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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.
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Journal of Physics: Conference Series
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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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CSP2020
Journal of Physics: Conference Series 1740 (2021) 012006
IOP Publishing
doi:10.1088/1742-6596/1740/1/012006
1
Non-equilibrium phenomena in magnetic multilayer
nanostructures and aging in magnetoresistance
M V Mamonova, P V Prudnikovand 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 TTc.
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 Rand spin down R. As a result, the magnetoresistance of the multilayer
structure is determined by the relation:
δ=(RR)2
4RR
=(JJ)2
4JJ
,(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+nis 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 ttwand 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 ttwwith 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= 4.4·1014 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 δon plateau are characterized by sequenced
increase of δfrom 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
Xin the multilayer magnetic structures are unknown for temperatures TsTc, 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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A Monte Carlo simulation of the non-equilibrium behavior of multilayer magnetic nanostructure Co/Cu(100)/Co consisting of alternating magnetic and nonmagnetic nanolayers is carried out. Analysis of calculated two-time autocorrelation function for structure relaxing from both high-temperature and low-temperature initial states reveals aging characterized by a slowing down of correlation characteristics with increase of the waiting time. It is shown that, in contrast to bulk magnetic systems, the aging effects in nanostructure arise not only at the ferromagnetic ordering temperature T c but also within a wide temperature range at T ≤ T c . For evolution from high-temperature initial state, the study of dependence of aging characteristics on thickness N of cobalt films reveals a weakening of the aging with increasing N at the critical temperatures T c (N) and an opposite tendency at temperatures T < T c (N) with strengthening of aging with increasing N for considered N ≤ 9 ML. This phenomenon is connected with increasing correlation and relaxation times in nanostructures when temperature is decreased. For case of the low-temperature initial state, it is shown that correlation times are two-three orders of magnitude smaller than those in the evolution from a high-temperature initial state at the same t w values. In this case, time behavior of the autocorrelation function doesn't depend considerably on temperature for T s ≤ T c and thickness N of cobalt films. Simulation of transport properties in Co/Cu(100)/Co structure permitted to calculate temperature dependence of its equilibrium magnetoresistance values. For the first time, it was revealed influence of non-equilibrium behavior on the magnetoresistance with demonstration of nontrivial aging effects. It has been shown that the magnetoresistance reaches plateau in asymptotic long-time regime with values , which depend on type of initial state, thickness of cobalt films, and temperature.
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