Spectrum of Andreev bound states in Josephson junctions with a ferromagnetic insulator

doi:10.1016/j.jmmm.2012.02.067

Journal of Magnetism and Magnetic Materials

Volume 324, Issue 21, October 2012, Pages 3467-3470

https://doi.org/10.1016/j.jmmm.2012.02.067 Get rights and content

Abstract

Ferromagnetic-insulator (FI) based Josephson junctions are promising candidates for a coherent superconducting quantum bit as well as a classical superconducting logic circuit. Recently the appearance of an intriguing atomic-scale $0 - π$ transition has been theoretically predicted. In order to uncover the mechanism of this phenomena, we numerically calculate the spectrum of Andreev bound states in a FI barrier by diagonalizing the Bogoliubov–de Gennes equation. We show that Andreev spectrum drastically depends on the parity of the FI-layer number L and accordingly the $π (0)$ state is always more stable than the 0 ( $π$ ) state if L is odd (even).

Introduction

The peculiarity of the proximity effect in superconductor/ferromagnetic-metal (S/FM) bilayers is the damped oscillation of the pair amplitude inside a FM [1], [2]. This anomalous proximity effect leads to the $π$ Josephson S/FM/S junction [3], [4] which has the opposite sign to the superconducting order parameter in two S electrodes in the ground state. Experimentally $π - junction$ was firstly observed by Ryazanov [5] and Kontos [6] and since then a lot of progress has been made in the physics of $π - junctions$ and now they are proving to be promising elements of superconducting classical and quantum circuits [7], [8], [9], [10].

On the other hands, recently a possibility of $π$ junction formation in a Josephson junction with a ferromagnetic insulator (FI) has been theoretically predicted [11], [12], [13], [14], [15], [16], [17], [18], [19], [20]. The $π$ junction using such an insulating barrier is very promising for future qubit [21], [22], [23], [24] and microwave [25] applications because of the low decoherence nature [26], [27]. More importantly, it has been shown that the ground state of S/FI/S junction alternates between 0- and $π - states$ when thickness of FI is increasing by a single atomic layer [16], [18]. In this paper in order to understand the physical mechanism of the anomalous atomic scale $0 - π$ transition, we will calculate the spectrum of the Andreev bound states in such systems. Based on this calculation, we will show that Andreev spectrum drastically depends on the parity of the FI layer number L and thence the $π (0)$ state is always more stable than the 0 ( $π$ ) state if L is odd (even).

In this paper we focused on the one dimensional s-wave junction with a FI barrier (Fig. 1(a)). It should be noted that the qualitatively same result can be obtained for two- or three-dimensional cases.

Section snippets

Model

Let us consider a one-dimensional tight-binding lattice of a superconductor/ferromagnetic-insulator/superconductor (S/FI/S) Josephson junction with L being the thickness or the numbers of the FI lattice sites as shown in Fig. 1(a). The lattice constant is set to be unity. Electronic states in a s-wave superconductor are described by the mean-field BCS Hamiltonian, $H_{S} = - t \sum_{n, n' \in S, σ} c_{n σ}^{†} c_{n' σ} + (2 t - μ_{s}) \sum_{n \in S, σ} c_{n σ}^{†} c_{n σ} + \frac{Δ}{2} \sum_{n \in S} (c_{n ↑}^{†} c_{n ↓}^{†} + c_{n ↑}^{†} c_{n ↓}^{†} + h . c .) .$ Here $c_{n σ}^{†}$ ( $c_{n σ}$ ) is the creation (annihilation) operator of

Andreev bound states and Josephson current

The Hamiltonian can be diagonalized by the Bogoliubov transformation. Due to the Andreev reflection at S/FI interfaces, the Andreev bound state is formed in the FI barrier (see Fig. 1(b)). Wave functions of the Andreev bound state decay far from the S/FI interface. In what follows, we focus on the subspace for spin- $↑$ electron and spin- $↓$ hole. In superconductors, the wave function of a bound state is given by $Ψ_{L} (n) = Φ_{L} [(\begin{matrix} u \\ v \end{matrix}) {Ae}^{- ikn} + (\begin{matrix} v \\ u \end{matrix}) {Be}^{{ik}^{⁎} n}],$ $Ψ_{R} (n) = Φ_{R} [(\begin{matrix} u \\ v \end{matrix}) {Ce}^{ikn} + (\begin{matrix} v \\ u \end{matrix}) {De}^{- {ik}^{⁎} n}] .$ Here A, B, C and D are amplitudes of

Numerical results

In this section, we show numerical results for the spectrum of Andreev bound states for a conventional S/I/S junction and an S/FI/S junction. In the calculation, we set $μ = μ_{s} = 2 t$ , and $Δ = 0.01 t$ .

Let us firstly consider Andreev bound states in an S/I/S junction. Fig. 3 shows the Andreev spectrum as a function of the thickness of the insulating barrier L. Due to the spin degeneracy, we have two Andreev levels for a given $ϕ$ and L. It is evident that the energy minimum is at $ϕ = 0$ irrespective of the

Summary

To summarize, we have theoretically studied the Andreev levels and the Josephson current in S/FI/S junctions by solving the Bogolubov–de Gennes equation in order to understand the physical mechanism of the atomic scale $0 - π$ transition. A characteristic and important feature for such systems is that the Andreev spectrum strongly depends on the parity of the thickness of the FI layer L. As a result, the junctions show the atomic scale $0 - π$ transition. Our finding suggests a way of understanding the

Acknowledgements

We would like to thank S. Kashiwaya and S. Nakamura for useful discussion. This work was supported by CREST-JST, the “Topological Quantum Phenomena” (No. 22103002) KAKENHI on Innovative Areas and a Grant-in-Aid for Scientific Research (No. 22710096) from MEXT of Japan.

References (42)

Y. Tanaka et al.
Physica C
(1997)
S. Kawabata et al.
Physica C
(2009)
S. Kawabata et al.
Physica E
(2010)
S. Kawabata et al.
Physica C
(2010)
S. Kawabata et al.
Physica E
(2011)
S. Kawabata et al.
Physica C
(2006)
S. Kawabata et al.
Physica E
(2007)
S. Kawabata et al.
Physica C
(2008)
G. Schön et al.
Physics Reports
(1990)
L.N. Bulaevskii et al.
JETP Letters
(1977)

A.I. Buzdin et al.

JETP Letters

(1982)

A.A. Golubov et al.

Reviews of Modern Physics

(2004)

A.I. Buzdin

Reviews of Modern Physics

(2005)

V.V. Ryazanov et al.

Physical Review Letters

(2001)

T. Kontos et al.

Physical Review Letters

(2002)

L.B. Ioffe et al.

Nature

(1999)

I. Petkovic et al.

Physical Review Letters

(2009)

V.M. Krasnov et al.

Physical Review B

(2007)

A.K. Feofanov et al.

Nature Physics

(2010)

J.C. Cuevas et al.

Physical Review B

(2001)

E. Zhao et al.

Physical Review B

(2004)

Cited by (13)

Microscopic study of the Josephson supercurrent diode effect in Josephson junctions based on two-dimensional electron gas
2023, Physical Review B
Microscopic study of the Josephson supercurrent diode effect in Josephson junctions based on two-dimensional electron gas
2023, arXiv
Anomalous Josephson Hall effect charge and transverse spin currents in superconductor/ferromagnetic-insulator/superconductor junctions
2020, Physical Review B
Anomalous josephson hall effect charge and transverse spin currents in superconductor/ferromagnetic insulator/superconductor junctions
2020, arXiv
Connection between zero-energy Yu-Shiba-Rusinov states and 0-π transitions in magnetic Josephson junctions
2018, Physical Review B
Superconducting mesa-structures with a spin filtering manganite interlayer
2018, EPJ Web of Conferences

View all citing articles on Scopus

View full text

Spectrum of Andreev bound states in Josephson junctions with a ferromagnetic insulator

Abstract

Introduction

Section snippets

Model

Andreev bound states and Josephson current

Numerical results

Summary

Acknowledgements

Physica C

Physica C

Physica E

Physica C

Physica E

Physica C

Physica E

Physica C

Physics Reports

JETP Letters

JETP Letters

Reviews of Modern Physics

Reviews of Modern Physics

Physical Review Letters

Physical Review Letters

Nature

Physical Review Letters

Physical Review B

Nature Physics

Physical Review B

Physical Review B