Table Of ContentCreation and Amplification of Electro-Magnon Solitons by Electric Field in
Nanostructured Multiferroics
R. Khomeriki1,2, L. Chotorlishvili1, B. A. Malomed3, J. Berakdar1
1Institut fu¨r Physik, Martin-Luther Universita¨t Halle-Wittenberg, D-06120 Halle/Saale, Germany
2Physics Department, Tbilisi State University, 0128 Tbilisi, Georgia
3Department of Interdisciplinary Studies, School of Electrical Engineering,
Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel
5 We develop a theoretical description of electro-magnon solitons in a coupled ferroelectric-
1 ferromagnetic heterostructure. The solitons are considered in theweakly nonlinear limit as a mod-
0 ulation of plane waves corresponding to two, electric- and magnetic-like branches in the spectrum.
2 Emphasis is put on magnetic-like envelope solitons that can be created by an alternating electric
n field. Itisshownalsothatthemagneticpulsescanbeamplifiedbyanelectricfieldwithafrequency
a close tothe band edge of themagnetic branch.
J
9 PACSnumbers: 85.80.Jm,75.78.-n,77.80.Fm
]
l Multiferroic materials, i.e., materials exhibiting cou- a weak magnetoelectric coupling. The theory is, how-
l
a pled order parameters, are in the focus of current re- ever, more general and can, in principle, be applied to
h
search. These systems offer not only new opportunities single-phase magnetoelectrics [13–15]. For the creation
-
s for applications but also provide a test ground for ad- of electro-magnon solitons, which is the subject of the
e
dressing fundamental issues regarding the interplay be- present work, a two-phase multiferroic structure is more
m
tweenelectronic correlations,symmetry, andthe interre- appropriate,as it allows to generate andmanipulate iso-
.
t lationbetweenmagnetismandferroelectricity[1–3]. Here lated FE or FM signals away from the interface.
a
m we address magnetoelectrics which possess a simultane- Bothsingle- andtwo-phasemultiferroics may be mod-
ousferroelectric-magneticresponse. Ainterestingaspect eledby aladderconsistingoftwoweaklycoupledchains:
-
d is the non-linear nature of the magnetoelectric excita- One chain is ferroelectric (FE) built out of unidimen-
n tion dynamics, which hints at the potential of these sys- sional electric dipole moments, P . The second chain
n
o tems for exploring nonlinear wave-localization phenom- is ferromagnetic (FM), composed of classical three-
c
[ ena, such as multicomponent solitons [4, 5], nonlinear dimensional magnetic moments, S~n, where n numbers
band-gap transmission [6, 7], and the interplay between the site in the lattice. Each chain is characterized by an
1 thenonlinearityandAndersonlocalization[8]. Inthispa- intrinsic nearest-neighbor coupling, and each P is cou-
v n
per weaimatexciting robustmagneticsignalsby means pledtoS~ viainter-chainweakmagnetoelectriccoupling.
4 n
6 of electric fields. Particularly, we consider a multifer- Foradiscussionofthemicroscopicnatureofthiscoupling
0 roic nano-heterostructure consisting of a ferromagnetic wereferto [16]. WeassumethedirectionofFEdipolesat
2 (FM) part deposited onto a ferroelectric (FE) substrate. some arbitraryangle with respectto FManisotropyaxis
0 As demonstrated experimentally, under favorable condi- ξ, as depicted in Fig. 1a. The magnetoelectric coupling
.
1 tions, a coupling between the ferroelectric and the ferro- willcausearearrangementofmagneticmoments. Letthe
0 magnetic order parameters may emerge (this coupling is newground-stateorderingdirectionofFMbe theaxisz,
5
referred to as the magnetoelectric coupling), thus allow- and φ is the angle between z and anisotropy axis ξ. The
1
: ing one to controlmagnetism (ferroelectricity)by means magnetic field~h(t) is applied along z, and θ is an angle
v of electric (magnetic) fields. Here we consider the case betweenz andFEmoments(seeFig. 1a). S (P )stands
i 0 0
X when the multiferroic structure is driven by an electric for the FM (FE) equilibriumconfiguration. We will con-
r fieldwithafrequencylocatedwithintheband-gapofthe siderperturbationsaroundtheequilibrium. Definingthe
a FE branch and in the band of the magnetic-excitation scaled dipolar deviations p ≡ (P − P )/P and the
n n 0 0
branch. Foraproperchoiceoftheelectric-fieldfrequency scaled magnetic variables ~s ≡ S~ /S , the Hamiltonian
n n 0
(that follows from the electro-magnonsoliton theory de- is written as
veloped below) it is possible to excite propagating mag-
netic solitons. In addition, we point out a possibility for N
H =H +H +H , H =−g˜ p sx, (1)
the amplification of weak magnetic signals, which sug- P S SP SP n n
gests the design of a digital magnetic transistor, where nX=1
the role of the pump is played by the electric field. N α˜ dp 2 α˜ β˜ α˜
H = 0 n + p2 + p4 + J (p −p )2,
Examples of the two-phase multiferroics under study P 2 (cid:18) dt (cid:19) 2 n 4 n 2 n+1 n
nX=1
[9–12], are BaTiO /CoFe O or PbZr Ti O /ferrites.
3 2 4 1−x x 3 N
The developed model will be applied to a system where H = −J˜~s ~s +D˜ (sx)2+D˜ (sy)2 ,
S n n+1 1 n 2 n
theFEandFMregionsarecoupledataninterfacewhith nX=1h i
2
z
ξ potential [17, 19] near the equilibrium state P0. HS
(a) S (b) stands the ferromagnetic contribution [20], where J is
0
the nearest-neighbor exchange coupling in the FM part.
φ θ P0 D˜1 = D˜S0cos2φ and D˜2 = D˜S0 are anisotropy con-
stants, and D is the uniaxial anisotropy constant along
x axis ξ (see Fig. 1b).
y
We operatewithdimensionlessquantities byusing the
scaling t → ω t with ω = α /α (for the examples
0 0 J 0
4 (c) shown below ω0 ∼ 1012 rad/spec). The other parameters
ncy 3 of the model g˜, α˜, β˜, J˜, D˜1, D˜2 are scaled with ω0.
e The scaled quantities are indicated by omitting the tilde
u
q 2
e superscript. The time evolution is governed by
Fr
1
∂sx
ω 00 1 2 3 ∂tn =−J syn szn−1+szn+1 −szn syn−1+syn+1 −
Wavenumber −2D sysz,(cid:2) (cid:0) (cid:1) (cid:0) (cid:1)(cid:3)
2 n n
∂sy
n =J sx sz +sz −sz sx +sx +
FIG. 1. (Color online) (a) A schematic of the multiferroic ∂t n n−1 n+1 n n−1 n+1
(cid:2) (cid:0) (cid:1) (cid:0) (cid:1)(cid:3)
ladder built of FE and FM chains. Arrows indicate the di- +2D sxsz −gp sz, (2)
1 n n n n
rections and the magnitudes of the electric dipole moments
d2p
and magnetic moments in the course of the soliton advance- n =−αp −βp3 +(p −2p +p )+gsx.
ment along the ladder. (b) Mutual orientations of FE and dt2 n n n−1 n n+1 n
FM ground-state vectors, in the framework of Hamiltonian As we are interested in small perturbations, sx, sy and
n n
(1). (c) Dispersion relations for FE (upper) and FM (lower)
p aremuchlessthanunityandtheapproximateequality
branches of the multiferroic composite. The arrow indicates n
sz =1−(sx)2/2−(sy)2/2 is justified.
theselected carrier frequency. n n n
We seek weakly nonlinear harmonic solutions to Eq.
(2),withafrequencyω andawavenumberk,intheform
ofacolumnvector(sx,sy,p )=Rexp[i(ωt−kn)]+c.c.,
n n n
where R is a set of complex amplitudes R ≡ (a,b,c),
where H stands for the linearized interfacial magne-
SP and c.c. stands for the complex conjugate. Neglecting
toelectric coupling between the FM and the FE chain
higher harmonics in Eq. (2), we find the set of nonlinear
[16]. H is FE part of the energy functional for N-
P algebraic equations
interacting FE dipole moments [17, 18]. Further, α˜ is
0
a kinetic coefficient; α˜J is the nearest-neighbor coupling Wˆ ∗R=Qnl, (3)
constant; α˜ and β˜are second- andforth-orderexpansion
coefficients of the Ginzburg-Landau-Devonshire (GLD) where the matrix and source are, respectively,
iω Jsin2(k/2)+2D 0
2
Wˆ = − Jsin2(k/2)+2D2 iω g ,
(cid:2) g (cid:3) 0 ω2−α−sin2(k/2)
b(|b|2+|a|2)(J −Jcosk+D )−b∗(b2+a2)(Jcosk−Jcos2k−D )
2 2
Qnl = −a(|b|2+|a|2)(J −Jcosk+D1)+a∗(b2+a2)(Jcosk−Jcos2k−D1). (4)
−3β|c|2c
The linear limit amounts to the set of linear homoge- Wecalladispersionbranchferroelectric(defining itsfre-
neous algebraic equations Wˆ ∗R = 0. The solvability quency ω and labeling the amplitude with index E), if
E
conditionDet Wˆ =0leadsto twobranchesofthe dis- it has |c|>|a| [the red curve in Fig. 1(c)], while a ferro-
persion relatio(cid:16)n ω((cid:17)k) which are shown in Fig. 1(c), with magneticbranch(ωM) is definedby the relation|c|<|a|
(the blue curve in the same figure).
the corresponding amplitude set, R = (a,b,c), where b
and c are expressed via the arbitrary constant a: b = Of a particular interest is the case when the system
−iaω/ Jsin2(k/2)+D , c = ga/ α+sin2(k/2)−ω . is excited at an edge (at the left one, for the sake of
2
(cid:2) (cid:3) (cid:2) (cid:3) definiteness), with a frequency ωs which falls into the
3
bandgap of FE mode and, simultaneously, the propa- Thevelocity,dispersion,andthenonlinearitycoefficients
gation band of the FM one, as shown by the arrow are
in Fig. 1(c). The dispersion relation with the fixed
frequency, ω = ω becomes then a cubic equation for v = ∂ωM, ω′′ =−∂2ωM, ∆= ∂ δωM(k,|a|2) .
s ∂k ∂k2 (cid:2) ∂[|a|2] (cid:3)
sin2(k/2). Forω belongingtothebandofFMmodeand
s (9)
bandgap of FE one, the cubic equation yields two com-
The carrier frequency of this soliton is defined by the
plex wavenumbers, associated with FE and FM modes,
dispersion relation [the lower blue curve in Fig. 1(c)]:
and a real one, corresponding to the FM mode. These
three wavenumbers determine a set of three orthogonal ω =ω (k)+δω k,|a|2 /2. (10)
s M M
eigenvectors, RE ≡ (aE,bE,cE), R−M ≡ (a−M,b−M,c−M) (cid:0) (cid:1)
and R+ ≡(a+ ,b+,c+), where the first two corre- Thus, one can generate both the FE evanescent (5)
M M M M
and FM solitonic (8) modes, driving the left edge of the
spond to complex FE and FM wavenumbers, k and
E
k−, respectively, while the last one is related to the real chainatthesamefrequency,ωs. Itispossibletoproduce
M
wavenumber, k+. In linear systems, the solutions with acombinationofthesesolutionstoo. Generallyinnonlin-
M ear systems, linear combinations of particular solutions
thecomplexwavenumbersareevanescentwaveslocalized
is not another solution but if the solutions are far sepa-
atthe left edge ofthe multiferroicchain. Thus, the solu-
tion for the vector function, Fn =(sxn,syn,pn), is rated,whichmakesinteractionsbetweenthemnegligible,
the linear combination
FEn =A(t)REeiωst−|kE|n+c.c. Fn =f1FEn +f2FMn − +FMn + (11)
FMn − =B(t)R−Meiωst−|kM−|n+c.c. (5) isstillasolutionofthenonlinearproblem. Intheweakly
nonlinear limit it is even possible to construct a solu-
where the amplitudes A(t) and B(t) may vary slowly in
tionforthe case whenparticularmodes overlap(i.e., the
time.
magneticsolitonislocatedneartheedge),addingatime-
As mentioned above, the solutions corresponding to
dependentphasetoeachterm(5)and(8)inthesum[26].
the complexwavenumbersarelocalizedatthe boundary.
For instance, one can consider an approximate solution
To examine the possibility of a solitonic self-localization
at the left edge of the ladder, n=0, in the form of
of the third solution with a real wavenumber (cf. Refs.
[21–23] for similar solutions in multiferroic models), we F0 =f1FE0eiΨE(t)+f2FM0 −eiΨM−(t)+FM0 +eiΨM+(t),
consider the nonlinear frequency shift produced by the (12)
small terms Qnl in (3). Assuming a shifted frequency, where, in the weakly-nonlinear limit, the phases Ψ are
ω +δω instead of ω, the matrix Wˆ is substituted by a proportional to the wave amplitudes. Hence the waves
modified one, Wˆ +δWˆ ,with the diagonalmatrix δWˆ ≡ do not gain significant phase shifts due to interaction
iδω · Diag(1,1,2ω). We also define a row vector L = effects, if their relative group velocity is not negligible.
(a′,b′,c′)whichsolvesfortheequationL∗Wˆ =0. Then, In this case, all phase shifts may be neglected.
multiplying both sides of Eq. (3) on L, we obtain the Our particular aim is to create an FM soliton by ex-
nonlinear plane-wave frequency shift: citing only the FE degree of freedom at the edge, i.e.,
sx = sy = 0. To this end, we choose A(t) = B(t) =
0 0
δω k,|a|2 =−i L∗Qnl L∗δWˆ ∗R . (6) sech a+vt ∆ 8ω′′ , seeking to impose the following
(cid:0) (cid:1) (cid:16) (cid:17)(cid:30)(cid:16) (cid:17) (cid:20) M r . (cid:21)
vector relation at the edge, n=0:
From these results, operating with the en-
velope function ϕ(n,t) defined from FMn + = (0,0,p )= f1RE+f2R−M+R+M eiωst +c.c. (13)
ϕ(n,t)ei(ωst−ikM+n) a+M, b+M, c+M , one can derive 0 (cid:0) cosh a+Mvt ∆/8ω(cid:1)′′
the nonlinear Schr(cid:0)¨odinger equat(cid:1)ion (NLS), cf. Refs. h p i
Using now the orthogonality of eigenvectors R, we
[24, 25] in the form:
readily get the appropriate expression for p :
0
∂ϕ ∂ϕ ∂2ϕ 2
2i(cid:18)∂t +v∂n(cid:19)+ω′′∂n2 +∆|ϕ|2ϕ=0, (7) R+M eiωst
p (t)= (cid:12) (cid:12) +c.c. (14)
0 (cid:12)(cid:12)c+M (cid:12)(cid:12)∗ cosh a+Mvt ∆/8ω′′
which gives rise to the respective envelope-soliton solu- (cid:0) (cid:1) h p i
tion with the FM-like localized mode being written as Further, it is possible to compute the coefficients f and
1
f , using the same orthogonality property:
2
FMn + = coshhae2+Mi(ωqst2−∆ωk′M+′(nn)−vt)iabc+M+M+M +c.c.. (8) f1 = (cid:12)(cid:12)|RRE+M|2(cid:12)(cid:12)2(cc+ME)∗∗; f2 = (cid:12)(cid:12)RR−M+M(cid:12)(cid:12)22(cid:0)cc+M−M(cid:1)∗∗. (15)
(cid:0) (cid:1) (cid:12) (cid:12) (cid:0) (cid:1)
(cid:12) (cid:12)
4
In numerical simulations, if we make p a function of
0
time as per Eq. (14), keeping the magnetic moments
pinned at the boundary, it is possible to excite the FE
and FM evanescent waves (5), and also propagating FM
soliton (8). These simulations correspond to an experi-
mental setup with pinned boundary conditions at both
FE and FM edges, and to the application of the elec-
tric field E(t) = p (t) according to Eq. (14) at the first
0
cell of the FE chain. In this way, one can realize the
excitation of magnetic solitons in the FM chain of the
multiferroicladderviaanelectric(ratherthanmagnetic)
field by virtue of the magnetoelectric coupling.
For an assessment of the above, we performed full nu-
merical simulations with the following values of the nor-
malized parameters
α=0.2; β =0.1; J =1; D =0.1; D =0.2; g =0.1.
1 2
FIG.3. (Coloronline)Resultsofnumericalsimulationsofthe
(16)
amplificationofamagneticsignalinthemultiferroicchain(a).
These values correspond to BaTiO /Fe [27, 28]. Fur-
3 The space-time evolution of excitations in the ferroelectric
thermore, we assume for the FE second and fourth or-
part. (b) The amplified propagation of the magnetic soliton
der potential coefficients α˜1/(a3FE) = 2.77·107 [Vm/C], in the ferromagnetic part. The arrow indicates the moment
α˜2/(a3FE)=1.7·108 [Vm5/C3], and for the FE coupling of theinjection of themagnetic signal. Inset (c) presentsthe
coefficient α˜ /(a3 )=1.3·108 [Vm/C], the equilibrium time dependence of the energy of the input magnetic signal
polarizationJP =FE0.265 [C/m2], and the coarse-grained at the FM edge, n = 0, while the energy of output signal is
0 measured at site n=200.
FE cell size a = 1 [nm]. The FM exchange interac-
FE
tion strength is J˜= 3.15·10−20 [J], the FM anisotropy
constant is D˜ = 6.75·10−21 [J], and the ME coupling FM, sx = sy = 0. The results are displayed in Fig. 2,
strength is g˜ ≈10−21 [Vm2]. 0 0
0 where the comparison of numerical simulations and ap-
We drive the left boundary of FE according to (14)
proximateanalyticalsolution (11) are shownat different
with a driving frequency ω = 0.4 and an amplitude
s moments of time [Figs. 2(a,b)].
a+ = 0.07 and apply pinned boundary conditions for
M Next,letusenvisagethepossibilityofamplifyingmag-
netic pulses: We apply a continuous electric signal with
the frequency ω which is slightly below the FM band
s
boundary, ω (0), keeping FM moments pinned at the
M
edge. In this setting, and for small driving amplitudes,
no energy is transmitted through the chain. Both FM
andFE modes areevanescentanddescribedbythe solu-
tions (5). A propagating FM soliton emerges only if the
electric-field amplitude attains the band gap transmis-
sion threshold [6, 29]: This happens if the amplitude is
largeenoughsothatasolutionofthenonlineardispersion
relation (10) for real wave number k exits. Then, if one
keeps the amplitude of the electric field just slightly be-
low this band gap threshold, a small-amplitude in-phase
magnetic signal coupled to FM chain allows to pass the
threshold. This gives rise to a large-amplitude FM soli-
ton propagating through the multiferroic, see Fig. 3. In
thisway,onecanrealizeanamplificationofthemagnetic
pulsesbyelectricfield. Itmayhappenthatalmostallthe
FIG. 2. (Color online) Numerical simulations of the creation
energyoftheelectricfieldwillbetransferredtotheferro-
of a ferromagnetic soliton by the electric field, using Eq. (2)
magnet chain, and the corresponding amplification rate
with parameters (16). (a,b) Show a comparison between the
analytical (solid lines) and the numerical (points) results for mayachievevaluesasmuchas∼100. Forinstance,inthe
thesoliton’sspatialprofileatdifferentmomentsoftime. The simulationspresentedinFig. 3wechoosethedrivingfre-
analytical solution is taken according to Eq. (11) with the quencyandthe amplitude ofthe electricfieldω =0.229
s
coefficients(15). (c,d): Thepropagation oftheFMsoliton in (that is ≈ 30 GHz in real units) and (p ) = 0.414,
0 max
theferroelectric and theferromagnetic layers, respectively. while the FM signal amplitude is (sx) =0.02.
0 max
5
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