Subgap modes in two-dimensional magnetic Josephson junctions Yinan Fang1Seungju Han2Stefano Chesi3 4and Mahn-Soo Choi2y 1School of Physics and Astronomy Yunnan University Kunming 650091 China

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Subgap modes in two-dimensional magnetic Josephson junctions

Yinan Fang,1Seungju Han,2Stefano Chesi,3, 4, ∗and Mahn-Soo Choi2, †

1School of Physics and Astronomy, Yunnan University, Kunming 650091, China

2Department of Physics, Korea University, Seoul 02841, South Korea

3Beijing Computational Science Research Center, Beijing 100193, People’s Republic of China

4Department of Physics, Beijing Normal University, Beijing 100875, People’s Republic of China

We consider two-dimensional superconductor/ferromagnet/superconductor junctions and inves-

tigate the subgap modes along the junction interface. The subgap modes exhibit characteristics

similar to the Yu-Shiba-Rusinov states that originate form the interplay between superconductivity

and ferromagnetism in the magnetic junction. The dispersion relation of the subgap modes shows

qualitatively diﬀerent proﬁles depending on the transport state (metallic, half-metallic, or insulat-

ing) of the ferromagnet. As the spin splitting in the ferromagnet is increased, the subgap modes

bring about a 0-πtransition in the Josephson current across the junction, with the Josephson current

density depending strongly on the momentum along the junction interface (i.e., the direction of the

incident current). For clean superconductor-ferromagnet interfaces (i.e., strong coupling between

superconductors and ferromagnet), the subgap modes develop ﬂat quasi-particle bands that allow

to engineer the wave functions of the subgap modes along an inhomogeneous magnetic junction.

PACS numbers: 74.50.+r; 74.25.Ha; 85.25.Cp

I. INTRODUCTION

The interplay between superconducting and ferromag-

netic order leads to unconventional pairing mechanisms

as well as exotic quantum states, such as the Yu-Shiba-

Rusinov (YSR) state bounded to a (classical) mag-

netic impurity,1–3the Fulde-Ferrell-Larkin-Ovchinnikov

(FFLO) states in ferromagnetic metals,4,5and the chiral

Majorana edge modes in topological superconductors.6,7

Understanding and controlling the delicate competition

between diﬀerent orders will undoubtedly beneﬁt the

development of quantum devices for various spintronic

applications.8

In a Josephson junction, the transport properties are

governed by subgap states below the superconducting en-

ergy gap, and these subgap states reﬂect the fate of the

competition between superconductivity and magnetism.

For example, consider a Josephson junction through a

quantum dot,9which can be regarded as a magnetic

impurity with strong quantum ﬂuctuations. The sub-

gap state induced by the impurity behaves like an An-

dreev bound state in the strong-coupling limit, where

the Kondo eﬀect10–12 dominates over superconductivity,

whereas it bears a closer resemblance to the YSR state

in the weak-coupling limit,13,14 where superconductivity

dominates over the Kondo eﬀect. Such change in charac-

ter of the subgap state results in a transition from neg-

ative to positive supercurrent across the junction, usu-

ally referred to as a quantum phase transition from a

0-junction to a π-junction.15,16

In a superconductor/ferromagnet/superconductor

(S/FM/S) junction, the nature of subgap states depends

on the transport properties of the ferromagnetic layer.

When the ferromagnetic layer is metallic, spin-dependent

Andreev subgap states play a dominant role: The ﬁnite

center-of-mass momentum of Cooper pairs which pene-

trate into the ferromagnetic metal causes an oscillatory

behavior in the proximity-induced pairing potential.8,17

Depending on the relative width of the ferromagnetic

layer with respect to the wave length of the oscillation,

the ground state of the S/FM/S junction may be

stabilized with either a 0 or πphase diﬀerence between

the two superconductors.18–20 In a recent paper,21

however, it was found that the YSR subgap states play

a more signiﬁcant role when the ferromagnet is a thin

insulator. The competition of superconductivity versus

magnetism induces a strong dependence of the YSR

state on the spin splitting in the ferromagnet, leading to

a 0-πtransition in the junction when the spin splitting

is increased.21

While so far most of the previous works studied one-

dimensional (1D) or quasi-1D junctions, i.e., narrow junc-

tions, in this work we consider two-dimensional S/FM/S

junctions (Fig. 1) and investigate the subgap modes along

the junction interface. We ﬁnd that, due to the in-

terplay between superconductivity and ferromagnetism

in the magnetic junction, the subgap modes inherit the

characteristics of the YSR states and lead to the follow-

ing intriguing properties that are hard to observe in nar-

row junctions: (i) The dispersion relation of the subgap

modes shows qualitatively diﬀerent proﬁles depending on

the transport state (metallic, half-metallic, or insulating)

of the ferromagnet. (ii) The subgap modes mediate the 0-

πtransition in the Josephson current across the junction,

induced by increasing the spin splitting in the ferromag-

net. They also determine a dependence of the Joseph-

son current density on the superconducting phase diﬀer-

ence which changes sharply with the momentum along

the junction interface (i.e., the direction of the incident

current). (iii) For clean superconductor-ferromagnet in-

terfaces (i.e., strong coupling between superconductors

and ferromagnet), the subgap modes develop ﬂat quasi-

particle bands that allow to engineer the wave functions

of the subgap modes along an inhomogeneous magnetic

arXiv:2210.04558v1 [cond-mat.supr-con] 10 Oct 2022





















FIG. 1: (color online) A schematic tight-binding

model for the two-dimensional superconduc-

tor/ferromagnet/superconductor junction. The empty

gray circles represent the lattice sites on the left and right su-

perconductors, whereas the red circles ﬁlled in yellow denote

sites in the ferromagnet. The homogeneous nearest-neighbor

tunnel coupling strengths within the superconducting and

ferromagnetic regions (tsc and tfm) are denoted by gray and

black solid links, respectively, and the coupling between

superconductors and ferromagnet (tc) is depicted by red

dashed links. The system consists of 2Nx+ 1 sites along the

x-direction and Nysites along the y-direction.

junction.

Apart from these results, which are interesting

from a fundamental point of view, we note that

several recent studies used scanning tunneling mi-

croscopy/spectroscopy to explore the exotic physics asso-

ciated with chains of magnetic atoms.22–25 This suggests

that the characterization of S/FM/S junction of inter-

mediate size, interpolating between the few impurities

and the continuum junction limits, can be useful also for

device applications.

The outline of the rest of our paper is as follows: In

Section II, we present the model of the S/FM/S junction.

In Section III, we discuss the characteristic properties

of the subgap modes as, in particular, their dispersion

relation and dependence on the superconducting phase

diﬀerence. Section IV is devoted to the 0-πtransition

of the magnetic Josephson junction and Section Vdis-

cusses the quasi-particle ﬂat bands of the subgap modes,

together with their eﬀect on the wave functions of the

subgap modes along inhomogeneous magnetic junctions.

Section VI concludes the paper.

II. MODELS

We consider a two-dimensional (2D) S/FM/S junction,

shown schematically in Fig. 1, and describe it with the

following tight-binding Hamiltonian26,27

H=ˆ

HL+ˆ

HR+ˆ

Hfm +ˆ

Htun.(1)

Terms ˆ

HL/Rare responsible for the left (L) and right (R)

superconductors, respectively, and are given by

HL/R=−tsc

x=1

y=1 X

σ=↑,↓

ˆc†

∓x,y,σ ˆc∓x,y+1,σ + h.c.

−tsc

Nx−1

x=1 X

ˆc†

∓x,y,σ ˆc∓x+1,y,σ + h.c.

+ ∆L/R

x=1 X

ˆc†

∓x,y,↑ˆc†

∓x,y,↓+ h.c.

−µsc

x=1 X

ˆc†

∓x,y,σ ˆc∓x,y,σ ,(2)

where ˆcx,y,σ is the electron annihilation operator for spin

σat site (x, y) of the superconductors, µsc is the Fermi en-

ergy of the superconductors, and ∆L/R= ∆sc exp[iϕL/R]

(∆sc >0) is the superconducting pairing potential of

each superconductor. Here, we impose periodic bound-

ary conditions in the y-direction (ˆcx,Ny+1,σ = ˆcx,1,σ for

all xand σ), and open boundary conditions in the x-

direction. We also assume identical superconductors on

both sides of the junction (|∆L|=|∆R|).In the follow-

ing, we measure energy from the Fermi level and denote

with ϕ=ϕL−ϕRthe phase across the junction. Term

Hfm describes the one-dimensional ferromagnet along the

y-direction (see Fig. 1) and is given by

Hfm =X

y"λMˆ

f†

y,↑ˆ

fy,↑−ˆ

f†

y,↓ˆ

fy,↓−µfm X

f†

y,σ ˆ

fy,σ

−tfm

σˆ

f†

y,σ ˆ

fy+1,σ + h.c.#,(3)

where ˆ

fy,σ is the electron annihilation operator for spin

σat site y(recall the periodic boundary condition,

fNy+1,σ =ˆ

f1,σ), µfm is the Fermi energy of the ferro-

magnet, and λMis the magnetic spin-splitting due to the

ferromagnetic order.48 Finally, the last term ˆ

Htun de-

scribes the tunnel coupling between the ferromagnet and

superconductors, that is,

Htun =−tc

x=±1X

σˆc†

x,y,σ ˆ

fy,σ + h.c..(4)

Throughout the paper, we will mainly discuss the nu-

merical results based on the above tight-binding model.

On the other hand, it turns out that some of the results

may be understood more transparently with a continuum

model28,29 governed by the Bogoliubov de Gennes (BdG)

Hamiltonian30–32

H=1

2Zd2rˆ

Ψ(r)†HBdG(r)ˆ

Ψ(r) (5)

for a magnetic Josephson junction between two super-

conductors (|x|> L/2) through a narrow ferromagnet

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

-1.5

-1

-0.5

0.5

1.5

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

-1.5

-1

-0.5

0.5

1.5

(a)

(d) (e) (f)

(b) (c)

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

-1.5

-1

-0.5

0.5

1.5

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

-1.5

-1

-0.5

0.5

1.5

/E∆

c sc

/t∆

c sc

/t∆

k a

5.2

= −

∆

0.1

= −

∆

5.2

= −

∆

0.1

= −

∆

4.8

= −

∆

0.7

= −

∆

4.8

= −

∆

0.7

= −

∆

FIG. 2: (color online) Energy spectrum of the subgap modes in the S/FM/S Josephson junction as a function of the transverse

wave number kyand tunnel coupling tcbetween the superconductor and ferromagnet. The upper panels (a–c) are the S/FM/S

Josephson junction with an insulating ferromagnet and the lower panels (d–f) are for the junction with a half-metallic ferromag-

net. In (a) and (d), the dispersions of isolated ferromagnets are shown as dashed curves for reference. The anti-crossing marked

by the red circle in (d) is further analyzed in Fig. 3; see also the main text. We used µfm =−5.2∆sc and λM=−0.1∆sc in the

upper panels and µfm =−4.8∆sc and λM=−0.7∆sc in the lower panels. Other parameters: Nx= 60, ϕ=π/4, µsc =−5∆sc,

and tsc =tfm = 5∆sc. The superconductor-ferromagnetic coupling is tc= 0.4∆sc for panels (a,d) and tc= 5∆sc for panels

(b,e).

(|x|< L/2) with

HBdG(r) = −~2∇2

2m−µ(x) + Vb(x)τzσ0

+ ∆scθ(x−L/2) (cos ϕRτx−sin ϕRτy)σ0

+ ∆scθ(−x−L/2) (cos ϕLτx−sin ϕLτy)σ0

+λMθ(L/2− |x|)τ0σz,(6)

where ˆ

Ψ(r) is the Nambu spinor,

Ψ(r) = ˆ

ψ↑(r)ˆ

ψ↓(r)ˆ

ψ↓(r)†−ˆ

ψ↑(r)†T,(7)

and σα(τα) for α= 0,1,2,3 represent the Pauli matrices

acting on the spin (particle-hole) subspace (α= 0 corre-

sponds to the identity matrix). In Eq. (6), the chemical

potential takes two diﬀerent values for the superconduc-

tors (¯µsc) and the ferromagnet (¯µfm), i.e.,

µ(x) = (¯µsc,|x|> L/2,

¯µfm,|x|< L/2,(8)

and λMquantiﬁes the magnetic spin splitting of the fer-

romagnet. The interface between the ferromagnet and

superconductors is modeled by δ-function tunnel barri-

ers

Vb(x) = l0V0δ(x−L/2) + l0V0δ(x+L/2),(9)

where l0is an arbitrary parameter, physically character-

izing the width of the (thin) potential barrier. In Ap-

pendix A, we discuss the correspondence between con-

tinuum and tight-binding models and obtain that, when

tctsc ≈tfm, the strength l0V0of the interface poten-

tial can be related to the tunneling amplitudes tsc, tcof

Eq. (1) as follows:

πtsc

tc−tc

tsc ≈r2m

¯µsc

λ0V0

~.(10)

Here, we have assumed that the Fermi wavenumber and

the lattice constant aof the tight-binding model satisfy

kFa≈π, where kF=√2m¯µsc/~. In Appendix A, we also

consider the relation between chemical potentials enter-

ing the two models. In particular, the diﬀerence between

µfm and ¯µfm depends on the conﬁnement energy in the

ferromagnetic strip, that is aﬀected in a nontrivial way

by both Land the strength of interface terms.

III. SUBGAP MODES

When a quasi-particle has energy lower than the super-

conducting gap, it cannot penetrate into the supercon-

ductors and only propagates inside the junction, that is,

within the ferromagnetic region of the S/FM/S junction.

Nonetheless, in a wide junction, the quasi-particle can

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Subgapmodesintwo-dimensionalmagneticJosephsonjunctionsYinanFang,1SeungjuHan,2StefanoChesi,3,4,andMahn-SooChoi2,y1SchoolofPhysicsandAstronomy,YunnanUniversity,Kunming650091,China2DepartmentofPhysics,KoreaUniversity,Seoul02841,SouthKorea3BeijingComputationalScienceResearchCenter,Beijing100193,People'...

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Subgap modes in two-dimensional magnetic Josephson junctions Yinan Fang1Seungju Han2Stefano Chesi3 4and Mahn-Soo Choi2y 1School of Physics and Astronomy Yunnan University Kunming 650091 China

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