Localized magnetic fields enhance the field-sensitivity of the gyrotropic resonance frequency of a magnetic vortex

2.5 MB·

by  Jasper Fried

Localized magnetic fields enhance the field-sensitivity of the gyrotropic resonance
frequency of a magnetic vortex
Jasper P. Fried1 and Peter J. Metaxas1,∗
1School of Physics, M013, University of Western Australia, 35 Stirling Hwy, Crawley WA 6009, Australia.
(Dated: January 12, 2016)
We have carried out micromagnetic simulations of the gyrotropic resonance mode of a magnetic
vortex in the presence of spatially localized and spatially uniform out-of-plane magnetic fields. We
6
show that the field-induced change in the gyrotropic mode frequency is significantly larger when
1
0 the field is centrally localized over lengths which are comparable to or a few times larger than the
2 vortex core radius. When aligned with the core magnetization, such fields generate an additional
confinement of the core. This confinement increases the vortex stiffness in the small displacement
n
limit,leadingtoaresonanceshiftwhichisgreaterthanthatexpectedforauniformout-of-planefield
a
ofthesameamplitude. Fieldsgenerated byuniformlymagnetized sphericalparticles havingafixed
J
separation from the disk are found to generate analogous effects except that there is a maximum
1
intheshiftatintermediateparticlesizeswherefieldlocalization andstrayfieldmagnitudecombine
1
optimally to generate a maximum confinement.

]
l
l I. INTRODUCTION plitude. It is shownthat this is due to anincrease in the
a
h vortexstiffnessasaresultoftheout-of-planemagnetiza-
- tionofthe corepreferentiallyaligningwiththe strongest
s A magnetic vortex is a curled magnetization configu-
e ration with an out-of-plane magnetized nano-scale sized part of the localized field. For the particular case of
m MNPs whose lower surfaces are separated from the disk
core1,2. Vortices arise spontaneously in (sub)micron-
by a fixed distance, we demonstrate that the frequency
. scale3 magnetic elements such as discs4 (as well as
t shift is characterized by a clear maximum at intermedi-
a square5 or triangular6 plates) and are of relevance for a
m ate particle sizes, a result of an optimized combination
range of applications ranging from radiofrequency signal
of the amplitude and localization of the stray field cre-
- generation7,8 and detection9 to cancer treatment10, data
d ated by the MNP. We also note that short range struc-
storage11andmagnonics12–14. Manyapplicationsexploit
n turaldisorder(aswellasdomain-generatedstrayfields32)
o the lowestfrequencymagneticexcitationofavortex,the can also pin vortices33 (with a potential for quantum
c gyrotropicmode15–18, which correspondsto an orbit-like depinning34,35) and that these defects have also been
[ motion of the vortex core about the disk’s center.
shown to be capable of modifying the gyrotropic mode

1 An important characteristic of the gyrotropic mode is frequency8,36–40.
v that its frequency, f , can be tuned by applying static
G
5 out-of-plane7,8,19,20 or in-plane6,21 magnetic fields. Uni-
7 form out-of-plane magnetic fields modify both the el-
5
ement’s magnetization configuration and the magneto- II. MICROMAGNETIC SIMULATION
2
static confinement of the core and, when sufficiently far
0
. below the saturationfieldofthe disk, induce a changein Our results were obtained using finite difference mi-
01 fG which is a linear function of the field strength7,8,19. cromagnetic simulations of the gyrotropic mode using
6 The combination of this field-linearity with the ability MuMax341. We will focus on simulation results for NiFe
1 to electrically probe vortex dynamics (by fabricating a discswithradiusR=192nm,thicknessL=30nm,satu-
: vortex-based spin torque nano-oscillator7,8,22, STNO), rationmagnetizationM =800kA/m,exchangestiffness
v S
has application not only for field-tunable electronic A =13 pJ/m, magnetic damping α=0.008,no intrin-
i ex
X oscillators7,8,20,23 but potentially also for intrinsically sicanisotropyandacellsizeof2×2×3.75nm3. Tobegin
r frequency-based sub-micron magnetic field sensors24,25. with,thesystemisinitializedwithavortex-likemagnetic
a The latter typically exploit the field dependence of the configurationandallowedtorelaxusingMuMax3’sinter-
output frequency of a STNO (e.g. see24–26 for non- nalrelaxationroutine whichtime evolvesthe magnetiza-
vortex based devices) for frequency-based24,27–30 field tion(without precession)using energy-andthentorque-
sensing. One potential application of such a sensor is in minimization as stopping criteria41. A transverse mag-
the development of frequency-based nano-scale devices netic field sinc pulse is then applied with an amplitude
to detect (biofunctionalized) magnetic nanoparticles30 of 2 mT, a time offset of 300 ps and a cut-off frequency
(MNPs) for in-vitro bio-sensing and point-of-care med- of 30 GHz. This induces a displacement of the vortex
ical diagnostics31. core(aswellassomehigherfrequencyexcitations21,42–46)
In this work we show that central, localized out-of- which is followed by a damped gyrotropic motion of the
plane fields (such as those generated by MNPs) produce core around the disk’s center. Fourier analysis of the
shifts in the gyrotropic frequency greater than those in- x-component of the spatially averaged magnetization is
ducedbyuniformout-of-planefieldshavingthesameam- used to extract f . Good agreement between MuMax3,
G
2
OOMMF47 andFinMag(derivedfromNmag48)wascon- out-of-plane fields aligned with the vortex core. As pre-
firmed for a number of test cases49. viouslyobserved,thefrequencyvarieslinearlywithH .
uni
We were able to quantitatively reproduce the simulated
(a) frequenciestowithin2%[bluecirclesinFig.1(a)]using15
κ
2πf = (1)
G
G
where κ is the vortex stiffness coefficient and G is the
gyroconstant. The remainder of this section is focused
on the extraction of κ and G to obtain the calculated
f =κ/(2πG) values shown in Fig. 1(a).
G
κ, the stiffness coefficient describes the harmonic scal-
ing of the vortex energy, W, with lateral in-plane dis-
placement, X, measured radially from the disk’s cen-
ter: W(X) = W(0) + 1κX2 + O(X4). For uniform
2
ornull out-of-plane fields, this confinement is dominated
(b) (c) by dipolar effects18 howeverdynamic exchange fields are
also present. For each simulation, we extracted κ from
a parabolic fit to the total energy of the system plotted
against the dynamic displacement of the core as mea-
sured during the field-pulse-induced gyrotropic motion
[e.g.Fig.1(b)]. As shownpreviously21, this dynamic ap-
proach, which analyzes the energy of the moving vor-
tex core, produces a more accurate prediction of the gy-
(d) (e) rotropicfrequency than a static method in which the to-
0mT 200mT talenergyiscalculatedfordisplacedcoresatequilibrium
which have been shifted by static in-plane fields. This
said, the static method will be instructive in visualizing
Halo
the influence of localized fields on the vortex stiffness.
Thegyroconstant,G,determinesthemagnitudeofthe
gyrovector, G = Geˆ , in the Thiele equation describing
Z
vortexdynamics15,50–52. The gyroconstantcanbe calcu-
lated from the vortex spin structure using
FIG. 1. (a) Simulated and calculated gyrotropic mode fre-
quencies,fG [thelattercalculatedusingEq.(1)],forspatially G= MsL m· dm × dm dxdy (2)
uniform out-of-plane fields. (b) Plot of the change in energy γ dx dy
ZZA (cid:18) (cid:19)
(relativetotheenergyatzerocoredisplacement)asafunction
of core displacement as measured during field-pulse-induced where γ is the gyromagnetic ratio and m is the unit-
gyrotropicmotioninzeroout-of-planefield. Thefirst25nsof length magnetization vector. Given that G acts along
motion are disregarded dueto thehigh frequencyspin waves the z-axis this equation can be shown to be equal to the
producedinthisperiod. Aquadraticfittotheenergyisshown often quoted15,51 G = M L/γ sin(θ)(∇θ×∇φ)dxdy
S
bytheredcurve. (c)Gyroconstantcalculatedconsideringdif-
where θ and φ are the polar and azimuthal angles of the
ferentradiifromthediskcenterforfourdifferentuniformout- RR
magnetizationrespectively. Theoretically,forazeroout-
of-plane fields. The gray dashed lines represent the value of
the gyroconstant calculated using19 G(H) = G(0)(1−cosθ) of-planefieldEq.(3)yieldsG=2πqpLMs/γ whereq =1
is the vorticity and p = ±1 is the core polarity aligned
where cosθ is taken at the magnetostatic halo surrounding
along ±eˆ .
the vortex core. (d) Out-of-plane magnetization for a cen- Z
tered (non-displaced) vortex along a slice through the disk’s Wenumericallycalculatedthegyroconstantassociated
centerinzeroout-of-planefieldand(e)underaspatiallyuni- with the static, non-displaced core for the studied H
uni
form 200 mT out-of-plane field. values using Eq. (3) by integrating a thickness-averaged
m53overtheentirediskarea. However,insertingthiscal-
culatedvalueofGintoEq.(1)ledtoavalueforf which
G
was significantlylower than the simulatedvalue. For ex-
III. UNIFORM OUT OF PLANE MAGNETIC ample,forauniformout-of-planefieldof200mTthegy-
FIELDS roconstantcalculatedusingthismethodis≈36%smaller
thanwhatisexpectedtakingthesimulatedfrequencyand
Wewillfirstconsiderthecaseofavortexinaspatially extractedκandsolvingforG(i.e.5.17×10−13kgs−1com-
uniform out-of-plane field19, H . Fig. 1(a) shows the pared to 7.03×10−13 kgs−1 = κ/2πf ). To attempt to
uni g
simulated gyrotropic frequency under various amplitude understand this discrepancy, we calculated G by consid-
3
eringonly the spinstructure withina givenradiusofthe (a)
disk center for four different H values. The resultant
uni
data is shown in Fig. 1(c). G reaches a clear maximum
when integrating over a radius close to the edge of the
vortex core (≈40 nm). Notably, at this point, the value
of G closely corresponds to the gyroconstant expected
from the simulated frequency and extracted stiffness co-
efficient. This peak is present for all out-of-plane field
amplitudes and is due to the ‘magnetostatic halo’ [see
Fig. 1(d)] surrounding the vortex core as a result of its
demagnetization field54. The drop-off in G at large radii
for H > 0 is due to the out-of-plane canting of spins
uni
near the disk edge [e.g. Fig. 1(e)]. Encouragingly, the G
calculated using this method corresponds closely to the (b) (c)
value predicted by the equation G(H)=G(0)(1−cosθ)
given by de Loubens et al19 for uniform out-of-plane
fields where cosθ is given as the polar angle of the mag-
netization at the disk’s edge. The value of G calculated
usingtheaboveexpressionforG(H)isshownbythegray
dashedlinesinFig.1(c)howevercosθhasbeenextracted
at the center of the magnetostatic halo. This result sug-
gests that althoughthere is a divergence of the magneti-
zation far away from the vortex core, it is the local spin (d) (e)
structure around the core which is relevant for the small
amplitude oscillations considered here (<1 nm).
In Fig. 1(c) we see that G reduces considerably when
H is increased and it is this reduction in G which is
uni
primarily responsible for the linear increase in f with
G
increasing H . In fact, κ is reduced at large H val-
uni uni
ues where it becomes easier to shift the vortex laterally.
Below, localized out-of-plane fields will also be shown to
increase f however that increase will be demonstrated FIG. 2. (a) Simulated and calculated gyrotropic mode fre-
G
to be primarily due to a increase in κ induced by the quencies,fG [thelattercalculated usingEq.(1)],inthepres-
ence of 200 mT amplitude Gaussian [Eq. (3)] out-of-plane
localized field.
fieldsofvariouswidths(thewidthisapproximately1.2times
thehalf-width-half-maximum(HWHM)oftheGaussian field
profile). The simulated frequency for a spatially uniform 200
IV. EFFECT OF SPATIALLY LOCALIZED
mT out-of-plane field is shown by the gray dashed line. (b)
FIELDS
Vortexcoredisplacementasafunctionofstaticin-planefield
for two Gaussian fields and zero out-of-plane field. (c) Stiff-
A. Gaussian fields nesscoefficientversusGaussianfieldwidth. (d)Gyroconstant
calculated considering different radii from thedisk centerfor
three different Gaussian field widths. (e) Gyroconstant de-
To study the effect of spatially localized out-of-plane
pendenceon Gaussian field width where G is calculated con-
fields on the gyrotropic frequency, we first consider cen-
sidering only the vortex structure within 40 nm of the disk’s
tralizedfieldswithatwo-dimensional(2D)Gaussian-like
center.
profile:
2
H =A exp − r . (3) the Gaussian field becomes more localized.
loc 0
" (cid:18)wloc(cid:19) # To begin to understand the above frequency behavior
(and confirm its link it to a H -induced confinement),
loc
Here w is a width parameter approximately equal to wefirstlookathowthevortexcoremoveslaterallyinre-
loc
1.2 times the half-width-half-maximum (HWHM) of the sponsetostatic,uniformin-planefields,H ,inthepres-
IP
field profile and r is the lateral distance from the cen- ence of a centrallylocalizedfield, H . By doing this we
loc
ter of the disk. µ A is fixed at +200 mT making H can explicitly probe the (static) confinement of the core
0 0 loc
aligned with the vortex polarity (p = +1). As shown in across the disc. In the complete absence of an out-of-
Fig. 2(a) localized Gaussian fields significantly increase plane field, the core position varies linearly with H at
IP
f as compared to the action of a spatially uniform 200 lowfields (lowdisplacements)[Fig.2(b)]. Forhigherdis-
G
mTout-of-planefield[graydottedlineinFig.2(a)]. Fur- placementstheresponsetofieldisslightlyweaker,consis-
thermore,this frequencyenhancementbecomes largeras tentwithanincreased(anharmonic55)confinementwhen
4
the core moves closerto the disk’s edge under the action the disk. For small w the equilibrium core structure
loc
of H . If we add a localized field however, the H - itself is changed [Fig. 4(a)]: the magnetostatic halo is
IP IP
inducedcoredisplacementisclearlylower,butonlyifthe less sharp and the core widens. Intermediate w values
loc
displacement is comparable or smaller than the HWHM [Figs. 4(b,c)] generate a clear out of plane magnetized
of the localized field. Indeed, at larger displacements, regionaroundthecorewhilelargew values[Fig.4(d)],
loc
the responsetoH is similarforbothlocalizedanduni- which lead to a broad field profile, result in a magneti-
IP
form out-of-plane fields. This result explicitly confirms zation profile which is similar to that seen for uniform
the H -induced confinement (or stiffening) of the vor- out-of-planefields [Fig. 1(e)]. For smalldisplacementsof
loc
texcorewhicharisesbecausekeepingthecorewithinthe thecoreinallofthesecaseshowever,theconfiningpoten-
central region minimizes the Zeeman energy associated tial nevertheless remains close to harmonic and we have
with the interaction between H and the vortex core again used the method previously described to calculate
loc
magnetization. (dynamic) values of κ. Consistent with the results seen
The influence of H on the H -induced shift is vi- in Figs. 2(a,b) and 3, κ increases strongly for narrower
loc IP
sualized directly in Fig. 3 where we compare the equi- localizationsoftheGaussianfield[asshowninFig.2(c)],
librium static positions of the vortex core (white) for again confirming the Hloc-induced confinement.
H = 12 mT in two cases: a vortex with no out-of-
IP
(a) (b)
planefield[Fig.3(a)]andavortexsubjectto aH field
loc 50nm 100nm
with w =25 nm [Fig. 3(b)]. The core has clearly been
loc
displaced a smaller distance for the case of a localized
field, with the core remaining within the strong part of
the H profile (visible as a broadout-of-planemagneti-
loc
zationcomponentatthedisk’scenter). Referenceimages
of the unshifted vortex core at H =0 mT are given in
IP
Figs. 3(c,d).
(c) (d)
0HIP=12mT 0HIP=0mT 200nm 350nm
d
l
e
i
f

P
O
O

o
N (a) (c)
FIG. 4. Plots of the out-of-plane magnetization for a slice
m through the disk center for a vortex in Gaussian fields of
n width, wloc of (a) 50 nm (b) 100 nm (c) 200 nm and (d)
5 350 nm.
2

= We alsocalculatedthe gyroconstantinthe presenceof

c the H fields, again considering different radii as done
o loc
wl (b) (d) previously. The resultant data are shown in Fig. 2(d)
for three Gaussian field profiles. Notably, the peak in
G which was clearly visible in Fig. 1(c) disappears (or
FIG. 3. Visualizations of the out-of-plane component of the
for w =200 nm becomes much less prominent) due to
disk magnetization (white) for: (a) µ0Huni = 0 mT and loc
the absence of a deep magnetostatic halo for the studied
µ0HIP=12 mT, (b) a Hloc with wloc=25 nm and µ0HIP=12
w values[Fig.4]. ForthetwobroaderGaussians,Gin-
mT, (c) µ0Huni = 0 mT and µ0HIP=0 mT and (d) a Hloc loc
creaseswith the consideredradius since the non-uniform
with wloc=25nm and µ0HIP=0 mT. Thelight bluelines ref-
erence the centered and HIP-displaced core positions in zero Hloc induces a canting which depends on the distance
out-of-plane field. In (a) and (c) the white part of the im- from the center of the disk [as seen in Figs. 4(b,c)]. For
age corresponds to the core. In (b) and (d) the out-of-plane the narrowest Gaussian, G becomes flat at large consid-
core magnetization is convoluted with the Hloc-induced out- eredradii. ThisisbecausethenarrowlocalizationofHloc
of-plane cantingin thedisk’s center(which also translates to leads to a quasi-null canting of m away from the center.
white coloring). However,Ggrowsquicklyatsmallandintermediatecon-
sidered radii, due to the H -induced broadening of the
loc
By looking at Fig. 3(d) and Fig. 4, the latter showing core [Fig. 4(a)]. In Fig. 2(e) we show the extracted G
profiles of the out-of-plane component of the magnetiza- values versus w where G was again calculated using a
loc
tion across the disk, one also sees that the presence of considered radius of 40 nm (i.e. analyzing the magneti-
H clearly modifies the magnetic configuration within zation in the core’s immediate vicinity). Like κ, G also
loc
5
increases at small w but to a lesser extent. ter being independent of the lateral disk dimensions57.
loc
TheextractedGanddynamicκyieldreasonablequan- Along these same lines, since fG is intrinsically smaller
titative agreement between the simulated frequencies atlargerdiskradii15,18,19,44 duetoaweakercoreconfine-
and the frequency predicted by Eq. (1) [red squares in ment, the relative frequency shift induced by a localized
Fig. 2(a)]. This said, the agreement is clearly best at field (i.e. as a percentage) will be higher for wider disks
largew . Thisisperhapsnotunexpectedhoweversince [see Fig. 6(b)].
loc
wider profiles result in weaker deformation of the mag- Analogous behavior to that shown in Fig. 2(a) was
netization near the core. We also emphasize that, in alsoseenwhenreducingthediskthickness. Theabsolute
contrast to the case of a uniform out-of-plane field the changeinfGdidhoweverreduceasthediskbecamethin-
growth in f at small w [Fig. 2(a)] is driven by in- ner[Fig.6(c)]. This reductionwasfoundtobe drivenby
G loc
creased confinement [i.e. κ in Fig. 2(c)] rather than by a reduced dynamic κ (confirmed at w =90 nm), indi-
loc
changes in G [Fig. 2(e)]. catingalowerGaussian-field-inducedcoreconfinementat
Upuntilnowwehaveconsideredonlythemodifications smallerthicknesses. Thisreducedconfinementisperhaps
to f induced by changing w . Fig. 5 however shows not surprising as decreasing the disk thickness leads to
G loc
data analogous to that in Fig. 1(a), demonstrating the a smaller vortexcore volume (the core is narrower57 and
change in fG induced when modifying the amplitude of its height reduces), loweringthe Hloc-associatedZeeman
the localized fields. The change to f per unit of field energy which drives the confinement.
G
amplitude, which can be thought of as a ‘field sensitiv-
ity’,isnotablymorethanfivetimeslargerforalocalized
Gaussianfieldwithwloc =50nmthanthatobservedfora B. Dipole fields and width-dependent Gaussian
spatiallyuniformfield. Consistentwiththeresultsshown fields
in Fig. 2(a), this sensitivity enhancement reduces as the
field profile is made broader (i.e. when w increases).
loc Localizedmagneticfieldscanalsobegeneratedbyuni-
formly magnetized, spherical MNPs. The in-plane com-
ponents of the field generated by a submicron particle
haspreviouslybeenusedtoshiftthevortexcoreposition
and probe the anharmonicity of the disk’s intrinsic con-
1.2 Uniform fining potential55. Here we consider the case of a field
1.1 wloc = 50 nm generated by a centralized MNP with radius RP whose
wloc = 80 nm lower surface is at a fixed distance (10 nm) from the
z) 1.0 wloc = 120 nm upper surface of the disk (i.e. the height of the center
H of the MNP from the top of the disk will be R +10
G P
( 0.9 nm). To minimize simulation time the field created by a
f
MNPwithsaturationmagnetizationM hasbeenmod-
0.8 S,P
elled as that of a dipole with moment 4πR3M where
3 P S,P
0.7 M =200 kAm−1. To confirm the validity of this sim-
S,P
plification, we determined the gyrotropic frequency ob-
0.6
servedwhenasolidferromagneticsphereofdiameter100
0 100 200 nm (Aex =13 pJ/m) was placed above the center of the
disk (in this case we used a smaller disk with R = 96
(Localized) out of plane field amplitude (mT)
nm to minimize simulation time and memory use) and
FIG.5. Simulatedgyrotropicmodefrequencyplottedagainst explicitly simulated it together with the disk in a 200
the amplitude of localized, Gaussian fields with different mT uniform out-of-plane field. When compared to the
widths. Data for a spatially uniform field [taken from simulated f in an equivalent dipole field there was a
G
Fig. 1(a)] is shown for comparison. relativelysmalldiscrepancyof≈2.6%whichwasfurther
reduced to 0.3% when adding a strong z-axis oriented
Finallyforthissection,wedemonstratethatthetrends anisotropy to the sphere (K = 107 J/m3). The lat-
1
observed in Fig. 2(a) apply to other disk geometries56. ter tends to fix the sphere’s magnetization in the out-of-
Fig. 6(a) shows the frequency increase (relative to f plane orientation, suggesting that the worse agreement
G
under H = 200 mT) as a function of w for three for the zero-anisotropysphere was due to changes in the
uni loc
different disk radii. The increase in f is found to be al- sphere’smagneticconfigurationinducedbythemagnetic
G
most independent of the disk radius. This demonstrates strayfield of the disk. We alsonote that the simulations
for these radii that the frequency increase (at least for belowwereperformedwithnoexternalout-of-planefield.
small oscillations in the presence of this strong 200 mT However,afieldwouldusuallyhavetobeappliedexperi-
localized field) is largely independent of the ‘intrinsic’ mentallyinthecaseofasuperparamagneticparticleand
core confinement which is defined by the disc geometry. indeed we found analogous results to those given below
Theincreaseinsteadresultsfromtheinteractionbetween [f versus R in Fig. 7(a)] for simulations in a 200 mT
G P
thelocalizedfieldandthevortexcore,thesizeofthelat- externalout-of-plane field(sufficient to induce the afore-
6
(a) (b) (c)
FIG. 6. The (a) absolute and (b) relative change in the gyrotropic frequency, fG (compared to its value in Huni = 200 mT)
as a function of Gaussian width for disks of radii R = 96, 192 and 384 nm and a constant thickness of L = 30 nm. (c) The
absolute change in fG for disks of thickness L= 7.5, 15 and 30 nm and a constant radius of R= 192 nm.
mentioned M value30). as R changes. Indeed, if we run simulations to deter-
S,P P
mine f in the presence of uniform out-of-plane fields
G
equivalent in magnitude to the MNP field (as calculated
(a)
inthe centerofthe disk)wesee amonotonicallyincreas-
ingf [redsquaresinFig.7(a)]withnopeak. Thislatter
G
growth in f is consistent with the bigger particles gen-
G
eratingbiggerfieldsandthusbiggerchangesinf [i.e.as
G
per Fig. 1(a)]. Instead, the peak in f observed for the
G
dipole fields atintermediateR canbe correlatedwith a
P
maximum in the confinement of the core, manifested as
apeakinκ [Fig.7(b)]. Note howeverthatthe maximum
f occurs at a slightly higher diameter than that which
G
leadstothemaximumκduetothediameterdependence
of G which increases [thus reducing f , as per Eq. (1)]
G
as the particle diameter becomes small [Fig. 7(c)].
Inthe insetofFig.7(a)weshowthe out-of-planecom-
ponent of the dipolar MNP field for particle diameters
(b) (c) located around the point at which the peak in f lies.
G
The peak in confinement can be understood as follows.
At small particle diameters (e.g. 2R = 50 nm), the
P
field is highly localized however the magnetic moment
of the MNP (and thus the stray field amplitude) is low.
This results in a weak confinement. A weak confine-
ment also occurs for large particles (e.g. 2R =400 nm)
P
which generate strong but nevertheless broad (and thus
weakly localized) spatial field profiles. Between these
two extrema however is some intermediate particle size
FIG.7. (a)Simulatedfrequencyasafunction ofMNPdiam-
(e.g.2R =135nm) wherethere is anoptimalcombina-
eter (2RP). The red squares shows the simulated frequency P
tion of field strength and localization which maximizes
in a uniform out-of-plane field of the same amplitude as the
stray fieldcreated bytheMNPascalculated at thedisk cen- the core confinement, thus leading to a large fG.
ter. Theinsetshowstheout-of-planecomponentofthedipole We can also reproduce the above tendencies using
field for three particle diameters below, above and near the Gaussian fields which have been scaled by a factor ∝
observed peak in fG. (b) Stiffness coefficient as a function wl3oc/(wloc+d)3. This scaling mimics some characteris-
of particle diameter. (c) Gyrovector calculated by consider- tics of the R -dependent dipole field. d is chosen to be
P
ing the spin structure within 40 nm of the disk center as a 25 nm as this is the distance used in our simulations be-
function of particle diameter.
tween the bottom of the particle and center of the disk.
The numerator in the above scaling factor describes the
InFig.7(a),weshowresultsobtainedfordipolesequiv- magnetic moment dependence on particle radius (here
alentto+z-magnetizedMNPsofvarioussizespositioned equivalent to w ) whereas the denominator describes
loc
above the disk as detailed above. In contrastto the case thefieldbehaviorasthedipolecentermovesfurtheraway
of the Gaussian field, f displays a maximum at some fromthe diskcenter due to anincreasingparticleradius.
G
intermediate particle size. This behavior cannot be ex- The simulated f values are shown in Fig. 8. Unlike the
G
plained simply by the stray field changing in magnitude resultsforGaussianfieldsofconstantamplitude, f now
G
7
exhibits apeak atintermediate w values,analogousto changesinthevortexgyrotropicfrequencythanthatgen-
loc
thefrequencybehaviorseenforMNPswhenchangingR erated by a spatially uniform out-of-plane field of the
P
[Fig.7(a)]. ThecorrespondingonedimensionalGaussian same amplitude. This behavior is consistent with an in-
field profiles are shown in the inset of Fig. 8 for three crease in the vortex stiffness as a result of the out-of-
different w values. Again, as we increase w we see a plane magnetization of the core preferentially aligning
loc loc
transitionfromaweak,highly localizedfieldto a strong, with the strongest part of the localized field which gen-
broadlylocalizedfieldwithmaximumconfinementoccur- eratesanadditionalvortexcoreconfinement. Inthecase
ring at an intermediate w . offields whichapproximatethosegeneratedby magnetic
loc
particlesofvaryingradius,thefrequencywasobservedto
be maximized for some intermediate particle size which
led to an optimized combination of field amplitude and
900 field localization. This may be relevant for vortex-based
MNP sensors exploiting changes of the gyrotropic fre-
quency induced by localized MNP fields58. We finally
note that this work has focused on very low amplitude
) 800 excitationsandhighlightthefactthatgyrotropicmotion
z
H outside of the strongest part of the localized field will
f (M 700 1Field (mT)8900 14255 agrele.sn3u9elrtfaotirneldawrbegyaekcaehmrapnclhgiteausndgineesosastcotiullfraaGttiioo(nanssmaorabogsuennrevdteizpdaintbinyoinnM)g.isniteest
25
0
-200 0 200
x-position (nm)
600
0 40 80 120 160 ACKNOWLEDGMENTS
Gaussian width (nm)
FIG. 8. Simulated frequency in the presence of a Gaussian This work was supported by the Australian Research
field whose amplitude has been modified to depend upon its Council’s Discovery Early Career Researcher Award
width and thusgenerate analogous data to that in Fig. 7(a). scheme(DE120100155),aresearchgrantfromtheUnited
Inset shows a central slice through the field profile for three StatesAirForce(AsianOfficeofAerospaceResearchand
different Gaussian widths. Development, AOARD), the University of Western Aus-
tralia (including the ECRFS, RCA and RDA schemes),
NeCTAR (National eResearch Collaboration Tools and
Resources, supported by the Australian Government
V. CONCLUSION through the National Collaborative ResearchInfrastruc-
ture Strategy) and by resources provided by the Pawsey
We have shown that the sensitivity of the vortex gy- Supercomputing Centre with funding from the Aus-
rotropic mode frequency to out-of-plane fields can de- tralianGovernmentandtheGovernmentofWesternAus-
pend strongly on the spatial localization of those fields. tralia. TheauthorsthankJoo-VonKim,ManuSushruth,
Centralized, out-of-plane magnetic fields localized over Mikhail Kostylev, Vincent Cros, Rebecca Carey, Maxi-
lengths which are comparable to or a few times larger milian Albert and Hans Fangohr for useful discussions,
than the vortex core radius induce significantly larger advice and/or assistance.
∗ [email protected] 7 V. S. Pribiag, I. N. Krivorotov, G. D. Fuchs, P. M. Bra-
1 T.Shinjo,T.Okuno,R.Hassdorf,K.Shigeto, andT.Ono, ganca, O. Ozatay, J. C. Sankey, D. C. Ralph, and R. A.
Science 289, 930 (2000). Buhrman, Nat. Phys.3, 498 (2007).
2 A. Wachowiak, J. Wiebe, M. Bode, O. Pietzsch, M. Mor- 8 A. Dussaux, B. Georges, J. Grollier, V. Cros, A. V.
genstern, and R.Wiesendanger, Science 298, 577 (2002). Khvalkovskiy, A. Fukushima, M. Konoto, H. Kubota,
3 K. Y.Guslienko, J. Nanosci. Nanotech. 8, 2745 (2008). K. Yakushiji, S. Yuasa, K. A. Zvezdin, K. Ando, and
4 R. P. Cowburn, J. Ferr´e, S. J. Gray, and J. A. C. Bland, A. Fert,Nat. Commun. 1, 8 (2010).
J. Appl.Phys.74, 1466 (1999). 9 A. S. Jenkins, R. Lebrun, E. Grimaldi, S. Tsunegi,
5 A. Vogel, T. Kamionka, M. Martens, A. Drews, K. W. P. Bortolotti, H. Kubota, K. Yakushiji, A. Fukushima,
Chou, T. Tyliszczak, H. Stoll, B. Van Waeyenberge, and G. de Loubens, O. Klein, S. Yuasa, and V. Cros,
G. Meier, Phys. Rev.Lett. 106, 137201 (2011). arXiv:1505.05358 , 2262 (2015).
6 S. Yakata, T. Tanaka, K. Kiseki, K. Matsuyama, and 10 D.Kim,E.Rozhkova,I.Ulasov,S.Bader,T.Rajh,M.Les-
T. Kimura, Sci. Rep.3, 3567 (2013). niak, and V.Novosad, Nat. Mater. 9, 165 (2010).
8
11 K. Yamada, S. Kasai, Y. Nakatani, K. Kobayashi, 35 R. Zarzuela, E. M. Chudnovsky, J. M. Hernandez, and
H. Kohno, A. Thiaville, and T. Ono, Nat. Mater. 6, 270 J. Tejada, Phys. Rev.B 87, 144420 (2013).
(2007). 36 R.L.Silva,A.R.Pereira,R.C.Silva,W.A.Moura-Melo,
12 R. Huber and D. Grundler, Proc. SPIE 8100, 81000D N.M.Oliveira-Neto,S.A.Leonel, andP.Z.Coura,Phys.
(2011). Rev.B 78, 054423 (2008).
13 D.-S. Han, A. Vogel, H. Jung, K.-S. Lee, M. Weigand, 37 A. Vansteenkiste, M. Weigand, M. Curcic, H. Stoll,
H. Stoll, G. Schtz, P. Fischer, G. Meier, and S.-K. Kim, G. Schtz, and B. Van Waeyenberge, New J. Phys. 11,
Sci. Rep. 3, 2262 (2013). 063006 (2009).
14 C. Behncke, M. H¨anze, C. F. Adolff, M. Weigand, and 38 R. L. Compton, T. Y. Chen, and P. A. Crowell, Phys.
G. Meier, Phys. Rev.B 91, 224417 (2015). Rev.B 81, 144412 (2010).
15 K. Y. Guslienko, B. A. Ivanov, V. Novosad, Y. Otani, 39 H. Min, R. D. McMichael, J. Miltat, and M. D. Stiles,
H. Shima, and K. Fukamichi, J. Appl. Phys. 91, 8037 Phys. Rev.B 83, 064411 (2011).
(2002). 40 T. Y. Chen, M. J. Erickson, P. A. Crowell, and
16 S.-B. Choe, Y. Acremann, A.Scholl, A. Bauer, A. Doran, C. Leighton, Phys.Rev.Lett. 109, 097202 (2012).
J. St¨ohr, and H. A.Padmore, Science 304, 420 (2004). 41 A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen,
17 B. A.Ivanov,J. Appl.Phys. 95, 7444 (2004). F. Garcia-Sanchez, and B. Van Waeyenberg, AIP Adv.
18 K.Y.Guslienko,X.F.Han,D.J.Keavney,R.Divan, and 4, 107133 (2014).
S. D.Bader, Phys. Rev.Lett.96, 067205 (2006). 42 B.A.IvanovandC.E.Zaspel,Appl.Phys.Lett.81,1261
19 G. de Loubens, A. Riegler, B. Pigeau, F. Lochner, (2002).
F. Boust, K. Y. Guslienko, H. Hurdequint, L. W. 43 B. A. Ivanov and C. Zaspel, Phys. Rev. Lett. 94, 027205
Molenkamp, G. Schmidt, A. N. Slavin, V. S. Tiberke- (2005).
vich,N.Vukadinovic, andO.Klein,Phys.Rev.Lett.102, 44 J. P.Park andP.A.Crowell, Phys.Rev.Lett.95, 167201
177602 (2009). (2005).
20 N. Locatelli, V. V. Naletov, J. Grollier, G. de Loubens, 45 C.E.Zaspel, B.A.Ivanov,J.P.Park, andP.A.Crowell,
V. Cros, C. Deranlot, C. Ulysse, G. Faini, O. Klein, and Phys. Rev.B 72, 024427 (2005).
A. Fert,Appl. Phys.Lett. 98, 062501 (2011). 46 F. G. Aliev, J. F. Sierra, A. A. Awad, G. N. Kakazei,
21 K. S. Buchanan, P. E. Roy, M. Grimsditch, F. Y. Fradin, D.-S. Han, S.-K. Kim, V. Metlushko, B. Ilic, and K. Y.
K.Y.Guslienko,S.D.Bader, andV.Novosad,Phys.Rev. Guslienko, Phys. Rev.B 79, 174433 (2009).
B 74, 064404 (2006). 47 M. J. Donahueand D. G. Porter, OOMMF User’s Guide,
22 J.-V. Kim, Solid State Phys. Solid State Phys., 63, 217 Version 1.0, Interagency Report NISTIR 6376,Tech.Rep.
(2012). (NationalInstituteofStandardsandTechnology,Gaithers-
23 R. Lebrun, N. Locatelli, S. Tsunegi, J. Grollier, V. Cros, burg, MD, 1999).
F.AbreuAraujo,H.Kubota,K.Yakushiji,A.Fukushima, 48 T. Fischbacher, M. Franchin, G. Bordignon, and H. Fan-
and S.Yuasa, Phys. Rev.App.2, 061001 (2014). gohr, IEEE Trans. Mag. 43, 2896 (2007).
24 P.M.Braganca,B.A.Gurney,B.A.Wilson,J.A.Katine, 49 Simulationsforan identicallysized disk(R=192 nm,t=
S. Maat, and J. R. Childress, Nanotechnol. 21, 235202 30nm)witha3×3×3nm3 cellsizewereruninOOMMF
(2010). for zero field (agreement with MuMax within 1.1%) and
25 R. J. Ryan, H. Xi, and I. Jin, Patent US8053244 B2 in the presence of a localized Hloc field (Sec. IVA) with
(2011). wloc = 100 nm (agreement within 1%). There was also
26 T.Srimani,A.N.Mukhopadhyay,K.Roy, andM.Sharad, goodagreementinthebarediskfrequencyusingtheeigen-
arXiv:1511.09072 (2015). mode solver in FinMag for a R = 96 nm disk however a
27 K. Mizushima, K. Kudo, T. Nagasawa, and R. Sato, J. larger meshing had to be used at the disk edges due to
Appl.Phys. 107, 063904 (2010). memory constraints.
28 M. Inoue, A. Baryshev, H. Takagi, P. B. Lim, K. Hata- 50 A. Thiele, Phys.Rev.Lett. 30, 230233 (1973).
fuku,J.Noda, andK.Togo,Appl.Phys.Lett.98,132511 51 D. L. Huber,Phys.Rev.B 26, 3758 (1982).
(2011). 52 B.A.IvanovandC.E.Zaspel,Phys.Rev.Lett.99,247208
29 S.Atalay,A.O.Kaya,V.S.Kolat,H.Gencer, andT.Izgi, (2007).
J. Supercond.Nov.Magn. 28, 2071 (2015). 53 We observed a 0.4% change in G across different layers of
30 P. J. Metaxas, M. Sushruth, R. Begley, J. Ding, R. C. thedisk.
Woodward, I. Maksymov, M. Albert, W. Wang, H. Fan- 54 Y. Gaididei, V. P. Kravchuk, and D. D. Sehka, Int. J.
gohr, A. Adeyeye, and M. Kostylev, Appl. Phys. Lett. Quant.Chem. 110, 83 (2010).
106, 232406 (2015). 55 O.V.Sukhostavets,B.Pigeau,S.Sangiao,G.deLoubens,
31 R. S. Gaster, D. A. Hall, C. H. Nielsen, S. J. Osterfeld, V. V. Naletov, O. Klein, K. Mitsuzuka, S. Andrieu,
H.Yu,K.E.Mach,R.J.Wilson,B.Murmann,J.C.Liao, F.Montaigne, andK.Y.Guslienko,Phys.Rev.Lett.111,
S. S.Gambhir, et al.,Nat. Med. 15, 1327 (2009). 247601 (2013).
32 G. Heldt, M. T. Bryan, G. Hrkac, S. E. Stevenson, R. V. 56 Other disk geometries were simulated using a cell size of
Chopdekar,J.Raabe,T.Thomson, andL.J.Heyderman, 3×3×3.75 nm3.
Appl.Phys. Lett. 104, 182401 (2014). 57 N. Usov and S. Peschany, J. Magn. Magn. Mater. 118,
33 T.Uhlig,M.Rahm,C.Dietrich,R.Hllinger,M.Heumann, L290 (1993).
D. Weiss, and J. Zweck, Phys. Rev. Lett. 95, 237205 58 P. Wohlhu¨ter, M. T. Bryan, P. Warnicke, S. Gliga,
(2005). S. E. Stevenson, G. Heldt, L. Saharan, A. K. Suszka,
34 R. Zarzuela, S. V´elez, J. M. Hernandez, J. Tejada, and C. Moutafis, R. V. Chopdekar, J. Raabe, T. Thomson,
V. Novosad, Phys.Rev.B 85, 180401(R) (2012). G. Hrkac, and L. J. Heyderman, Nat. Comms. 6, 7836
(2015).