Electric field driven magnetic domain wall motion in ferromagnetic-ferroelectric heterostructures

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by  Ben Van de Wiele

Electric field driven magnetic domain wall motion in ferromagnetic-ferroelectric
heterostructures
1 2 3 3
Ben Van de Wiele , Lasse Laurson , K´evin J. A. Franke , and Sebastiaan van Dijken
1Department of Electrical Energy, Systems and Automation, Ghent University, Ghent B-9000, Belgium
2COMP Centre of Excellence, Department of Applied Physics,
Aalto University, P.O. Box 11100, FIN-00076 Aalto, Espoo, Finland and
3NanoSpin, Department of Applied Physics, Aalto University School of Science, P.O. Box 15100, FI-00076 Aalto, Finland
We investigate magnetic domain wall (MDW) dynamics induced by applied electric fields in
4
ferromagnetic-ferroelectric thin-film heterostructures. In contrast to conventional driving mecha-
1
nismswhereMDWmotionisinduceddirectlybymagneticfieldsorelectriccurrents,MDWmotion
0
arises here as a result of strong pinning of MDWs onto ferroelectric domain walls (FDWs) via lo-
2
cal strain coupling. By performing extensivemicromagnetic simulations, we findseveral dynamical

n regimes,includinginstabilitiessuchasspinwaveemissionandcomplextransformationsoftheMDW
a structure. In all cases, the time-averaged MDW velocity equals that of the FDW, indicating the
J
absence of Walker breakdown.

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1 PACSnumbers: 75.78.Fg,75.30.Gw,75.78.Cd

]
i
c Magnetic domain wall (MDW) dynamics in nanoscale velocity. Thus, the sharp decrease of the average veloc-
s ferromagnetic wires and strips, as well as in thin films, ity associatedwith Walkerbreakdown[6, 12–15] inmag-
-
l is a subject of major technological importance for the netic field and electric current driven MDW dynamics is
r
t operation of potential future magnetic memory [1] and absent. For small FDW velocities, the MDW coupled
m
logic devices [2, 3]. While current efforts to construct to it is found to follow the moving FDW nearly qua-
. such devices mostly focus on spin-polarized electric cur- sistatically, without significant changes in the internal
t
a rents [4, 5] or applied magnetic fields [6] as the driv- MDW structure. Above a threshold velocity, this close-
m
ing force, a promising low-power alternative has been to-quasistatic behavior breaks down, with various insta-
- demonstrated in recent experiments [7] where electric bilities occurring depending on the MDW type (charged
d
n fieldsmovetheMDWsinferromagnetic-ferroelectrichet- or uncharged) and the material parameters. For un-
o erostructures. In such configurations, the MDWs in the charged MDWs these instabilities comprise oscillatory
c ferromagnetic layer are strongly pinned onto ferroelec- MDW motions or complicated transformations of the
[
tric domain walls (FDWs) in the ferroelectric sublayer MDWstructure,whiletheemissionofregularspinwaves

1 via elastic interactions. Consequently, when an applied is observed for charged MDWs.
v electricfielddisplacestheFDWs,theMDWsaredragged
9 In ferromagnetic-ferroelectric thin-film heterostruc-
along.
5 tures, the anisotropy of the ferromagnetic layer is later-
9 Fromafundamentalphysicspointofview,thequestion
allymodulatedvialocalstraintransferfromferroelectric
3 of the nature of this driving protocol is very important.
domains and inverse magnetostriction. If polarization
.
1 Indeed, the electric field driving mechanism of MDWs rotation between ferroelectric domains is less than 180◦
0 differs substantially from the more conventional driving
(i.e. if the domain pattern consists of FDWs that are
4 modes where either a magnetic field or spin-polarized
both ferroelectric and ferroelastic), minimization of the
1
electriccurrentareusedtomoveMDWs. Whiletheeffect
: anisotropyenergycanleadtofullimprintingofferroelec-
v of an applied electric field on magnetic field [8–10] and
tricdomainsintotheferromagneticlayer[16–19]. Impor-
Xi spin-polarizedcurrent[11]drivenMDWmotionhasbeen tantly, abrupt rotation of the ferroelectric polarization
considered,the nature offully electric fielddrivenMDW
r at FDWs and the concurrentinstant change of magnetic
a dynamics in ferromagnetic-ferroelectric heterostructures
anisotropyin the adjacentferromagnetstrongly pins the
remains elusive.
MDWsontotheirferroelectriccounterparts. Thereverse
InthisLetter,wepresentadetailednumericalstudyof effect, i.e. modulation of the ferroelectric properties due
electric field driven MDW dynamics, including the short to magnetizationreversalinthe ferromagneticfilm, does
time scale details which have so far not been accessible not occur due to a pronounced asymmetry in the strain
◦
experimentally. We consider two different 90 MDWs, coupling mechanism. The maximum strain that can be
one being magnetostatically charged and the other un- transferred from a ferroelectric domain is given by the
charged. Our results highlight the different nature of elongation of the structural unit cell. For archetypical
theelectricfielddrivingmechanismascomparedtowell- tetragonal BaTiO3 at room temperature, this is 1.1%.
known magnetic field and electric current driven MDW Strain transfer from a ferromagnetic material to a ferro-
motion: ForallappliedFDWvelocities,theMDWmoves electric layer via magnetostriction, on the other hand, is
along with a time-averaged velocity equal to the FDW typically several orders of magnitude smaller [20]. As a
2
y
(a)
x
v v
AB AB K
Ku Ku u
(b)
M M (d) (e)
x y
(c)
FIG.1. (coloronline) (a)Sketchofthemicromagnetic simulation geometry. Thesystemisdiscretizedintwodimensionsusing
finite difference cells of 3.125×3.125×15nm3. The micromagnetic structures of the equilibrium uncharged and charged 90◦
MDWs are shown in (b) and (c), respectively, along with the color code used. Each arrow represents the locally averaged
magnetization over a 12.5×12.5×15nm3 volume, i.e. the scale of the plots in (b) and (c) is identical. (d) Evolution of the
maximum out-of-plane magnetization component and MDW width with AB velocity. (e) Lagging behind distance in the
quasistatic regime with vAB < vth [27]. For the reference material parameters used here, vth ≈ 850 m/s for the uncharged
MDW and vth≈1200 m/s for the charged MDW.
result,ferromagneticeffectsonthedynamicsoftheferro- these values to investigate their effect on the observed
electric sub-system can be safely neglected, and electric MDWdynamics. By consideringperiodicboundarycon-
field induced MDW propagation is accurately modeled ditionsinbothx-andy-direction,wemimicMDWprop-
using a moving magnetic anisotropy boundary (AB) in agation in an infinite film. The simulation window is
the ferromagnetic layer. restricted to two 6.4µm wide domains with orthogonal
In our micromagnetic simulations, an AB separating anisotropy axes over a length of 200nm, see Fig. 1(a).
◦
two regions of uniform uniaxial anisotropies is moved The resulting 90 MDWs can be of two different types,
with constant velocity v (Fig. 1(a)), corresponding magnetostatically charged or uncharged, depending on
AB
to the propagationvelocity of the underlying FDW. The the magnetization directions in the domains [26]. The
◦
anglebetweenthemagneticanisotropyaxesis90 ,which widthsofthesetwoMDWtypesdiffersubstantiallyasil-
in practice can be obtained by strain coupling to in- lustratedbytheirmicromagneticstructuresinFigs. 1(b)
plane domains of BaTiO3 [16, 17]. In an experiment, and(c). Weconsiderbothcasesseparatelyandshowthat
v would be controlled by the magnitude of an ap- their dynamic behavior is very different.
AB
plied electric field [21–23]. Coarse-grained Monte Carlo
For small imposed v , both the charged and un-
AB
simulations indicate that ferroelectric DW speeds of up
charged MDW follow the motion of the AB. While the
to several km/s are possible [24]. Our simulations are
magnetization of the uncharged DW is completely in-
performedwiththeGPU-basedmicromagneticsimulator
plane at rest, an out-of-plane magnetization component
MuMax [25]. To study the time evolution of the magne-
developsandthe DWwidthreduceswithincreasingv
AB
tizationM(r,t),wesolvetheLandau-Lifshitz(LL)equa-
(Fig. 1(d)). These deformations are similar to those ob-
tion,
served during magnetic field and current driven MDW
∂M γ motion in magnetic nanowires and strips, where the ap-
=− M×H (1)
∂t 1+α2 eff pearance of an out-of-plane magnetization component
− αγ M×(M×H ), and a narrowing of the MDW are precursors of Walker
M (1+α2) eff breakdown[6, 12–15]. In addition, the uncharged MDW
s
where H is the effective magnetic field (with con- lags slightly behind the AB by a distance that increases
eff
with v (Fig. 1(e)). For higher AB speeds, a break-
tributions from the exchange, anisotropy and demag- AB
down of quasistatic MDW motion occurs at a threshold
netizing fields), and γ is the gyromagnetic ratio.
velocity v . In comparison,the internalstructure of the
The reference sample in this study is a 15nm thick th
chargedMDW is muchmorerobustagainstdynamic de-
Co60Fe40 layer on top of a BaTiO3 substrate, corre-
formations, as illustrated by the negligible out-of-plane
sponding to the experimental system in [16], with sat-
uration magnetization M =1.7×106A/m, uniaxial mag- magnetization and nearly constant MDW width in Fig.
s
netic anisotropy strength K =1.7×104J/m3, exchange 1(d).
u
constant K =2.1×10−11J/m, and damping constant The dynamics of the uncharged MDW above the
ex
α=0.015. All material parameters are varied around threshold velocity v depends on the material param-
th
3
1.0
(a) (c)
600
0.9
400
m]
200
n 0.8
ce [ 0
n
a
st -200 0.7
di
-400
0.6
-600
0.5
0.5 1 1.5 2 2.5 3 M [M]
(b) time [ns] x s
v=0 M M v = v
x y AB
FIG. 2. (color online) (a) Local x-component of the magnetization of an uncharged MDW displaying “oscillatory” behavior
abovevth. ThepositionofthedomainwallisshowninthemovingframeworkoftheAB,withvAB =1000m/s. Theuncharged
MDW is initially moving at a reduced velocity of about 900 m/s for the first 2 ns, after which it is abruptly pulled back to
the AB. (b) Magnetization structure of an uncharged MDW that transformed into a “charged complex” MDW (v = 0) and
a charged MDW (v = vAB). (c) Phase diagrams showing the effects of the micromagnetic parameters on vth and the type
of MDW dynamics (“quasistatic”, “oscillatory”, or “charged complex”, see text for details). From top-left to bottom-right
the parameters that are varied with respect to their reference values are the exchange constant, uniaxial magnetic anisotropy
strength, damping constant, and thesaturation magnetization.
eters. Two possible scenarios are observed in our sim- the MDW. The average speed of an initially uncharged
ulations. In the first, termed here as “oscillatory”, the MDWthatisstronglycoupledtoaFDWequalstheFDW
MDWfirstlagsincreasinglybehindtheABuntilamaxi- velocity, and it moves either in an “oscillatory” fashion
mumdistanceisreached(about200nmforthereference or as a charged MDW after the transformation.
parameters). Atthatpoint,theMDWisabruptlypulled The dependence of the threshold velocity v and the
th
back to the AB (a process that also involves the emis- type of dynamic behavior on the micromagnetic param-
sion of spin waves), after which the MDW starts to lag eters are illustrated by the phase diagrams of Fig. 2(c).
behind the AB again (Fig. 2(a)). This cycle of events The maximum distance between a moving uncharged
is repeated continuously. The second scenario, labeled MDWandtheABinthesmallimposedv regime(Fig.
AB
as “charged complex”, is significantly different. In this 1(e)), i.e. the “stiffness” of the pinning potential that is
case, the initially uncharged MDW transforms by creat- created by the AB, determines the dynamics above v :
th
ingtwooppositelychargedMDWs. Afterthis,oneofthe Material parameters that give rise to a large lagging be-
charged MDWs follows the AB, while a MDW complex hind distance in the low velocity regime lead to “oscil-
comprising the original uncharged MDW and the other latory” behavior above v , while small MDW-AB dis-
th
charged MDW is left behind. The MDW complex sub- tances belowv imply a “chargedcomplex”scenariofor
th
sequently slows down and eventually stops moving since v >v (Fig2(c)). Thethresholdvelocityv strongly
AB th th
it is no longer pinned onto a moving FDW (the AB in dependsontheexchangeconstant,withweakerexchange
our simulations). This behavior is somewhat analogous interactions resulting in smaller v . Thus, in order to
th
to recent observations of a break-up of the compact do- experimentallyobservenon-trivialhighvelocitydynamic
main wall structure in wide submicrometer wires [28]. effects in ferromagnetic-ferroelectric heterostructures, it
An example of the resulting configuration is shown in would be favorable to use magnetic materials that com-
Fig. 2(b). This process obeys the principle of charge bine a small exchange constant and large magnetostric-
conservation, i.e. the net charge of the newly created tion,withthelatterneededinordertohaveasufficiently
charged MDWs is zero. Although the deformations of strongpinningoftheMDWsontotheirferroelectriccoun-
anunchargedMDW belowvth arereminiscentofWalker terparts.
breakdown, the physical consequences above v are dif-
th The dynamic behavior of charged MDWs is different
ferentcomparedtomagneticfieldorcurrentdrivenmag-
from that of uncharged MDWs at high AB velocities.
netic systems. In the latter cases, an abrupt decrease
ThemuchlargerwidthofthechargedMDWimpliesthat
in the time-averaged MDW speed is often observed be-
itcanmovefastwithoutencounteringmajorinstabilities.
yond breakdowndue to magnetization precession within
Thisisduetothemuchlowerrateofspinrotationneeded
4
which we obtained by considering different AB veloci-
(a)
2
u.]1.5 spinwavestructure ties. The dependence on material parameters (same val-
0.1 [a. 1 PSDbehindtheDW ues as in Fig. 2(c)) is summarized in Fig. 3(b). The
D PSDinfrontoftheDW
S0.5 threshold velocity for spin wave emission corresponds to
P
0.05 00 50 100 the minimum spin wave phase velocity vp,min. At this
ω[rad/ns] velocity, the spin waves moving in front and behind the
Mz 0 MDW have the same wavenumber and frequency. From
2
u.]1.5 Fig. 3(b) it is clear that the exchange constant Kex has
-0.05 [a. 1 a large influence on v . The saturation magnetiza-
D p,min
-0.1 PS0.05 tionMsdoesbarelyinfluencevp,min,butitinfluencesthe
0.00 0.03 0.06 corresponding wave number. The dispersion properties
k[nm-1] ofthe spinwavesemitted infrontofthe MDW aremuch
-600 -400 -200 0 200 more affected than those behind the MDW when K
distancefromAB[nm] ex
and M are varied. The strength of the uniaxial mag-
s
netic anisotropyK andthe magnetic damping constant
(b) u
α have no effect on the emission spectra. The indepen-
dence of K is caused by the cancellation of two oppos-
u
ing effects, one related to the emission of spin waves (a
strongerABpumpsmoreenergyintothespinwaves)and
another due to the stiffness of the medium in which the
spin waves propagate (larger anisotropy implies a stiffer
medium). Thenegligibleinfluence ofαonthe dispersion
properties of spin waves is well known and often utilized
in micromagnetic studies.
To summarize, we have studied electric field induced
MDW motion in ferromagnetic-ferroelectricheterostruc-
tures. The driving force, which can be modeled as an
anisotropy boundary moving in the ferromagnetic layer,
provides a mechanism for MDW dynamics exhibiting
FIG. 3. (Color online) (a) Typical spin waves emitted by properties that are fundamentally different from mag-
a moving charged MDW for vAB > vp,min along with the netic fieldandelectriccurrentdrivenMDW motion. De-
corresponding spatial and temporal power spectral densities pendingontheMDWtypeandmaterialparameters,spin
(PSD) of the waves. Here, thereference material parameters
wave emission and MDW transformations are found for
and vAB = 1800m/s are used. (b) Phase velocity vp versus highdrivingvelocities. Duetotherobustnessofthisdriv-
wavenumberk. Thematerialpropertiesofthereferencesam-
ing mechanism,manifestedby the absenceof the Walker
pleareused,exceptfortheparameterthatisvariedusingthe
same values as in Fig. 2(c). The threshold velocity for spin breakdown,electricfielddrivenMDWmotioncouldopen
waveemission corresponds to theminimum of thecurves. vp upexcitingopportunitiesinthedesignoflowpowerspin-
equalsvAB anddeterminesthewavevectorofthewavemov- tronics applications such as magnetic memory and logic
ing in front (largest k) and behind the MDW (smallest k). devices.
to move the wide charged MDW, as compared to the Acknowledgments. We thank Mikko Alava for
narrow uncharged MDW. The charged MDW starts to a critical reading of the manuscript. This work has
emit spin waves when vAB exceeds the minimum spin been supported by the Flanders Research Foundation
wave phase velocity. A typical wave profile of a charged (B.V.d.W.), the AcademyofFinlandthroughaPostdoc-
MDWatvAB =1800m/sisshowninFig. 3(a). Thespin toralResearcher’sProject(L.L., project no. 139132),an
waves that are emitted have a phase velocity vp which AcademyResearchFellowship(L.L.,projectno. 268302),
equals vAB, but their well-defined wave number k, and an Academy project (S.v.D., project no. 260361) and
thus their frequency, is different in front and behind the via the Centres of Excellence Program (L.L., project
moving MDW (insets of Fig. 3(a)). This behavior is no. 251748), as well as by the Theory Programme of
similar to recent numerical observations of fast MDW Helsinki Insitute of Physics (L.L.), by the Finnish Doc-
dynamics in magnetic nanotubes [29]. toralPrograminComputationalSciences(K.J.A.F.)and
The emittedspinwavesinfrontandbehindthe MDW by the European Research Council (S.v.D., ERC-2012-
are characterized by dispersion graphs (v versus k), StG 307502-E-CONTROL)
p
5
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