One Analytical Approach of Rashba-Edelstein Magnetoresistance in 2D Materials Wibson W. G. Silva and Jos e Holanda

2025-05-02 0 0 349.14KB 12 页 10玖币

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One Analytical Approach of Rashba-Edelstein

Magnetoresistance in 2D Materials

Wibson W. G. Silva and Jos´e Holanda∗

Programa de P´os-Gradua¸c˜ao em Engenharia F´ısica, Universidade Federal Rural de

Pernambuco, 54518-430, Cabo de Santo Agostinho, Pernambuco, Brazil

E-mail: ∗joseholanda.silvajunior@ufrpe.br

Abstract. We study analytically the Rashba-Edelstein magnetoresistance (REMR)

in a structure made from an insulator ferromagnet, such as yttrium iron garnet (YIG),

and a 2D material (2DM) with direct and inverse Rashba-Edelstein eﬀects, such as SLG

and MoS2. Our results represent an eﬃcient way of analyzing the Rashba-Edelstein

eﬀects.

To ArXiv

1. Introduction

2D spintronics has gained an important meaning in data storage technologies, in

many those cases, the 2D materials are non-magnetic, however, magnetism can be

induced at the interface of those materials. Among some methods, two of them are

broadly applied to induce magnetism on 2D material. The ﬁrst method is to introduce

vacancies or adding atoms producing spin polarization [1, 2, 3]. The other one is

to induce magnetism of the adjacent magnetic materials via the magnetic proximity

eﬀect [4, 5, 6, 7]. Recently was discovered that 2D magnetic van der Waals crystals

have intrinsic magnetic ground states at the atomic scale, providing new opportunities

in the ﬁeld of 2D spintronics [8, 9]. Furthermore, it was discovered that several

materials as single-layer graphene (SLG) and molybdenum disulﬁde (MoS2) [10, 11, 12]

can also be used for spin-charge current conversion [13, 14, 15, 16, 17, 18]. Due to

their layered structures, MoS2and SLG can be easily prepared with one or several

atomic layers to explore the transport properties. SLG and semiconducting MoS2

have 2D electronic states that are expected to exhibit remarkable pseudospin and spin-

momentum locking, respectively [10, 19, 20, 21, 22]. These are essential ingredients

arXiv:2210.03854v1 [cond-mat.mtrl-sci] 8 Oct 2022

Wibson W. G. Silva and Jos´e Holanda...................................................................2

for the charge-to-spin current conversion by the direct Rashba-Edelstein eﬀect (REE)

or for spin-to-charge current conversion by the inverse Rashba Edelstein eﬀect (IREE).

Another fundamental ingredient is the broken inversion symmetry at material surfaces

and interfaces [1, 2, 3, 23, 24, 25, 26, 27].

Although the change of electrical resistance of ferromagnets has been studied for

a long time, providing a fundamental understanding of spin-dependent transport in

diﬀerent structures [24, 25, 26], the transport properties of 2D materials still present

themselves as a challenge. One of the most important eﬀects in spin-dependent transport

is the spin Hall magnetoresistance (SMR) [27, 28, 29]. In 3D materials, the SMR is

explained by the spin-current reﬂection and reciprocal spin-charge conversion caused

by the simultaneous action of the spin Hall eﬀect (SHE) [30, 31, 32] and inverse spin

Hall eﬀect (ISHE) [33]. The challenge is to explore the magnetoresistance induced in

2D materials [34, 35]. In this paper, we present a study based on direct and inverse

Rashba-Edelstein eﬀects that describes the magnetoresistance in 2D materials, which is

called of Rashba-Edelstein magnetoresistance (REMR).

2. 2D materiais in contact with a magnetic insulator

The REMR is induced by the simultaneous action of direct and inverse Rashba-Edelstein

eﬀects and therefore a nonequilibrium proximity phenomenon. The magnetoresistance

study was carried out with arrangement as illustrated in Fig. 1 below. The eﬀects

2DM

Figure 1. (online color) Illustration of the sample structure used to study the Rashba-

Edelstein magnetoresistance (REMR).

of the spin current in 2D materials are very important for phenomena of transport.

Considering the Ohm’s law for 2D materials with direct and inverse Rashba-Edelstein

eﬀects and therefore a nonequilibrium proximity phenomenon can be understood by

the relation between thermodynamic driving force and currents that reﬂects Onsager’s

Wibson W. G. Silva and Jos´e Holanda...................................................................3

reciprocity by the symmetry of the response matrix:







JSx

JSy

JSz







R2D







1 ˆx×ˆy×ˆz×

λREE ˆx×1

λREE 0 0

λREE ˆy×01

λREE 0

λREE ˆz×0 0 1

λREE













−∇µC/e

−∇µSx/2e

−∇µSy/2e

−∇µSz /2e







,(1)

where e=|e|is the electron charge, R2Dis the resistance of 2D material, µCis the

charge chemical potential, ~µSis the spin accumulation, ~

JCis the charge current density

and ~

JSis the spin current density. The direct Rashba-Edelstein is represented by the

lower diagonal elements that generate the spin currents in the presence of an applied

current density, which generates an electric ﬁeld, in the following chosen to be in the

ˆxdirection ~

E=Exˆx=−ˆx(∂xµC/e). On the other hand, the inverse Rashba-Edelstein

eﬀect is governed by element above the diagonal that connect the gradients of the spin

accumulations to the charge current density. The spin accumulation ~µSis obtained from

the spin-diﬀusion equation in the 2D materials

∇2~µS=~µS

λ2

,(2)

where λSD is the spin-diﬀusion length. Spin accumulation is always due to spin diﬀusion,

which even for a 2D material such as graphene has spin diﬀusion in the z-direction. For

2D materials with thickness l2Din the ˆxdirection the solution of equation (2) is

~µS(z) = ~pe−z/λ2D+~qez/λ2D,(3)

where the constant column vectors ~p and ~q are determined by the boundary conditions

at the interfaces. According to Eq. (2), the spin current in 2D materials consists of spin

diﬀusion process. For a system homogeneous in the x-y plane, the spin current density

ﬂowing in the ˆzdirection is

S(z) = −1

2eR2DλREE ∂z~µSz −JREE

SO ˆy, (4)

where JREE

SO =Ex/R2DλREE is the bare Rashba-Edelstein current, i. e., the spin current

generated directly by the REE and λREE is the REE length. At the interfaces z=l2D

and z= 0 the boundary conditions demand that ~

S(z) is continuous. The spin current at

z=l2Dinterface vanishes, ~

S(z=l2D) = ~

J2D

S= 0. On the other hand, in general at the

magnetic interface the spin current density ~

JF M

Sis governed by the spin accumulation

and spin-mixing conductance [36], such that:

JF M

S( ˆm) = grˆm× ˆm×~µS

e!+gi ˆm×~µS

e!,(5)

where ˆm= (mx, my, mz)Trepresents a unit vector along the magnetization and

g↑↓ =gr+igithe complex spin-mixing interface conductance per unit length and

resistance. It is agreed that grcharacterizes the eﬃciency of the interfacial spin transport

and the imaginary part gican be interpreted as an eﬀective exchange ﬁeld acting on the

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摘要：

OneAnalyticalApproachofRashba-EdelsteinMagnetoresistancein2DMaterialsWibsonW.G.SilvaandJoseHolandaProgramadePos-Graduac~aoemEngenhariaFsica,UniversidadeFederalRuraldePernambuco,54518-430,CabodeSantoAgostinho,Pernambuco,BrazilE-mail:joseholanda.silvajunior@ufrpe.brAbstract.Westudyanalyticallyt...

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One Analytical Approach of Rashba-Edelstein Magnetoresistance in 2D Materials Wibson W. G. Silva and Jos e Holanda

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