The classical two-sublattice Landau–Lifshitz–Bloch equation for all temperatures
Micromagnetic modeling has proved itself as a widely used tool, complimentary in many respects to experimental measurements. The Landau–Lifshitz equation provides a basis for this modeling, especially where the dynamical behaviour is concerned. However, this approach is strictly valid only for zer...
Збережено в:
| Дата: | 2015 |
|---|---|
| Автори: | , , , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2015
|
| Назва видання: | Физика низких температур |
| Теми: | |
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/128077 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | The classical two-sublattice Landau–Lifshitz–Bloch equation for all temperatures / P. Nieves, U. Atxitia, R.W. Chantrell, O. Chubykalo-Fesenko // Физика низких температур. — 2015. — Т. 41, № 9. — С. 949–955. — Бібліогр.: 39 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-128077 |
|---|---|
| record_format |
dspace |
| spelling |
nasplib_isofts_kiev_ua-123456789-1280772025-06-03T16:29:05Z The classical two-sublattice Landau–Lifshitz–Bloch equation for all temperatures Nieves, P. Atxitia, U. Chantrell, R.W. Chubykalo-Fesenko, O. К 80-летию уравнения Ландау–Лифшица Micromagnetic modeling has proved itself as a widely used tool, complimentary in many respects to experimental measurements. The Landau–Lifshitz equation provides a basis for this modeling, especially where the dynamical behaviour is concerned. However, this approach is strictly valid only for zero temperature and for high temperatures must be replaced by a more thermodynamically consistent approach such as the the Landau– Lifshitz–Bloch (LLB) equation. Here we review the recently derived LLB equation for two-sublattice systems and extend its derivation for temperatures above the Curie temperature. We present comparison with many-body atomistic simulations and show that this equation can describe the ultra-fast switching in ferrimagnets, observed experimentally This work was supported by the EU Seventh Framework Programme FP7/2007-2013 under grant agreement No. 281043, FEMTOSPIN. P.N. and O.C-F also acknowledge support from the Spanish Ministry of Economy and Competitiveness under the grant MAT2013-47078-C2-2-P. U.A. acknowledges support from the European Community’s Seventh Framework Programme (FP7/2007-2013) under the Curie Zukunftskolleg Incoming Fellowship Programme (Grant No. 291784), University of Konstanz. 2015 Article The classical two-sublattice Landau–Lifshitz–Bloch equation for all temperatures / P. Nieves, U. Atxitia, R.W. Chantrell, O. Chubykalo-Fesenko // Физика низких температур. — 2015. — Т. 41, № 9. — С. 949–955. — Бібліогр.: 39 назв. — англ. 0132-6414 PACS: 75.10.Hk, 75., 75.78.–n https://nasplib.isofts.kiev.ua/handle/123456789/128077 en Физика низких температур application/pdf Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
К 80-летию уравнения Ландау–Лифшица К 80-летию уравнения Ландау–Лифшица |
| spellingShingle |
К 80-летию уравнения Ландау–Лифшица К 80-летию уравнения Ландау–Лифшица Nieves, P. Atxitia, U. Chantrell, R.W. Chubykalo-Fesenko, O. The classical two-sublattice Landau–Lifshitz–Bloch equation for all temperatures Физика низких температур |
| description |
Micromagnetic modeling has proved itself as a widely used tool, complimentary in many respects to experimental
measurements. The Landau–Lifshitz equation provides a basis for this modeling, especially where the
dynamical behaviour is concerned. However, this approach is strictly valid only for zero temperature and for
high temperatures must be replaced by a more thermodynamically consistent approach such as the the Landau–
Lifshitz–Bloch (LLB) equation. Here we review the recently derived LLB equation for two-sublattice systems
and extend its derivation for temperatures above the Curie temperature. We present comparison with many-body
atomistic simulations and show that this equation can describe the ultra-fast switching in ferrimagnets, observed
experimentally |
| format |
Article |
| author |
Nieves, P. Atxitia, U. Chantrell, R.W. Chubykalo-Fesenko, O. |
| author_facet |
Nieves, P. Atxitia, U. Chantrell, R.W. Chubykalo-Fesenko, O. |
| author_sort |
Nieves, P. |
| title |
The classical two-sublattice Landau–Lifshitz–Bloch equation for all temperatures |
| title_short |
The classical two-sublattice Landau–Lifshitz–Bloch equation for all temperatures |
| title_full |
The classical two-sublattice Landau–Lifshitz–Bloch equation for all temperatures |
| title_fullStr |
The classical two-sublattice Landau–Lifshitz–Bloch equation for all temperatures |
| title_full_unstemmed |
The classical two-sublattice Landau–Lifshitz–Bloch equation for all temperatures |
| title_sort |
classical two-sublattice landau–lifshitz–bloch equation for all temperatures |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2015 |
| topic_facet |
К 80-летию уравнения Ландау–Лифшица |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/128077 |
| citation_txt |
The classical two-sublattice Landau–Lifshitz–Bloch equation for all temperatures / P. Nieves, U. Atxitia, R.W. Chantrell, O. Chubykalo-Fesenko // Физика низких температур. — 2015. — Т. 41, № 9. — С. 949–955. — Бібліогр.: 39 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT nievesp theclassicaltwosublatticelandaulifshitzblochequationforalltemperatures AT atxitiau theclassicaltwosublatticelandaulifshitzblochequationforalltemperatures AT chantrellrw theclassicaltwosublatticelandaulifshitzblochequationforalltemperatures AT chubykalofesenkoo theclassicaltwosublatticelandaulifshitzblochequationforalltemperatures AT nievesp classicaltwosublatticelandaulifshitzblochequationforalltemperatures AT atxitiau classicaltwosublatticelandaulifshitzblochequationforalltemperatures AT chantrellrw classicaltwosublatticelandaulifshitzblochequationforalltemperatures AT chubykalofesenkoo classicaltwosublatticelandaulifshitzblochequationforalltemperatures |
| first_indexed |
2025-11-24T10:17:23Z |
| last_indexed |
2025-11-24T10:17:23Z |
| _version_ |
1849666509269368832 |
| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 9, pp. 949–955
The classical two-sublattice Landau–Lifshitz–Bloch
equation for all temperatures
P. Nieves1, U. Atxitia2, R.W. Chantrell3, and O. Chubykalo-Fesenko1
1Instituto de Ciencia de Materiales de Madrid, CSIC, Cantoblanco, Madrid 28049, Spain
E-mail: oksana@icmm.csic.es
2Fachbereich Physik and Zukunftskolleg, Universität, Konstanz D-78457, Germany
3Department of Physics, University of York, York YO105DD, U.K.
Received April 21, 2015, published online July 24, 2015
Micromagnetic modeling has proved itself as a widely used tool, complimentary in many respects to experi-
mental measurements. The Landau–Lifshitz equation provides a basis for this modeling, especially where the
dynamical behaviour is concerned. However, this approach is strictly valid only for zero temperature and for
high temperatures must be replaced by a more thermodynamically consistent approach such as the the Landau–
Lifshitz–Bloch (LLB) equation. Here we review the recently derived LLB equation for two-sublattice systems
and extend its derivation for temperatures above the Curie temperature. We present comparison with many-body
atomistic simulations and show that this equation can describe the ultra-fast switching in ferrimagnets, observed
experimentally.
PACS: 75.10.Hk Classical spin models;
75. Magnetic properties and materials;
75.78.–n Magnetization dynamics.
Keywords: Landau–Lifshitz–Bloch equation, micromagnetic modeling, ferromagnets.
1 Introduction
The importance of the Landau–Lifshitz (LL) equation
for modelling of magnetic materials is difficult to overes-
timate since it is the benchmark equation for static and
dynamical simulations both in fundamental and applied
nanomagnetism. Recent advances in synchrotron meas-
urement techniques and Kerr magneto-optics [1] have al-
lowed to measure the magnetization dynamics which is
almost always modeled by the LL equation [2]. Magnetic
recording simulations is also based on the LL approach to
simulate the recording dynamics. However, the standard
micromagnetics is essentially a zero-temperature approach
although the temperature dependence can be included
in macroscopic parameters such as the saturation magneti-
zation or anisotropy. Temperature fluctuations can be also
included in LL-based micromagnetics, following W.F.
Brown [3], leading to the stochastic LL equation. Howev-
er, this approach is known to seriously overestimate the
Curie temperature [4].
Several recent technological applications such as heat-
assisted magnetic recording, thermally assisted MRAM or
spincaloritronics have shown the need to generalize the
micromagnetic approach for high temperatures. This need
was especially stimulated by the discovery of the ultra-fast
magnetization dynamics [5] and its pure heat-driven origin
[6,7].
The standard LL equation cannot be used at the micro-
magnetic level for temperatures close to the Curie tempera-
ture basically due to the fact that it conserves the magneti-
zation magnitude, effectively truncating the high-frequency
spin waves, thereby neglecting the fluctuations of the mag-
netization length at high temperatures. A more thermody-
namically consistent approach was introduced by D. Ga-
ranin [8,9] who derived the Landau–Lifshitz–Bloch (LLB)
equation for ferromagnets. The derivation has two coun-
terparts: (i) the classical derivation assuming atomic local-
ised spins governed by the stochastic LL equation-based
dynamics and the Fokker–Planck equation [8] (ii) the quan-
tum derivation assuming paramagnetic spin interacting
with a phonon environment within the density matrix ap-
proach [9]. In both cases the ferromagnetic character was
taken into account within the mean-field approximation
(MFA). The LLB equation essentially interpolates between
© P. Nieves, U. Atxitia, R.W. Chantrell, and O. Chubykalo-Fesenko, 2015
P. Nieves, U. Atxitia, R.W. Chantrell, and O. Chubykalo-Fesenko
the dynamics governed by the LL equation at low tempera-
tures and by the Landau–Ginzburg free energy near the
critical temperature. In comparison to the LL equation it
contains an additional term, responsible for longitudinal
relaxation of the magnetization magnitude, which is infi-
nitely fast at low temperatures (and thus the LLB equation
reduces to the LL one) and slows down at high tempera-
tures, especially approaching the Curie temperature. LLB-
based micromagnetics has been successfully tested via the
corresponding atomistic large-scale simulations [10,11]
and was successfully used to model the ultra-fast magneti-
zation dynamics in ferromagnets such as Ni [12], Gd [13],
and FePt [14].
One should also note an alternative macroscopic ap-
proach suggested by V. Bar’yakhtar [15], based on the
symmetry principle which was recently also used to model
ultra-fast dynamics [16]. Also the M3TM model [17], pro-
posed by Koopmans as well as the self-consistent Bloch
(SCB) equation [18] were used for this purpose. It should
be noted that from the quantum LLB equation with spin
= 1/ 2S one can recover the SCB and M3TM equations [19].
The LLB equation could be further generalized to take into
account additional terms like the inhomogeneous exchange
field and, thus, it could be put in the form similar to the
Bar’yakhtar one.
From the point of view of ultra-fast dynamics, two-
component materials such as ferrimagnetic alloys like
GdFeCo [20], TbFeCo [21], TbFe [22] are especially im-
portant, since an all optical switching (AOS) has been ob-
served in these materials using an intense ultrashort pulse
of circularly polarized light [23] and linearly polarized
light [20]. Although initially it was thought that the inverse
Faraday effect is solely responsible for the magnetization
reversal in these materials [23], later it was shown that the
reversal also occurs without its presence [20]. The reversal
was explained [6,7] by pure thermodynamical reasons,
involving the angular momentum transfer between the two
sublattices [24,25]. More recently, T. Ostler et al. [6] used
a multi-spin atomistic approach based on the Heisenberg
model showing that the switching occurs without any ap-
plied field or even with the field up to 40 T applied in the
opposite direction. The predictions for the heat-driven re-
versal were confirmed in several experiments in magnetic
thin films and dots using linearly polarized pulses. [26–28]
Furthermore, I. Radu et al. [20] used the same atomistic
model for the magnetization dynamics to simulate GdFeCo
and compared the simulation results to the experimental
data measured by the element-specific x-ray magnetic cir-
cular dichroism (XMCD). They unexpectedly found that
the ultrafast magnetization reversal in this material, where
spins are coupled antiferromagnetically, occurs by way of
a transient ferromagnetic-like state. The different ultra-fast
spin dynamics in multi-component alloys has been also
observed in ferromagnetic materials such as permalloy
FeNi [29].
The latter experiments demonstrated the necessity to
have a macroscopic equation where the different magneti-
zation dynamics of the sublattices and their ultra-fast angu-
lar momentum transfer and relaxation can be taken into
account separately. One of the first modeling of ultra-fast
angular momentum transfer has been done by Mentink
et al. [24] using the Bar’yakhtar equation for ferrimagnets.
Initially this equation took into account only the longitudi-
nal relaxation processes but more recently a more complete
treatment to include the precessional motion was also pre-
sented [30].
Simultaneously, the LLB equation for ferrimagnetic al-
loys have been derived by U. Atxitia et al. [31]. Here we
present a review of this classical equation and generalise it
for two-component magnets (including the ferromagnetic
two-sublattice systems). Additionally, in the present work
we extend its derivation for temperatures above the Curie
temperature. In order to show the viability of the equation
we present its comparison with many-body atomistic simu-
lation. We also show that the equation successfully de-
scribes the switching process in ferrimagnetic alloys.
2. The LLB equation for two-component magnets
2.1. General derivation
The derivation of the classical LLB for two sublattice
magnets has the same starting point as for the ferromagnet-
ic classical LLB equation [8]. Namely, we start with the
usual atomistic approach where it is assumed that the dy-
namics of the atomic magnetic moment =i i iµ µμ s is gov-
erned by the Langevin dynamics based on the stochastic
LL equation, which in terms of the unitary vector is reads
= ( )] [ [ ( )] ,i
i i i i i i i
d
dt
γ × + − γλ × × +
s
s H ζ s s H ζ (1)
where λ is the coupling to the bath parameter, γ is the
giromagnetic ratio, iH is the external magnetic field and
the components of the stochastic Langevin field ( )tζ are
given by
2( ) ( ) = ( )i j ij
Tt t t tα σ
ασ
λ′ ′〈ζ ζ 〉 δ δ δ −
γµ
(2)
where the indices i and j stand for spin numbers and the
indices α and σ for their components , ,x y z and T is the
bath's temperature.
The Fokker–Planck equation [32] corresponding to ma-
ny-spin Eq. (1) was calculated in Ref. 8. Using the evolu-
tion of the probability function, governed by this equation,
one obtains an equation for thermal average of the spin
polarisation, i.e. the reduced magnetization = i iν ∈ν〈 〉m s
(where ν denotes the sublattice) in a paramagnetic state.
For the treatment of ferro (ferri) magnet, the external field
is substituted by the mean field. A detailed discussion about
the mean-field approximation (MFA) for a disordered
ferrimagnet can be found in Ref. 33.
950 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 9
The classical two-sublattice Landau–Lifshitz–Bloch equation for all temperatures
The corresponding set of coupled LLB equations for
each sublattice magnetization νm has the following form,
see details in Ref. 31:
0,
, 2= [ ] 1MFAd
dt m
ν νν
ν ν ν ν ν
ν
−γ × −Γ − −
m mm
m H m
0,
, 2
[ [ ]]
m
ν ν ν
ν ⊥
ν
× ×
− Γ
m m m
(3)
where
0, = ( ) , MFAL ν
ν ν ν ν ν
ν
ξ ≡ βµ
ξ
ξ
m ξ H . (4)
Here | |ν νξ ≡ ξ , ( ) coth ( ) 1/L ξ = ξ − ξ is the Langevin func-
tion, 1/ Bk Tβ = , Bk is the Boltzmann constant and
,
, , ,
( )
= , = 1
( ) 2 ( )
N
N
L
L L
νν ν
ν ν ν ⊥
ν ν ν
Λ ξ ξ
Γ Λ Γ − ′ξ ξ ξ
(5)
describe parallel and perpendicular relaxation, respective-
ly, , = 2 /N Bk Tν ν ν νΛ γ λ µ is the characteristic diffusion
relaxation rate or, for the thermo-activation escape prob-
lem, the Néel attempt frequency. ( ) = /L dL d′ ξ ξ is the de-
rivative of the Langevin function. The mean fields have the
following forms:
0, 0,=MFA J Jν νκ
ν ν κ ν
ν ν
+ +
µ µ
H m m h (6)
where 0, =J x zJν ν νν, 0, =J x zJνκ κ νκ, z is the number of
nearest neighbours in the ordered lattice, Jνν and Jνκ are
the Heisenberg intra and inter-sublattice exchange interac-
tion parameters, xν and = 1x xκ ν− are the concentrations
of the sublattices ν and κ , respectively. The field
,= aν ν+h H H contains the external applied field (H) and
the anisotropy field ( ,a νH ).
In Eq. (3) the first (precession) and the last (transverse
relaxation) terms have forms, similar to the LL equation
and turn to it if = 1mν and 0,νm is proportional to the ef-
fective field, acting on the sublattice. The latter is true if
the external (such as the field) perturbations are not varied
too fast.
On the other hand, if only longitudinal processes are
considered, Eq. (3) becomes
, 0,= ( ),
dm
m m
dt
ν
ν ν ν−Γ −
(7)
which coincides with the self-consistent Bloch equation
[18]. In spite of the fact that the form of Eq. (7) is similar
to the well known Bloch equation, the quantity 0,m ν is not
the equilibrium magnetization and it changes dynamically
through the dependence of the mean field given by Eq. (6).
Moreover, the rate parameter ,νΓ
contains highly non-
linear terms in 0,m ν and 0,m κ . The MFA field Eq. (6) con-
tains the homogeneous exchange field only ( = 0k mode).
The inhomogeneous exchange field could be also taken
into account, however, normally we include it within the
many-body micromagnetic approach, based on the LLB.
Finally, the equilibrium solution of Eq. (3), ,em ν , coin-
cides with the self-consistent solution of the Curie–Weiss
equations with the MFA field
( ) .L ν
ν ν
ν
= ξ
ξ
ξ
m (8)
In the absence of external magnetic field and anisotropy
field ( = 0νh ) one finds from Eq. (8) that the Curie tem-
perature ( cT ) of the system is given by
2
0, 0, 0, 0, 0, 0,| | 4
= .
6c
B
J J J J J J
T
k
ν κ ν κ νκ κν+ + − +
(9)
Equation (3) is ready for modeling. However, for ana-
lytical estimations a more closed form of the longitudinal
relaxation in the LLB equation is convenient. Particularly,
it is convenient to express the final form of the LLB equa-
tion in terms of physically measurable quantities such as
the longitudinal susceptibilities 0,|| = ( / )Hm Hν →νχ ∂ ∂
which can be evaluated independently or even measured.
To this end, further approximations are made. Namely,
assuming that the longitudinal homogeneous exchange
field is large in comparison to the other fields, hν , in
Eq. (6) and adiabatic (quasi stationary) processes and after
an expansion around the equilibrium up to the fourth order,
one arrives from Eq. (3) to (see Ref. 31)
eff ,
eff , 2= [ ]
d
dt m
ν ννν
ν ν ν ν ν
ν
⋅
−γ × + γ α −
m Hm
m H m
eff ,
2
[ [ ]]
,
m
ν ν νν
ν ⊥
ν
× ×
−γ α
m m H
(10)
where the effective field is given by
0,
eff , ,= a
J νκ
ν ν κ
ν
+ + −
µ
H H H Π
, ,
1 1( ) (| ) | | |e em m
m
ν
ν ν κ κ
νν νκ ν
− − − τ − τ Λ Λ
m
(11)
where ,a νH is the anisotropy field, H is the applied field,
2= [ [ ]] / mκ ν ν κ ν− × ×Π m m m , = ( ) / mν ν κ κτ ⋅m m ,
, , , ,= ( ) /e e e emν ν κ κτ ⋅m m and
0,1
,||
,||
1= 1
J νκ−
νν κ
ν ν
Λ + χ χ µ
, 0,1 | |J νκ−
νκ
ν
Λ =
µ
. (12)
The longitudinal susceptibility can be calculated in the
MFA as [34]
0, 0,
,||
0, 0, 0, 0,
(1 )
(1 )(1 )
L J L L J L
J L J L J L J L
κ ν νκ κ ν ν κ κ
ν
ν ν κ κ κν ν νκ κ
′ ′ ′ ′µ β β +µ β − β
χ =
′ ′ ′ ′− β − β − β β
. (13)
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 9 951
P. Nieves, U. Atxitia, R.W. Chantrell, and O. Chubykalo-Fesenko
where ,= ( )eL Lν ν′ ′ ξ with , 0, , 0, ,= ( | | ) .e e eJ m J mν ν ν νκ κξ β +
The damping parameters are
0, , 0, ,
2 1= , = 1
e eJ J
ν νν
⊥ ν
ν ν
λ
α α λ − β β
, (14)
where 0, , 0, 0, , ,= | | ( / )e e eJ J J m mν ν νκ κ ν+ . Note that we
have checked that all the expressions above are the same
for two-sublattice ferrimagnetic and ferromagnetic alloys.
The LLB equation in the form given by Eq. (10) is very
useful for analytic estimations and predictions since the
relaxation terms have closed expressions. Particularly, one
can estimate several asymptotic behaviours of the relaxa-
tion times. Namely, for relatively low temperatures and not
too strongly coupled alloy we can estimate the longitudinal
relaxation time as
||
ex
, ,
1 ,
2 e em H
ν
ν ν ν ν
τ ≈
γ λ
(15)
where
0, ,ex
, ,= ,e
e e
JH mν
ν ν
νµ
(16)
is the homogeneous exchange field evaluated at the equi-
librium. Note that in this approximation, the relaxation
time is independent on the sign of the coupling between
sublattices (ferro or antiferro).
Close to the Curie temperatures one can prove that the
susceptibilities diverge, namely ,|| 1/ | |cT Tνχ ∝ − which is
the main source for the critical slowing down of the longi-
tudinal relaxation times near the Curie temperature. How-
ever, in weakly coupled ferrimagnets, only the material
with the largest exchange value slows down at the common
Curie temperature [35].
2.2. Classical LLB equation for two-component magnets
above cT
In the previous work [31] the derivation of the LLB
equation given by Eq. (10) was presented for temperatures
below the critical temperature cT . However, in ultrafast
magnetization dynamics the electronic temperature can
easily exceed cT . Therefore, it is interesting to analyze the
applicability of Eq. (10) for temperatures above cT . When
cT T→ the longitudinal susceptibility goes to infinity
,||( νχ → ∞ ) and the equilibrium magnetization to zero
,( 0em ν → ), as a consequence at = cT T the quantities ννΛ
and the damping parameters να
and ν
⊥α are undefined
a priori. However, the quantities ννΛ , να
and ν
⊥α are con-
tinuous functions at = cT T . In order to see this fact we can
rewrite these quantities in a form suitable for temperatures
very close to cT . To this end we notice that
0, 0,
0, 0,
,||
0, 0,
0, 0,
(3 )
,
6 (6 )
(3 )
,
3 (6 )( )
B c
c
B c B c
B c
c
B c B c
k T J J
T T
k T k T J J
k T J J
T T
k T k T J J
ν κ κ νκ
ν κ
ν
ν κ κ νκ
ν κ
µ − +µ
− − εχ µ − +µ
− − −ε
(17)
and
, 0,
, 0,
3
( ),
| |
e B c
c
e
m k T J
T T
m J
κ ν
ν νκ
−
+ ε ≈
(18)
where = ( ) /c cT T Tε − . Note that it follows from Eq. (9)
that 0, < 3 B cJ k Tν .
With the help of Eqs. (17) and (18) we can rewrite
Eqs. (12) and (14) as
___________________________________________________
0, 0, 0,
0, 0,
0,1
0, 0, 0,
0, 0,
6 ( )(6 ) 3
,
(3 )
3
( ) , =
3 ( )(6 ) 3
,
(3 )
B c B c B c
c
B c
B c
c
B c B c B c
c
B c
k T T k T J J k T J
T T
k T J J
k T J
T T T
k T T k T J J k T J
T T
k T J J
ν κ ν
ν κ κ νκ ν
ν−
νν
ν
ν κ ν
ν κ κ νκ ν
− − − −
+ µ − +µ µ
−Λ
µ
− − − −
+
µ − +µ µ
(19)
______________________________________________
and
2
= ( ), .
3 c
c
T
T T
T
ν ν ν
⊥
λ
α α + ε
(20)
Thus, above cT the longitudinal and transverse damping
parameters are equal and coincide with the expression [8]
for the classical LLB equation of a ferromagnet above cT ,
since at these temperatures the system becomes paramag-
netic at the equilibrium.
In order to check the validity of Eq. (10) for temperatures
above cT we consider a disordered ferrimagnet like GdFeCo
with parameters given in Table 1 and a rare-earth concen-
Table 1. Table with the parameters of GdFeCo used in the clas-
sical LLB equation. The exchange parameters are obtained through
a renormalization of the atomistic exchange parameters given in
Ref. 7 in order to obtain the same cT as in the atomistic approach.
FeCo Gd Gd–FeCo
(Joule)zJνκ 202.99·10− 201.19·10− 201.04·10−−
νλ 0.02 0.02 –
( )Bνµ µ 1.92 7.63 –
(Joule)dν 248.0·10− 248.0·10− –
1 1(rad·s ·Oe )− −
νγ 71.76·10 71.76·10 –
952 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 9
The classical two-sublattice Landau–Lifshitz–Bloch equation for all temperatures
tration of 25% ( = 0.25Gdx ). We consider zero external
magnetic field ( = 0H ) and uniaxial anisotropy in each
sublattice given by , ,= 2( / )A z zd mν ν ν νµH e where dν is
the atomistic anisotropy constant. In Fig. (1) we present the
calculated magnetization dynamics under a temperature-
step with initial temperature 0 = 500T K, below = 565cT K
and the final temperature = 650fT K, above cT . We inte-
grate both “paramagnetic” LLB equation with the MFA
expression for ferrimagnet, given by Eq. (3) and the “final”
approximate ferrimagnetic LLB equation, given by Eq. (10),
obtaining a good agreement. Above cT the quantity να
was
evaluated using Eq. (20), while since the final temperature
was far from cT the quantity ννΛ was evaluated using
Eq. (12) with 1/ 3Lν′ = , which comes immediately from
the fact that at this temperature , 0em ν = , instead of Eq. (19).
2.3. Comparison between the classical LLB equation
and atomistic simulations
In this subsection we compare the classical LLB equa-
tion for two-component magnets with many-body atomis-
tic simulations based on the stochastic LL equation (1). For
this task, we perform atomistic simulations for GdFeCo
ferrimagnetic compound, where the ultra-fast switching
has been observed. The parameters are taken from Ref. 7.
On the other hand, in the LLB model, as usual, we rescale
exchange parameters in order to obtain the same critical
temperatures cT as in atomistic model. These parameters
are presented in Table 1.
Firstly, we compare the magnetization relaxation dy-
namics under a temperature step using both approaches. In
Fig. 2 we show the magnetization dynamics obtained with
atomistic simulations (square), LLB equation given by
Eq. (3) called “paramagnet+MFA” (solid line) and LLB
equation given Eq. (10) called “final” (dash line) for an
electronic temperature-step where the temperature is varied
in step from 0 = 100T K to = 300fT K. Both LLB equa-
tions and atomistic simulation produce the same relaxatinal
dynamics. In Fig. 3 we show the same comparison but for
a higher electronic temperature-step with the final tempera-
ture = 500fT K which is higher than the Curie temperature
for Gd. We observe that in this case the time evolution
of the magnetization obtained using atomistic simulation
and the LLB equation given by Eq. (3) is very similar,
however the magnetization dynamics obtained using the
LLB equation given by Eq. (10) leads to a faster longitudi-
nal relaxation of Gd. This discrepancy may be related to
a large deviation from the equilibrium achieved due to
Fig. 1. (Color online) Magnetization dynamics obtained from
LLB equation given by Eq. (3) (solid lines) called “paramag-
net+MFA” and LLB equation given Eq. (10) (dash lines) called
“final” for a temperature-step where the initial temperature is
0 = 500T K which is below = 565cT K and the final temperature
is = 650fT K which is above cT .
Fig. 2. (Color online) (Top) Low electronic temperature step
where the initial temperature is 0 = 100T K and the final tempera-
ture is = 300 К.fT (Bottom) Magnetization dynamics obtained
using atomistic simulation (squares), LLB equation given by
Eq. (3) (solid lines) called “paramagnet+MFA” and LLB equa-
tion given Eq. (10) (dash lines) called “final” for a low tempera-
ture-step.
Fig. 3. (Color online) (Top) High electronic temperature step
where the initial temperature is 0 = 100T K and the final tempera-
ture is = 500 К.fT (Bottom) Magnetization dynamics obtained
using atomistic simulation (squares), LLB equation given by
Eq. (3) (solid lines) called “paramagnet+MFA” and LLB equa-
tion given Eq. (10) (dash lines) called “final” for a high tempera-
ture-step.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 9 953
P. Nieves, U. Atxitia, R.W. Chantrell, and O. Chubykalo-Fesenko
the high electronic temperature-step, in such situations the
accuracy of some approximations in the derivation of
Eq. (10) may be not sufficient and higher order corrections
should be included.
Finally, we show that the LLB equation describes switch-
ing in ferrimagnetic FeCoGd under the laser pulse heating.
The electronic temperature dynamics is modelled using the
two-temperature model [36] with parameters taken from
Ref. 7 and the laser pulse fluence 40 mJ/cm2. The electron-
ic temperature dynamics is presented in the upper panel in
Fig. 4 and the switching of GdFeCo obtained by atomistic
simulations is presented in the bottom panel in Fig. 4. In
the LLB simulations, however, a pure longitudinal motion
does not produce any torque on magnetization. This mo-
tion becomes unstable close to the point at which the mag-
netization of one of the sublattices is zero [25]. However,
even in this unstable situation numerically there is no
torque acting on the magnetization. This situation is similar
to the integration of the LL equation with field parallel to
the anisotropy where a small angle between them should
be used in order to move the system from the point where
the torque is zero. Similar to this, for the LLB equation one
needs to use a small angle between sublattice magnetiza-
tions (or alternatively the stochastic LLB equation should
be used). The results are presented in Fig. 5 where we see
that a switching of the sublattices occurs using the same
electronic temperature dynamics as in the atomistic ap-
proach showed in Fig. 4. Note that the stochastic nature of
the switching and/or a small angle between the sublattice
magnetizations naturally occur in the atomistic modelling.
As was pointed out in Ref. 25, in order to switch the mag-
netization with a ultrashort laser pulse it is necessary that
angular momentum is transferred from the longitudinal to
the transverse magnetization components. This conclusion
is also supported by atomistic modelling.
3. Conclusions
We presented a novel LLB equation for two-component
alloys which can model the separate dynamics of their
components in the whole temperature range. The new equa-
tion constitutes an important step forward in the descrip-
tion of the dynamics of two-coomponent alloys, such as
ferrimagnets which are traditionally modelled using two
coupled macroscopic LL equations. The two-component
LLB equation has been already successfully applied to
model FeCoGd [31] and more recently to FeNi [37] show-
ing that sublattices have distinct dynamics, in agreement
with experimental findings. Also the FMR and exchange
modes in ferrimagnets and their temperature dependence
are better understood within this approach [38].
These equations can serve in the future as a basis for
multiscale modeling in two-component systems at high
temperatures and/or ultrafast timescales, in the same way
as the LLB equation for ferromagnets [39]. This also opens
a possibility for novel micromagnetic modeling of ultrafast
and/or temperature-driven dynamics in large structures,
such as sub-micron and micron-size dots, stripes, nanowires
etc., made of two-components ferro or ferrimagnetic alloys.
Acknowledgements
This work was supported by the EU Seventh Frame-
work Programme FP7/2007-2013 under grant agreement
No. 281043, FEMTOSPIN. P.N. and O.C-F also acknow-
ledge support from the Spanish Ministry of Economy and
Competitiveness under the grant MAT2013-47078-C2-2-P.
U.A. acknowledges support from the European Community’s
Seventh Framework Programme (FP7/2007-2013) under
the Curie Zukunftskolleg Incoming Fellowship Programme
(Grant No. 291784), University of Konstanz.
1. J. Stöhr and H.C. Siegmann, Magnetism, Springer-Verlag,
Berlin, Heidelberg (2006).
2. http://math.nist.gov/oommf.
Fig. 4. (Color online) (Top) Electronic temperature dynamics
(green line). (Bottom) Magnetization response to this electronic
temperature dynamic obtained using atomistic simulation.
Fig. 5. (Color online) (Top) Electronic temperature dynamics
(green line). (Bottom) Magnetization response to this electronic
temperature dynamic obtained using the LLB equation given
Eq. (3) where the lattices magnetizations are deviated by 10°
from the antiparallel state.
954 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 9
The classical two-sublattice Landau–Lifshitz–Bloch equation for all temperatures
3. W.F. Brown, Phys. Rev. 130, 1677 (1963).
4. G. Grinstein and R.H. Koch, Phys. Rev. Lett. 90, 207201
(2003).
5. E. Beaurepaire, J.-C. Merle, A. Daunois, and J.Y. Bigot,
Phys. Rev. Lett. 76, 4250 (1996).
6. T.A. Ostler, J. Barker, R.F.L. Evans, R.W. Chantrell,
U. Atxitia, O. Chubykalo-Fesenko, S. El Moussaoui, L. Le
Guyader, E. Mengotti, L.J. Heyderman, F. Nolting, A.
Tsukamoto, A. Itoh, D. Afanasiev, B.A. Ivanov, A.M.
Kalashnikova, K. Vahaplar, J. Mentink, A. Kirilyuk, Th.
Rasing, and A.V. Kimel, Nature Commun. 3, 666 (2012).
7. J. Barker, U. Atxitia, T.A. Ostler, O. Hovorka, O.
Chubykalo-Fesenko, and R.W. Chantrell, Sci. Rep. 3, 3262
(2013).
8. D. Garanin, Phys. Rev. B 55, 3050 (1997).
9. D.A. Garanin, Physica A 172, 470 (1991).
10. O. Chubykalo-Fesenko, U. Nowak, R.W. Chantrell, and D.
Garanin, Phys. Rev. B 74, 094436 (2006).
11. U. Atxitia, O. Chubykalo-Fesenko, N. Kazantseva, D.
Hinzke, U. Nowak, and R.W. Chantrell, Appl. Phys. Lett. 91,
232507 (2007).
12. U. Atxitia, O. Chubykalo-Fesenko, J. Walowski, A. Mann,
and M. Munzenberg, Phys. Rev. B 81, 174401 (2010).
13. M. Sultan, U. Atxitia, O. Chubykalo-Fesenko, A. Melnikov,
and U. Bovensiepen, Phys. Rev. B 85, 184407 (2012).
14. J. Mendil, P. Nieves, O. Chubykalo-Fesenko, J. Walowski,
T. Santos, S. Pisana, and M. Münzenberg, Sci. Rep. 4, 3980
(2014).
15. V.G. Bar’yakhtar, Zh. Exp Teor. Fiz. 94, 196 (1988).
16. M. Dvornik, A. Vansteenkiste, and B. Van Waeyenberge,
arXiv:1402.3487.
17. B. Koopmans, G. Malinowsk, F. Dalla Longa, D. Steiauf,
M. Fhnle, T. Roth, M. Cinchetti, and M. Aeschlimann,
Nature Mater. 9, 259 (2010).
18. L. Xu and S. Zhang, Physica E 45, 72 (2012).
19. P. Nieves, D. Serantes, U. Atxitia, and O. Chubykalo-
Fesenko, Phys. Rev. B 90, 104428 (2014).
20. I. Radu, K. Vahaplar, C. Stamm, T. Kachel, N. Pontius, H.A.
Dürr, T.A. Ostler, J. Barker, R.F.L. Evans, R.W. Chantrell,
A. Tsukamoto, A. Itoh, A. Kirilyuk, Th. Rasing, and A.V.
Kimel, Nature 472, 205 (2011).
21. T.Y. Cheng, J. Wu, M. Willcox, T. Liu, J.W. Cai, R.W.
Chantrell, and Y.B. Xu, Magnetics, IEEE Transactions 48,
3387 (2012).
22. A. Hassdenteufel, B. Hebler, C. Schubert, A. Liebig, M.
Teich, M. Helm, M. Aeschlimann, M. Albrecht, and R.
Bratschitsch, Adv. Mater. 25, 3122 (2013).
23. K. Vahaplar, A.M. Kalashnikova, A.V. Kimel, D. Hinzke,
U. Nowak, R. Chantrell, A. Tsukamoto, A. Itoh, A. Kitilyuk
and Th. Rasing, Phys. Rev. Lett. 102, 117201 (2009).
24. J.H. Mentink, J. Hellsvik, D.V. Afanasiev, B.A. Ivanov,
A. Kirilyuk, A.V. Kimel, A.V.O. Eriksson, M.I. Katsnelson,
and Th. Rasing, Phys. Rev. Lett. 108, 057202 (2012).
25. U. Atxitia, T. Ostler, J. Barker, R.F.L. Evans, R.W. Chantrell,
and O. Chubykalo-Fesenko, Phys. Rev. B 87, 224417 (2013).
26. S. Alebrand, M. Gottwald, M. Hehn, D. Steil, M. Cinchetti,
D. Lacour, E.E. Fullerton, M. Aeschlimann, and S. Mangin,
Appl. Phys. Lett. 101, 162408 (2012).
27. A.R. Khorsand, M. Savoini, A. Kirilyuk, A.V. Kimel,
A. Tsukamoto, A. Itoh, and Th. Rasing, Phys. Rev. Lett. 108,
127205 (2012).
28. A. Kirilyuk, A.V. Kimel, and Th. Rasing, Rep. Prog. Phys.
76, 026501 (2013).
29. S. Mathias et al., Proc. Natl. Acad. Sci. USA 109, 479
(2012).
30. V.G. Bar’yakhtar, V.I. Butrim, and B.A. Ivanov, JETP Lett.
98, 289 (2013)
31. U. Atxitia, P. Nieves, and O. Chubykalo-Fesenko, Phys. Rev.
B 86, 104414 (2012).
32. H. Risken, The Fokker-Planck Equation: Methods of Solu-
tions and Applications, Springer (1989).
33. T. Ostler, R.F.L. Evans, R.W. Chantrell, U. Atxitia, O.
Chubykalo-Fesenko, I. Radu, R. Abrudan, F. Radu, A.
Tsukamoto, A. Itoh, A. Kirilyuk, T. Rasing, and A. Kimel,
Phys. Rev. B 84, 024407 (2011).
34. In Ref. 31. The longitudinal susceptibility was defined as
,|| =0( / )z Hzm Hν νχ ≡ ∂ ∂ where = | |mν νm . However, here
we use a more physically meaningful and convenient
definition which is ,|| , =0( / )z z Hzm Hν νχ ≡ ∂ ∂ , as result the
expression of the longitudinal susceptibility given by
Eq. (13) is slightly different than in Ref. 31.
35. O.J. Suarez, P. Nieves, D. Laroze, D. Altbir, and O.
Chubykalo-Fesenko, submitted to Phys. Rev. B.
36. M.I. Kaganov, I.M. Lifshitz, and L.V. Tanatarov, Sov. Phys.
JETP 4, 173 (1957).
37. D. Hinzke, U. Atxitia, K. Carva, P. Nieves, O. Chubykalo-
Fesenko, P. Oppeneer, and U. Nowak, Phys. Rev. B (2015),
accepted.
38. F. Schlickeiser, U. Atxitia, S. Wienholdt, D. Hinzke, O.
Chubykalo-Fesenko, and U. Nowak, Phys. Rev. B 86,
214416 (2012).
39. N. Kazantseva, D. Hinzke, U. Nowak, R.W. Chantrell, U.
Atxitia, and O. Chubykalo-Fesenko, Phys. Rev. B 77, 184428
(2008).
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 9 955
1 Introduction
2. The LLB equation for two-component magnets
2.1. General derivation
2.2. Classical LLB equation for two-component magnets above
2.3. Comparison between the classical LLB equation and atomistic simulations
3. Conclusions
Acknowledgements
|