Magnetic vortex echoes: application to the study of arrays of magnetic nanostructures

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by  F. Garcia

Magnetic vortex echoes: application to the study of arrays of magnetic nanostructures
F. Garcia1, J.P. Sinnecker2, E.R.P. Novais2, and A.P. Guimar˜aes2∗
1Laborat´orio Nacional de Luz S´ıncrotron, 13083-970, Campinas, SP, Brazil and
2Centro Brasileiro de Pesquisas F´ısicas, 22290-180, Rio de Janeiro, RJ, Brazil
(Dated: January 18, 2012)
Weproposetheuseofthegyrotropicmotionofvortexcoresinnanomagnetstoproduceamagnetic
echo, analogous to the spin echo in NMR. This echo occurs when an array of nanomagnets, e.g.,
nanodisks, is magnetized with an in-plane (xy) field, and after a time τ a field pulse inverts the
2
1 core magnetization; the echo is a peak in Mxy at t = 2τ. Its relaxation times depend on the
inhomogeneity, on the interaction between the nanodots and on the Gilbert damping constant α.
0
Its feasibility is demonstrated using micromagnetic simulation. To illustrate an application of the
2
echoes, we have determined the inhomogeneity and measured the magnetic interaction in an array

n of nanodisks separated by a distance d, findinga d−n dependence,with n≈4.
a
J
PACSnumbers: 75.70.Kw,75.78.Cd,62.23.Eg,76.60.Lz

7
1
The interest in magnetic vortices and their properties

]
and applications has grown steadily in the last years[1–
l
al 4]. Vortices have been observed, for example, in disks
h and ellipses having sub-micron dimensions[5]. More re-
- cently, the question of the intensity of the coupling be-
s
e tweenneighbordiskswithmagneticvortexstructureshas
m attracted an increasing interest[6–9].
t. Atthevortexcorethemagnetizationpointsperpendic-
a
ularlytotheplane: thischaracterizesitspolarity,p=+1
m
for the +z direction and p = 1 for z. The direc-
- − −
tionofthemomentsinthevortexdefinesthecirculation:
d
n c = 1 for clockwise (CW) direction, and +1 for CCW.
−
o If removed from the equilibrium position at the center
c of the nanodisk by, for example, an in-plane field, and
[
thenlefttorelax,avortexcorewillperformagyrotropic

1 motion, withangularfrequency ω,givenforthin disks[3]
v by ω (20/9)γM β (β =h/R is the aspect ratio)[10].
G s
3 ≈
5 We propose in this paper that, manipulating the dy-
5 namic properties of the vortex in an analogousway as it
3 is done in Nuclear Magnetic Resonance (NMR), a new
1. phenomenon results, the magnetic vortex echo (MVE), FIG. 1. (Color online) Diagram showing the formation of
0 similar to the spin echo observed in NMR[11]. This new magnetic vortex echoes; thedisks are described from a refer-
2 echomayprovideinformationonfundamentalproperties enceframe that turnswith theaverage translation frequency
1 of arrays of nanodisks, e.g., their inhomogeneity and in- ω0: a)diskswithin-planemagnetizationsMi alongthesame
: direction (defined by the white arrows); b) after a time τ
v teractions. Despite the fact that applications of vortices
the disks on the left, center and right have turned with fre-
i
X necessarilyinvolvearrays,mostoftherecentpublications quencies, respectively, lower, higher and equal to ω0; c) the
r deal with the analysis of individual nanodisks or arrays polarities of the vortex cores are reversed, and the ωi of the
a with a few elements. Therefore, the possibility of char- vortex cores (and of theMi) changesign, and d) after a sec-
acterizing large arrays is of much interest. In this paper ondintervalτ thecores(and Mi)areagain aligned, creating
theecho.
we have examined the motion of vortex cores in an ar-
ray of nanometric disks under the influence of a pulsed
magnetic field, using micromagnetic simulation.
is the same: p =+1. This is notnecessaryfor ourargu-
i
Let us consider an array of nanodisks where the vor- ment, but, if required, the system can be prepared; see
tex cores precess with a distribution of angular frequen- ref. [12] and the references therein.
cies centered on ω , of width ∆ω, arising from any type Since the direction of rotation of the magnetic vortex
0
of inhomogeneity (see note [10]). We assume that the cores after removal of the in-plane field is defined exclu-
frequencies vary continuously, and have a Gaussian dis- sively by p, all the cores will turn in the same direction;
tribution P(ω) with mean square deviation∆ω. To sim- as the vortex core turns, the in-plane magnetization of
plifywecanassumethatthe polarizationofeveryvortex the nanodot also turns.
2
one can derive the total in-plane magnetization:
1 = 10 nm, = 30 ns, = 0
0 ∞ eh−21(ω∆−ωω20)2i
-1 a My(t)=My(0) cos(ωt)dω (1)
Z−∞ ∆ω√2π
= 20 nm, = 10 ns, = 40 ns, = 0.001
1
m) an integral that is[16] the Fourier transform of the func-
-14 0A/-01 b tion P(ω); using T2∗ =1/∆ω:
1
M ( 1 = 20 nm, = 20 ns, = 0 1 t2
M (t)=M (0) exp cos(ω t) (2)
0 y y (cid:18)−2T2∗2(cid:19) 0
-1 c This result shows that the total magnetization tends to
1 = 10 nm, = 20 ns, = 0.005 zero, as the different contributions to My(t) get gradu-
ally out of phase. This decay is analogous to the free
0
induction decay (FID) in NMR; its characteristictime is
-1 d ∗
T =1/∆ω.
2
0 10 20 30 40 50 60 70 80 90 After a time τ, the angle rotated by each vortex core
t (ns)
will be ωτ; if at t=τ we invertthe polarities of the vor-
FIG.2. (Coloronline)Micromagneticsimulationofmagnetic ticesinthe array,usinganappropriatepulse,themotion
vortex echoes, for 100 nanodisks, with d = infinity, and a)
ofthecoreswillchangedirection(i.e.,ω ω),andone
σ = 10nm, τ = 30ns, α = 0; b) σ = 20nm, τ = 10ns →−
obtains:
and τ = 40ns (two pulses), and α = 0.001; c) σ = 20nm,
τ = 20ns, α = 0; d) σ = 10nm, τ = 20ns, α = 0.005. The ∞ −1(ω−ω0)2
inversion pulses (Bz =−300mT) are also shown (in red). M (t τ)=M (0) e 2 ∆ω2 cos[ω(τ t)]dω (3)
y y
− Z−∞ ∆ω√2π −
The magnetization at a time t>τ is then:
0.10
My(t)=My(0) e[−21(t−T22∗τ2)2]e(−tT−2τ)cos(ω0t) (4)
0.08
ThismeansthatthemagnetizationcomponentM (t)in-
y
creases for τ < t < 2τ, reaching a maximum at a time
0.06
-1 s) t=2τ: this maximumis the magnetic vortex echo, anal-
n
T (20.04 ogous to the spin echo observed in magnetic resonance,
1/ which has important applications in NMR, including in
Magnetic Resonance Imaging (MRI)[14, 15] (Fig. 1). In
0.02 thecaseoftheNMRspinecho,themaximumarisesfrom
the refocusing of the in-plane components of the nuclear
magnetization.
0.00
In Eq. 4 we have included a relaxation term contain-
0.000 0.001 0.002 0.003 0.004 0.005 ing the time constant T - also occurring in NMR -, to
2
Damping constant account for a possible decay of the echo amplitude with
time; its justification will be given below.
FIG. 3. (Color online) Variation of the inverse of the re-
laxation times T2 (diamonds) obtained by fitting the curves Inthearrayofnanodisks,therewillbeinprincipletwo
of echo intensity versus time interval τ to M0exp(−τ/T2), contributionstothedefocusingofthemagnetization,i.e.,
as a function of α, for simulations made with D = 250nm, two mechanisms for the loss of in-plane magnetization
σ =10nm, separation infinite; the continuous line is a linear memory: 1) the spread in values of β and H (see note
least squares fit.
[10]), producing an angular frequency broadening term
∆ω, and 2) irreversible processes that are characterized
∗
by a relaxation time T : thus 1/T =∆ω+1/T .
2 2 2
Let us assume that all the vortex cores have been dis- The second contribution is the homogeneous term
placedfromtheirequilibriumpositionsalongthepositive whose inverse, T , is the magnetic vortex transverse re-
2
x axis[13], by a field B. The total in-plane magnetiza- laxationtime,analogoustothespintransverserelaxation
tion (that is perpendicular to the displacement of the time (or spin-spin relaxation time) T in magnetic reso-
2
core) will point along the y axis, therefore forming an nance. The irreversible processes include a) the interac-
angle θ = 0 at t = 0. Using the approach employed in tion between the disks, which amounts to random mag-
0
the descriptionof magnetic resonance(e.g., see [14, 15]), netic fields that will increase or decrease ω of a given
3
′ ′
disk,producingafrequencyspreadofwidth∆ω =1/T , we have introduced a Gaussian distribution of diame-
2
and b) the loss in magnetization (of rate 1/T ) arising ters, centered on 250nm and mean square deviation σ;
α
fromthe energydissipationrelatedto the Gilbert damp- σ = 10nm corresponds to ∆ω 1.6 108s−1. The
≈ ×
ing constant α. Identifying T to the NMR longitudinal disks were placed at random on the square lattice. The
α
′
relaxation time T , one has [14]: 1/T =1/T +1/2T . initial state of the disks (p = +1 and c = 1) was pre-
1 ∗2 2 α −
Therefore the relaxation rate 1/T is given by: pared by applying an in-plane field of 25mT; the po-
2
larity was inverted with a Gaussian pulse of amplitude
1 1 1 1
=∆ω+ =∆ω+ + (5) B = 300mT, with width 100ps. The results for the
T∗ T T′ 2T z −
2 2 2 α case d= were simulations made on the disks one at a
∞
1/T∗ isthereforethe totalrelaxationrateofthe in-plane time, adding the individual magnetic moments µi(t).
2
magnetization, composed of a) ∆ω, the inhomogeneity We have successfully demonstrated the occurrence of
term,andb)1/T ,thesumofalltheothercontributions, the magnetic vortex phenomenon, and have shown its
2
containing1/T′,duetotheinteractionbetweenthedisks, potential as a characterization technique. The simula-
2
and 1/T , the rate of energy decay. The vortex cores tions have confirmed the occurrence of the echoes at the
α
will reach the equilibrium position at r = 0 after a time expected times (t = 2τ). For different values of σ, the
∗
t Tα, therefore there will be no echo for 2τ Tα. T2 time, and consequently the duration of the FID and
∼The vortex echo maximum at t = 2τ, from≫Eq. 4, is the width ofthe echoaremodified (Fig. 2a,2c); increas-
M (2τ) exp( τ/T ); one should therefore note that ing α results in a faster decay of the echo intensity as a
y 2
the maxi∝mum m−agnetization recovered at a time 2τ de- function of time (Fig. 2a, 2d). We have also obtained
creasesexponentiallywith T , i.e., this maximum is only multiple echoes, by exciting the system with two pulses
2
affectedby the homogeneouspartofthe totaldecayrate (Fig. 2b)[18].
given by Eq. 5. In other words, the vortex echo cancels Fig. 3showsthedependenceofT2 onαforσ =10nm;
the loss inM due to the inhomogeneity∆ω, butit does essentially the same result is obtained for σ = 20nm,
y
not cancel the decrease in My due to the interaction be- sinceT2 doesnotdependon∆ω (Eq. 5). Takingalinear
tween the nanodisks (the homogeneous relaxation term approximation, 1/T2 = Aα, and since for d =infinity
1/T2′), or due to the energy dissipation (term 1/2Tα). there is no interaction between the disks, 1/T2 =1/2Tα,
Note also that if one attempted to estimate the inho- and therefore:
mogeneityofanarrayofnanodotsusinganothermethod, 1
=2Aα (6)
for example, measuring the linewidth of a FMR spec- T
α
trum, one would have the contribution of this inhomo-
From the least squares fit (Fig. 3), A = 1.6 1010s−1.
geneitytogetherwiththe othertermsthatappearinEq.
×
Thisrelationcanbeusedtodetermineexperimentallyα,
5, arising from interaction between the dots and from
measuring T with vortex echoes, for an array of well-
the damping. On the other hand, measuring the vor- 2
separated disks.
tex echo it would be possible to separate the intrinsic
Recently some workers have analyzed the important
inhomogeneity from these contributions, since T can be
2
problemoftheinteractionbetweendisksexhibitingmag-
measured separately, independently of the term ∆ω. T
2 netic vortices,obtaining thatitvarieswith a d−n depen-
can be measured by determining the decay of the echo
dence: Vogel and co-workers [6], using FMR, obtained
amplitude for different values of the interval τ.
for a 4 300 array a dependence of the form d−6, the
The Fourier transform of either the vortex free induc-
×
same found by Sugimoto et al. [8] using a pair of disks
tiondecayorthetimedependenceoftheechoM (t)gives
y
excitedwith rfcurrent. Junget al. [7]studying a pairof
the distribution of gyrotropic frequencies P(ω).
nanodisks with time-resolved X-ray spectroscopy, found
For the experimental study of vortex echoes, the se-
n = 3.91 0.07 and Sukhostavets et al. [9], also for
quence of preparation (at t=0) and inversion fields (at
±
a pair of disks, in this case studied by micromagnetic
t=τ) should of course be repeated periodically, with a
simulation, obtained n = 3.2 and 3.7 for the x and y
period T T . As in pulse NMR, this will produce
α
≫ interaction terms, respectively.
echoes on every cycle, improving the S/N ratio of the
∗ As a first approximation one can derive the depen-
measuredsignals. Also note that the time T canbe ob-
2 ∗
denceofthe contributionto 1/T relatedtothe distance
tained either fromthe initial decay (FID, Eq. 2) or from 2
between the disks as:
the echo (Eq. 4), but T can only be obtained from the
2
MVE.
In order to demonstrate the MVE, we have performed T∗ =B+Cd−n (7)
micromagnetic simulations of an assembly of 100 mag- 2
netic nanodisks employing the OOMMF code[17]. The From our simulations, and using Eq. 7 we found, from
simulated system was a square array of 10 10 disks, the best fit, that this interactionvaries asd−n, with n=
×
thickness 20nm, with distance d from center to center. 3.9 0.1, in a good agreement with [7] and reasonable
±
In order to simulate the inhomogeneity of the system, agreement with Sukhostavets et al.[9].
4

2 m) systems.
6.5 14 ment x 10(A 000...012 ccoielTslahCbeoNraPautqtih,oonCr;AswPweEoauSrle,dFaAlliskoPeEintRodJetb,htFaeAndkPtoEGSt.hMPe..BB.raFziioliranfoargtehne-
6.0 o
5.5 gnetic m--00..21
ns) Ma 0 10 20 30 40 50 60 70 ∗ Author to whom correspondence should be
T* (25.0 Time (ns) addressed: [email protected]
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4.5
(Springer, Berlin, 2009)
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-4 -11 -4
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FIG. 4. (Color online) Variation of the relaxation times
T2∗ versus d−4 for an array of 10×10 magnetic nanodisks [5] EM.aRrt.inPs.,FN.oGvaairsc,iPa,.aLnadndAe.rPos.,GAu.imGa.rSa˜e.sB,Jar.bAopsap,l.MP.hyDs..
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110, 053917 (2011)
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G. Meier, Phys.Rev.Lett. 105, 037201 (2010)
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Yu, D.-S. Han, A. Vogel, L. Bocklage, G. Meier, M.-
Y. Im, P. Fischer, and S.-K. Kim, Sci. Rep. 59, 1
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′
Determining 1/T has allowed us to obtain the inten-
2 [8] S. Sugimoto, Y. Fukuma, S. Kasai, T. Kimura, A. Bar-
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Eq. 7 in Eq. 5, we can obtain the expression for the Appl. Phys.Express 4, 065003 (2011)
interaction as a function of d: [10] The sources of inhomogeneity are the spread in radii, in
thicknessorthepresenceofdefects.Anexternalperpen-
1 1 dn dicularfieldHaddsacontributiontoω[19],ω=ωG+ωH,
T′ = B+Cd−n −Aα−∆ω ≈ C −Aα−∆ω; (8) with ωH =ω0 p(H/Hs), where p is the polarity and Hs
2 | | the field that saturates the nanodisk magnetization. A
distribution ∆H is another source of thespread ∆ω
(approximationvalidfor d small). InFig. 4 we showthe
[11] E. L. Hahn,Phys.Rev.80, 580 (1950)
resultsofthesimulationswithσ =10nmandα=0.001.
[12] R. Antos, M. Urbanek, and Y. Otani, J. Phys.: Conf.
Assuming n = 4 and making a linear squares fit, we Series 200, 042002 (2010)
obtained B =6.15 10−9s, C = 4.03 10−35s m4. [13] Ifthediskshavedifferentcirculations(c=±1)thecores
× − ×
A new phenomenon, the magnetic vortex echo, anal- willbedisplacedinoppositedirections,buttheeffectwill
ogous to the NMR spin echo, is proposed and demon- be the same, since the Mi will all point along the same
direction.
strated here through micromagnetic simulation. Appli-
[14] C. P. Slichter, Principles of Magnetic Resonance, 3. ed.
cationsofthemagneticvortexechoincludesthemeasure-
(Springer, Berlin, 1990)
ment of the inhomogeneity, such as, distribution of di- [15] A.P.Guimara˜es,MagnetismandMagneticResonancein
mensions,aspectratios,defects, andperpendicularmag- Solids (John Wiley & Sons, New York,1998)
netic fields and so on, in a planar array of nanodisks or [16] T. Butz, Fourier Transformation for Pedestrians
ellipses; it may be used to study arrays of nanowires or (Springer, Berlin, 2006)
nanopillars containing thin layers of magnetic material. [17] Available from http://math.nist.gov/oommf/
[18] These echoes, however, are not equivalent to the stimu-
These properties cannot be obtained directly, for exam-
lated echoes observed in NMR with two 90o pulses[11]
ple, from the linewidth of FMR absorption.
[19] G. de Loubens, A. Riegler, B. Pigeau, F. Lochner,
The MVE is a tool that can be used to evaluate the
F. Boust, K. Y. Guslienko, H. Hurdequint, L. W.
interactionbetweentheelementsofalargearrayofnano- Molenkamp, G. Schmidt, A. N. Slavin, V. S. Tiberke-
magnets with vortex ground states. It can also be used vich,N.Vukadinovic,andO.Klein,Phys.Rev.Lett.102,
to determine the Gilbert damping constant α in these 177602 (2009)