^{1}, M. G. Khusainov

^{2}and Y. N. Proshin

^{3,a)}

### Abstract

The phase diagrams of few-layered nanosystems consisting of dirty superconducting (S) and ferromagnetic (F) metals are investigated within the framework of the modern theory of the proximity effect taking into account the boundary conditions. The F/S tetralayer and pentalayer are shown to have considerably richer physics than the F/S bi- and trilayer (due to the interplay between the 0 and phase superconductivity and the 0 and phase magnetism and nonequivalence of layers) and even the F/S superlattices. It is proven that these systems can have different critical temperatures and fields for different S layers. This predicted decoupled superconductivity is found to manifest itself in its most striking way for F/S tetralayer. It is shown that tetralayer is the most promising candidate for use in superconducting spin nanoelectronics.

Y. P. and M. K. are grateful to the Max-Planck-Institute for the Physics of Complex Systems (Dresden) for providing excellent conditions for fruitful work. Y. P. thanks Prof. Fulde and the participants of his seminar for the productive discussion of the results presented in this paper. This work was supported in part by RFBR (No. 04-02-16761, No. 05-02-16369).

I. INTRODUCTION

II. FEW-LAYERED F/S STRUCTURES

A. Simple few-layered systems (bi- and trilayers) and superlattice

B. Tetra- and pentalayers

III. DISCUSSION OF THE PHASE DIAGRAMS OF LAYERED F/S SYSTEMS

A. Decoupled superconductivity in the F/S tetra- and pentalayer

IV. CONCLUSIONS

### Key Topics

- Superconductivity
- 80.0
- Antiferromagnetism
- 38.0
- Superlattices
- 36.0
- Ferromagnetism
- 32.0
- F region
- 31.0

## Figures

The geometry of various few-layered systems. Bilayer F/S (a), two trilayers (b) and (c), tetralayer (d), and pentalayer (e) are shown. Vertical arrows show the directions (in-line) of the magnetizations that play the role of the magnetic order parameter. Here , and thicknesses of outer F layers and S layers are equal to and , where and are thicknesses of inner F and S layers, respectively.

The geometry of various few-layered systems. Bilayer F/S (a), two trilayers (b) and (c), tetralayer (d), and pentalayer (e) are shown. Vertical arrows show the directions (in-line) of the magnetizations that play the role of the magnetic order parameter. Here , and thicknesses of outer F layers and S layers are equal to and , where and are thicknesses of inner F and S layers, respectively.

The phase diagrams of the F/S nanostructures for the following values of parameters: , , , , and . In the figure is the reduced temperature, and is the reduced F layer thickness. The phase diagram of the F/S bilayer with one possible state (00) (a); the phase diagram of the F/S/F trilayer for the FM (00) and configurations (b); the phase diagram of the S/F/S trilayer with competition between 0 phase and phase of superconductivity (c); the phase diagram of the F/S superlattice where the thicknesses of all F layers equal , and the thicknesses of all S layers equal (d).

The phase diagrams of the F/S nanostructures for the following values of parameters: , , , , and . In the figure is the reduced temperature, and is the reduced F layer thickness. The phase diagram of the F/S bilayer with one possible state (00) (a); the phase diagram of the F/S/F trilayer for the FM (00) and configurations (b); the phase diagram of the S/F/S trilayer with competition between 0 phase and phase of superconductivity (c); the phase diagram of the F/S superlattice where the thicknesses of all F layers equal , and the thicknesses of all S layers equal (d).

The phase diagrams of the tetralayer for the same set of parameters as in Fig. 2. The curves for the outer layer are denoted using letters *with a prime*. The letters *without a prime* indicate the curves for the inner S layer. All denotations of curves correspond to those introduced in Eqs. (20) and (21). In the panels (a) and (b) the arrows show the difference between the states which are discussed in the paper. The phase diagram for the FM configuration of the magnetizations of both F layers (a); the phase diagram for the AFM configuration (b); the combined phase diagram of the tetralayer (c); the generalized phase diagram of the tetralayer (d). Vertical arrows show the direction of the magnetization in the corresponding ferromagnetic layer. The letters S and N stand for the superconducting and normal states of the superconducting layers, respectively.

The phase diagrams of the tetralayer for the same set of parameters as in Fig. 2. The curves for the outer layer are denoted using letters *with a prime*. The letters *without a prime* indicate the curves for the inner S layer. All denotations of curves correspond to those introduced in Eqs. (20) and (21). In the panels (a) and (b) the arrows show the difference between the states which are discussed in the paper. The phase diagram for the FM configuration of the magnetizations of both F layers (a); the phase diagram for the AFM configuration (b); the combined phase diagram of the tetralayer (c); the generalized phase diagram of the tetralayer (d). Vertical arrows show the direction of the magnetization in the corresponding ferromagnetic layer. The letters S and N stand for the superconducting and normal states of the superconducting layers, respectively.

The phase diagrams of the pentalayer with the same parameters as in Figs. 2 and 3. All denotations of curves and states correspond to those introduced in the last part of Sec. II. The phase diagram for the FM configuration of the magnetizations of all F layers (a); the phase diagram for the AFM configuration (b); the phase diagram for the FMAFM configuration (c); the generalized phase diagram of the pentalayer (the designations follow those used in Fig. 3d) (d).

The phase diagrams of the pentalayer with the same parameters as in Figs. 2 and 3. All denotations of curves and states correspond to those introduced in the last part of Sec. II. The phase diagram for the FM configuration of the magnetizations of all F layers (a); the phase diagram for the AFM configuration (b); the phase diagram for the FMAFM configuration (c); the generalized phase diagram of the pentalayer (the designations follow those used in Fig. 3d) (d).

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