=Paper=
{{Paper
|id=Vol-2642/paper5
|storemode=property
|title=Computer simulations of the ferromagnetic-antiferromagnetic bilayer system
|pdfUrl=https://ceur-ws.org/Vol-2642/paper5.pdf
|volume=Vol-2642
|authors=Sergey V. Belim
}}
==Computer simulations of the ferromagnetic-antiferromagnetic bilayer system==
<pdf width="1500px">https://ceur-ws.org/Vol-2642/paper5.pdf</pdf>
<pre>
                  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




                                                            1
                                           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



                    0,30
              m


                    0,25



                    0,20



                    0,15



                    0,10



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


                                                         R




    Figure 3: Dependence of ferromagnetic film order parameter on interaction value at film boundary R
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