Novel Andreev bound states in nanostructured superconductors

Shi-Hsin Lin1, Milorad V. Milošević2, François Peeters3, and Boldizsár Jankó4

1 Dept. of Physics, Univ. of Notre Dame, Notre Dame IN, 46556 USA; e-mail: albert.shihsin.lin@gmail.com 2 Dept. Fysica, Univ. Antwerpen, Groenenborgerlaan 171, B-2020 Antwerpen; email: Milorad.Milosevic@ua.ac.be 3 Dept. Fysica, Univ. Antwerpen, Groenenborgerlaan 171, B-2020 Antwerpen; email: francois.peeters@ua.ac.be 4 Institute for Theoretical Sciences & Dept. of Physics, Univ. of Notre Dame, Notre Dame IN, 46556 USA; e-mail: bjanko@nd.edu

Abstract −We investigate the electronic structure of a superconductor(SC) in proximity of nanoscale ferromagnetic(FM) disc (SC/FM hybrids) and find that the superconductor order parameter has its minimum under the edge of the disc, thus forming a sombrero-like profile. The quasiparticle dispersion for such an order parameter is calculated with quasi-classical Bogoliubov-de Gennes equations. The result shows that the energy versus angular momentum dispersion has a minimum at finite non-zero angular momentum. The low-lying superconducting excitations of this SC/FM nanohbryid are Andreev bound states that possess a large non-zero angular momentum.

1 Introduction

Confinement on nanometer scale has profound influence on the electronic degrees of freedom in nanostructured materials. One of the nanoengineering approaches is to utilize hybrid structures. By combining different classes of materials with different magnetic properties, for example, diluted magnetic semiconductor (DMS), ferromagnet, and superconductor (SC), a wide variety of superconducting order parameter landscapes can be achieved.

In our previous work, we investigated Zeeman-localized bound states on the DMS/SC hybrid system. We predicted previously [1] the appearance of these bound states when a dilute magnetic semiconductor is exposed to an external magnetic field showing nanoscale inhomogeneity. Such an inhomogeneous magnetic field is created, for example, by a superconductor vortex. We also found [2] that the spin polarized Zeeman bound states are very robust, and they remain bound to the external field inhomogeneity throughout the experimentally relevant region of impurity concentration and scattering strength.

Here we are exploring a complementary system of a superconductor in the proximity of nanoscale magnets. The nanomagnets generate highly inhomogeneous magnetic field, which, in turn, can be used to design the spatial dependence of the superconducting order parameter.

2 Results

We focus our attention to the regime when the magnetic field generated by the nanomagnet is small

enough not to cause flux penetration in the superconductor, but large enough to cause substantial spatial modulation in the superconducting order parameter. Specifically, we study the case of a cylindrical nanomagnet with magnetization perpendicular to superconductor-magnet interface. Our numerical Ginzburg-Landau calculations[3] indicate that the local superconducting order parameter shows a substantial decrease along a ring with a radius comparable to the radius of the magnetic cylinder, forming a sombrero-like order parameter landscape.

If the suppression of the superconducting order parameter is large enough, it leads to trapped quasiparticles called Andreev bound states. We expect that in the SC/FM hybrid structure we propose there will be Andreev bound states trapped in the circular region right under the edge of the FM disc.

To confirm our expectation and illustrate the implications of such structure, we consider a simplified model of a step-wise order SC parameter first. This model reflects the qualitative structure of the realistic potential under a magnetic dot, but it can also describe accurately a different hybrid: a superconducting film with a ring-shaped etching that is filled by a normal metal. In order to describe the electronic properties of our model system, we perform quasiclassical Bogoliubov-de Gennes calculations analytically and numerically as well. The results show the existence of trapped quasiparticles. Furthermore, we find an unusual energy versus angular momentum dispersion for these trapped states. In contrast to a classical system, the low energy excitations do not correspond zero angular momentum. Instead, we find a minimum in the energy dispersion that is located near a large angular momentum. Low energy excitations with large momentum quantum number were encountered long before in the context of superfluid 4He, where the low energy bosonic excitation (rotons) has finite nonzero linear momentum. However, the physics of the fermionic dispersion in our system is quite different. It can be understood from the quasiclassical picture of the quasiparticle moving along a linear trajectory in a ring shape chamber. The quasiparticle with larger impact factor than the inner radius resides in a wider potential well than the quasiparticle with

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vanishing impact factor. The impact parameter is proportional to the angular momentum. Hence the angular momentum of the low-lying Andreev bound states is non-zero and proportional to the radius of the ring.

For a realistic system, say, a Pb superconducting film with a Co magnetic disk on top, our numerical Ginzburg-Landau calculations predict a smooth sombrero-like order parameter under the stray magnetic field of the magnet. The numerically generated order parameter was subsequently used in the Bogoliubov-de Gennes calculation to obtain the dispersion. Despite the lack of step-like features in the order parameter, the quasiparticle dispersion still has a local minimum at finite non-zero angular momentum. We therefore predict for the realistic FM/SC hybrid system a novel type of low-lying Andreev bound states with large angular momentum.

One of the effect of the stray magnetic field is that it shifts the eigenvalues for different chirality. Indeed, chiral symmetry is broken in our system and the lowest-lying Andreev bound states have a well-defined chirality.

We would like emphasize that the nanoscale hybrid structures we discussed in this paper allow for designable quasiparticle parameters. Indeed, several energy dispersion paramterers can be varied and designed by changing the fabrication parameters. As mentioned earlier, the angular momentum corresponding to lowest energy excitation depends on the radius of the FM disc. Furthermore, the depth of the order parameter well in the superconductor depends on the strength of the stray magnetic field, which in turn can be varied by the thickness of the magnet. Thus, by varying the geometry and materials parameters of the hybrid structures, the excitations can be custom-tailored to specific needs and potential applications.

3 Conclusion

We predict the existence a novel type of Andreev bound state in a SC/FM hybrid system with superconductivity suppressed in a ring-shaped geometry. The low-lying Andreev bound states have an unusual energy versus angular momentum dispersion with a global minimum at a non-zero angular momentum. The low-lying excitations are chiral and carry large, tunable angular momentum.

Acknowledgments

We would like to thank Dr. George Crabtree, Dr. Csaba Gyorgy and Dr. Wai-Kwong Kwok for numerous useful conversation. The work was supported by the Flemish Science Foundation (FWO-Vl), the Belgian Science Policy (IAP), the U. S.

Department of Energy, Office of Science, Office of Basic Energy Sciences, under contract W-31-109-Eng-38 and NSF-NIRT ECS-0609249.

References

[1] M.I.Berciu, T.G. Rappoport, B. Janko, Nature, 435 (7038): 71-75 (2005); T.G. Rappoport, M. Berciu, B. Janko, Phys. Rev. B 1-12 Art. No. 094502, p. 1-12 (2006) [2] S.-H. Lin, M.I.Berciu, T.G. Rappoport, B. Janko, J. Appl. Phys. 107, Art. No: 034307 (2010). [3]See for example M.V. Milosevic et al. Appl. Phys. Letters , 96 , Art. No: 032503 (2010).

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