probing microscopic magnetic dots and their periodic arrays with ferromagnetic resonance g.n....
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Probing microscopic magnetic dots and their periodic arrays with ferromagnetic resonance
G.N. Kakazei Institute of Materials Science of Madrid, CSIC
Institute of Magnetism National Academy of Sciences of Ukraine, Kiev
Tim MewesMINT/Department of Physics and Astronomy, University of Alabama, USA
Phil WigenDepartment of Physics, Ohio State University, Columbus, USA
Ferromagnetic Resonance
H0
µ
Hrf
natural response
driven response
Precession and relaxation of M in response to an applied field H.
Lorentzian absorption line typical of FMR showing microwave power absorption as a function of swept bias field.
ω = 2πf = γB
0
1HM
t
M
Ferromagnetic resonance spectrometer
X-band : 9-10 GHz
K-band : 24-25 GHz
Q-band 34-36 GHz
)(sinsincoscos2
)coscossinsin(sin22
1222
Hus
HHs
KKM
HME
Calculation of the angular dependence for thin film
0E
s
us M
KMHHH
24 where,cossin)sin( effeffH
22
22
22
2
2
sin
1
EEE
M s
1st : Zeeman; 2nd : demagnetization energy
3rd : out-of-plane anisotropy 4th : in-plane uniaxial anisotropy
If in-plane uniaxial anisotropy is absent or negligible y
x
zz’
Ms
H
θ1
θH
From this equation we can find equilibrium θ can be found for each particular H
cos)cos(
2cos)cos(
eff
eff
2
HH
HH
H
H
g=2.2
Ms (from SQUID)=1.3103 emu/cm3
Ku=-4.3 105 erg/cm3
easy plane type
Micro-fabrication process (lift-off)
Exposed Areas
Double Layer Resist Spin-coating
EB Patterning
Resist Development
EB Deposition (1 A/sec) (FeNi target)
Lift-off process FeNi circular dot
Obtained circular dot array (AFM and SEM)
Si substrate
Permalloy,1 m,
t = 50 nm
Magnetic Vortex State in circular ferromagnetic dot
Magnetization reversal due to formation of the magnetic vortex state in circular dot
-150 -100 -50 0 50 100 150
-1.0
-0.5
0
0.5
1.0 (e)
(c)
(d)
(b) (a)
Mag
net
izat
ion
, M/M
s
Field, mT
H
(a) (b) (c) (d) (e)
H
“Vortex” nucleation
annihilation
Ferromagnetic resonance vs. Brillouin light scattering (BLS)
FMR advantagesHigh speed (1 minute per spectra with standard Bruker EMX-300 ESR
spectrometer)High resolution (position of the mode can be determined with an
accuracy of few Oersteds)External field can be applied at the arbitrary angle to the sample normal
(in BLS only in the sample plane)It is possible to study relaxation processes by measuring resonance
linewidth
FMR disadvantagesLower sensitivity (for patterned 20 nm thick Permalloy minimal sample
size is 0.2 mm x 0.2 mm)Fixed frequencyNot all types of modes can be exited in FMR studies
Rectangular arrays of magnetostatically coupled Ni dots
0.25 m0.8 m
1 m
0.05 m 0.10 m
It is known that Magnetostatic interaction in rectangular dot arrays affects the nucleation and annihilation fields, as well as the initial susceptibility.
Ni_1 Ni_2 Ni_3 Ni_4
Diameter (μm) 1 1 1 1
Thickness (nm) 70 70 70 70
Interdot distance a (μm)
1 1 1 1
Interdot distance b (μm)
0.05 0.1 0.25 0.8
Ni_1 Ni_2 Ni_3 Ni_4
Ni dots and continuous film – out-of-planeFMR angular dependence
210 / 101.9
kOe, 1.87/3
4 films, continuousFor
Gs 480 ,
cmdin
MH
πN
MHMNH
s
dem
ssdem
film. nuous
-conti for the than less 10%~ kOe, 1.65
ratio.aspect dot a is / where
)5.0)/8(ln(2 cylinder For
)31(4)(4
σ
yyxx
xxxxzzdem
H
rtb
bbNN
NNNN
Ni dots – FMR in-plane angular dependence
0 90 180 270 360
1.8
1.9
2.0
Res
onan
ce f
ield
Hr, k
Oe
Angle , degree
Ni_1 (Hin-plane~120 Oe)Ni_3 (Hin-plane~60 Oe)
Ni_4 (Hin-plane~ 0 Oe)
2
5.1
2
2
, If
minmax
max
maxminmax
HH
HH
HHHHH
HH
planein
planein
0.05 m 0.25 m
0 90 180 270 360
1.8
1.9
2.0
Res
onan
ce f
ield
Hr, k
Oe
Angle , degree0 90 180 270 360
1.8
1.9
2.0
Res
onan
ce f
ield
Hr, k
Oe
Angle , degree
Modeling: an array of magnetostatically coupled dots
HumHm Kww 2cos2/
k
k
2cos8
, 22
21
2
k
kRJkRf
TTK
yxu
kx=2/(2R+dx)
ky =2/(2R+dy)
2R
dy
dxFor rectangular array of cylindrical dots we can obtain the following decomposition of the magnetostatic energy density (in units of M2 and normalized per unit dot volume):
0.0 0.2 0.4 0.6 0.80
25
50
75
100
125
In-p
lane
ani
sotr
opy
fiel
d 2K
U/M
, Oe
Interdot distance dx, m
where δ=d/R (dx=d) is the normalized inter-dot distance, =L/R, J1(x) is the Bessel fun-ction, φk and φH are the polar angles of the vectors k and H, respectively. The function Ku( , δ) < 0 have sense of uniaxial anisotropy induced by interdot coupling with an easy magnetization axis parallel to the shor-test period Tx of the rectangular dot array (φH =0).K.Yu. Guslienko et.al. Phys. Rev. B 65, 024414 (2002).
In-plane anisotropy in square array of Py dotsPy square lattice of closely packed circular dots, 1/1.1 µm
Four-fold anisotropy (FFA) fields for square arrays are significantly smaller than in-plane uniaxial anisotropy fields in rectangular arrays. However it is was not clear what is a source of FFA. In completely magnetized square arrays noanisotropy should appear at all.
0 45 90 135 180 225 270 315 360
1060
1080
1100
1120
1100
1120
1140
1160
(b)
Res
onan
ce f
ield
(O
e)
Azimuthal angle (degrees)
(a)
Dependencies of the mean FMR field Hr,av and 4th order anisotropy constant H4 on the interdot spacing a in the square
array of circular dots.
1.0 1.5 2.0 2.5
1.10
1.12
1.14
Hr,
av (
Oe)
Center-to-center distance a (m)
1.0 1.5 2.0 2.50
5
10
15
20
25
Four
-fol
d an
isot
ropy
fie
ld (
Oe)
Center-to-center distance (m)
Correlations between 4-fold anisotropy and FMR resonance linewidth in square arrays of circular magnetic dots
0 30 60 90 120 150 1801090
1100
1110
1120
1130
Res
onan
ce f
ield
(O
e)
Angle between [10] direction and applied field (degree)
Evidence of 8-fold anisotropy for closely packed square array of Py dots
0 45 90 135 180 22526
28
30
32
34
36
Experimental data Fit axcos(4+bxsin(8R
eson
ance
line
wid
th (
Oe)
Azimuthal angle (degree)
Ferromagnetic resonance vs. Brillouin light scattering (BLS)
FMR advantagesHigh speed (1 minute per spectra with standard Bruker EMX-300 ESR
spectrometer)High resolution (position of the mode can be determined with an
accuracy of few Oersteds)External field can be applied at the arbitrary angle to the sample
normal (in BLS only in the sample plane)It is possible to study relaxation processes by measuring resonance
linewidth
FMR disadvantagesLower sensitivity (for patterned 20 nm thick Permalloy minimal sample
size is 0.2 mm x 0.2 mm)Fixed frequencyNot all types of modes can be exited in FMR studies
H = 2πMs ≈ 3 kOe
H = 0
BLS studies of Ni nanowires in nanoporous alumina
H ~ 2 kOe
Z. K. Wang, M. H. Kuok, S. C. Ng, D. J. Lockwood, M. G. Cottam, K. Nielsh, R. B. Wehrsphon, and U. Gosele, Phys. Rev. Lett. 89, 027201 (2002).
1 μm
30 nm
Spin wave resonance in normally magnetized Ni dots(experiment)
5000 6000 7000
Mic
row
ave
abso
rbtio
n de
riva
tion
(a.u
.)
Magnetic Field (Oe)
6000 8000
Mic
row
ave
abso
rbtio
n de
riva
tion
(a.u
.)
Magnetic Field (Oe)
Ni witness continuous film Array of Ni dots (Ni_1)
When the applied field was close to normal of the sample, additional sharp resonant peaks were observed below the main ferromagnetic resonance peak. No sight of such periodic SW spectra was found for the reference Ni continuous film, which supports the idea that in dots SW modes with discrete frequencies due to restricted sample geometry will dominate.
Spin wave resonance in circular Ni dots
For all the dot samples the differences ΔHr between neighbor modes, excluding the main one, were approximately the same ( ~ 250 Oe), indicating that the interdot interactions (different for each sample) do not affect magnetostatic spin wave modes in perpendicular geometry.
0.8 m
1 m
0.10 m
Two dimensional Bessel function modes of a disc
(1,0) (2,0) (3,0) (4,0)
Phase of the transverse (rf) moment
For perpendicularly saturated cylindrical dots these modes have circular symmetry and are spatially non-uniform due to the non-ellipsoidal dot shape. Condition for the existing of standing waves between specific turning points leads to the quantization of the observed resonance fields.
0.0 0.2 0.4 0.6 0.8 1.0-1.00
-0.75
-0.50
-0.25
0.00
0.25
0.50
0.75
1.00
-1.00
-0.75
-0.50
-0.25
0.00
0.25
0.50
0.75
1.00
()()
SW
eig
enfu
ncti
ons,
mn(
r)
Radius vector, /R
()
Eff
ecti
ve d
emag
neti
zing
fac
tor
-N(r
)
Comparing theory and experiment
Comparison of measured and calculated FMR resonance peak
positions: (a) patterned Ni film, parameters used
in calculation are Ms = 484 G, γ/2π = 3.05 MHz/Oe, L=70 nm, R=500 nm, H = 1.84 kOe, A=8×10-7 erg/cm;
(b) patterned NiFe film, parameters used in calculation are Ms=830 G, γ/2π = 2.96 MHz/Oe, L = 50 nm, R=500 nm, H=0, A=1.43×10-6 erg/cm.
1 2 3 4 5 6 7 8
10
11
12
1 2 3 4 5 6 7 84
5
6
7
b) Permalloy patterned film
experimental data calculations
Res
onan
ce f
ield
(kO
e)
Resonance mode number
a) Ni patterned film
experimental data calculations
Angular dependence of SWR in Py circular dots
8 10
= 6o
Magnetic Field (Oe)
10 12
Magnetic Field (kOe)
= 0o
10 12
= 4o
Magnetic Field (kOe)
Angular dependence of SWR in Py circular dots (contin.)
6 8
= 10o
Magnetic Field (kOe)3 4 5
= 20o
Magnetic Field (kOe)
1200 1500 1800
= 90o
Magnetic Field (kOe)
Out-of-plane angular dependence of the main peak
0 15 30 45 60 75 900
2
4
6
8
10
12
14
Mai
n pe
ak r
eson
ance
fie
ld (
kOe)
Angle (degree)
Experimental data calculations for Kittel uniform mode calculations for J
01 mode R∂m/∂n+dmR=0 (1)
pinned d=, mR=0
unpinned d=0, ∂m/∂nR=0
d is pinning parameter [1,2]. We can assume from experimental results that for perpendicularly magnetized dot (θ = 0o) pinned boundary conditions are realized, but for θ > 7o uniform Kittel mode perfectly describes the experiment. For Kittel mode pure UNPINNED boundary conditions are valid. So, in the range 0o < θ < 7o d varies from to 0 and, consequently, the basic mode varies from J0(K), where K is the first Bessel function root = 2.4, to J0(0)=Kittel mode. Between these points K can be determined as a root of (1) with corresponding d.[1] K.Yu. Guslienko, S.O. Demokritov, B. Hillebrands, A.N.Slavin, PRB 66, 132402 (2002)[2] R. Zivieri, R.L. Stamps PRB 73, 144422 (2006)
Transition from J0 to J1 near θ =0?
In-plane SWR in square array of Py dots
1200 1500 1800
[10]
[11]M
W a
bsor
btio
n de
riva
tion
(a.u
.)
Magnetic Field (Oe)
Py square lattice of closely packed circular dots, 1/1.1 µm
It is important to note that in-plane additional spin-wave modes for [10] and [11]are shifted in a different way with respect to the main one, indicating that interdot interactions DO AFFECT magnetostatic spin wave modes in in-plane geometry.
0 45 90 135 180
1100
1200
1300
1400
1500
1600 4th peak
3rd peak
2nd peak
Re
son
an
ce fi
eld
in-plane angle
1st peak
Ferromagnetic resonance vs. Brillouin light scattering (BLS)
FMR advantagesHigh speed (1 minute per spectra with standard Bruker EMX-300 ESR
spectrometer)High resolution (position of the mode can be determined with an
accuracy of few Oersteds)External field can be applied at the arbitrary angle to the sample normal
(in BLS only in the sample plane)It is possible to study relaxation processes by measuring resonance
linewidth
FMR disadvantagesLower sensitivity (for patterned 20 nm thick Permalloy minimal sample
size is 0.2 mm x 0.2 mm)Fixed frequencyNot all types of modes can be exited in FMR studies
Resonance linewidth
0 30 60 900
100
200
300
400
Res
onan
ce li
new
idth
H
pp (
Oe)
Angle (degree)
continuous film patterned film
For patterned film at perpendicular the linewidth is considerably smaller. This is because the density of spin-waves that the resonance can scatter into is much reduced, kmin = 2π/d.At parallel the internal fields will be much more inhomogeneous leading to a stronger coupling to higher order spin-waves and a shorter relaxation time. As a function of angle from the perpendicular the inhomogeneities gradually increases and the linewidth increases.
MRFM Spectrometer
Sputter Coated NiFe Tip
Bias Field Solenoid (± 300 G)
Bias Field H(permanent magnet)
H
DetectionCantilever(with H)
RF Microstrip ()
Sample on substrateModulation Field, Hmod
Fiber opticinterferometer
xy
z
Force
J. A. Sidles, Appl. Phys. Letts. 59, 2854 (1991)D. Rugar, C. S. Yannoni and J. A. Sidles, Nature, 360, 563 (1992)
Resonance Slice
Resonance Slice = Hres
Ftip = m(r,t)dH/dz
M Htip(r)
Happ < Hres
h(t)
Hint > Hres
Bint < Bres
FMRFM spectra of circular dot array
Spatially resolved FMRFM
Conclusions
FMR has shown its very high effectiveness in determination of different types of anisotropy in rectangular and square arrays of circular submicron dots
It is shown that, even under strong enough FMR field H, the dipolar coupling in the square array of circular magnetic microdots is able to produce some continuous deformation of uniformly magnetized state, depending on the in-plane field orientation.
We observed the multi-resonance FMR spectra of laterally confined spin wave modes in nickel and Permalloy circular dot arrays. These spectra are quantitatively described by a simple dipole-exchange theory of spin wave dispersion in a perpendicularly magnetized film.
The global FMR properties of circular submicron dots determined using magnetic resonance force microscopy are in a very good agreement with results obtained using X-band conventional FMR and with theoretical description.
AcknowledgementV.O. Golub, E.V. TartakovskayaInstitute of Magnetism, Kiev, UkraineP.C. HammelDepartment of Physics, Ohio State University, USAA.N. SlavinDepertment of Physics, Oakland University, USAYu.G. Pogorelov, M. CostaDepartment of Physics, University of Porto, PortugalK.Yu. Guslienko, V. NovosadMaterials Science Division, Argonne National Laboratory, Y. OtaniFrontier Research System, RIKEN, JapanS. BatraSeagate Research, Pittsburgh, USA
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