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JOURNAL OF APPLIED PHYSICS 111, 07D112 (2012)
Semi-implicit integration scheme for Landau–Lifshitz–Gilbert-Slonczewski
equation
A. Giordano,1 G. Finocchio,1,a) L. Torres,2 M. Carpentieri,3 and B. Azzerboni1
1
Department of Fisica della Materia e Ingegneria Elettronica, University of Messina,
Salita Sperone 31, 98166 Messina, Italy
2
Department of Fı́sica Aplicada, Universidad de Salamanca, Plaza de la Merced, Salamanca 37008, Spain
3
Department of Elettronica, Informatica e Sistemistica, University of Calabria, Via P. Bucci 42C, I-87036,
Rende (CS), Italy
(Presented 3 November 2011; received 6 October 2011; accepted 29 October 2011; published
online 27 February 2012)
This paper shows how to implement a semi-implicit algorithm based on the Adams-Bashforth
algorithm as a predictor, and a second order Adams-Moulton procedure as a corrector in the
Landau–Lifshitz–Gilbert-Slonczewski equation. We compare the results with a Runge-Kutta
scheme of the 5th order, while for the standard problem #4 (and, in general, for the LLG equation)
the computational speeds are of the same order, and we found better performance when the thermal
C 2012 American Institute of
fluctuations or the spin-polarized currents are taken into account. V
Physics. [doi:10.1063/1.3673428]
The Landau-Lifshitz-Gilbert (LLG) equation describes
the relationship between the magnetic field and the magnetization of a system at the micrometric and nanometric scale.1
The time variation of the magnetization is due to the sum
of two terms: (i) the conservative one which describes the
magnetization precession around an effective magnetic field,
and (ii) the dissipative term which describes the relaxation
processes toward an equilibrium configuration.2,3 In the simplified scenario of macrospin approximation (one domain for
the whole ferromagnet), the LLG equation can be also solved
analytically.4 In general, nucleation processes are involved
in the magnetization dynamics (for example magnetization
switching driven by a magnetic field pulse) and consequently
it is necessary to solve the LLG equation in a full micromagnetic framework numerically.2 Here, we discuss the finitedifference-formulation in time domain of the LLG equation
using two different time solvers: the Runge-Kutta (RK5)
described in detail in a recent work5 by our group and the
semi-implicit Adams-Bashforth-Moulton method, based on
the use of the Adams-Bashforth algorithm as a predictor
with a second order Adams-Moulton procedure as a corrector
(AB3M2). Together with the comparison of the magnetization dynamics for the standard micromagnetic problem #4
(STD#4),6 we also studied the numerical properties of the
dynamical response of the magnetization in the presence of
thermal fluctuations and spin-transfer-torque.7 To take into
account the spin-transfer torque effects in the micromagnetic
scenario, the LLG-Slonczewski (LLGS) has to be solved,
which is the LLG equation with an addition term. This additional term can be seen as a damping term which, depending
on the sign of the current, can act as negative damping compensating for the losses due to the intrinsic magnetic dissipation. The spin-polarized current (SPC) can induce either
magnetization switching or persistent magnetization oscillation. Here, we studied, in detail, the efficiency in terms of
the speed of the two algorithms in the presence of a nonuniform magnetization precession due to a non-uniform
injection of an SPC (vortex-antivortex pair dynamics)8 and,
in a non-autonomous case of the LLGS, together with a bias
SPC density, which induces persistent magnetization oscillation, a microwave current is applied at a frequency where the
injection locking phenomenon is achieved.9 Our numerical
results show that the RK5 is stable for a larger time step
compared with the AB3M2 (for example, in the STD#4
DtRK5 ¼ 1.25 ps and DtAB3M2 ¼ 0.57 ps) for both the LLG and
the LLGS, however, the computational times are of the same
order for the LLG equation, while the AB3M2 is faster than
the RK5 in the presence of the thermal fluctuation or SPC
because it computes two times the demagnetization field for
each time step against the 5 times of the RK5. For all of the
cases we studied in this paper, we also find that the integration time which can be used for the LLGS is smaller than the
one used for the LLG in the STD#4 for both algorithms.
The integration techniques of Euler, Heun, and RungeKutta are called single step methods because they use only
the information from the ith iteration to predict the solution
at the i þ 1th iteration. In contrast, the Adams-BashforthMoulton algorithms also need information from the iterations preceding the ith. In particular, for the AB3M2 yiþ1
¼ f ðt; yi ; yi1 ; yi2 Þ where yi1 and yi2 at the beginning of
the algorithm have to be computed with a single step
method. In the specific case of the integration of the LLGS
equation, we solve the following equation:10
a)
where m, mp , and heff are the magnetization of the free
(change in time) and the reference layer (fixed in time) and
Author to whom correspondence should be addressed. Electronic mail:
gfinocchio@unime.it.
0021-8979/2012/111(7)/07D112/3/$30.00
ð1 þ a2 Þdm
¼ ðm heff Þ a m ðm heff Þ bjeðm; mp Þ
c0 MS dt
(1)
m ðm mp Þ aðm mp Þ ;
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V
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Giordano et al.
J. Appl. Phys. 111, 07D112 (2012)
the effective field, respectively, a is the damping parameter,
c0 and MS are the gyromagnetic ratio and the saturation
magnetization, j and eðm; mp Þ are the current density given
by j ¼ jDC þ jAC and the polarization function defined by
Slonczewski.7 Here, b is computed as b ¼ ec2 Mjl2Bdj being the
0
s
thickness of the free layer, lB is the Bohr magneton, and e is
the electron charge. For the effective field, we take into
account the standard contributions: exchange, magnetostatic,
anisotropy, and external. For the case of spin-valves, we also
include the Oersted field due to the SPC and the dipolar coupling with the pinned layer.
By applying the AB3M2 to Eq. (1) at the ith iteration,
we first have to compute the magnetization from the predictor algorithm (Adams-Bashforth) as,
miþ1;pred ¼ mi þ ab3
0
dmi
dmi1
dmi2
þ ab3 1
þ ab3 2
;
dt
dt
dt
(2)
and then the final value of the magnetization of the i þ 1th
iteration is computed by applying the corrector algorithm
(Bashforth-Moulton) as,
dmiþ1;pred
dt
dmi
dmi1
þ am2 1
þ am2 2
;
dt
dt
miþ1 ¼ miþ1;corr ¼ mi þ am2
0
(3)
where the coefficients are ab3_0 ¼ 23.0/12.0, ab3_1 ¼ 16.0/
12.0, ab3_2 ¼ 5.0/12.0, am2_0 ¼ 5.0/12.0, am2_1 ¼ 8.0/12.0,
and am2_2 ¼ 1.0/12.0; all of these coefficients should be
multiplied for the integration time step, Dt.
First, we show the results of the micromagnetic simulations for the standard problem #4.6 Fig. 1(a) compares
the time domain traces obtained for the normalized average
y-components of the magnetization hmYi (the main critical
component in this switching process) for computations
obtained using the RK5 (Dt ¼ 1.25 ps) and from the AB3M2
(Dt ¼ 0.57 ps). As can be observed, the perfect agreement is
achieved, indicating the correct implementation of the
AB3M2 method in the micromagnetic framework. Even if
the integration time step of the RK5 is larger, the total computational time of each iteration of the AB3M2 is smaller
because the demagnetizing field is only computed two times
against the necessary five times in the RK5 (basically, the
computational time of the demagnetizing field is 95% of
the total computational time). By comparing the (STD#4) for
the two integration algorithms, they have approximately the
same speed. As will be discussed next, the AB3M2 is faster
than the RK5 when the effect of the thermal fluctuations or
the presence of a spin-polarized current is included in the
micromagnetic model.
First, we analyze the effects of the thermal fluctuations
for T ¼ 300 K in the same switching process of the STD#4.
The thermal field is computed as an additive stochastic
contribution to the effective field. It is a Gaussian process, n,
characterized
by a zero mean andffi unit variance with amplipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
tude, 1=Ms 2akB T=l0 c0 DVMs Dt, where kB is the Boltzmann constant, DV is the volume of the computational
cubic cell, Dt is the simulation time step, and T is the
FIG. 1. (Color online) Comparison of the time domain traces computed
using the AB3M2 and RK5 for the STD#4; (a) hmYi at T ¼ 0 K, (b) hmXi at
T ¼ 300 K (averaged over 50 iterations) computed with the integration time
step where the two algorithms are not non-convergent compared with hmXi
at T ¼ 0 K, and (c) hmXi at T ¼ 300 K computed with the integration time
step where the two algorithms are non-convergent.
temperature.11 The main effect of the thermal field is
the reduction of the maximum integration time step in both
algorithms with respect to the T ¼ 0 K case. Figure 1(b)
shows the averaged x-component of the magnetization
(hmXi) computed over 50 iterations and at 300 K for the
RK5 (Dt ¼ 1.25 ps) and the AB3M2 (Dt ¼ 0.57 ps) compared
with the one at T ¼ 0 K, as can be observed in the computations obtained with the time step used for the STD#4 at
T ¼ 0 K, which yields unphysical results (the reversal process
is not fully achieved). We performed a systematic study to
identify the correct integration time step finding for the RK5,
Dt ¼ 0.56 ps, and for the AB3M2, Dt ¼ 0.34 ps. Figure 1(c)
compares the simulation results for those latter Dt and, as
can be observed, both algorithms converge to the same average (50 iterations) time trace (hmXi). Our results indicate
that in the presence of thermal effects the speedup of the
AB2M3 over the RK5 is 1.5.
Finally, we studied the performance of the AB3M2
integration scheme in Eq. (3), when the current density is
different from zero. We addressed the magnetization dynamics in two different spin-valve devices: (i) the nano-point
contact geometry studied in Ref. 8 with a non-uniform injection of a bias SPC via a nano-aperture (the numerical parameters are the same as Ref. 8, with the external field,
H ¼ 20 mT and jDC ¼ 3 108 A/cm2), and (ii) a standard
spin-valve where a bias SPC and a “weak” microwave current, jAC ¼ jM sinð2pfAC tÞ, are applied (in this latter case the
dynamics is non-autonomous) (all of the physical parameters
are reported in Ref. 9). We studied the magnetization
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Giordano et al.
J. Appl. Phys. 111, 07D112 (2012)
for the AB3M2, Dt ¼ 0.51 ps (the speedup of AB2M3 over
RK5 is 1.3).
In summary, we have discussed the application of a
semi-implicit time integration scheme based on the AdamsBashforth algorithm as a predictor, and a second order
Adams-Moulton procedure as a corrector, which can be used
as the solver in micromagnetic simulations for the LLGS
equation. While for the standard LLG equation, the computational time of the AB3M2 is of the same order of the RK5, in
the presence of thermal fluctuations or the SPC in the micromagnetic model, we find a better performance of the AB3M2
with respect to the RK5, also obtaining a speedup of 1.5 in
processes where the magnetization dynamics are characterized by a strongly non-uniform configuration.
ACKNOWLEDGMENTS
FIG. 2. (Color online) A comparison between the time traces of the average
x-component of the magnetization computed with the AB3M2 and the RK5
(a) in the case of vortex-antivortex dynamics, and (b) in the case of nonautonomous dynamics.
This work was supported by the Spanish Project under
Contract Nos. MAT2008-04706/NAN and SA025A08.
1
W. F. Brown, Micromagnetics, Wiley, New York, 1963.
E. Cardelli, E. Della Torre, E. Pinzaglia, J. Appl. Phys. 93, 6647, (2003).
3
G. Bertotti, Hysteresis in Magnetism (Academic, New York, 1998).
4
C. Serpico, I. D. Mayergoyz, and G. Bertotti, J. Appl. Phys. 93, 6903
(2003).
5
A. Romeo, G. Finocchio, M. Carpentieri, L. Torres, G. Consolo, and B.
Azzerboni, Physica B 403, 464 (2008).
6
See http://www.ctcms.nist.gov/~rdm/mumag.org.html for information,
specification and reports, about the micromagnetic standard problem #4.
7
J. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).
8
G. Finocchio, O. Ozatay, L. Torres, R. A. Buhrman, D. C. Ralph, and B.
Azzerboni, Phys. Rev. B 78, 174408 (2008).
9
G. Finocchio, I. Krivorotov, X. Cheng, L. Torres, and B. Azzerboni, Phys.
Rev. B 83, 134402 (2011).
10
G. Finocchio, M. Carpentieri, B. Azzerboni, L. Torres, E. Martinez, and L.
Lopez-Diaz, J. Appl. Phys. 99, 08G522 (2006).
11
W. F. Brown, Jr., Phys. Rev. 130, 1677 (1963).
2
dynamics for jDC ¼ 3.5 107 A/cm2, jAC ¼ jM sinð2pfAC tÞ
(jM ¼ 1 107 A/cm2, fAC ¼ 3.75 GHz), where the injection
locking mechanism is achieved. Also, in the presence of the
SPC the Dt of both algorithms are smaller than the ones used
in the STD#4, however, in terms of the total computational
speed the AB2M3 is faster, at least for all the dynamical
processes we studied in this paper. Our results are summarized in Figs. 2(a) and 2(b), where one can observe the comparison of the average x-component of the magnetization in
the two cases; in the case (i) for the RK5, Dt ¼ 0.83 ps and
for the AB3M2, Dt ¼ 0.51 ps (the speedup of AB2M3 over
RK5 is 1.5), while in case (ii) for the RK5, Dt ¼ 0.98 ps and