                  Computer simulations of the
          ferromagnetic-antiferromagnetic bilayer system

                                               Sergey V. Belim1,2
                                       1
                                         Omsk State Technical University
                                    11 Mira avenue, 644050, Omsk, Russia
                                          2
                                            Siberian State Automobile
                                             and Highway University
                                     5 Mira avenue, 644080, Omsk, Russia
                                                 sbelim@mail.ru


                                                       Abstract
                      Computer simulation of bilayer spin system is performed. The ﬁrst
                      layer is a ferromagnetic ﬁlm. The second layer is an antiferromagnetic
                      ﬁlm. The exchange interaction acts between the ﬁlms. The Ising model
                      is used for modeling. The Wolf cluster algorithm is used for calcula-
                      tions. The dependence of the bias ﬁeld on the exchange interaction
                      between the ﬁlms is determined. The dependence is the linear.


1    Introduction
Computer modeling of spin systems is actively used to identify patterns in the magnetic substances behavior.
Bilayer systems consisting of ferromagnetic and antiferromagnetic ﬁlms are actively used in spintronics devices.
The main property of such systems is to magnetize the ferromagnetic ﬁlm with an antiferromagnetic ﬁlm [1, 2, 3].
This bias is based on the exchange interaction at the boundary of the two ﬁlms. Bias allows you to ﬁx the state
of the ferromagnet. Layered antiferromagnets are actively used in this role [4].
   The displacement of the ferromagnetic ﬁlm hysteresis loop is due to the presence of bias [5, 6, 7, 8]. The
oﬀset value is called the bias ﬁeld [9, 10, 11, 12]. Experimental exchange bias found in compounds F eF2 /F e and
M nF2 /F e [13, 14, 15]. In this article, computer simulation of exchange bias in a two-layer FM/AFM structure
within the Ising model is performed. Research carried out by computer modeling, which has proven itself well
in similar tasks [16, 17, 18].

2    System model
The bilayer system is modeled as a cellular automat. Each cell can take one of two values +1/2 or −1/2. The
state of the cell number i denotes Si . State of cell corresponds to value of spin in real magnetic substance. Such
systems correspond to the Ising model. We’re looking at a rectangular lattice with two layers. The dimensions
of each layer along the OX and OY axes are L. L is the number of cells arranged in one direction. Along the
OZ axis, both ﬁlms have a thickness d of cells. The system is considered a thin ﬁlm if inequality d ≪ L is
performed. Periodic boundary conditions are used along the OX and OY axes to simulate endless ﬁlms. The
system geometry is shown in Figure 1.
Copyright ⃝
          c by the paper’s authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
In: Sergei S. Goncharov, Yuri G. Evtushenko (eds.): Proceedings of the Workshop on Applied Mathematics and Fundamental
Computer Science 2020 , Omsk, Russia, 30-04-2020, published at http://ceur-ws.org


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                                           Figure 1: The system geometry.
   Each cell communicates only with its nearest neighbors. Two ﬁlms belong to diﬀerent types of magnetic
substances. The amount of cell interaction is diﬀerent. In the ﬁrst ﬁlm, the amount of interaction is indicated
by Ja (Ja > 0). Cells of one layer interact with energy −Ja . Cells in adjacent layers interact with the value Ja .
Cells of one layer are energetically advantageous to have the same value. Cells of adjacent layers are energetically
advantageous to have opposite values. Such model corresponds to layered antiferromagnets. In the second ﬁlm,
all cells interact with the same value −J0 . Cells of the second layer are energetically advantageous to have the
same values.Such model corresponds to ferromagnetic materials. Between cells at the boundary of two ﬁlms,
interaction occurs with energy −J. It is energetically advantageous to have the same meanings. Complete
ordering of the system is hindered by the thermal molecules movement. Thermal motion is modeled by random
forces. It puts cells at an energy disadvantage. Intensity of thermal motion is determined by temperature t.
   For computer modeling of the system, it is necessary to deﬁne its Hamiltonian.
                                 ∑                        ∑                    ∑                    ∑
                       H = Ja           (−1)b Si Sj − J         Si Sj − J 0            Si Sj + hv       Si .
                                0≤z<d                     z=d                 d≤z<2d


   Only the nearest neighbors are summed. The external magnetic ﬁeld hv is also taken into account here. b = 1
if Si and Sj are in the same layer. b = 0 if Si and Sj are located in adjacent layers.
   We use relative values in computer modeling.

                                        Ra = Ja /J0 , R = J/J0 , h = hv /J0 .

  We write Hamiltonian using relative quantities.
                                    ∑              ∑         ∑           ∑
                       H/J0 = Ra         Si Sj − R   Si Sj −   Si Sj + h   Si .
                                        0≤z<d              z=d           d≤z<2d


  We introduce a relative temperature.

                                                      T = kt/J0 .

   k is Boltzmann’s constant.
   We performed a computer simulation for the two thin ﬁlms system with dimensions L × L × d. Computer
modeling was carried out using the Wolf cluster algorithm.
   The order parameter is entered to describe the collective behavior of the system. Diﬀerent order parameters
are used for diﬀerent ﬁlms. The average spin value m is used for the ferromagnetic ﬁlm.


                                                                2
                                                              
                                                     ∑           /
                                           m=               Si  (L2 d).
                                                  d≤z<2d


    The diﬀerence in the spins of the even and odd layers was used for the antiferromagnetic ﬁlm.
                                                                          
                                           ∑                  ∑              /
                             ma =                  Si −                Si  (L2 d).
                                    0≤z<d,z mod2=0       0≤z<d,z mod2=1


   The phase transition can occur in the system when the temperature changes. During the phase transition, spin
values are ordered. Fourth order Binder cummulants [19] are used to determine the phase transition temperature.

                                                ⟨m4 ⟩               ⟨m4a ⟩
                                      U =1−             , Ua = 1 −          .
                                               3⟨m ⟩
                                                  2   2            3⟨m2a ⟩2

   Averaging by system states is indicated by angle brackets. Binder cummulants depend on temperature.
Finite dimensional scaling theory[19] shows that Binder cummulants of systems with diﬀerent sizes intersect at
one point. The crossing point corresponds to the phase transition temperature. Phase transition temperatures
vary from ﬁlm to ﬁlm. The phase transition temperature depends on the intensity of the interaction between
the spins.
   The computer modeling purpose is to determine the eﬀect of the antiferromagnetic ﬁlm on the ferromagnetic
ﬁlm. The antiferromagnetic ﬁlm can be replaced by an external magnetic ﬁeld. This ﬁeld is called the bias ﬁeld.
The task is to determine the dependence of the bias ﬁeld on the interaction intensity at the ﬁlm boundary.

3    Results of computer simulation
Systems with a linear size from L = 16 to L = 32 with a step of ∆L = 4 were investigated in a computer
experiment. We considered ﬁlms with thickness d = 4. The antiferromagnetic ﬁlm had a parameter Ra = 2.0.
The temperature of the phase transitions was determined in the ﬁrst experiment. Phase transition temperatures
TC = 3.51 and TN = 7.02 are obtained for ferromagnetic and antiferromagnetic ﬁlms respectively. Temperature
T = 4.0 is selected for further experiments. The antiferromagnetic ﬁlm is ordered and the ferromagnetic ﬁlm is
disordered at this temperature.
   The isolated ferromagnetic ﬁlm in the external magnetic ﬁeld was simulated in the ﬁrst experiment. The
magnetic ﬁeld varied from h = 0.0 to h = 2.0 in increments ∆h = 0.1. Dependence of ferromagnetic ﬁlm order
parameter on external magnetic ﬁeld is shown in Figure 2.
   The bilayer system in the zero external magnetic ﬁeld was simulated in the second experiment. Interaction at
the boundary of two ﬁlms varied from R = 1.0 to R = 2.0 in increments ∆R = 0.1. Dependence of ferromagnetic
ﬁlm order parameter on interaction value at ﬁlm boundary R shown in Figure 3.
   The hypothesis about the linear dependence the bias ﬁeld Hbias on the interaction at the border of ﬁlms R
was tested.

                                                    Hbias = bR.

   We applied the least squares method to determine the coeﬃcient a. We obtained the values of these values
from the computer experiment results.

                                           a = 0.23,    Hbias = 0.23R.

4    Conclusion
The computer experiment was performed for the study the bilayer systems. It has been shown that the eﬀect
of the antiferromagnetic layer on the ferromagnetic layer is similar to the external magnetic bias ﬁeld. The
dependence of the bias ﬁeld on the amount of interaction at the layer boundary is obtained. Dependency is
linear.


                                                         3
                     0,8


                     0,7


                     0,6


                     0,5
              m

                     0,4


                     0,3


                     0,2


                     0,1


                     0,0

                           0,0         0,5            1,0           1,5            2,0


                                                         H


           Figure 2: Dependence of ferromagnetic ﬁlm order parameter on external magnetic ﬁeld
5   Acknowledgements
The reported study was funded by RFBR, project number 20-07-00053.

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                                                     4
                    0,50


                    0,45


                    0,40


                    0,35


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              m


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