Table Of ContentInfluence of the magnetoelectric coupling on the electric field induced magnetization
reversal in a composite non-strained multiferroic chain
Paul P. Horley1, Alexander Sukhov2, Chenglong Jia2, Eduardo Mart´ınez1 and Jamal Berakdar2
1Centro de Investigacio´n en Materiales Avanzados (CIMAV S.C.),
Chihuahua/Monterrey, 31109 Chihuahua, Mexico
2Institut fu¨r Physik, Martin-Luther Universita¨t Halle-Wittenberg, 06120 Halle (Saale), Germany
(Dated: January 24, 2012)
Westudytheoreticallythemultiferroicdynamicsinacompositeone-dimensionalsystemconsisting
2 of BaTiO3 multiferroically coupled toan iron chain. Themethod treats themagnetization and the
1 polarization as thermodynamic quantities describable via a combination of the Landau-Lifshits-
0 Gilbert andtheGinzburg-Landaudynamicscoupled viaan additionalterm inthetotalfree energy
2 density. This term stems from the multiferroic interaction at the interface. For a wide range of
n strengths of this coupling we predict the possibility of obtaining a well-developed hysteresis in the
a ferromagneticpartofthesysteminducedbyanexternalelectricfield. Thedependenceofthereversal
J modes on the electric field frequency is also investigated and we predict a considerable stability of
3 themagnetization reversal for frequencies in therange of 0.5−12 [GHz].
2
PACSnumbers: 85.80.Jm,75.78.-n,77.80.Fm,75.60.Ej,77.80.Dj
]
i
c
s I. INTRODUCTION we consider the rhombohedral phase17 at zero Kelvin
- with the corresponding ferroelectric potential. Under
l
r these circumstances it is possible to neglect the thermal
t Multiferroics (MFs), i.e. materials that possess ferro-
m stochastic effects on the effective fields. To improve the
magnetic and ferroelectric properties, have recently at-
. tracted significant research1–3 as they hold the promise accuracy of the model, we also included the long-range
t interactions for both the FE and the FM materials and
a ofqualitatively new device concepts suchaselectric field
m induced magnetization switching at low heat dissipa- the effect of the FE depolarizing field.
- tion. In addition to single-phase MFs4, i.e. materials
d
with ferroelectric (FE) and ferromagnetic (FM) or an-
n II. THEORETICAL FORMULATION
tiferromagnetic (AFM) ordering (e.g. BiFeO 5), com-
o 3
posite MFs6 are in the focus of current research. These
c
[ are systems realized as heterostructures of a wide range We consider a chain composed of FE (BaTiO3) and
of different materials7. This diversity is hoped to com- FM (Fe) sites as shownin Fig. 1. The aim is to describe
1 pensates for the scarcity of single-phase MFs for room- the evolution of the magnetization (polarization) under
v
temperature applications and to offer new routs to the the influence of external fields. We adopt a Ginzburg-
0
4 engineering of the magnetoelectric coupling and its en- Landau phenomenology based on coarse grained order
7 hancement, e.g. via an appropriate multilayer stacking. parameters that formally result from an averaging of
4 A typical example for such composite FE/FM junctions the relevant microscopic quantities over an appropri-
1. is BaTiO3/Fe that was predicted8 and studied9 theoret- ate cell. These cells are in our calculations for the
20 iizceadllya.nEdxcphearrimacetnetraizlleyd.B10a,1T1iOIn3/aFdedwitaiosns,uaccreescsefunltlypurebalil-- wumhoelea3FcEha=inac3FuMbe=s (ac3al=led5×he5re×aft5er[nsmit3e]s.)Wofeewquilallshvoowl-
1 cation reported on a considerable coupling between the results for a chain formed by five FE and five FM sites.
v: iron and the barium titanate which renders possible a Calculations performed for longer chains15 showed that
i change of the magnetization of the iron layer when an the total magnetization reversal is hardly achievable for
X electric field is applied, even at a room temperature.7 a chain that has more than ten magnetic sites.
r Another important issue concerns the significant varia- The coarse grained total free energy14,15 reads
a
tion of the tunnel magnetoresistance depending on the
F =a3F +a3F +E . (1)
polarization direction of BaTiO layer.7,12 All these ex- TOT FE FM CON
3
perimentalfindings areverypromisingstepstowardsthe
The free energy density of the ferroelectric
creationofmemoryelements13 basedoncompositeMFs,
which could be written by electric field pulses. Prob- F =F +F +FFE +FFE +F (2)
FE GLD DEP CPL DDI EXT
lems appearing on the way to achieve this goal include
the optimization of the geometry of FE/FM layers and includes the Ginzburg-Landau-Devonshire term
the improvement of the interaction between them. This F 18,19
GLD
paper is dedicated to the latter issue.
Incontrasttoourearlierstudies,14–16wherewefocused FGLD = αFE1(Pi2x+Pi2y+Pi2z)
on a more simple tetragonal phase of barium titanate in i
X(cid:2)
which the perovskite exists at room temperature,17 here +β (P4 +P4 +P4)+γ (P6 +P6 +P6)
FE1 ix iy iz FE1 ix iy iz
2
+β (P2P2 +P2P2 +P2P2)+γ P2P2P2 where µ is the susceptibility constant. The coupling
FE2 ix iy iy iz ix iz FE3 ix iy iz 0
betweentheferroelectricandtheferromagneticparts23is
+γ P4(P2 +P2)+P4(P2 +P2)+P4(P2 +P2)
FE2 ix iy iz iy ix iz iz ix iy duetothemobilespin-polarizedelectronsaccumulatedat
(3)
(cid:0) (cid:1)(cid:3) theinterfaceinordertoscreentheelectricpolarizationin
with the expansion coefficients α ,β and theFEpart.24 Achangeofthe accumulatedspindensity
FE1 FE1,2
γ . (e.g. duetoachangeofthe electricpolarization)willact
FE1−3
The depolarizing energy density F 20 (cf. Fig. 1) with a torque on the magnetization. This is however a
DEP
reads surface effect restricted to the region in the vicinity of
theinterface(whichisthereasonwhyweareconsidering
1 2λ P P
F = ( M ix ix) (4) short chains). In other words, only surface cells (those
DEP
2 aN εDLε0 withindex 1inFig. 1)willparticipateinthis coupling25
i
X
which contributes with the interaction energy
which involves the dead layer permittivity ε , the di-
DL
electricconstantinvacuumε0,themetalscreeninglength E =a3λP~ ·M~ . (11)
CON 1 1
λ ,theFEcellsizea,andthecellnumberN. Theferro-
M
eisl2e1ctric nearest neighbors coupling energy density FCFPEL neTtizhaettioimneM~dynoafmtichseofintdhievpidoulaarlizsaitteiosnaP~rei aonbdtathineemdabgy-
j
FFE =κ (P −P )2 propagating the Landau-Khalatnikov (LKh)26,27 equa-
CPL FE (i+1)x (i)x
tion
i
X(cid:2)
+(P −P )2+(P −P )2
(i+1)y (i)y (i+1)z (i)z dP~ 1 δF
γ i =E~ ≡− TOT, (12)
where κFE is the ferroelectric coupling consta(cid:3)nt. The ν dt FEi a3 δP~
i
dipole-dipole interactions FFE is
DDI
and the Landau-Lifshitz-Gilbert (LLG)28,29 equation
1 P~ ·P~ −3(P~ ·~e )(~e ·P~ )
FFE = i k i ik ik k .
DDI 4πεFEε0 Xi6=k" n3ik # ddM~tj =−1+γα2 [M~j ×H~FMj]
(5) FM
n
Here ε is the ferroelectric permittivity, ~e is a unit α
FE ik + FMM~ (M~ ·H~ )−α M H~ . (13)
vectorparalleltothelinejoiningthecentersofthedipoles M j j FMj FM S FMj
P~ and P~ and n is the distance (measured in units of S (cid:27)
i k ik
ThecoefficientsenteringintotheLKharetheviscosity
a) between the two dipoles.
TheenergytermFEXT stemsfromtheappliedelectric constantγν andtheeffectiveelectricfieldE~FEi. TheLLG
field E~ equation involves the gyromagnetic ratio γ, the Gilbert
damping coefficient α , the saturation magnetization
FM
FEXT =− E~ ·P~i. (6) M , and the effective field H~ =− 1 δFTOT.
i S FMj a3 δM~
X j
From a computational point of view the appropriate
Analogouslytheferromagneticpartischaracterizedby
choice of the cell size a is important. If it is cho-
the coarse grained free energy density
sen too small, the coarse graining procedure to obtain
FFM =FANI+FXFCMG+FDFDMI (7) the macroscopic quantities P~i and M~j becomes ques-
tionable and one faces in addition problems with the
which consists of the (uniaxial) magnetocrystalline
superpara-magnetic/electric limits. For a cell too large,
anisotropy contribution22
a multi-domain state sets in. The cell size used in our
K calculation is therefore a = 5 [nm] for both FE and
F =− 1M2 (8)
ANI M2 xj FM parts of the chain. The material parameters were
Xj S set as those of barium titanate at T = 0 [K]. Specif-
ically, we choose α = −1.275 × 108 [V·m/C],30
withtheanisotropyconstantK andthesaturationmag- FE1
1 β = −2.045 × 109 [V·m5/C3],30 β = 3.230 ×
ntieotnizhaatisonthMe fSo.rmT2h2e nearest-neighbor exchange interac- 10F8E1[V·m5/C3],30 γFE1 = 9.384 × 109FE[V2 ·m9/C5],30
γ = 4.470 × 109 [V·m9/C5],30 γ = 4.919 ×
FE2 FE3
FXFCMG =− j a2AMS2M~j ·M~j+1 (9) 100.499[9V[·Cm/m9/2C],53]2,κ30FEγν==21..054××101−058[V[V··mm·/sC/]C,2]1,31εFPES ==
X 164.33 The depolarizing energy density F includes
DEP
whereAistheinteractionconstant,andthedipole-dipole the permittivity of the metallic electrodes, or the so
interaction is called “dead layer”,34 which can be lower than the per-
mittivity of the material. This parameter is difficult to
µ M~ ·M~ −3(M~ ·~e )(~e ·M~ )
FFM = 0 j l j jl jl l (10) measure35. Inthe numericalcalculationweusedthe rea-
DDI 4π n3
j6=l" jl # sonable value εDL = εFE/2 = 82. The material of the
X
3
FIG. 1. Schematics of the composite multiferroic structure
formed of five ferroelectric and five ferromagnetic sites. The
initial state for both theFE polarization and themagnetiza-
tion is chosen as random.
FM layer is taken as iron with the following parameters
at T = 0 [K]: α = 0.5,36 K = 4.8 × 104 [J/m],37
FM 1
M =1.71×106 [A/m],37 A=2.1×10−11 [J/m].37
S
A special role in the multiferroic dynamics is played
by the coupling strength λ. We estimate the strength
of the magnetoelectric coupling parameter related to
the BaTiO /Fe-interface using the proposed ab-initio
3
expression8,38 α = µ ∆M/E , where the surface mag-
S 0 C
FIG. 2. The difference in the time scale for the ferroelectric
netoelectric coupling α is defined as the change of the
S
(upper panel) and the ferromagnetic (lower panel) reversals.
surface magnetization ∆M for the electric coercive field
E . The coercive field can be estimated as E ≈ The FE/FM coupling constant is taken as λ=10 [s/F].
C C
P /(ε ε ). Keeping in mind that the coupling energy
s FE 0
(11) might also be expressed through the induced mag-
sion, causing a fast and a well-developed reversal. On
netization ∆M at the interface and the net ferromag-
0 the contrary,the magnetizationreversalrequiresa much
netic magnetization M as E = −J/(M2)∆M~ ·M~,
CON S 1 larger time up to a nanosecond. As one can see from
we finally obtain λ = Jα /(µ M2a4ε ε ). For α =
S 0 S FE 0 S the plot, the re-orientation of the magnetization vector
0.2 · 10−17 [T m2/V]9 the last expression yields λ ≈
involves a heavy precession with considerable deviations
0.063[s/F].Thisnumericalvaluedidnotresultinanysiz-
of the mA and mA components from zero. These oscil-
able interactionbetweenFE andFM layersin the model y z
lations have a higher frequency at the beginning of the
considered. We attribute this situation to the difficulty
reversal process, gradually lowering the frequency and
in the numerical definition of λ, because the expression
theamplitudetilltheequilibriumstateisreached. Natu-
for the coupling constantdepends stronglyon the size of
rally,suchadistincttimescaleofthesystemcomponents
theinterfacecella. Underthesecircumstances,the most
poses a significant problem for a proper modeling. To
adequate approach was to consider λ as a variable, with
ensure the accurate numerical solution, one should keep
the aim to obtain the general picture of magnetization
the integration time step at femtoseconds, which drasti-
reversal induced by an electric field as a function of the
callyincreasesthe numberofintegrationstepsneededto
magnetoelectric coupling.
follow the system dynamics on time scales adequate to
As in our previous studies14,15 the MF-chain will be observethe hysteresiscurvesformedunderthe fieldvari-
characterized by the averaged total polarization p~A = ation with GHz frequencies. This large number of steps
(NPS)−1 iP~i and the averaged net magnetization definitelywillbeanissueforthecalculationsoflargesys-
m~A =(NM )−1 M~ . tems composed of hundreds of particles due to the need
PS j j
To reverse the FM part via the FE part of the chain, toevaluatethelongrangedipole-dipoleinteractionfields
P
we apply the harmonic electric field E (t) = E sin(ωt) in the ferroelectric and the ferromagnetic parts. In our
x 0
with the amplitude E =8×107 [V/m]. As a frequency case, the LKh and the LLG equations were integrated
0
we choose at first ω/(2π) = 2 [GHz]. The time profiles with the Heun method using time step ∆t = 2 [fs].
of the field reversal for the ferroelectric and the ferro-
magnetic parts of the system are given in Fig. 2. As
III. NUMERICAL RESULTS AND
one can see, the ferroelectric part re-polarizes quickly
DISCUSSIONS
withinapproximately300[fs]. Thereversalmodifiesonly
the x-component of the polarization, with a very minor
variation of the z-component in the vicinity of the point To study the influence of the coupling constanton the
where pA changes its sign. This happens because under behavior of the system, it is necessary to select param-
x
anegativebiasthe polarizationvectorofthe pre-contact eters that allow an easy and a reliable characterization
diminishes in magnitude to zero, and then reverses the of the hysteresis curve. We propose to use the values of
direction along the field. The remaining cells flip their theaveragedpolarizationandthemagnetization(pA and
polarization vectors along the x-axis without a preces- mA, respectively) at a zero applied field, as well as the
4
FIG. 3. The dependence of the multiferroic reversal on the strength of FE/FM interaction λ. The hysteresis curves are
presented for: a) λ = 0.1 [s/F], b) λ = 0.255 [s/F], c) λ = 2 [s/F], d) λ = 12 [s/F], e) λ = 45.5 [s/F]. The hystograms in
the lower part of the figure depict the remanence (pAR, mAR) end coercitivity (ECFE, ECFM) for the averaged hysteresis curves
obtained for ferroelectric and ferromagnetic parts of the chain. The applied electric field is characterized by theamplitude E0
= 80 [MV/m] and frequency ω/(2π) = 2 [GHz].
fieldvaluesE forwhichthemagnetizationorthepolar- bands instead of the thin dark curves. The light-colored
C
izationareequaltozero. Ifthehysteresisloopisproperly noise for the positive fields on the coercitivity diagrams
formed, these points will define the remanence and the correspondtothe pointswhenthe polarization/magne-
coercitivity ofthe hysteresisloop. If the hysteresisis ab- tizationchangessignbeforereachingthestablehysteresis
sent,itwillbecomeimmediatelynotable,forexample,by curve. Insomecases,such“self-adjustment”ofthephase
the existenceofasingle remanencepointorthe presence trajectory takes place during the several first cycles of
of various zeros of magnetization / polarization curve. the electric field, which complicates the definition of an
To collect the data shown in Fig. 3 we integrated the exact threshold that would allow to remove this noise.
system for 15 full field cycles and plotted the density of On the other hand, as we are studying steady hysteresis
pA = pA(E = 0), mA = mA(E = 0) and EFE, EFM curves observed during the 15th field cycle, such initial
R R C C
as a function of the coupling parameter λ. These plots noise is irrelevantandcanbe easily neglected. The most
are shown in the bottom part of Fig. 3. This approach characteristiccaseswithFEandFMhysteresiscurvesil-
allows to see if the hysteresis curve is well-defined and lustratedinthe upperpartofthe figurearemarkedwith
repeatable, which will result in sharp and dark lines on arrows in remanence and coercitivity diagrams.
the corresponding plot. If a hysteresis loop is unstable
and varies from cycle to cycle, one obtains the grayish As seen from the figure the coupling strength λ < 0.8
[s/F]is insufficientto reversethe magnetization. Forthe
5
smallvalueλ=0.1[s/F](Fig. 3a)thevalueofmAiscon- ifests itself by rounded ”corners” of the hysteresis curve
x
stant and matches the saturation magnetization of the (Fig. 4b)thatbecomesmorepronouncedforafrequency
material, while the ferroelectric part manages to reach around 7 [GHz]. At this value, one can also observe a
an entire hysteresis cycle. The steps at the edges of the slight decrease in the magnetic remanence (Fig. 4c).
hysteresis are caused by the discussed peculiarity of FE Whenω/(2π)exceedsthecharacteristicfrequencyofiron
reversalmechanism that involves a vanishing of the first 2γK /[(1+α2 )M µ ]=7.89[GHz],thebehaviorofthe
1 FM S 0
site before the field achieves the reversal value. For a systemchangesdrastically. TheFMpartdoesnotfollow
stronger coupling λ = 0.255 [s/F] (Fig. 3b) the torque the fast FE dynamics so that the curve of the magnetic
rendered by the polarization “kicks” the magnetization hysteresis features several ”breaks” and its coercitivity
with a frequency that is similar to the FM precession slightlydropsdown. Itisessentialtostressthat,because
frequency, resulting in a periodic variation of mA, mA we are dealing with a complex multi-site system driven
x y
and mA. These oscillation modes may be interesting as by FE/FM interaction, the changes of the magnetiza-
z
a way to achieve a resonant magnetization precession. tiondynamicsarenoticeableforthe frequencyofelectric
However,theydonotallowtoreachasaturationmagne- fields slightly lower than the characteristic frequency of
tizationandtoformthehysteresisloop,whichisrequired iron. The magnetization reversal is accompanied with a
for the operation as memory devices. strong deviation of my and mz components (Fig. 4d).
A further increase of the frequency destabilizes the sys-
The full ferromagnetichysteresisloopemergesfor λ>
temevenmore,withthemagnetizationhysteresislooking
0.8 [s/F], quickly reaching the full saturation value M .
S
“skewed”andthecoercitivityvalues“jumping”backand
Due to the fact that the FM part is much slower than
forth between two branches(Fig. 4e). It should be men-
FE,themagnetichysteresishasalargercoercitivity,how-
tionedthattheFMremanencelowersconstantlyforafre-
ever. Thisis mostclearlyseenforthecaseofarelatively
quency ω/(2π) > 6.5 [GHz], signaling that the achieved
weak coupling λ = 2 [s/F] (Fig. 3c), where FM dynam-
operationmodesarenotverypromisingforapplications.
ics definitely can not follow FE dynamics at the desired
speed. We observe pronounced oscillations in the mA Also it is important to stress that the considered value
y
of the coupling constant λ = 12 [s/F], while allowing a
and mA components upon the field-induced reversal. It
z good influence of the FE over the FM layer, provides a
is worth mentioning that the FE part of the system is
minimal feedback that can be seen in a lack of the dras-
not influenced much by the variation of the magnetiza-
tic changes of the remanence and the coercitivity for the
tion for these values of λ, for the ferroelectric hystere-
FE part of the system. It goes without saying that such
sis exhibits only a minor enhancement of the remanence
a pronounced unidirectional connection is an important
and has almost the same coercitivity. An increase in the
featureforadevicethatisaimedtocontrolthemagnetic
magneto-couplingstrenght λ to 12 [s/F] (Fig. 3d) seems
dynamics with an electric field via FE/FM coupling.
to yield desirable results: the magnetization hysteresis
Basedonthese simulations, we wouldrecommend to use
is saturated and narrow, with a fast relaxation of the
the described compound multiferroic system under low
mA and mA components after the full reversal of mA.
y z x GHz frequency of the applied electric fields.
As the FE/FM feedback becomes more pronounced, the
hysteresis of FE shows a larger coercitivity and rema-
nence. The observed “mirror-symmetry” of the FE and
IV. CONCLUSIONS
FM hysteresis loops has its origin in the opposite signof
the coupling terms in the LKh and LLG equations. For
a large coupling λ = 45.5 [s/F] (Fig. 3e) both hystere- Wereportedonafull-scalemagnetizationreversalina
siscurvesdeteriorate,asthe ferromagneticpartstartsto composite multiferroic chain using an applied harmonic
“hold” the FE sites, hindering their reversaluntil a con- electricfieldwithfrequencies0.5−12[GHz]. Thedynam-
siderable electric field is applied. A further increase of ics and in particular the reversal depend sensitively on
the coupling strength destroys the hysteresis of both the the strength of the magneto-electric coupling λ. Hence,
FE and FM layers as λ exceeds 60 [s/F]. tracing the dynamics should deliver information on this
Toaddressthefrequencydependence ofthe composite coupling. For a weak coupling λ < 0.8 [s/F], the ferro-
multiferroic reversal, we calculated the remanence and electricpartshowswell-definedhysteresisloopwhichmay
coercitivity diagrams (Fig. 4) for the frequency range result in periodic oscillations of the ferromagnetic part.
ω/(2π) =0.5-12 [GHz]. As one can see from the figure, AsaturationmagnetizationM maynotbereachedhow-
S
for a low frequency ω/(2π) < 7.2 [GHz] the system is ever. A magnetic hysteresis opens when λ grows above
well-tuned, featuring a complete magnetization reversal 0.8 [s/F]. For a coupling constant strength in the ranges
triggered by the ferroelectric component of the struc- 10−20[s/F],themultiferroicsystemhasanoptimalper-
ture. The increase in the frequency results in a linear formancewithbothFEandFMhysteresiscurvesfeatur-
increase of the coercetivity field for the ferromagnetic ing a high remanence and a low coercitivity. It is im-
hysteresis. The low-frequencyhysteresishasa sharpand portanttohighlightthattherangeofλthatcorresponds
a well-defined shape (Fig. 4a). The slow response of to the FM hysteresis with the most definite shape does
the FM to the faster oscillations of the FE polarization not vary with the frequency of the applied electric field.
becomes noticeable under the increase of ω, which man- Whenthecouplingconstantbecomestoostrong(exceed-
6
FIG.4. ThefrequencydependenceoftheFE/FMreversalforλ=12[s/F]. Themostcharacteristic hysteresiscurvesaregiven
for the following values of thefield frequency ω/(2π): a) 0.5 [GHz], b) 4 [GHz], c) 7 [GHz], d) 9 [GHz] and e) 11.4 [GHz].
ing 40 [s/F]) the system does not show any hysteresis. ferent in general. Hence, in addition to the relevance for
Thisdegradationiscausedbythefactthatthe ferroelec- application, an attractive feature of studying the com-
tric dynamics is strongly disturbed by the ferromagnetic posite dynamics is that it may deliver some details on
layer. the underlying multiferroic coupling mechanisms. Stud-
We considered here one type of magnetoelectric cou- ies along this line are currently underway.
V. ACKNOWLEDGEMENTS
pling based on a screening model at the interface of the
FE/FM parts. A different type of magnetoelectric cou-
pling mechanism, e.g. a stress-strain coupling or a dy- The authors gratefully acknowledge M. Alexe and I.
namical Dzyaloshinskii-Moriya coupling39, may qualita- Vrejoiu for fuitful discussions. This work has been sup-
tively alter the coupled dynamics (even when the cou- ported by the grants of CONACYT as Basic Science
pling strengths are comparable), for the corresponding Projects 129269 and 133252 (Mexico), by the German
functional forms entering the equationofmotion aredif- Research Foundation SU 690/1-1 and by the SFB 762.
1 M. Fiebig, J. Phys.D: Appl.Phys. 38, R123 (2005). 4 J.-P. Rivera, Eur. Phys.J. B 71, 299 (2009).
2 W. Eerenstein, N.D.Mathur, J.F. Scott, Nature 442, 759 5 J.Wang,J.B.Neaton,H.Zheng,V.Nagarajan,S.B.Ogale,
(2006). B. Liu, D. Viehland, V. Vaithyanathan, D.G. Schlom,
3 R. Ramesh, N.A.Spaldin, Nat. Mater. 6, 21 (2007). U.V. Waghmare, N.A. Spaldin, K.M. Rabe, M. Wuttig,
7
R. Ramesh, Science299, 1719 (2003). 23 C.A.F.Vaz,J.Hoffman,C.H.Ahn,R.Ramesh,Adv.Mat.
6 C.-W.Nan,M.I.Bichurin,S.Dong,D.Viehland,G.Srini- 22, 2900 (2010).
vasan, J. Appl.Phys.103, 031101 (2008). 24 T.Cai,S.Ju,J.Lee,N.Sai,A.A.Demkov,Q.Niu,Z.Li,
7 S. Valencia, A. Crassous, L. Bocher, V. Garcia, X. Moya, J. Shi, and E. Wang,Phys. Rev.B 80, 140415(R) (2009).
R. O. Cherifi, C. Deranlot, K. Bouzehouane, S. Fusil, A. 25 J.-W. Lee, N. Sai, T. Cai, Q. Niu, A.A. Demkov, Phys.
Zobelli, A. Gloter, N.D. Mathur, A. Gaupp, R. Abrudan, Rev.B 81, 144425 (2010).
F. Radu, A. Barth´el´emy, and M. Bibes, Nature Materials 26 L.D. Landau, I.M. Khalatnikov, Dokl. Akad. Nauk SSSR
DOI:10.1038/NMAT3098 (2011). 46, 469 (1954).
8 C.-G. Duan, S.S. Jaswal, E.Y. Tsymbal, Phys. Rev. Lett. 27 S. Sivasubramanian, A. Widom, Y.N. Srivastava, Ferro-
97, 047201 (2006). electrics 300, 43 (2004).
9 M. Fechner, I.V. Maznichenko, S. Ostanin, A. Ernst, J. 28 L.D.Landau,E.M.Lifshitz,Phys.Z.Sowjet.8,153(1935).
Henk,I. Mertig, Phys.Stat. Solidi B 247, 1600 (2010). 29 T.L.Gilbert,Phys.Rev.100,1243(1955)(abstractonly);
10 S. Sahoo, S. Polisetty, C.-G. Duan, S. S. Jaswal, E. Y. T.L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).
Tsymbal, and C. Binek, Phys. Rev.B 76, 092108 (2007). 30 Taken from AppendixA of Ref. [20].
11 H.L. Meyerheim, F. Klimenta, A. Ernst, K. Mohseni, S. 31 J. Hlinka, Ferroelectrics 349, 49 (2007).
Ostanin,M.Fechner,S.Parihar,I.V.Maznichenko,I.Mer- 32 Calculated from the minimization of the given GLD-free
tig,andJ.Kirschner,Phys.Rev.Lett.106,087203(2011). energy.
12 V. Garcia, M. Bibes, L. Bocher, S. Valencia, F. Kronast, 33 Calculated from the minimization of the given GLD-free
A. Crassous, X. Moya, S. Enouz-Vedrenne, A. Gloter, D. energy as themain diagonal element of the dielectric con-
Imhoff, C. Deranlot, N.D. Mathur, S. Fusil, K. Bouze- stant tensor.
houane, and A. Barth´el´emy, Science 327, 1106 (2010). 34 A.M.Bratkovsky,A.P.Levanyuk,J.Comp.Theor.Nanosc.
13 M. Bibes, A. Barth´el´emy, Nat. Mater. 7, 425 (2008). 6, 465 (2009).
14 A. Sukhov, C.-L. Jia, P.P. Horley, and J. Berakdar, J. 35 D.J. Kim, J.Y. Jo, Y.S. Kim, Y.J. Chang, J.S. Lee, J.-
Phys.: Cond. Matt. 22, 352201 (2010). G. Yoon, T.K. Song, and T.W. Noh, Phys.Rev. Lett.95,
15 C.-L. Jia, A. Sukhov, P.P. Horley, and J. Berakdar, J. 237602 (2005).
Phys.: Confer. Ser. 303, 012061 (2011). 36 The damping coefficient varies with the size of the com-
16 A.Sukhov,P.P.Horley,C.-L.Jia, andJ.Berakdar,Ferro- putationalcell. Thevalueused inthispaperwas obtained
electrics (at press). by“equalization”oftheitemsappearingintherighthand
17 W.J. Merz, Phys.Rev.76, 1221 (1949). side of equation (13), ensuring numerical stability of the
18 V.L. Ginzburg, Usp. Fiz. Nauk 38, 490 (1949). system and a reasonable relaxation of the magnetization
19 A.F. Devonshire, Philos. Mag. 40, 1040 (1949). vector.
20 Physics of Ferroelectrics, K. Rabe, Ch. H. Ahn and J.-M. 37 Magnetism and magnetic materials, J.M.D. Coey, (Cam-
Triscone (Eds.), (Springer, Berlin 2007). bridge University Press, Cambridge 2010), pp. 385.
21 J. Hlinka,P. M´arton, Phys.Rev.B 74, 104104 (2006). 38 C.-G. Duan, J.P. Velev, R.F. Sabirianov, Z. Zhu, J. Chu,
22 Physics of Ferromagnetism, S. Chikazumi, (Oxford Uni- S.S. Jaswal, and E.Y. Tsymbal, Phys. Rev. Lett. 101,
versity Press Inc., New York2002). 137201 (2008).
39 C.L. Jia, J. Berakdar, J. Superconductivity and Novel
Magnetism (at press) arXiv:1101.2067.
