completely because of the absence of the magnetization induced field component along the cylinder axis. These simple estimates allow us to exclude the effect of magnetization on assuming that and .
Iii Vortex states in thin-walled superconducting cylinder
Now we proceed with calculations of the critical temperature dependence on the external magnetic field for two examples of multiply connected mesoscopic hybrid S/F systems. The first one is a ferromagnetic cylindrical filament (core) surrounded by a thin-walled superconducting shell (see Fig. 1a). The second one is a cylindrical cavity in a bulk ferromagnet covered by a thin layer of superconducting material (see Fig. 1b). External magnetic field is assumed to be parallel to the cylinder axis () for the both cases.
iii.1 Ferromagnetic filament covered by a cylindrical superconducting shell
Consider a superconducting cylindrical shell of a thickness surrounding a cylinder (core) of a ferromagnetic metal. Here is the radius of the F core, and is the outer radius of the S shell (see Fig. 1a). Naturally, to observe a pronounced influence of the proximity effect on the transition temperature, the thickness of the S shell must be smaller than the superconducting coherence length .
where is the flux of the external magnetic field threading the circle of radius in the units of the flux quantum
Here is the characteristic length scale of the order parameter variation in the F metal. In the dirty limit, the parameter determines both the length scale of oscillations and the decay length for the Cooper pair wave function in a ferromagnet. The boundary conditions (4) and (5) for Eq. (9) take the form
where is the flux of the external magnetic field enclosed in the F cylinder in the units of flux quantum , and
For , the variations of the functions and in the superconducting shell are small: , . Therefore, we can average Eq. (8) over the thickness of the S shell, using the boundary conditions (13) to integrate the term . Finally, we obtain the following expression:
where is the digamma function. The depairing parameter of the mode
is responsible for the superconductivity destruction in the shell in the applied magnetic field , due to both the orbital and exchange effects. As usual, the critical temperature of a superconductivity nucleation in the shell is determined by the maximal value :
We start our numerical analysis from the case of zero external magnetic field focusing on the effect of the S/F interface resistance on the behavior of . Taking into account the asymptotic expressions for the Kummer’s function for we obtain the parameter in a simplified form:
The states with angular momenta are degenerated for and have the same critical temperature .
In Fig. 2 we present examples of dependencies of the critical temperature on the F cylinder radius for different values of the S/F interface transparency. We see that for a small F cylinder radius only the state with appears to be energetically favorable. The influence of the proximity effect is weak and the critical temperature is close to . For a vortex state with the value is suppressed because of a large orbital effect. The increase in the radius results in a decrease in for the state with and reduce the kinetic energy of supercurrents for . At the same time, the damped-oscillatory behavior of the superconducting order parameter in a ferromagnet becomes important, since the diameter of the F cylinder becomes comparable with the period of the order parameter oscillations (). As a result, for the vortex state with becomes more energetically favorable due to the exchange interaction. It is interesting to note that at small F cylinder radius () the critical temperature of the vortex free mode with decreases with an increase of the interface barrier , if the condition is fulfilled. Such counterintiutive behavior is explained by the enchancement of the pair-breaking role of the exchange field due to the increase in the time of the Cooper pair stay in the F metal in the presence of the interface barrier Buzdin-RMP05 ().
Figure 3 shows examples of dependences of the critical temperature on the external magnetic field , obtained from Eqs. (16), (17), (18) for different values of F cylinder radius . Due to the symmetry of the phase boundary , we present here the curves for positive values of the external field only. The phase boundary exhibits Little-Parks oscillations, indicating transitions between the states with different angular momenta of the superconducting order parameter. For a small F cylinder radius (Fig. 3a,b) the influence of the exchange interaction is weak and the phase boundary exhibits undamped quasiperiodic oscillations as function of magnetic field and resembles the curve for a mesoscopic disk-shaped superconductor Buisson-PL90 (); Jadallah-PRL99 (). It means that the S ring induces superconductivity in the thin F filament due to proximity, and the behavior of the S/F hybrid system under consideration is similar to the one for a superconducting cylinder (or disk) with an inhomogeneous order parameter.
With an increase in the F cylinder radius one can clearly observe a shift of the main maximum towards nonzero values. To explain this shift we note that the highest critical temperature for a given magnetic flux corresponds to the states with the angular momentum close to the integer part of the value. Exactly at the points the orbital term in the depairing parameter (17) is cancelled and the dependendence of vs at these points is determined only by the exchange effect. On the other hand, for this exchange part of the depairing parameter is minimal and the corresponding is maximal for a state with a nonzero vorticity (see Fig. 2). Thus, the main maximum shifts to a certain nonzero value. To support this explanation we have performed simulations of the critical temperature for the points , i.e. when the orbital effect is cancelled. The results are shown in panels (c, d) of Fig. 3 by dashed lines. These curves describe a large scale (as compared to ) behavior of the phase boundary and clearly demonstrate the shift of the main maximum. With the increase in the F cylinder radius and the interface barrier this main maximum shift appears to increase.
Note, that a similar shift of the main maximum has been observed in mesoscopic S/F hybrid structures without the proximity effect Lange-PRL03 (); Schildermans-PRB08 () and in the quasi-two-dimensional organic conductors Uji-Nature01 (); Balicas-PRL01 (). In the latter case, an applied magnetic field compensates the exchange fields of the paramagnetic ions and neutralizes the destructive action of these fields Jaccarino-PRL62 ().
iii.2 Cylindrical cavity in a bulk ferromagnet covered by a superconductor
As a second example we consider a superconducting shell of a thickness covering a cylindrical cavity of the radius in a ferromagnetic material (see Fig. 1b). For simplicity, we consider only rather thin S shells with which allows us to assume the variations of the functions and in the superconductor to be small. The solution of Eqn. (8) in F metal can be expressed via the confluent hypergeometric function of the second kind Abramowitz-Handbook (),
where is the flux of the external magnetic field enclosed in the cavity in the units of the flux quantum , and
Assuming that the variations of the functions and in the superconducting shell are small (, ) we can average Eq. (9) over the thickness of the S shell, using the boundary conditions (21) to integrate the term . Substitution of the solution
into the self-consistency equation (9) results in the equation for the critical temperature of the state with a vorticity (15), where the depairing parameter of the mode is determined by the following expression:
Figure 4 shows examples of dependencies of the critical temperature on the external magnetic field , obtained from Eqs. (16), (22), (24) for two values of the cylindrical cavity radius . Similarly to the previous subsection this phase boundary resembles the curve for the Little-Parks oscillations. Slow envelopes of the oscillating curve clearly demonstrate the shift of the main maximum towards nonzero values.
To sum up, we have analysed the behavior of the Little-Parks oscillations of the critical temperature on an external magnetic field in multiply connected S/F systems affected by the exchange interaction. As an example, we have considered mesoscopic thin-walled superconducting cylindrical shell placed in electrical contact with a ferromagnet. The phase-transition line and order parameter structure have been studied on the basis of linearized Usadel equations. We have demonstrated that the exchange field provokes the switching between the superconducting states with different vorticities: the interplay between the oscillations of due to the Little–Parks effect and the oscillations due to the exchange field results in breaking of the strict periodicity of the dependence. We have also observed a slow modulation of the amplitude of the quasiperiodic oscillations. With an increase in the superconducting shell radius the envelope of the oscillating phase transition line exhibits a shift of the main maximum to finite external magnetic field values. This shift is explained by the increase in the critical temperature of the superconducting states with nonzero vorticities due to the exchange interaction. The above effects strongly depend on the S/F interface barrier strength. We have shown that a decrease in the S/F transparency may stimulate the states with higher vorticities.
This work was supported, in part, by the Russian Foundation for Basic Research, by International Exchange Program of Universite Bordeaux I, by French ANR project ”ELEC-EPR”, by the ”Dynasty” Foundation, and by the program of LEA Physique Theorique et Matiere Condensee.
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