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Micromagnetic simulation of an antiferromagnetic particle
Computational Materials Science 97 (2015) 42–47
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Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Micromagnetic simulation of an antiferromagnetic particle N. Ntallis, K.G. Efthimiadis ⇑ Department of Physics, Aristotle University, 54124 Thessaloniki, Greece
a r t i c l e
i n f o
Article history: Received 25 July 2014 Received in revised form 5 October 2014 Accepted 7 October 2014
Keywords: Micromagnetic simulation Finite element method Antiferromagnet
a b s t r a c t A continuum micromagnetic model is derived, describing an antiferromagnet. Using the finite element method, magnetization curves are calculated for a spherical uniaxial particle, varying the particle’s size and the anisotropy field strength. Different magnetization processes appear by increasing the size of the particle. For large particles nucleation and expansion of a reversed domain is observed, separated by an almost 90° wall. An estimation of the single domain radius Rc is made. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction
2. The simulation model
Antiferromagnetic nanoparticles are of great interest for new composite magnetic materials [1–6], which can be used in applications like spin valves [7,8] and random access memories (MRAM) [5,9], so examining the magnetic behavior of antiferromagnets is of great importance. Micromagnetism is a widely used theoretical model, due to its capability of predicting the magnetic behavior of a continuous medium. According to this model, the magnetization vector is treated as a continuous function in space [10]. The Finite Difference Method (FDM) [11,12] and the Finite Element Method (FEM) [13–15], both with their drawbacks and advantages, are the two basic techniques in computational micromagnetism. FEM has the main advantage that it can handle easily large particles and complex geometries. Extensive work has been done, by means of FEM, on the coupling of an antiferromagnet with a ferromagnet [16–19], but not for the antiferromagnet itself. This paper is organized as follows. First, a continuous micromagnetic model is presented for a magnetic system consisting of two sublattices. Then, in order to test the validity of the model, simulation, under the absence of dipole interactions, of a spherical uniaxial antiferromagnetic particle by FEM is curried out, in order to be compared with known analytical calculations. After the validation, magnetization curves are simulated, under dipole field interactions, while varying the size and the anisotropy constant of the particle.
An antiferromagnet is composed of two sublattices (A and B) of equal magnetization norm, Ms and is characterized by a negative exchange integral between them. In an atomistic model, for each atom a unique magnetic moment is defined. In a continuous approximation, as the micromagnetic model, the magnetization vector is considered a continuous function in space, so the magne! ! tization of each sublattice, M A and M B must be defined at every point in space. The evolution of the magnetization vectors is governed by the damped Partial Differential Equations (PDE) [20]
⇑ Corresponding author. Tel.: +30 2310998065. E-mail address: [email protected] (K.G. Efthimiadis). URL: http://users.auth.gr/kge (K.G. Efthimiadis). http://dx.doi.org/10.1016/j.commatsci.2014.10.010 0927-0256/Ó 2014 Elsevier B.V. All rights reserved.
s
! ! ! @M A 1 ! ¼ 2 M A H Aeff M A and @t Ms ! ! 1 ! ¼ 2 M B H Beff M B ; Ms
s
! @M B @t
! where H Ieff is the effective field acting on sublattice I ¼ fA; Bg and can be found from the variational derivative of the micromagnetic energy with respect to the magnetic polarization [10]. The main contributions are the exchange, magnetocrystalline, dipole, and Zeeman energies. Summing up for all the above contributions, an ! effective field H Ieff can be defined at every point in space as
!I ! ! ! ! H eff ¼ H Iex þ H Ik þ H d þ H : !I H ex is the exchange field acting on sublattice I. The exchange energy is described by a Heisenberg Hamiltonian
Ex ¼
1 X ! ! J Si Sj ; 2 i;j ij
N. Ntallis, K.G. Efthimiadis / Computational Materials Science 97 (2015) 42–47
where i runs for all atoms in the particle and j for the first neighbors of each atom. In the micromagnetic model discrete atoms cannot be defined. As already mentioned, at each position ~ r i inside the particle the magnetization of each sublattice is defined, and in the neighborhood ~ rj around this position, each magnetization component can be approximated by a Taylor expansion as
M Ik ð~ rj Þ
¼
M Ik ð~ ri Þ
þ ð~ rj ~ ri Þ r
M Ik ð~ ri Þ
2 o ~ rj ~ ri n 2 I þ r Mk ð~ ri Þ ; 2
k ¼ fx; y; zg: For a symmetric distribution of neighborhood atoms the first order terms vanish. For an isotropic one, the off diagonal second order terms vanish too. Under these assumptions, the exchange energy takes the form: Eex ¼
z
Z
2M2s v
2 6 4
n ! o n ! o 3 ! ! ! ! 2 2 J AA M A M A þ a2 r2 M A þ J BB M B M B þ a2 r2 M B þ 7 n ! o n ! o 5dV; ! ! ! ! 2 2 þJ AB M A M B þ a2 r2 M B þ J BA M B M A þ a2 r2 M A
where the integration takes place in the particle’s volume, v is the atomic volume, z is the number of the first neighbors and a is the first neighbors distance. J AB ¼ JBA are the antiferromagnetic exchange integrals and JAA ¼ J BB are the ferromagnetic ones. The latter interactions must be considered seriously and cannot be omitted, because in a continuum approximation there is no validation of the magnetization norm conservation rule. The terms of the form !I !I M M can be omitted, because they contribute as constants in the energy expression. The exchange field for the two sublattices is
n ! o n !o !A ! H ex ¼ kM B þ ‘2AA r2 M A ‘2AB r2 M B and n !o n ! o !B ! H ex ¼ kM A þ ‘2BB r2 M B ‘2BA r2 M A ; 2lo M2s v is a negative dimensionless mean field rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi constant, ‘AA ¼ ‘BB ¼ a zJ AA = 4lo M2s v is a ferromagnetic exchange rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi length and ‘AB ¼ ‘BA ¼ a zjJAB j= 4lo M2s v is an antiferromagnetic where k ¼ zJ AB
.
exchange length. In order for the antiferromagnetic order to exist, the two sublattices must be ferromagnetically stable, so the antiferromagnetic exchange length cannot be larger than the ferromagnetic one. In the limiting case, which was used in the following simulation results, the two lengths have the same value ‘x . Assuming a uniaxial anisotropy for each sublattice, with different strength and orientation, the magnetocrystalline energy can be written as
Ek ¼
Z
2
4K A 1
!
43
!
r2 / ¼ r ðM A þ M B Þ: The Zeeman energy is written as
Ez ¼ lo
Z
! !A !B H ðM þ M ÞdV
!
where H is the external applied field. The governing equations are supplemented by the boundary ! ! ^ ¼ 0 and @M B =@ n ^ ¼ 0, where n ^ represents the conditions @M A =@ n normal unit vector on the surface of the particle. The six PDE for the magnetization components and the one for the magnetic scalar potential are solved simultaneously through the FEM by directly applying the weak form, which is derived by the Galerkin method. The Poisson equation is solved by applying a mapped infinite element scheme [22,23], thus an outer domain surrounding the particle is used. Magnetization is interpolated using 3rd order Lagrange elements in the domain of a magnetic particle. For the magnetic potential, 2nd order Lagrange elements have been used, both on the magnetic particle domain and on the surrounding outer domain. The time integration was performed by a variable step size and order BDF method [24] and the algebraic system of linear equations was solved with PARDISO [25]. More information about the method used can been found in [20]. 3. Simulation results Magnetization curves were calculated for a spherical uniaxial antiferromagnetic particle with the above-mentioned magnetic parameters for variable radius and anisotropy constant. The exter^ , which coinnal field was set parallel to the easy axis direction u cides with x-axis, and was varied from zero to the maximum value in 106 s. As initial condition at all simulations the magnetiza! ! ^ and M B ¼ M s u ^, tion of the two sublattices was set to M A ¼ Ms u resulting in a zero net magnetic moment. In order to check the validity of the model, simulations were made under the absence of dipole field interactions. In this context, analytic atomistic models describe magnetization process by spin flip and spin flop transitions [26,27]. Fig. 1 shows calculated magnetization curves for spin flop (Fig. 1(left)) and spin flip (Fig. 1(right)) regions. In the first case at spin flop field, Hsf , the magnetization vectors of the two lattices jump coherently to opposite angles (sharp increment in Fig. 1(left)) with respect to the external field axis. This state is shown in Fig. 2. Then both lattices gradually rotate towards
3 !A !2 !B !2 M M B B 5dV ^ u þ K2 u Ms Ms ^A
^ I is the direcwhere K I1 is the first order anisotropy constant and u tion of easy axis. The anisotropy field for each sublattice is
!I 2K I1 I ^ I I ^ M u ^: u Hk ¼ lo M s For the simulations presented in this work, both the anisotropy constants and easy axis directions were assumed to be the same for the two lattices. For the dipole interactions the energy takes the form:
Ed ¼
lo 2
Z
! !A !B H d ðM þ M ÞdV
! where H d is the demagnetizing field and is equal to the gradient of the magnetic scalar potential for which the Poisson equation holds [21]
Fig. 1. Magnetization curves for a spherical uniaxial antiferromagnetic particle with k ¼ 1 and R=‘x ¼ 1 without dipole interactions. On the left are spin flop cases and on the right spin flip cases.
44
N. Ntallis, K.G. Efthimiadis / Computational Materials Science 97 (2015) 42–47
Fig. 2. Magnetization image for both sublattices of a spherical uniaxial antiferromagnetic particle just after the spin flop field. The gray scale represents the parallel to the external field magnetization component (which in both sublattices ffi 0:4Ms ), whereas the arrows the normal one.
the external field direction (constant slope in Fig. 1(left)) until saturation is reached. In the spin flip case, the magnetization of the lattice that is set initially antiparallel to the external field jumps parallel to it directly to the saturation state. By varying the anisotropy field Hk, the spin flop field Hsf was calculated (Fig. 3) and is found to fit a relation of the form
Fig. 3. Calculated spin flop field of a spherical uniaxial antiferromagnetic particle for different values of the anisotropy field. The dotted line represents the analytical relation, provided by atomistic models. For these calculations dipole interactions are not taken into account.
Fig. 4. Calculated magnetization curve in the easy axis direction for spherical uniaxial antiferromagnetic particles with different size. All the particles have Hk ¼ 0:2Ms and k ¼ 1.
Hsf ¼ Ms
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Hk Hk 2k : Ms Ms
The same relation is also provided analytically by atomistic models [26,27] and this was the validation criterion of our calculations. In order to investigate the effect of the dipole interactions, magnetization curves were calculated by varying the particle’s radius R with respect to exchange length ‘x (Fig. 4). Different magnetization processes appear and they are generally very fast. As shown in Fig. 4, in every magnetization curve there is a critical field Hn
Fig. 5. Nucleation field as a function of the particle’s size for different anisotropy fields. In all cases k ¼ 1.
N. Ntallis, K.G. Efthimiadis / Computational Materials Science 97 (2015) 42–47
45
Fig. 7. Inhomogeneous curling in an antiferromagnetic particle with Hk ¼ 0:5Ms , k ¼ 1 and R=‘x ¼ 50. The gray scale represents the absolute value of the magnetization component parallel to the external field. Fig. 6. Representation of the curling mode for a spherical antiferromagnetic particle with Hk ¼ 0:2M s , k ¼ 1 and R=‘x ¼ 30. The field is in the x direction. The curves show the variation of longitudinal (up) and normal (down) to the field magnetization component over the particle’s diameter normal to the applied field.
(nucleation field) where the magnetization departs from the initial zero value. For small particles (R < 20‘x ) the magnetization process is similar to the one with no dipole interactions, ie coherent rotation of the magnetization vectors. Generally, Hn is found larger
Fig. 8. Calculated magnetization curve of an antiferromagnetic particle with Hk ¼ 0:2M s , k ¼ 1 and R=‘x ¼ 50. The nucleation and the expansion of reversed domains are shown for both sublattices. The gray scale represents the absolute value of the magnetization component parallel to the external field.
46
N. Ntallis, K.G. Efthimiadis / Computational Materials Science 97 (2015) 42–47
Fig. 9. Calculated magnetization curve of an antiferromagnetic particle with Hk ¼ 0:2Ms , k ¼ 1 and R=‘x ¼ 100. The gray scale represents the absolute value of the magnetization component parallel to the external field.
Fig. 10. Size dependence of the spin flop field for a spherical uniaxial antiferromagnetic particle with different values of the anisotropy field. For these calculations dipole interactions are taken into account. In all cases k ¼ 1.
than the associated Hsf, indicating the fact that in this regime the dipole interactions try to sustain the state of the antiparallel orientation of the two lattices. As shown in Fig. 5, Hn is almost constant up to a critical radius Rc, where a decrement starts. The decrement arises from the fact that the magnetization process becomes incoherent. The transition becomes sharper by increasing the anisotropy field and the estimation of the coherent radius limit is more evident. Rc is estimated close to 12‘x , independently of the anisotropy constant. For R > Rc deviation from zero magnetization takes place by a curling mode normal to applied field for both lattices (Fig. 6). This is an initial incoherent transition leading to the spin flop state at H0sf ð Hn Þ. The accentuation of the field has the meaning that dipole interactions are now taken into account. The curling is more intense at the sublattice that is initially antiparallel to the applied field. This is due to the higher Zeeman energy of this sublattice.
For higher anisotropy the distribution of the curling is not homogenous in the whole particle’s volume. As the normal to applied field magnetization component becomes unfavourable because of the anisotropy energy, the curling takes place only in a fraction of the particle’s volume, i.e. in a spherically shaped region near the particle’s surface (Fig. 7). In the center of the particle a spherical domain is sustained, where the magnetization vectors are pointing in the direction of the external field. In this case the transition to the spin flop state is also not homogenous for both sublattices. By further increment of the particle size, immediately with the occurrence of the curling mode, nucleation of a reversed magnetic domain appears in each sublattice (Fig. 8). The orientation of the reversed domains differs for the two lattices by 180°. In each sublattice there are two magnetic domains with relative orientation close to 90°. By increasing the field strength, expansion of the nucleated reversed domains takes place. As the particle’s size
N. Ntallis, K.G. Efthimiadis / Computational Materials Science 97 (2015) 42–47
increases, this expansion process is active for a larger field range, and under almost constant susceptibility (Fig. 4). Further increment of the external field to the critical value H0sf , the reversed domains extend to the whole particle’s volume and a spin flop state appears. Then, homogenous rotation takes place until saturation. In Fig. 9 the same magnetization process is shown for a particle with R=‘x ¼ 100. It is shown that in the initial state of the magnetization process the curling mode evolves in the nucleation of reversed domains. Regarding the size dependence of the H0sf , for R < Rc is almost constant, but for larger particles is complicated. Initially it shows a decrement, due to appearance of the curling mode as the only magnetization process, whereas when the curling mode is followed by domain nucleation it increases (Fig. 10). 4. Conclusions A micromagnetic model describing an antiferromagnet is derived. Under the absence of dipole interaction FEM calculations agree with atomistic analytical models. It was shown that dipole interactions have a drastic role in the magnetization process. For small particles with radius less than the critical radius Rc 12‘x coherent rotation of magnetization vectors leads to the spin flop or the spin flip state. For particles with larger radius, curling mode is the mechanism that leads incoherently to the spin flop or spin flip state. As the particle size increases further, reversed domain nucleation and expansion is observed in each sublattice. The reversed domains extend to the whole particle’s volume and a spin flop state appears.
47
References [1] J. Sort, J. Nogués, X. Amils, S. Suriñach, J.S. Muñoz, M.D. Baro, Appl. Phys. Lett. 75 (1999) 3177. [2] J. Sort, S. Suriñach, J.S. Muñoz, M.D. Baro, J. Nogués, G. Chouteau, V. Skumryev, G.C. Hadjipanayis, Phys. Rev. B 65 (2002) 174420. [3] J. Nogués, J. Sort, V. Langlais, V. Skumryev, S. Suriñach, J.S. Muñoz, M.D. Baro, Phys. Rep. 422 (2005) 65. [4] T.A. Anhøj, C.S. Jacobsen, S. Mørup, J. Appl. Phys. 95 (2004) 3649. [5] M. Vasilakaki, K.N. Trohidou, D. Peddis, D. Fiorani, R. Mathieu, M. Hudl, P. Nordblad, C. Binns, S. Baker, Phys. Rev. B 88 (2013) 140402(R). [6] M. Vasilakaki, K.N. Trohidou, Phys. Rev. B 79 (2009) 144402. [7] J. Nogués, I.K. Schuller, J. Magn. Magn. Mater. 192 (1999) 203. [8] A.E. Berkowitz, K. Takano, J. Magn. Magn. Mater. 200 (1999) 552. [9] R.L. Comstock, J. Mater. Sci.: Mater. Electron. 13 (2002) 509. [10] J.M.D. Coey, Magnetism and Magnetic Materials, Cambridge University Press, New York, 2009. [11] M. d’Aquino, C. Serpico, G. Miano, J. Comp. Phys. 209 (2005) 730. [12] O. Bottausio, M. Chiampi, A. Manzin, IEEE Trans. Magn. 44 (2008) 3149. [13] J. Fiddler, T. Schrefl, J. Phys. D: Appl. Phys. 33 (2000) R135. [14] D. Suess, V. Tsiantos, T. Schrefl, J. Fidler, W. Scholz, H. Forster, R. Dittrich, J.J. Miles, J. Magn. Magn. Mater. 248 (2002) 298. [15] R. Hertel, J. Magn. Magn. Mater. 249 (2002) 251. [16] J.G. Zhu, Y. Zheng, X. Lin, J. Appl. Phys. 81 (1997) 4336. [17] M. Kirschner, D. Suess, T. Schrefl, J. Fidler, J.N. Chapman, IEEE Trans. Magn. 39 (2003) 2735. [18] Y. Zheng, Y. Wu, T. Chong, IEEE Trans. Magn. 36 (2000) 3158. [19] D. Suess, T. Schrefl, W. Scholz, J.-V. Kim, R.L. Stamps, J. Fidler, IEEE Trans. Magn. 38 (2002) 2397. [20] N. Ntallis, K.G. Efthimiadis, J. Magn. Magn. Mater. 363 (2014) 152. [21] J.D. Jackson, Classical Electrodynamics, Willey, New York, 1975. [22] Q. Chen, A. Konrad, IEEE Trans. Magn. 33 (1997) 663. [23] O.C. Zienkiewicz, C. Emson, P. Bettes, J. Numer. Methods Eng. 19 (1983) 393. [24] T.E. Hull, W.H. Enright, B.M. Fellen, A.E. Sedgwick, SIAM J. Numer. Anal. 9 (1971) 603. [25]. [26] R. Skomski, Simple Models of Magnetism, Oxford University Press, New York, 2008. [27] D. Fiorani, A.M. Tesla, L. Suber, E. Agostinelli, Phys. Rev. B 68 (2003) 212408.
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Micromagnetic simulation of an antiferromagnetic particle N. Ntallis, K.G. Efthimiadis ⇑ Department of Physics, Aristotle University, 54124 Thessaloniki, Greece
a r t i c l e
i n f o
Article history: Received 25 July 2014 Received in revised form 5 October 2014 Accepted 7 October 2014
Keywords: Micromagnetic simulation Finite element method Antiferromagnet
a b s t r a c t A continuum micromagnetic model is derived, describing an antiferromagnet. Using the finite element method, magnetization curves are calculated for a spherical uniaxial particle, varying the particle’s size and the anisotropy field strength. Different magnetization processes appear by increasing the size of the particle. For large particles nucleation and expansion of a reversed domain is observed, separated by an almost 90° wall. An estimation of the single domain radius Rc is made. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction
2. The simulation model
Antiferromagnetic nanoparticles are of great interest for new composite magnetic materials [1–6], which can be used in applications like spin valves [7,8] and random access memories (MRAM) [5,9], so examining the magnetic behavior of antiferromagnets is of great importance. Micromagnetism is a widely used theoretical model, due to its capability of predicting the magnetic behavior of a continuous medium. According to this model, the magnetization vector is treated as a continuous function in space [10]. The Finite Difference Method (FDM) [11,12] and the Finite Element Method (FEM) [13–15], both with their drawbacks and advantages, are the two basic techniques in computational micromagnetism. FEM has the main advantage that it can handle easily large particles and complex geometries. Extensive work has been done, by means of FEM, on the coupling of an antiferromagnet with a ferromagnet [16–19], but not for the antiferromagnet itself. This paper is organized as follows. First, a continuous micromagnetic model is presented for a magnetic system consisting of two sublattices. Then, in order to test the validity of the model, simulation, under the absence of dipole interactions, of a spherical uniaxial antiferromagnetic particle by FEM is curried out, in order to be compared with known analytical calculations. After the validation, magnetization curves are simulated, under dipole field interactions, while varying the size and the anisotropy constant of the particle.
An antiferromagnet is composed of two sublattices (A and B) of equal magnetization norm, Ms and is characterized by a negative exchange integral between them. In an atomistic model, for each atom a unique magnetic moment is defined. In a continuous approximation, as the micromagnetic model, the magnetization vector is considered a continuous function in space, so the magne! ! tization of each sublattice, M A and M B must be defined at every point in space. The evolution of the magnetization vectors is governed by the damped Partial Differential Equations (PDE) [20]
⇑ Corresponding author. Tel.: +30 2310998065. E-mail address: [email protected] (K.G. Efthimiadis). URL: http://users.auth.gr/kge (K.G. Efthimiadis). http://dx.doi.org/10.1016/j.commatsci.2014.10.010 0927-0256/Ó 2014 Elsevier B.V. All rights reserved.
s
! ! ! @M A 1 ! ¼ 2 M A H Aeff M A and @t Ms ! ! 1 ! ¼ 2 M B H Beff M B ; Ms
s
! @M B @t
! where H Ieff is the effective field acting on sublattice I ¼ fA; Bg and can be found from the variational derivative of the micromagnetic energy with respect to the magnetic polarization [10]. The main contributions are the exchange, magnetocrystalline, dipole, and Zeeman energies. Summing up for all the above contributions, an ! effective field H Ieff can be defined at every point in space as
!I ! ! ! ! H eff ¼ H Iex þ H Ik þ H d þ H : !I H ex is the exchange field acting on sublattice I. The exchange energy is described by a Heisenberg Hamiltonian
Ex ¼
1 X ! ! J Si Sj ; 2 i;j ij
N. Ntallis, K.G. Efthimiadis / Computational Materials Science 97 (2015) 42–47
where i runs for all atoms in the particle and j for the first neighbors of each atom. In the micromagnetic model discrete atoms cannot be defined. As already mentioned, at each position ~ r i inside the particle the magnetization of each sublattice is defined, and in the neighborhood ~ rj around this position, each magnetization component can be approximated by a Taylor expansion as
M Ik ð~ rj Þ
¼
M Ik ð~ ri Þ
þ ð~ rj ~ ri Þ r
M Ik ð~ ri Þ
2 o ~ rj ~ ri n 2 I þ r Mk ð~ ri Þ ; 2
k ¼ fx; y; zg: For a symmetric distribution of neighborhood atoms the first order terms vanish. For an isotropic one, the off diagonal second order terms vanish too. Under these assumptions, the exchange energy takes the form: Eex ¼
z
Z
2M2s v
2 6 4
n ! o n ! o 3 ! ! ! ! 2 2 J AA M A M A þ a2 r2 M A þ J BB M B M B þ a2 r2 M B þ 7 n ! o n ! o 5dV; ! ! ! ! 2 2 þJ AB M A M B þ a2 r2 M B þ J BA M B M A þ a2 r2 M A
where the integration takes place in the particle’s volume, v is the atomic volume, z is the number of the first neighbors and a is the first neighbors distance. J AB ¼ JBA are the antiferromagnetic exchange integrals and JAA ¼ J BB are the ferromagnetic ones. The latter interactions must be considered seriously and cannot be omitted, because in a continuum approximation there is no validation of the magnetization norm conservation rule. The terms of the form !I !I M M can be omitted, because they contribute as constants in the energy expression. The exchange field for the two sublattices is
n ! o n !o !A ! H ex ¼ kM B þ ‘2AA r2 M A ‘2AB r2 M B and n !o n ! o !B ! H ex ¼ kM A þ ‘2BB r2 M B ‘2BA r2 M A ; 2lo M2s v is a negative dimensionless mean field rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi constant, ‘AA ¼ ‘BB ¼ a zJ AA = 4lo M2s v is a ferromagnetic exchange rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi length and ‘AB ¼ ‘BA ¼ a zjJAB j= 4lo M2s v is an antiferromagnetic where k ¼ zJ AB
.
exchange length. In order for the antiferromagnetic order to exist, the two sublattices must be ferromagnetically stable, so the antiferromagnetic exchange length cannot be larger than the ferromagnetic one. In the limiting case, which was used in the following simulation results, the two lengths have the same value ‘x . Assuming a uniaxial anisotropy for each sublattice, with different strength and orientation, the magnetocrystalline energy can be written as
Ek ¼
Z
2
4K A 1
!
43
!
r2 / ¼ r ðM A þ M B Þ: The Zeeman energy is written as
Ez ¼ lo
Z
! !A !B H ðM þ M ÞdV
!
where H is the external applied field. The governing equations are supplemented by the boundary ! ! ^ ¼ 0 and @M B =@ n ^ ¼ 0, where n ^ represents the conditions @M A =@ n normal unit vector on the surface of the particle. The six PDE for the magnetization components and the one for the magnetic scalar potential are solved simultaneously through the FEM by directly applying the weak form, which is derived by the Galerkin method. The Poisson equation is solved by applying a mapped infinite element scheme [22,23], thus an outer domain surrounding the particle is used. Magnetization is interpolated using 3rd order Lagrange elements in the domain of a magnetic particle. For the magnetic potential, 2nd order Lagrange elements have been used, both on the magnetic particle domain and on the surrounding outer domain. The time integration was performed by a variable step size and order BDF method [24] and the algebraic system of linear equations was solved with PARDISO [25]. More information about the method used can been found in [20]. 3. Simulation results Magnetization curves were calculated for a spherical uniaxial antiferromagnetic particle with the above-mentioned magnetic parameters for variable radius and anisotropy constant. The exter^ , which coinnal field was set parallel to the easy axis direction u cides with x-axis, and was varied from zero to the maximum value in 106 s. As initial condition at all simulations the magnetiza! ! ^ and M B ¼ M s u ^, tion of the two sublattices was set to M A ¼ Ms u resulting in a zero net magnetic moment. In order to check the validity of the model, simulations were made under the absence of dipole field interactions. In this context, analytic atomistic models describe magnetization process by spin flip and spin flop transitions [26,27]. Fig. 1 shows calculated magnetization curves for spin flop (Fig. 1(left)) and spin flip (Fig. 1(right)) regions. In the first case at spin flop field, Hsf , the magnetization vectors of the two lattices jump coherently to opposite angles (sharp increment in Fig. 1(left)) with respect to the external field axis. This state is shown in Fig. 2. Then both lattices gradually rotate towards
3 !A !2 !B !2 M M B B 5dV ^ u þ K2 u Ms Ms ^A
^ I is the direcwhere K I1 is the first order anisotropy constant and u tion of easy axis. The anisotropy field for each sublattice is
!I 2K I1 I ^ I I ^ M u ^: u Hk ¼ lo M s For the simulations presented in this work, both the anisotropy constants and easy axis directions were assumed to be the same for the two lattices. For the dipole interactions the energy takes the form:
Ed ¼
lo 2
Z
! !A !B H d ðM þ M ÞdV
! where H d is the demagnetizing field and is equal to the gradient of the magnetic scalar potential for which the Poisson equation holds [21]
Fig. 1. Magnetization curves for a spherical uniaxial antiferromagnetic particle with k ¼ 1 and R=‘x ¼ 1 without dipole interactions. On the left are spin flop cases and on the right spin flip cases.
44
N. Ntallis, K.G. Efthimiadis / Computational Materials Science 97 (2015) 42–47
Fig. 2. Magnetization image for both sublattices of a spherical uniaxial antiferromagnetic particle just after the spin flop field. The gray scale represents the parallel to the external field magnetization component (which in both sublattices ffi 0:4Ms ), whereas the arrows the normal one.
the external field direction (constant slope in Fig. 1(left)) until saturation is reached. In the spin flip case, the magnetization of the lattice that is set initially antiparallel to the external field jumps parallel to it directly to the saturation state. By varying the anisotropy field Hk, the spin flop field Hsf was calculated (Fig. 3) and is found to fit a relation of the form
Fig. 3. Calculated spin flop field of a spherical uniaxial antiferromagnetic particle for different values of the anisotropy field. The dotted line represents the analytical relation, provided by atomistic models. For these calculations dipole interactions are not taken into account.
Fig. 4. Calculated magnetization curve in the easy axis direction for spherical uniaxial antiferromagnetic particles with different size. All the particles have Hk ¼ 0:2Ms and k ¼ 1.
Hsf ¼ Ms
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Hk Hk 2k : Ms Ms
The same relation is also provided analytically by atomistic models [26,27] and this was the validation criterion of our calculations. In order to investigate the effect of the dipole interactions, magnetization curves were calculated by varying the particle’s radius R with respect to exchange length ‘x (Fig. 4). Different magnetization processes appear and they are generally very fast. As shown in Fig. 4, in every magnetization curve there is a critical field Hn
Fig. 5. Nucleation field as a function of the particle’s size for different anisotropy fields. In all cases k ¼ 1.
N. Ntallis, K.G. Efthimiadis / Computational Materials Science 97 (2015) 42–47
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Fig. 7. Inhomogeneous curling in an antiferromagnetic particle with Hk ¼ 0:5Ms , k ¼ 1 and R=‘x ¼ 50. The gray scale represents the absolute value of the magnetization component parallel to the external field. Fig. 6. Representation of the curling mode for a spherical antiferromagnetic particle with Hk ¼ 0:2M s , k ¼ 1 and R=‘x ¼ 30. The field is in the x direction. The curves show the variation of longitudinal (up) and normal (down) to the field magnetization component over the particle’s diameter normal to the applied field.
(nucleation field) where the magnetization departs from the initial zero value. For small particles (R < 20‘x ) the magnetization process is similar to the one with no dipole interactions, ie coherent rotation of the magnetization vectors. Generally, Hn is found larger
Fig. 8. Calculated magnetization curve of an antiferromagnetic particle with Hk ¼ 0:2M s , k ¼ 1 and R=‘x ¼ 50. The nucleation and the expansion of reversed domains are shown for both sublattices. The gray scale represents the absolute value of the magnetization component parallel to the external field.
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N. Ntallis, K.G. Efthimiadis / Computational Materials Science 97 (2015) 42–47
Fig. 9. Calculated magnetization curve of an antiferromagnetic particle with Hk ¼ 0:2Ms , k ¼ 1 and R=‘x ¼ 100. The gray scale represents the absolute value of the magnetization component parallel to the external field.
Fig. 10. Size dependence of the spin flop field for a spherical uniaxial antiferromagnetic particle with different values of the anisotropy field. For these calculations dipole interactions are taken into account. In all cases k ¼ 1.
than the associated Hsf, indicating the fact that in this regime the dipole interactions try to sustain the state of the antiparallel orientation of the two lattices. As shown in Fig. 5, Hn is almost constant up to a critical radius Rc, where a decrement starts. The decrement arises from the fact that the magnetization process becomes incoherent. The transition becomes sharper by increasing the anisotropy field and the estimation of the coherent radius limit is more evident. Rc is estimated close to 12‘x , independently of the anisotropy constant. For R > Rc deviation from zero magnetization takes place by a curling mode normal to applied field for both lattices (Fig. 6). This is an initial incoherent transition leading to the spin flop state at H0sf ð Hn Þ. The accentuation of the field has the meaning that dipole interactions are now taken into account. The curling is more intense at the sublattice that is initially antiparallel to the applied field. This is due to the higher Zeeman energy of this sublattice.
For higher anisotropy the distribution of the curling is not homogenous in the whole particle’s volume. As the normal to applied field magnetization component becomes unfavourable because of the anisotropy energy, the curling takes place only in a fraction of the particle’s volume, i.e. in a spherically shaped region near the particle’s surface (Fig. 7). In the center of the particle a spherical domain is sustained, where the magnetization vectors are pointing in the direction of the external field. In this case the transition to the spin flop state is also not homogenous for both sublattices. By further increment of the particle size, immediately with the occurrence of the curling mode, nucleation of a reversed magnetic domain appears in each sublattice (Fig. 8). The orientation of the reversed domains differs for the two lattices by 180°. In each sublattice there are two magnetic domains with relative orientation close to 90°. By increasing the field strength, expansion of the nucleated reversed domains takes place. As the particle’s size
N. Ntallis, K.G. Efthimiadis / Computational Materials Science 97 (2015) 42–47
increases, this expansion process is active for a larger field range, and under almost constant susceptibility (Fig. 4). Further increment of the external field to the critical value H0sf , the reversed domains extend to the whole particle’s volume and a spin flop state appears. Then, homogenous rotation takes place until saturation. In Fig. 9 the same magnetization process is shown for a particle with R=‘x ¼ 100. It is shown that in the initial state of the magnetization process the curling mode evolves in the nucleation of reversed domains. Regarding the size dependence of the H0sf , for R < Rc is almost constant, but for larger particles is complicated. Initially it shows a decrement, due to appearance of the curling mode as the only magnetization process, whereas when the curling mode is followed by domain nucleation it increases (Fig. 10). 4. Conclusions A micromagnetic model describing an antiferromagnet is derived. Under the absence of dipole interaction FEM calculations agree with atomistic analytical models. It was shown that dipole interactions have a drastic role in the magnetization process. For small particles with radius less than the critical radius Rc 12‘x coherent rotation of magnetization vectors leads to the spin flop or the spin flip state. For particles with larger radius, curling mode is the mechanism that leads incoherently to the spin flop or spin flip state. As the particle size increases further, reversed domain nucleation and expansion is observed in each sublattice. The reversed domains extend to the whole particle’s volume and a spin flop state appears.
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