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Non-equilibrium phenomena in magnetic multilayer nanostructures and

aging in magnetoresistance

To cite this article: M V Mamonova et al 2021 J. Phys.: Conf. Ser. 1740 012006

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Journal of Physics: Conference Series 1740 (2021) 012006

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doi:10.1088/1742-6596/1740/1/012006

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Non-equilibrium phenomena in magnetic multilayer

nanostructures and aging in magnetoresistance

M V Mamonova†, P V Prudnikov‡and V V Prudnikov§

Dostoevsky Omsk State University, Pr. Mira 55A, Omsk 644077, Russia

E-mail: †mamonova mv@mail.ru, ‡prudnikovpv@omsu.ru, §prudnikv@mail.ru

Abstract. A Monte Carlo simulation of the non-equilibrium behavior of multilayer magnetic

structures Co/Cu(100)/Co and Pt/Co/Cu(100)/Co/Pt characterizing diﬀerent types of

magnetic anisotropy is realized. Simulation of transport properties gives possibility to reveal

a nontrivial aging eﬀects in the magnetoresistance of these structures and inﬂuence of initial

states on two-time dependence of magnetoresistance.

The artiﬁcial created multilayer magnetic superlattices has become of great interest for wide

range of applications based on the phenomena of the giant magnetoresistance (GMR) and the

tunneling magnetoresistance. Devices based on the GMR eﬀect are widely used as read heads

of hard disks, memory devices, sensors, etc [1, 2]. The magnetic properties of ultrathin ﬁlms

and superstructures are sensitive to the eﬀects of anisotropy generated by the crystal ﬁeld of a

substrate or nonmagnetic layers. The multilayer magnetic structure Co/Cu(100)/Co extensively

usable in active elements of spintronic devices is characterized by anisotropy of ”easy” magnetic

plane type with magnetization oriented in plane of cobalt ﬁlm. The structure Pt/Co/Cu/Co/Pt

with cobalt ﬁlms coated by ultrathin platinum ﬁlms is characterized already by anisotropy of

”easy” magnetic axis type with magnetization oriented perpendicularly to plane of cobalt ﬁlm.

As it has been shown in [3], Pt/Co bilayer possess giant energy of magnetic anisotropy and high

Curie temperatures attaining 500 K in ultrathin ﬁlms. Combination of high Curie temperature

in cobalt ﬁlms and perpendicular magnetic anisotropy generated in Pt/Co bilayer makes possible

to increase signiﬁcantly magnetoresistance in Pt/Co/Cu/Co/Pt structure in comparison with

Co/Cu/Co structure [4].

The nanoscale periodicity in magnetic multilayer structures gives rise to the mesoscopic

eﬀects of the strong spatial spin correlation with the slow relaxation dynamics of

magnetization accompanying the quenching of the system in the non-equilibrium state. The

experimental investigations of relaxation [5] revealed magnetic aging in a Co/Cr-based magnetic

superstructure. We have performed in [6, 7, 8, 9] a numerical Monte Carlo simulation of the

non-equilibrium behavior of the multilayer Co/Cr/Co and Co/Cu/Co magnetic structures and

revealed the aging eﬀects, which are characterized by slowing down of correlation and relaxation

processes with an increase of a waiting time tw. In contrast to the bulk magnetic systems, where

the slow dynamics and aging eﬀects manifest themselves near the critical point [10], the aging

in magnetic superstructures is occurred within a wide range of temperatures at T≤Tc.

The non-equilibrium behavior of a system is realized via its transition at the starting instant t0

from the initial state at temperature T0to the state with temperature Tsdiﬀering from T0. The

evolution of systems with slow dynamics depends on its initial state for times ttrel(Ts), where

trel(Ts) is a relaxation time at temperature Ts. Various initial states exert noticeable inﬂuence

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on time dependence of characteristic functions in systems with slow dynamics [10, 11, 12]. In

this connection, the non-equilibrium behavior of the system depends on whether it evolves from

a high-temperature T0> Tsor a low temperature T0< Tsinitial state.

In this paper, a Monte Carlo simulation of the non-equilibrium behavior of multilayer

magnetic structures Co/Cu(100)/Co and Pt/Co/Cu(100)/Co/Pt characterizing diﬀerent types

of magnetic anisotropy is carried out. We study manifestation of non-equilibrium behavior of

these structures in aging properties of their magnetoresistance. We plan to reveal inﬂuence of

initial states on two-time dependence of the magnetoresistance in nanostructures with diﬀerent

thicknesses of ferromagnetic ﬁlms.

We realize in this work a Monte Carlo study of the non-equilibrium behavior of a multilayer

magnetic structure (Fig. 1 a) consisting of ferromagnetic ﬁlms separated by nonmagnetic metal

layer.

Figure 1. The model of the multilayer structure (a) consisting of two ferromagnetic ﬁlms separated

by a nonmagnetic metal ﬁlm. Land Nare the linear sizes of the ﬁlms; J1and J2are the exchange

integrals. Dependence of the anisotropy parameter ∆(N) on the thickness of the ﬁlm Nin ML’s (b). The

circles and squares correspond to experimental data for Ni/Cu(001) Co/Cu(001) [13]. The diamonds

correspond to experimental data for Ni(111)/W(110) [14]. The solid curve is obtained by approximation

of experimental data.

We consider the array consisting of ferromagnetic ﬁlms with the thicknesses N= 3, 5, 7,

9 in units of monatomic layers (ML). The exchange integral J1determining the interaction

between the neighboring spins is assumed to be J1/kBT= 1, whereas that for the interlayer

interaction is J2=−0.1J1. The sign of J2is negative because the thickness of nonmagnetic

spacers in multilayer structures exhibiting the giant magnetoresistance eﬀect is tuned such

that the interlayer exchange interaction eﬀectively provides antiferromagnetism. Owing to

this interaction, the magnetization of the neighboring ferromagnetic layers have opposite

orientations.

The physical properties of ultrathin ﬁlms based on Fe, Co, and Ni can be described by the

anisotropic Heisenberg model [15, 16] with the Hamiltonian given by the expression

H=−X

<i,j>

Jij [(Sx

iSx

j+Sy

iSy

j) + (1 −∆1(N))Sz

iSz

j],(1)

which corresponds to Co/Cu(100)/Co structure with the in plane magnetization. The

Hamiltonian in the form

H=−X

<i,j>

Jij [(1 −∆2(N))(Sx

iSx

j+Sy

iSy

j) + Sz

iSz

j] (2)

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corresponds to Pt/Co/Cu(100)/Co/Pt structure with the out of plane magnetization. Spin

~

Si= (Sx

i, Sy

i, Sz

i) is determined as the classical unit vector at the i-th site of a face-centered

cubic (fcc) lattice for Co ﬁlms. ∆1,2(N) are parameters characterizing the eﬀective inﬂuence of

anisotropy generated by the crystal ﬁeld of the Cu(100) substrate on magnetic properties of Co

ﬁlm subject to its thickness N. The dependence of the anisotropy parameter ∆1,2(N) on the

ﬁlm thickness Nis presented in Fig. 1 b.

At the beginning, we calculated the equilibrium characteristics of the multilayer magnetic

structure as the magnetization of the ﬁlms m1,2with the aim to determine the critical

temperatures Tc(N) of the ferromagnetic phase transition in ﬁlms with diﬀerent thicknesses N.

For most accurate determination of the critical temperatures we used the method of intersection

of curves for temperature dependencies of the Binder cumulant U4(N, T , L) for structures with

ﬁlms of diﬀerent linear sizes L= 24, 36, and 48. Metropolis algorithm was used for updating spin

conﬁgurations. During simulation, 105MCS/s were discarded for equilibration of spin system,

and then measured equilibrium quantities are averaged over 105MCS/s with 500 runs.

We determined for magnetic structures with ﬁlm thicknesses N= 3, 5, 7, and 9 ML the

following values of magnetic ordering temperatures: for structures with anisotropy characterized

by the in plane magnetization Tc(N= 3) = 2.3108(22), Tc(N= 5) = 2.7342(21), Tc(N=

7) = 2.9072(26), and Tc(N=9)=3.0020(6) and for structures with the out of plane

magnetization Tc(N= 3) = 2.5590(14), Tc(N= 5) = 3.0340(15), Tc(N= 7) = 3.1820(13),

and Tc(N= 9) = 3.2784(15).

Simulation of transport properties in Co/Cu/Co and Pt/Co/Cu/Co/Pt structures with

current perpendicular to plane (CPP) using methodology [17, 18] have permitted in [4] to

calculate temperature dependence of their equilibrium CPP-magnetoresistance values with

demonstration that magnetoresistance in Pt/Co/Cu/Co/Pt structures is higher than in

Co/Cu/Co structures with the same thickness N. We have used for calculation of the CPP

magnetoresistance the two-current Mott model to describe the resistance of diﬀerent conduction

channels [19]. It was introduced the resistance of an ferromagnetic ﬁlm for two groups of electrons

with spins up R↑and spin down R↓. As a result, the magnetoresistance of the multilayer

structure is determined by the relation:

δ=(R↑−R↓)2

4R↑R↓

=(J↑−J↓)2

4J↑J↓

,(3)

where J↑,↓=en↑,↓hV↑,↓iis the current density. Here, n↑,↓is the density of electrons with x

(or z) components of spin moment equal to +1/2 and −1/2 (axis xis the quantization axis

determined by orientation of magnetization in plane of ﬁlms for Co/Cu(100)/Co structure and

the quantization axis zfor Pt/Co/Cu(100)/Co/Pt structure with out of ﬁlm plane orientation of

the magnetization), n=n↑+n↓is the total electron density and hV↑,↓iare the averaged velocities

of electrons with corresponding spin projections. The electron densities with spin up and down

can be expressed through the magnetization of the ﬁlm n↑,↓/n = (1 ±m)/2 determined in the

process of the Monte Carlo simulation of magnetic properties of the structure. The averaged

electron velocity hV↑,↓ican be expressed through the electron mobility and the external electric

ﬁeld intensity E, and after that through the probability of electron displacement in unit time

(corresponding to one Monte Carlo step per spin) from unit cell ito a neighbouring unit cell in

the direction of the electric ﬁeld with averaging over all ﬁlm unit cells:

hV↑,↓i=µ↑,↓E=e

TEexp −∆Ei,↑,↓

T,(4)

where µis the electron mobility and ∆Eicharacterizes the change of system energy connected

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with electron jump from i-cell to a neighbouring cell. Ei,↑,↓is determined by relation

Ei,↑,↓=∓J1X

j6=i

Sx

j(nj,↑−nj,↓) + Sx

i(ni,↑−ni,↓)(5)

in case of magnetization orientation in plane of cobalt ﬁlms for Co/Cu/Co structure and following

relation

Ei,↑,↓=∓J1X

j6=i

Sz

j(nj,↑−nj,↓) + Sz

i(ni,↑−ni,↓)(6)

for Pt/Co/Cu/Co/Pt structure with the magnetization oriented by perpendicularly to plane of

ferromagnetic ﬁlms.

At the next stage of this work, we study of inﬂuence of non-equilibrium behavior of the

multilayer magnetic structures on their magnetoresitance with realization of evolution from

both high-temperature and low-temperature initial states. We calculate two-time dependence of

the magnetoresistance δ(t, tw) on observation time t−twand waiting time tw. The waiting time

twcharacterizes the time between a sample preparation in non-equilibrium initial state and the

beginning of measurement of its magnetoresistance.

Figure 2. Time dependence of the CPP-magnetoresistance in Co/Cu(100)/Co (a) and

Pt/Co/Cu(100)/Co/Pt (b) with the thicknesses N= 5 −9 ML’s of the cobalt ﬁlms at temperatures

Ts=Tc(N)/4 for diﬀerent waiting times tw= 50, 100, 200, 400 and 1000 MCS/s with evolution from

the low-temperature initial state with T0= 0.

As an example, we present in Fig. 2 calculated time dependence of the magnetoresistance

δ(t, tw) in Co/Cu(100)/Co and Pt/Co/Cu(100)/Co/Pt structures with the cobalt ﬁlm

thicknesses N= 3, 5, 7, and 9 ML on observation time t−twwith evolution from the low-

temperature completely ordered initial state with T0= 0 at temperatures Ts=Tc(N)/4. Values

of the magnetoresistance δ(t, tw) were averaged over 250 runs for N= 3 ML and N= 5 ML and

500 runs for N= 7 ML and 9 ML. The magnetoresistance demonstrates dependence on waiting

time twas general criterion of aging and that δ(t, tw) reaches a plateau with asymptotical values

δ∞(N, T ), which depend on thickness Nof cobalt ﬁlms, temperature and type of anisotropy

in ferromagnetic ﬁlms. So, values δ∞(N, T ) are higher for structures Pt/Co/Cu/Co/Pt with

easy-axis anisotropy than for structures Co/Cu/Co with easy-plane anisotropy and with the

same thickness Nof cobalt ﬁlms. As can be seen from Fig. 2, diﬀerence of values δ∞(N, T ) for

Pt/Co/Cu and Co/Cu grows up with increase of cobalt ﬁlm thickness N. Also, it was revealed

that values of δ∞(N, T ) obtained for case of evolution of system from the low-temperature initial

state agree very well with equilibrium values of the magnetoresistance δ(eq)(N, T ).

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Figure 3. Time dependence of the magnetoresistance in Co/Cu/Co (a) and Pt/Co/Cu/Co/Pt (b)

with diﬀerent thicknesses Nof the cobalt ﬁlms at temperatures Ts=Tc(N)/4 with evolution from the

high-temperature initial state with T0Tc(N).

We demonstrate in Fig. 3 calculated time dependence of the magnetoresistance δ(t, tw)

in Co/Cu(100)/Co and Pt/Co/Cu(100)/Co/Pt structures with evolution from the high-

temperature completely disordered initial state with T0Tc(N) at temperatures Ts=Tc(N)/4.

Comparison of obtained curves for δ(t, tw) in Fig’s. 2 and 3 shows that asymptotical values

δ∞(N, T ) on plateau for the low-temperature completely ordered initial state are higher than

values δ∞(N, T ) for case of evolution from the high-temperature completely disordered initial

state. Therefore, values δ∞(N, T ) for the high-temperature initial state diﬀer from equilibrium

values of the magnetoresistance and lower these values. The comparison of δ(t, tw) in Fig’s. 2

and 3 shows that the time dependence of magnetoresistance in Co/Cu structures with easy-plane

anisotropy reaches a plateau for times about 1 000 - 3 000 MCS/s while in Pt/Co/Cu structures

with easy-axis anisotropy for longer times about 3 000 - 6 000 MCS/s.

Figure 4. Time dependence of the magnetoresistance in Pt/Co/Cu/Co/Pt with thickness N= 5 ML

of the Co ﬁlms at temperature Ts=Tc(N= 5)/4'241.8 K with evolution from (a) the intermediary

initial states with T(ht)

0= 3Tc(N= 5)/8'362.7 K and T(lt)

0=Tc(N= 5)/8'120.8 K and (b) extreme

initial states with T(ht)

0Tc(N= 5) and T(lt)

0= 0.

Also, we considered inﬂuence of intermediary initial states with 0 < T0< Tcon time

dependence of the magnetoresistance. As an example, we present in Fig. 4 calculated δ(t, tw)

in Pt/Co/Cu(100)/Co/Pt structure with the cobalt ﬁlm thicknesses N= 5 ML with initial

temperatures T(ht)

0= 3Tc(N= 5)/8'362.7 K and T(lt)

0=Tc(N= 5)/8'120.8 K realized

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at quenched temperature Ts=Tc(N= 5)/4'241.8 K. Note that the temperature scale was

deﬁned through the value of the exchange integral J1= 4.4·10−14 erg corresponding to the bulk

cobalt. This value of J1is calculated with the use of the well known mean-ﬁeld approximation.

These initial temperatures T(ht)

0and T(lt)

0are the high-temperature and low-temperature states,

consequently, in relation to quenched temperature Ts. Also, we inserted in Fig. 4 for comparison

curves of δ(t, tw) for extreme initial temperatures T0Tc(N= 5) and T0= 0. We can see

that asymptotical values of the magnetoresistance δ∞on plateau are characterized by sequenced

increase of δ∞from values for T(ht)

0Tcand Ts< T (ht)

0< Tcto 0 < T (lt)

0< Tsand T(lt)

0= 0.

We connect these eﬀects with inﬂuence of the eﬀective temperature Teﬀ =T/X∞, where X∞

is the asymptotic value of the ﬂuctuation-dissipation ratio (FDR) [20]. Non-equilibrium critical

dynamics of the most statistical model systems is characterized by X∞<1 [10]. Values of

X∞in the multilayer magnetic structures are unknown for temperatures Ts≤Tc, but we can

use information about temperature dependence of the FDR with X∞(T)<1 and Teﬀ (T)> T

obtained in paper [21] for the 2D XY model. Some community of non-equilibrium properties

of the 2D XY model and the multilayer nanostructures permits to declare that Teﬀ (Ts)> Ts

and, consequently, values of the magnetoresistance on plateau δ∞(N, Teﬀ ) must be less than

equilibrium value of the magnetoresistance for Ts< Teﬀ.

Realized in the present paper Monte Carlo study of the non-equilibrium behavior of

Co/Cu(100)/Co and Pt/Co/Cu(100)/Co/Pt nanostructures has revealed nontrivial aging

eﬀects in the magnetoresistance δ(t, tw) and signiﬁcant inﬂuence of initial states on the

magnetoresistance. It has been shown that the magnetoresistance reaches plateau in

asymptotical long-time regime with values δ∞(N, T ), which depend on type of initial state,

thickness of cobalt ﬁlms, temperature and type of magnetic anisotropy in nanostructures.

Acknowledgments

This work was supported by the Russian Foundation for Basic Research, project No. 20-32-

70189, by the Ministry of Education and Science of Russian Federation in the framework of

the state assignment No. 0741-2020-0002, and the Council for Grants of the President of the

Russian Federation, project No. MD-2229.2020.2.

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