Nonlinear features of the superconductorferromagnetsuperconductor 0Josephson junction in ferromagnetic resonance region Aliasghar Janalizadeh1 Ilhom R. Rahmonov234 Sara A.

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Nonlinear features of the superconductor–ferromagnet–superconductor ϕ0Josephson

junction in ferromagnetic resonance region

Aliasghar Janalizadeh1, Ilhom R. Rahmonov2,3,4, Sara A.

Abdelmoneim 5, Yury M. Shukrinov2,3,4, and Mohammad R. Kolahchi1

1Department of Physics, Institute for Advanced Studies in Basic Sciences (IASBS), P.O. Box 45137-66731, Zanjan, Iran

2BLTP, JINR, Dubna, Moscow Region, 141980, Russia

3Dubna State University, Dubna, 141980, Russia

4Moscow Institute of Physics and Technology, Dolgoprudny, 141700, Moscow Region, Russia

5Physics department, Menoﬁya University, Faculty of Science, 32511, Shebin Elkom,Egypt

(Dated: October 4, 2022)

We demonstrate the manifestations of the nonlinear features in magnetic dynamics and IV-

characteristics of the ϕ0Josephson junction in the ferromagnetic resonance region. We show that

at small values of system parameters, namely, damping, spin-orbit interaction, and Josephson to

magnetic energy ratio, the magnetic dynamics is reduced to the dynamics of the scalar Duﬃng os-

cillator, driven by the Josephson oscillations. The role of increasing superconducting current in the

resonance region is clariﬁed. Shifting of the ferromagnetic resonant frequency and the reversal of

its damping dependence due to nonlinearity are demonstrated by the full Landau-Lifshitz-Gilbert-

Josephson system of equations, and in its diﬀerent approximations. Finally, we demonstrate the

negative diﬀerential resistance in the IV–characteristics, and its correlation with the foldover eﬀect.

I. I. INTRODUCTION

The coupling of superconducting phase diﬀerence with

magnetic moment of ferromagnet in the ϕ0junction leads

to a number of unique features important for supercon-

ducting spintronics, and modern information technology

[1–5]. It allows to control the magnetization preces-

sion by superconducting current and aﬀects the current–

voltage (IV) characteristics by magnetic dynamics in the

ferromagnet, in particular, to create a DC component in

the superconducting current [6–8]. A remarkable mani-

festation of such coupling is the possibility to stimulate

a magnetization reversal in the ferromagnetic layer by

applying current pulse through the ϕ0-junction [3, 9–13].

There are two features of our Josephson junction that

come into play in our study. One is the broken inver-

sion symmetry in the weak link of the Josephson junc-

tion, when the link is magnetic, which introduces an ex-

tra phase in the current—-phase relation, preventing it

from being antisymmetric. Such Josephson junctions are

named ϕ0junctions [1], and examples exist such as MnSi

and FeGe. Second is the nonlinear property of the system

that makes for an anomalous resonance behavior [14].

We couple such a Josephson junction to the model

that describes the magnetodynamics in thin ﬁlms or

heterostructure, to form the Landau-Lifshitz-Gilbert-

Josephson model (LLGJ)[14–16]. It is shown that for

a particular set of parameters, the coupled equations

reduce to the dynamics of a Duﬃng oscillator [14].

The cubic nonlinearity in this oscillator has applications

in describing several eﬀects in other models too [17].

One being the resonance eﬀects in the antiferromagnetic

bimeron in response to an alternating current, which has

applications in the detection of weak signals [15, 18, 19].

The Gilbert damping term is added phenomenologi-

cally to the Landau—-Lifshitz model, to reproduce the

damping of the precessing magnetic moment. Gilbert

damping is important in modeling other resonance fea-

tures too, as its temperature dependence aﬀects them

[20, 21], and in return in the superconducting correla-

tions that aﬀect it [22]. The magnetization precession

in the ultra thin Co20F e60B20 layer stimulated by mi-

crowave voltage under a large angle, needs modeling by

Duﬃng oscillator too. This gets help from the so called

foldover features, again due to nonlinearity [16, 23, 24].

The consequences of the nonlinear nature of the cou-

pled set of LLGJ system of equations in the weak cou-

pling regime was demonstrated recently in Ref. [14]. We

showed in this regime, where the Josephson energy is

small compared to the magnetic energy, the ϕ0Joseph-

son junction is equivalently described by a scalar non-

linear Duﬃng equation. An anomalous dependence of

the ferromagnetic resonant frequency (FMR) with the

increase of the Gilbert damping was found. We showed

that the damped precession of the magnetic moment is

dynamically driven by the Josephson supercurrent, and

the resonance behavior is given by the Duﬃng spring.

The obtained results were based on the numerical simu-

lations. The role of dc superconducting current, and the

state with negative diﬀerential resistance (NDR) in IV-

characteristic were not clariﬁed. Also, the eﬀects of the

Josephson to magnetic energy ratio and the spin-orbit

coupling (SOC) were not investigated at that time.

In the present paper, we study the nonlinear aspects

of the magnetic dynamics and IV-characteristics of the

ϕ0Josephson junction in the ferromagnetic resonance re-

gion. We compare description of the anomalous damp-

ing dependence (ADD) exhibited by full LLGJ system

of equations with approximated equations and demon-

strate the Duﬃng oscillator features in the small param-

eter regime. Eﬀects of the Josephson to magnetic energy

ratio, and the spin-orbit coupling on the ADD, referred

to earlier as the α-eﬀect [14] are demonstrated. By de-

riving the formula which couples the dc superconduct-

arXiv:2210.00366v1 [cond-mat.supr-con] 1 Oct 2022

ing current and maximal amplitude of magnetization we

discuss the correlation of superconducting current and

the negative diﬀerential resistance in the resonance re-

gion. Finally, we discuss the experimentally important

features by emphasizing the details of the magnetization

dynamics and the IV-characteristics of the ϕ0junction.

We have shown that in the limit of small system pa-

rameters; that is, the Josephson to magnetic energy ra-

tio G, the damping α, and the spin-orbit coupling r, the

dynamics is given by the Duﬃng spring [14]. We focus

on the shift in resonance and the eﬀects of nonlinear in-

teractions. We give semi-analytic models to explain our

results in various limits.

The paper is organized as follows. In Section II we

outline the theoretical model and discuss the methods

of calculations. The ferromagnetic resonance and ef-

fects of system parameters on the anomalous damping

dependence are considered in Subsection A of Section

III. In Subsection B we present analytical description of

the dynamics and IV-characteristics of the ϕ0junction

at small system parameters. Manifestation of the nega-

tive diﬀerential resistance in IV-characteristics through

the foldover eﬀect is discussed. We compare the de-

scription of the anomalous damping dependence by full

LLGJ system of equation with approximated equation,

and show how the Duﬃng oscillator captures the non-

linearities in the small parameter regime in Subsection

C. We present results on the critical damping and de-

rive the formula which couples the dc superconducting

current and maximal amplitude of magnetization in the

ferromagnetic layer. Finally, in Section IV we concludes

the paper.

II. II. MODELS AND METHOD

The following section is closely related to our work

in [13]. The ϕ0junction [6, 12, 25] that we study is shown

in Fig.1. The current-phase relation in varphi0junction

has the form Is=Icsin (ϕ−ϕ0), where ϕ0=rMy/M0,

Mydenotes the component of magnetic moment in ˆydi-

rection, M0is the modulus of the magnetization. The

physics of ϕ0Josephson juncton is determined by system

of equations which consists of Landau-Lifshits-Gilbert

(LLG), resistively capacitively shunted junction (RCSJ)

model expression with current-phase relation (Is) de-

scribed above, and Josephson relation between phase dif-

ference and voltage.

The dynamics of the magnetic moment Mis described

by the LLG equation [26]

dt =−γM×Heff +α

M0M×dM

dt ,(1)

where Mis the magnetization vector, γis the gyromag-

netic relation, Hef f is the eﬀective magnetic ﬁeld, αis

Gilbert damping parameter, M0=|M|.

Figure 1. Schematic view of SFS ϕ0Josephson junction. The

external current applied along x direction, ferromagnetic easy

axis is along z direction.

In order to ﬁnd the expression for the eﬀective mag-

netic ﬁeld we have used the model developed in Ref.[6],

where it is assumed that the gradient of the spin-orbit

potential is along the easy axis of magnetization taken to

be along ˆz. In this case the total energy of the system

can be written as

Etot =−Φ0

2πϕI +Es(ϕ, ϕ0) + EM(ϕ0),(2)

where ϕis the phase diﬀerence between the supercon-

ductors across the junction, Iis the external current,

Es(ϕ, ϕ0) = EJ[1 −cos (ϕ−ϕ0)], and EJ= Φ0Ic/2π

is the Josephson energy. Here Φ0is the ﬂux quantum,

Icis the critical current, r=lυso/υFl= 4hL/~υF,L

is the length of Flayer, his the exchange ﬁeld of the

Flayer, EM=−KVM2

z/(2M2

0), the parameter υso/υF

characterizes a relative strength of spin-orbit interaction,

Kis the anisotropic constant, and Vis the volume of the

ferromagnetic (F) layer.

The eﬀective ﬁeld for LLG equation is determined by

Heﬀ =−1

∂Etot

∂M

=ΩF

γGr sin ϕ−rMy

M0b

y+Mz

M0b

z(3)

where ΩF=γK/M0is frequency of ferromagnetic reso-

nance and G=EJ/(KV) determines the ratio of Joseph-

son energy to magnetic one.

In order to describe the full dynamics ϕ0junction the

LLG equations should be supplemented by the equation

for phase diﬀerence ϕ, i.e. equation of RCSJ model for

bias current and Josephson relation for voltage. Accord-

ing to the extended RCSJ model, which takes into ac-

count derivative of ϕ0phase shift, the current ﬂowing

through the system in underdamped case is determined

I=~C

d2ϕ

dt2+~

2eR dϕ

dt −r

dMy

dt (4)

+Icsin ϕ−r

My.

where Iis the bias current, Cand Rare the capacitance

and resistance of Josephson junction respectively. The

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Nonlinearfeaturesofthesuperconductor{ferromagnet{superconductor'0JosephsonjunctioninferromagneticresonanceregionAliasgharJanalizadeh1,IlhomR.Rahmonov2;3;4,SaraA.Abdelmoneim5,YuryM.Shukrinov2;3;4,andMohammadR.Kolahchi11DepartmentofPhysics,InstituteforAdvancedStudiesinBasicSciences(IASBS),P.O.Box45137...

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Nonlinear features of the superconductorferromagnetsuperconductor 0Josephson junction in ferromagnetic resonance region Aliasghar Janalizadeh1 Ilhom R. Rahmonov234 Sara A.

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