- Email: [email protected]
Relaxation times and cell size in nonzero-temperature micromagnetics
ARTICLE IN PRESS
Physica B 372 (2006) 277–281 www.elsevier.com/locate/physb
Relaxation times and cell size in nonzero-temperature micromagnetics Markus Kirschnera,, Thomas Schreflb, Gino Hrkaca, Florian Dorfbauera, Dieter Suessa, Josef Fidlera a
Vienna University of Technology, Institute of Solid State Physics, Wiedner Haupstr. 8-10/138, A-1040 Vienna, Austria b University of Sheffield, Department of Engineering Materials, Sheffield, S1 3JD, UK
Abstract In nonzero-temperature micromagnetics the intrinsic magnetic parameters depend on the computational cell size. For large cells the experimentally measured, only temperature dependent intrinsic properties can be used. For simulations on an atomistic level the experimentally measured zero-temperature values can be applied. In between, the intrinsic magnetic properties follow scaling laws which can be derived from Metropolis Monte Carlo simulations. Equilibrium magnetization states and thermally activated switching processes of small ferromagnetic cubes were calculated. With proper scaling of the material parameters the numerical results were found to be almost independent of the computational cell size. r 2005 Elsevier B.V. All rights reserved. PACS: 75.30.m; 75.40.Gb; 75.40.Mg Keywords: Coarse-graining; Relaxation times; Nonzero-temperature micromagnetics
1. Introduction Micromagnetic simulations at nonzero temperatures depend on the computational cell size. Thermal effects become relevant with decreasing dimensions of magnetic devices. The importance of high frequency spin waves necessitates small computational cells to find correct results [1]. Both conditions require nonzero-temperature simulations with small computational cells. Micromagnetics is a continuum theory. The intrinsic magnetic parameters represent thermal averages over a volume significantly larger than the unit cell of the material [2]. Thus, one can use the experimentally measured, temperature dependent values for the spontaneous magnetization and the magneto-crystalline anisotropy for large computational cells, M S;1 and K 1;1 , respectively. These material parameters are valid unless the number of atoms within a computational cell becomes too small (less than about 20 000 atoms per cell or 10 nm cell size) [3]. Then, Corresponding author. Tel.: +43 1 58801 13729; fax: +43 1 58801 13898. E-mail address: [email protected] (M. Kirschner).
0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.10.066
statistical averages over the cell volume, M S;cell and K 1;cell , will differ from the bulk properties and will depend not only on the temperature, but also on the cell size, i.e. on the discretization length. The cell size dependent material properties to be applied to small computational cells at a given temperature can be calculated by means of atomistic Metropolis Monte Carlo (MC) simulations. Contrary to the coarse-graining methods applied in Refs. [4–6], we propose to use the experimentally measured, only temperature dependent parameters for large computational cells and apply corrections to these parameters at least if the computational cell size approaches the exchange length [7]. In this work, we perform atomistic MC simulations of exchange-coupled anisotropic Heisenberg spins in an external field and extract a cell size dependent spontaneous magnetization M S;cell and anisotropy constant K 1;cell after equilibration. We will show that these material parameters should be used in nonatomistic simulations with small cells. Otherwise, physical properties, such as the equilibrium magnetization parallel to the field axis, hM z i, depend on the computational cell size. The error of about 2.4% as compared with the correct atomistic result at 1.5 nm cell
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size and a temperature of T=T C ¼ 0:34 can then be sufficiently reduced to less than 0.8%. To get rid of the remaining small deviations it is necessary to use a rescaled exchange constant Acell . After introducing our coarse-graining procedure for M S;cell , K 1;cell , and Acell , we present the results of stochastic Landau–Lifshitz–Gilbert (LLG) simulations of small ferromagnetic particles (6 nm) and investigate the influence of the discretization length on the relaxation time. We will show that the relaxation times strongly decrease with decreasing size of the computational cell for cell size independent parameters. This behavior can also be improved by applying our coarse-graining procedure to the dynamic process of magnetic switching. 2. Coarse-graining procedure We use a 3D discrete Heisenberg model of exchangecoupled magnetic moments. The total energy of the system reads E ¼ ADx
N3 X X
3
ð1 mi mj Þ K 1 Dx
i¼1 hiji
N3 X
ðmi z^ Þ2
i¼1 3
m0 M S Dx H
N3 X
mi .
ð1Þ
i¼1
A is the exchange constant, Dx denotes the computational cell size (regular cubic lattice), N is the number of magnetic moments in one direction, and mi ¼ Mi =M S is a unit vector on lattice site i with the Cartesian components mi;a . The sum hiji is carried out over all nearest-neighbor pairs, K 1 denotes the uniaxial anisotropy constant, and H is the external field parallel to the z-axis. The anisotropy axis is parallel to the external field axis. To compute the material parameters as function of temperature and cell size, we first performed Metropolis MC calculations on an atomistic level (Dx ¼ a, where a is the atomistic lattice constant) with the atomistic material parameters M S;0 , K 1;0 , and A0 . Throughout this paper the material parameters are chosen as m0 M S;0 ¼ 1:76 T, A0 ¼ 1:3 1011 J=m, K 1;0 ¼ 4:5 105 J=m3 , and a ¼ 0:376 nm. Temperatures are given in units of the mean field critical temperature T C ¼ 884:4 K. For this purpose, periodic boundary conditions were applied in all directions. A central averaging cube (Fig. 1) with n3 atomistic moments ðn ¼ 2 . . . 30Þ and an edge length of Dx ¼ na was used to derive the (normalized) spontaneous magnetization as a function of cell size * +1=2 3 X M 2a (2) M S;cell ¼ a¼1
with 3
Ma ¼
n M S;0 X mi;a 3 n i¼1
(3)
∆x = a
∆ x = na
Coarse graining
Averaging cube
MS,0
MS,cell
K1,0
K1,cell
A0
Acell
Fig. 1. Illustration of the applied coarse-graining procedure. Atomistic MC simulations (left hand side) yield hM z i and the cell size dependent values M S;cell and K 1;cell . Acell then follows numerically from a constant hM z i. The cell size dependent material parameters are used for nonatomistic simulations (right hand side).
and hM z i, the averaged magnetization per moment parallel to the easy axis and field axis, for a given temperature T=T C . hM x i and hM y i vanish due to symmetry reasons. A system size of N ¼ 60 ensures that the physical properties within the central averaging cube neither depend on the system size nor on the boundary conditions. So, we start at an atomistic level with lattice constant Dx ¼ a and evaluate rescaled material parameters for nonatomistic simulations with a discretization length of Dx ¼ na. The atomistic spins within an averaging cube can then be replaced by one nonatomistic moment with the appropriate rescaled parameters. The MC algorithm starts with a completely ordered ferromagnetic configuration parallel to the external field. One MC step is defined as N 3 trial steps, i.e. on average every magnetic moment is considered once per MC step (for details see Ref. [3]). We discard the first 104 MC steps and use the following 105 steps for averaging. The atomistic MC simulations show a cell size independent mean value hM z i, since Eq. (3) is linear in the Cartesian components mi;a . The cell size dependent spontaneous magnetization M S;cell is found to decrease with increasing cell size Dx according to the Bloch-like scaling law M S;cell ðDx; TÞ ¼ M S;1 ðTÞ
a 3=2 þ ðM S;0 M S;1 ðTÞÞ . Dx
ð4Þ
M S;1 ðTÞ is the experimentally found spontaneous magnetization for large cells and depends on A and K 1 . Fig. 2 summarizes the results of the atomistic MC simulations as a function of the averaging cube size Dx for T=T C ¼ 0:34 and 0.17, respectively, and an external field of m0 H ¼ 0:1 T. The scaling law Eq. (4) with the exponent 32 is found to be valid for a wide range of the exchange constant (at least for 1013 J=mpAp2 1011 J=m) and holds for temperatures up to T=T C 0:75. The anisotropy constant has an impact only on M S;1 . The effect of the external field strength on M S;cell is very small. For the chosen material parameters the difference in M S;cell between simulations at zero field
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1 4 T/TC = 0.17 Acell / A0
0.96 MS,cell / MS,0
T/ TC = 0.34
0.92 T/TC = 0.34
3 lex≈ 3.25 nm
parabolic behavior
2
0.88 2
4
atomistic
6 ∆ x (nm)
8
10
1 2 atomistic
Fig. 2. Spontaneous magnetization M S;cell as function of the averaging cell size Dx for T=T C ¼ 0:34 and 0.17, respectively, resulting from atomistic MC simulations. The graphs are normalized to the atomistic spontaneous magnetization M S;0 . The solid lines represent fitting curves according to the Bloch-like scaling law Eq. (4). M S;cell decreases with increasing cell size due to thermal fluctuations and finally approaches the experimentally found M S;1 . hM z i=M S;0 ¼ 0:887 for T=T C ¼ 0:34 and hM z i=M S;0 ¼ 0:945 for T=T C ¼ 0:17.
and 0.5 T is less than 0.1% for apDxp11 nm and thus negligible for most cases. The reason for a decreasing saturation magnetization for an increasing number of atoms within the averaging cell is thermal fluctuation of the magnetic moments. Thus, partial cancellation of the spin components takes place in Eq. (3) leading to a smaller M S;cell . The scaling law for the anisotropy constant K 1 arises from the condition that the anisotropy field should not depend on the computational cell size, H ani;0 ¼
2K 1;0 2K 1;cell ¼ H ani;cell , m0 M S;0 m0 M S;cell
(5)
yielding K 1;cell ðDx; TÞ ¼ K 1;0
M S;cell . M S;0
(6)
Thus, in our coarse-graining procedure the anisotropy constant scales like the spontaneous magnetization (Eq. (4)). To test our method, we replaced the n3 atomistic spins within cubes with the edge length Dx ¼ na by one nonatomistic moment and looked at the magnetization of the central cube. The results of our simulations point out that the material parameters have to be corrected for small cells according to Eqs. (4) and (6). Otherwise physical properties, such as the equilibrium magnetization hM z i, will depend on the computational cell size Dx. For instance, MC and LLG simulations using M S;1 and K 1;1 show a deviation of hM z i of the central moment of about 2.4% at Dx ¼ 1:5 nm and T=T C ¼ 0:34 as compared with the correct atomistic result. Making use of the cell size dependent spontaneous magnetization and anisotropy, hM z i becomes nearly cell size independent with a deviation of less than 0.8% from the results obtained by atomistic MC simulations even at Dx ¼ 1:5 nm. Additionally, we
4
6
8
10
∆ x (nm)
Fig. 3. Exchange constant Acell as a function of the computational cell size Dx, derived for T=T C ¼ 0:34. The numerical results of Acell reveal an inflection point near the exchange length l ex . Above the exchange length, Acell behaves parabolically, leading to nonvanishing exchange fields Hex for large cells (Eq. (7)).
found an excellent agreement between the results of the nonatomistic MC and LLG simulations for all parameters and cell sizes. To get rid of the remaining small deviations it is necessary to use a rescaled exchange constant Acell , which can be derived numerically via the condition of a constant hM z i. Fig. 3 shows Acell for T=T C ¼ 0:34 and an external field of m0 H ¼ 0:1 T. Acell increases with increasing cell size and finally behaves parabolically above the exchange qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi length l ex ¼ 2A=m0 M 2S 3:25 nm (calculated with the atomistic parameters). This leads to nonvanishing exchange fields X 2Acell Hex ¼ mj (7) 2 m0 M S;cell Dx hiji for large cells because Hex is proportional to Acell =Dx2 . Actually, the exchange length also depends on the cell size, l ex ¼ l ex;cell . With this, Eq. (7) reads l ex;cell 2 X Hex ¼ M S;cell mj . (8) Dx hiji Fig. 4 shows a comparison between 1=Dx2 and ðl ex;cell =DxÞ2 , both normalized to 1 at Dx ¼ a. The constant behavior of the prefactor above Dx 7 nm means that the exchange length increases proportionally to the discretization length and thus preserves the ferromagnetic behavior of the material. But since the aim of this work is to give corrections for the material parameters for small computational cells, the interesting part of Acell is near and below the inflection point. 3. Relaxation times of small particles The obtained scaling laws for M S , K 1 , and A were derived from equilibrium magnetization properties and
ARTICLE IN PRESS M. Kirschner et al. / Physica B 372 (2006) 277–281
280
1
2.2 T/ TC = 0.34
0.01
2
atomistic
4 ∆ x (nm)
6
8
Fig. 4. Comparison between the cell size dependent prefactor ðl ex;cell =DxÞ2 and 1=Dx2 , both normalized to 1 at Dx ¼ a. The constant behavior above 7 nm avoids unphysical disappearance of the ferromagnetic character for large computational cells.
represent statistical averages over a large number of MC steps. Thus, it is a priori not justified to use the results of the previous section for highly dynamic processes such as magnetization switching. But the following points out that our simple coarse-graining procedure is also able to improve numerical results of dynamic behavior. We performed stochastic LLG simulations of cubic particles with uniaxial anisotropy and investigated the dependence of the averaged relaxation time on the number of exchange coupled subcubes or discretization length Dx, respectively. The total energy of the particles is given in Eq. (1). The edge length was 6 nm, the damping constant was 1, and open boundary conditions were applied. For the stochastic time integration of the Langevin equation with multiplicative white-noise term in Stratonovich interpretation [8], the Heun method was used with a time step of Dt ¼ 0:1 ps. The initial magnetization pointed upwards parallel to the z-axis and the anisotropy direction. At an external field of H=H ani ¼ 0:84 applied in the negative z-direction thermal activation drives the magnetization out of the z-axis and finally causes the particle to switch. For each cell size 2000 switching events were used to calculate the averaged switching time. The results of the switching simulations for T=T C ¼ 0:34 are given in Fig. 5. The relaxation time as a function of the discretization length is found to decrease with decreasing Dx when cell size independent material parameters are used, such as the atomistic material parameters M S;0 , K 1;0 and A0 . Fig. 5 points out that this cell size dependence can be improved by applying the previously derived scaling laws for the parameters M S;cell , K 1;cell and Acell . Then, the switching time nearly behaves constantly. Contrary to equilibrium properties as discussed in the previous section, the scaling of the exchange constant has a strong impact on dynamic processes and thus on the relaxation time. Scaling of M S and K 1 but still using A0 resulted in the dashed curve in Fig. 5.
23
83 43
1.8
constant parameters rescaled, but still A0 rescaled
1.6
1/∆x2 2
13
1.4
0.23
0.1
H/Hani = -0.84
0.63
switching time (10-10 s)
normalized (lex,cell / ∆x)2
T/TC = 0.34
1
2
3 4 ∆ x (nm)
5
6
Fig. 5. Switching times as a function of the computational cell size Dx. The numbers in the diagram indicate the number of subcubes. Simulations with the constant system parameters M S;0 , K 1;0 , and A0 resulted in a decreasing switching time for decreasing cell size. This behavior can be improved by using our scaling laws. Then, the difference between the lowest and highest switching time reduces from 0:63 1010 to 0:23 1010 s. Scaling of M S and K 1 but still using A0 resulted in the dashed curve.
The cell-size correct results for the switching time, however, increase for very small cell sizes, which indicates that high frequency spin waves may slow down thermally induced switching. Simulations with Dx ¼ a ¼ 0:376 nm (163 subcubes) resulted in switching times about 2.5 times larger than those of Fig. 5. But stochastic LLG simulations on an atomistic level are a special case, anyway, and should be handled carefully. 4. Conclusion Micromagnetism is a continuum theory to describe magnetization processes on a characteristic length scale of several atomic distances. However, in order to resolve domain walls or high frequency spin waves the computational cell size has to be small. Then the intrinsic magnetic properties which have to be assigned to the magnetic moment of each cell depend both on temperature and on the number of atoms within the cell. As a consequence, the intrinsic magnetic properties have to be scaled accordingly. The combination of Monte Carlo calculations and stochastic LLG simulations proposed in this paper is an easy way to treat coarse-graining in micromagnetics. The cell size dependent intrinsic magnetic properties are calculated by Metropolis Monte Carlo simulations in advance of the nonzero-temperature LLG simulations. The method proved successfully for both simulations of equilibrium properties and simulations of thermally driven magnetization reversal processes. Acknowledgements This work was supported by the Austrian Science Fund (Y132-N02).
ARTICLE IN PRESS M. Kirschner et al. / Physica B 372 (2006) 277–281
References [1] M. Igarashi, M. Hara, Y. Suzuki, A. Nakamura, Y. Sugita, IEEE Trans. Magn. 39 (5) (2003) 2303. [2] W.F. Brown, Phys. Rev. 58 (1940) 736. [3] M. Kirschner, T. Schrefl, F. Dorfbauer, G. Hrkac, D. Suess, J. Fidler, J. Appl. Phys. 97 (2005) 10E301. [4] V.V. Dobrovitsky, M.I. Katsnelson, B.N. Harmon, J. Magn. Magn. Mater. 221 (2000) 235.
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[5] V.V. Dobrovitsky, M.I. Katsnelson, B.N. Harmon, Phys. Rev. Lett. 90 (6) (2003) 067201. [6] G. Grinstein, R.H. Koch, Phys. Rev. Lett. 90 (20) (2003) 207201. [7] V. Tsiantos, W. Scholz, D. Suess, T. Schrefl, J. Fidler, J. Magn. Magn. Mater. 242–245 (2002) 999. [8] W. Scholz, T. Schrefl, J. Fidler, J. Magn. Magn. Mater. 233 (2001) 296.
Physica B 372 (2006) 277–281 www.elsevier.com/locate/physb
Relaxation times and cell size in nonzero-temperature micromagnetics Markus Kirschnera,, Thomas Schreflb, Gino Hrkaca, Florian Dorfbauera, Dieter Suessa, Josef Fidlera a
Vienna University of Technology, Institute of Solid State Physics, Wiedner Haupstr. 8-10/138, A-1040 Vienna, Austria b University of Sheffield, Department of Engineering Materials, Sheffield, S1 3JD, UK
Abstract In nonzero-temperature micromagnetics the intrinsic magnetic parameters depend on the computational cell size. For large cells the experimentally measured, only temperature dependent intrinsic properties can be used. For simulations on an atomistic level the experimentally measured zero-temperature values can be applied. In between, the intrinsic magnetic properties follow scaling laws which can be derived from Metropolis Monte Carlo simulations. Equilibrium magnetization states and thermally activated switching processes of small ferromagnetic cubes were calculated. With proper scaling of the material parameters the numerical results were found to be almost independent of the computational cell size. r 2005 Elsevier B.V. All rights reserved. PACS: 75.30.m; 75.40.Gb; 75.40.Mg Keywords: Coarse-graining; Relaxation times; Nonzero-temperature micromagnetics
1. Introduction Micromagnetic simulations at nonzero temperatures depend on the computational cell size. Thermal effects become relevant with decreasing dimensions of magnetic devices. The importance of high frequency spin waves necessitates small computational cells to find correct results [1]. Both conditions require nonzero-temperature simulations with small computational cells. Micromagnetics is a continuum theory. The intrinsic magnetic parameters represent thermal averages over a volume significantly larger than the unit cell of the material [2]. Thus, one can use the experimentally measured, temperature dependent values for the spontaneous magnetization and the magneto-crystalline anisotropy for large computational cells, M S;1 and K 1;1 , respectively. These material parameters are valid unless the number of atoms within a computational cell becomes too small (less than about 20 000 atoms per cell or 10 nm cell size) [3]. Then, Corresponding author. Tel.: +43 1 58801 13729; fax: +43 1 58801 13898. E-mail address: [email protected] (M. Kirschner).
0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.10.066
statistical averages over the cell volume, M S;cell and K 1;cell , will differ from the bulk properties and will depend not only on the temperature, but also on the cell size, i.e. on the discretization length. The cell size dependent material properties to be applied to small computational cells at a given temperature can be calculated by means of atomistic Metropolis Monte Carlo (MC) simulations. Contrary to the coarse-graining methods applied in Refs. [4–6], we propose to use the experimentally measured, only temperature dependent parameters for large computational cells and apply corrections to these parameters at least if the computational cell size approaches the exchange length [7]. In this work, we perform atomistic MC simulations of exchange-coupled anisotropic Heisenberg spins in an external field and extract a cell size dependent spontaneous magnetization M S;cell and anisotropy constant K 1;cell after equilibration. We will show that these material parameters should be used in nonatomistic simulations with small cells. Otherwise, physical properties, such as the equilibrium magnetization parallel to the field axis, hM z i, depend on the computational cell size. The error of about 2.4% as compared with the correct atomistic result at 1.5 nm cell
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278
size and a temperature of T=T C ¼ 0:34 can then be sufficiently reduced to less than 0.8%. To get rid of the remaining small deviations it is necessary to use a rescaled exchange constant Acell . After introducing our coarse-graining procedure for M S;cell , K 1;cell , and Acell , we present the results of stochastic Landau–Lifshitz–Gilbert (LLG) simulations of small ferromagnetic particles (6 nm) and investigate the influence of the discretization length on the relaxation time. We will show that the relaxation times strongly decrease with decreasing size of the computational cell for cell size independent parameters. This behavior can also be improved by applying our coarse-graining procedure to the dynamic process of magnetic switching. 2. Coarse-graining procedure We use a 3D discrete Heisenberg model of exchangecoupled magnetic moments. The total energy of the system reads E ¼ ADx
N3 X X
3
ð1 mi mj Þ K 1 Dx
i¼1 hiji
N3 X
ðmi z^ Þ2
i¼1 3
m0 M S Dx H
N3 X
mi .
ð1Þ
i¼1
A is the exchange constant, Dx denotes the computational cell size (regular cubic lattice), N is the number of magnetic moments in one direction, and mi ¼ Mi =M S is a unit vector on lattice site i with the Cartesian components mi;a . The sum hiji is carried out over all nearest-neighbor pairs, K 1 denotes the uniaxial anisotropy constant, and H is the external field parallel to the z-axis. The anisotropy axis is parallel to the external field axis. To compute the material parameters as function of temperature and cell size, we first performed Metropolis MC calculations on an atomistic level (Dx ¼ a, where a is the atomistic lattice constant) with the atomistic material parameters M S;0 , K 1;0 , and A0 . Throughout this paper the material parameters are chosen as m0 M S;0 ¼ 1:76 T, A0 ¼ 1:3 1011 J=m, K 1;0 ¼ 4:5 105 J=m3 , and a ¼ 0:376 nm. Temperatures are given in units of the mean field critical temperature T C ¼ 884:4 K. For this purpose, periodic boundary conditions were applied in all directions. A central averaging cube (Fig. 1) with n3 atomistic moments ðn ¼ 2 . . . 30Þ and an edge length of Dx ¼ na was used to derive the (normalized) spontaneous magnetization as a function of cell size * +1=2 3 X M 2a (2) M S;cell ¼ a¼1
with 3
Ma ¼
n M S;0 X mi;a 3 n i¼1
(3)
∆x = a
∆ x = na
Coarse graining
Averaging cube
MS,0
MS,cell
K1,0
K1,cell
A0
Acell
Fig. 1. Illustration of the applied coarse-graining procedure. Atomistic MC simulations (left hand side) yield hM z i and the cell size dependent values M S;cell and K 1;cell . Acell then follows numerically from a constant hM z i. The cell size dependent material parameters are used for nonatomistic simulations (right hand side).
and hM z i, the averaged magnetization per moment parallel to the easy axis and field axis, for a given temperature T=T C . hM x i and hM y i vanish due to symmetry reasons. A system size of N ¼ 60 ensures that the physical properties within the central averaging cube neither depend on the system size nor on the boundary conditions. So, we start at an atomistic level with lattice constant Dx ¼ a and evaluate rescaled material parameters for nonatomistic simulations with a discretization length of Dx ¼ na. The atomistic spins within an averaging cube can then be replaced by one nonatomistic moment with the appropriate rescaled parameters. The MC algorithm starts with a completely ordered ferromagnetic configuration parallel to the external field. One MC step is defined as N 3 trial steps, i.e. on average every magnetic moment is considered once per MC step (for details see Ref. [3]). We discard the first 104 MC steps and use the following 105 steps for averaging. The atomistic MC simulations show a cell size independent mean value hM z i, since Eq. (3) is linear in the Cartesian components mi;a . The cell size dependent spontaneous magnetization M S;cell is found to decrease with increasing cell size Dx according to the Bloch-like scaling law M S;cell ðDx; TÞ ¼ M S;1 ðTÞ
a 3=2 þ ðM S;0 M S;1 ðTÞÞ . Dx
ð4Þ
M S;1 ðTÞ is the experimentally found spontaneous magnetization for large cells and depends on A and K 1 . Fig. 2 summarizes the results of the atomistic MC simulations as a function of the averaging cube size Dx for T=T C ¼ 0:34 and 0.17, respectively, and an external field of m0 H ¼ 0:1 T. The scaling law Eq. (4) with the exponent 32 is found to be valid for a wide range of the exchange constant (at least for 1013 J=mpAp2 1011 J=m) and holds for temperatures up to T=T C 0:75. The anisotropy constant has an impact only on M S;1 . The effect of the external field strength on M S;cell is very small. For the chosen material parameters the difference in M S;cell between simulations at zero field
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279
1 4 T/TC = 0.17 Acell / A0
0.96 MS,cell / MS,0
T/ TC = 0.34
0.92 T/TC = 0.34
3 lex≈ 3.25 nm
parabolic behavior
2
0.88 2
4
atomistic
6 ∆ x (nm)
8
10
1 2 atomistic
Fig. 2. Spontaneous magnetization M S;cell as function of the averaging cell size Dx for T=T C ¼ 0:34 and 0.17, respectively, resulting from atomistic MC simulations. The graphs are normalized to the atomistic spontaneous magnetization M S;0 . The solid lines represent fitting curves according to the Bloch-like scaling law Eq. (4). M S;cell decreases with increasing cell size due to thermal fluctuations and finally approaches the experimentally found M S;1 . hM z i=M S;0 ¼ 0:887 for T=T C ¼ 0:34 and hM z i=M S;0 ¼ 0:945 for T=T C ¼ 0:17.
and 0.5 T is less than 0.1% for apDxp11 nm and thus negligible for most cases. The reason for a decreasing saturation magnetization for an increasing number of atoms within the averaging cell is thermal fluctuation of the magnetic moments. Thus, partial cancellation of the spin components takes place in Eq. (3) leading to a smaller M S;cell . The scaling law for the anisotropy constant K 1 arises from the condition that the anisotropy field should not depend on the computational cell size, H ani;0 ¼
2K 1;0 2K 1;cell ¼ H ani;cell , m0 M S;0 m0 M S;cell
(5)
yielding K 1;cell ðDx; TÞ ¼ K 1;0
M S;cell . M S;0
(6)
Thus, in our coarse-graining procedure the anisotropy constant scales like the spontaneous magnetization (Eq. (4)). To test our method, we replaced the n3 atomistic spins within cubes with the edge length Dx ¼ na by one nonatomistic moment and looked at the magnetization of the central cube. The results of our simulations point out that the material parameters have to be corrected for small cells according to Eqs. (4) and (6). Otherwise physical properties, such as the equilibrium magnetization hM z i, will depend on the computational cell size Dx. For instance, MC and LLG simulations using M S;1 and K 1;1 show a deviation of hM z i of the central moment of about 2.4% at Dx ¼ 1:5 nm and T=T C ¼ 0:34 as compared with the correct atomistic result. Making use of the cell size dependent spontaneous magnetization and anisotropy, hM z i becomes nearly cell size independent with a deviation of less than 0.8% from the results obtained by atomistic MC simulations even at Dx ¼ 1:5 nm. Additionally, we
4
6
8
10
∆ x (nm)
Fig. 3. Exchange constant Acell as a function of the computational cell size Dx, derived for T=T C ¼ 0:34. The numerical results of Acell reveal an inflection point near the exchange length l ex . Above the exchange length, Acell behaves parabolically, leading to nonvanishing exchange fields Hex for large cells (Eq. (7)).
found an excellent agreement between the results of the nonatomistic MC and LLG simulations for all parameters and cell sizes. To get rid of the remaining small deviations it is necessary to use a rescaled exchange constant Acell , which can be derived numerically via the condition of a constant hM z i. Fig. 3 shows Acell for T=T C ¼ 0:34 and an external field of m0 H ¼ 0:1 T. Acell increases with increasing cell size and finally behaves parabolically above the exchange qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi length l ex ¼ 2A=m0 M 2S 3:25 nm (calculated with the atomistic parameters). This leads to nonvanishing exchange fields X 2Acell Hex ¼ mj (7) 2 m0 M S;cell Dx hiji for large cells because Hex is proportional to Acell =Dx2 . Actually, the exchange length also depends on the cell size, l ex ¼ l ex;cell . With this, Eq. (7) reads l ex;cell 2 X Hex ¼ M S;cell mj . (8) Dx hiji Fig. 4 shows a comparison between 1=Dx2 and ðl ex;cell =DxÞ2 , both normalized to 1 at Dx ¼ a. The constant behavior of the prefactor above Dx 7 nm means that the exchange length increases proportionally to the discretization length and thus preserves the ferromagnetic behavior of the material. But since the aim of this work is to give corrections for the material parameters for small computational cells, the interesting part of Acell is near and below the inflection point. 3. Relaxation times of small particles The obtained scaling laws for M S , K 1 , and A were derived from equilibrium magnetization properties and
ARTICLE IN PRESS M. Kirschner et al. / Physica B 372 (2006) 277–281
280
1
2.2 T/ TC = 0.34
0.01
2
atomistic
4 ∆ x (nm)
6
8
Fig. 4. Comparison between the cell size dependent prefactor ðl ex;cell =DxÞ2 and 1=Dx2 , both normalized to 1 at Dx ¼ a. The constant behavior above 7 nm avoids unphysical disappearance of the ferromagnetic character for large computational cells.
represent statistical averages over a large number of MC steps. Thus, it is a priori not justified to use the results of the previous section for highly dynamic processes such as magnetization switching. But the following points out that our simple coarse-graining procedure is also able to improve numerical results of dynamic behavior. We performed stochastic LLG simulations of cubic particles with uniaxial anisotropy and investigated the dependence of the averaged relaxation time on the number of exchange coupled subcubes or discretization length Dx, respectively. The total energy of the particles is given in Eq. (1). The edge length was 6 nm, the damping constant was 1, and open boundary conditions were applied. For the stochastic time integration of the Langevin equation with multiplicative white-noise term in Stratonovich interpretation [8], the Heun method was used with a time step of Dt ¼ 0:1 ps. The initial magnetization pointed upwards parallel to the z-axis and the anisotropy direction. At an external field of H=H ani ¼ 0:84 applied in the negative z-direction thermal activation drives the magnetization out of the z-axis and finally causes the particle to switch. For each cell size 2000 switching events were used to calculate the averaged switching time. The results of the switching simulations for T=T C ¼ 0:34 are given in Fig. 5. The relaxation time as a function of the discretization length is found to decrease with decreasing Dx when cell size independent material parameters are used, such as the atomistic material parameters M S;0 , K 1;0 and A0 . Fig. 5 points out that this cell size dependence can be improved by applying the previously derived scaling laws for the parameters M S;cell , K 1;cell and Acell . Then, the switching time nearly behaves constantly. Contrary to equilibrium properties as discussed in the previous section, the scaling of the exchange constant has a strong impact on dynamic processes and thus on the relaxation time. Scaling of M S and K 1 but still using A0 resulted in the dashed curve in Fig. 5.
23
83 43
1.8
constant parameters rescaled, but still A0 rescaled
1.6
1/∆x2 2
13
1.4
0.23
0.1
H/Hani = -0.84
0.63
switching time (10-10 s)
normalized (lex,cell / ∆x)2
T/TC = 0.34
1
2
3 4 ∆ x (nm)
5
6
Fig. 5. Switching times as a function of the computational cell size Dx. The numbers in the diagram indicate the number of subcubes. Simulations with the constant system parameters M S;0 , K 1;0 , and A0 resulted in a decreasing switching time for decreasing cell size. This behavior can be improved by using our scaling laws. Then, the difference between the lowest and highest switching time reduces from 0:63 1010 to 0:23 1010 s. Scaling of M S and K 1 but still using A0 resulted in the dashed curve.
The cell-size correct results for the switching time, however, increase for very small cell sizes, which indicates that high frequency spin waves may slow down thermally induced switching. Simulations with Dx ¼ a ¼ 0:376 nm (163 subcubes) resulted in switching times about 2.5 times larger than those of Fig. 5. But stochastic LLG simulations on an atomistic level are a special case, anyway, and should be handled carefully. 4. Conclusion Micromagnetism is a continuum theory to describe magnetization processes on a characteristic length scale of several atomic distances. However, in order to resolve domain walls or high frequency spin waves the computational cell size has to be small. Then the intrinsic magnetic properties which have to be assigned to the magnetic moment of each cell depend both on temperature and on the number of atoms within the cell. As a consequence, the intrinsic magnetic properties have to be scaled accordingly. The combination of Monte Carlo calculations and stochastic LLG simulations proposed in this paper is an easy way to treat coarse-graining in micromagnetics. The cell size dependent intrinsic magnetic properties are calculated by Metropolis Monte Carlo simulations in advance of the nonzero-temperature LLG simulations. The method proved successfully for both simulations of equilibrium properties and simulations of thermally driven magnetization reversal processes. Acknowledgements This work was supported by the Austrian Science Fund (Y132-N02).
ARTICLE IN PRESS M. Kirschner et al. / Physica B 372 (2006) 277–281
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