MAGNETO-OPTICAL MEASUREMENTSOF COLLECTIVE SPIN DYNAMICS

OF TWO-DIMENSIONAL ARRAYS OFFERROMAGNETIC NANOELEMENTS

BIVAS RANA and ANJAN BARMAN*

Thematic Unit of Excellence on Nanodevice Technologyand Department of Condensed Matter Physics

and Material Sciences, S. N. Bose National Centrefor Basic Sciences, Block JD, Sector III,

Salt Lake, Kolkata 700 098, India*abarman@bose.res.in

Received 6 February 2013Accepted 27 March 2013Published 30 May 2013

Magnetic nanodot arrays are interesting systems for future applications in nanotechnologyincluding patterned magnetic media, magnonic crystals, magnetic logic, sensors, STNOs andbiomedical applications. All applications require the knowledge base of magnetization processes ofmagnetic nanodot arrays at various time and length scales. Here, we review the present status ofexperimental studies of picosecond precessional magnetization dynamics in magnetic nanodotarrays. We discuss the fabrication methods of magnetic nanodot arrays and excitation anddetection methods of precessional dynamics by optical means. We further discuss the all-opticalexcitation and detection of precessional dynamics in Ni80Fe20 (permalloy) nanodot arrays withwidth between 200 nm and 50 nm and with interdot separation between 50 nm and 400 nm. Atransition from strongly collective dynamics to completely isolated dynamics through variousweakly collective regimes, variation of precession frequency and damping with the interdot sep-aration, e®ects of dipolar and quadrupolar interdot interaction, e®ects of the variation of dot sizeon the dynamics of single elements and arrays, and anisotropy of collective dynamics have beenthoroughly studied by experimental and micromagnetic simulation results. Finally, we discuss thefuture directions in the research on the dynamics of magnetic nanodot arrays.

Keywords: Nanomagnetism; magnetization dynamics; time-resolved magneto-optical Kerr e®ect;spin waves.

1. Introduction

Though magnetism is a very old topic of researchamong di®erent scienti¯c disciplines, nanomagnet-ism has become a topic of interest for modern sci-enti¯c research during past few decades. The reason

behind that is not only the fundamental scienti¯c

interest but also the tremendous potential to use the

nanomagnets in modern and future nanotechnology.

The scienti¯c communities have already proved

the potential of nanomagnets to use them as the

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building blocks for a range of multidisciplinary ap-plications in modern and future nanotechnologieslike nonvolatile magnetic memory,1�3 magnetic datastorage,4�6 magnetic recording heads,7 magneticresonance imaging,8 biomedicine and biotechnol-ogy.9,10 They also have potential applications in spinlogic devices,11�13 spin torque nano-oscillators(STNOs),14,15 and magnonic crystals.16

Magnetic recording was invented more than hun-dred years ago by Valdemar Poulsen.17 Since then, ithas played a key role in the development of di®erentkinds of nonvolatile storage devices like data, audioand video storage. The ¯rst magnetic hard disk driveof storage capacity 5MB was developed by IBM in1956. In modern disk drives, the storage capacity hasincreased by more than 20 million-fold in the quest toincrease the storage density and to reduce the cost.Conventional recording media uses a single layer thingranular ¯lms of hcp Co-based alloys where the easyaxis ofmagnetization lies in-plane of the ¯lm.18 As thestorage capacity increases, the grain size also decrea-ses and the magnetization of the grains becomes un-stable due to superparamagnetic e®ects.19,20 Thermalstability can be improved by reducing crystal-lographic defects and introducing materials withhigher anisotropy. Now, in a longitudinal magneticmedia, the magnetization of grains is oriented ran-domly, which increases the signal-to-noise ratio. Toreduce this kind of e®ect, sometimes the easy axes ofthe grains are set along a track direction. This is calledoriented longitudinal media.21 Antiferromagneticallycoupled (AFC) media are another kind of novelmagnetic media. AFC media consist of two magneticlayers, which are AFC through a thin nonmagneticlayer.22,23 In AFC media, the writing ¯eld is lowerthan the longitudinal media and stability can beoptimized by controlling the thickness of the layers.In a perpendicular recording media, the easy axis ofmagnetization is perpendicular to the ¯lm surface.24

The thermal stability of perpendicular media isgreater than the longitudinal media. Therefore, thestorage density can be increased by decreasing thegrain size. The heat-assisted magnetic recording(HAMR) can be used for longitudinal recording orperpendicular recording. In HAMR, the switching¯elds of the grains are reduced by heating them withlaser. Then the medium is quickly cooled back to thenormal temperature to store the data.21 In all theabove recording media, a number of grains are usedas a single bit.25,26 In patternedmagneticmedia, eachbit consists of a lithographically patterned island,

which has much larger volume than a single grain.Therefore, their thermal stability is higher than thegranular media.

Giant magnetoresistance (GMR) is observed inthin-¯lm structures composed of alternating ferro-magnetic and nonmagnetic conductive layers.27,28

Modern recording head of hard-disk drives (HDD)are based on the phenomenon of GMR with current-in-plane (CIP) geometry.7 In CIP geometry, current°ow parallel to the layers. To overcome few limi-tations of CIP, a new type of sensor geometry calledcurrent-perpendicular (CPP) geometry can be used.In CPP geometry, current °ow perpendicular to theplane of the layers. Magnetic tunnel junction (MTJ),where two ferromagnetic thin-¯lm layers are sepa-rated by a thin insulating nonmagnetic barrier,also makes read-head sensors because of the largesignal at room temperature,29 which comes fromlarge tunnel-magetoresistance ratios (TMRs). But,MTJs are accompanied by large noise.30 The con-ventional solid state memories like dynamic randomaccess memory (DRAM) and the static randomaccess nemory (SRAM) rely on the electric charges.Therefore, they are volatile in nature and capable ofnanosecond access times in both read and writeoperations.1 The major nonvolatile memory usedtoday is FLASH but it has slow write access time andpoor bit cyclability. Ferroelectric RAM (FeRAM)is another kind of nonvolatile memory, which hasthe potential to have high speed performance similarto DRAM. But it has many processes of integra-tion complexities. Magnetoresistive random accessmemories (MRAM) is nonvolatile memory whicho®ers very high read-write speed by consuming verylow power.31 They are based on GMR elements orMTJs. However, the basic MRAM bits su®er fromhalf-select problem, which limits the bit size to about180 nm. In order for MRAM to be a productivetechnology, bit size need to reduce down to 65 nm,which requires the spin transfer torque (STT)switching and a thorough investigation of thedynamics of arrays of sub-100 nm MRAM bits atvarious timescale is required for further advancementof technology.

Conventional logic devices are based on transis-tors. The ongoing increment of the number ofdevices per unit area needs shrinkage of metal-oxide-semiconductor ¯eld e®ect transistor (MOSFET).This will cause the power dissipation problem andwill increase the di±culties in interconnection wir-ing.32 A new type of technology called \spintronics"

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are promising an alternative route as a solution tothis problem.33 In spintronics, both the spin andcharge of electrons are used for logic and memoryoperations. The spintronic devices promise highspeed and nonvolatile devices with low power con-sumption.33,34 There are many reports on magneticlogic devices. Few reports have been published onmagnetic logic based on a single electron transistorarchitecture.35 The logic devices can also be fabri-cated using MTJs.36,37 Magnetic nanowires showvery high shape anisotropy where the magnetizationprefers to align along the long axis. Binary data canbe stored on the basis of direction of magnetization.During the change in the magnetization direction,the domain wall propagates through nanowires. Thepropagation of domain wall can be controlled byan externally applied magnetic ¯eld. Logic devicesbased on the domain walls have also been repor-ted.11 Spin waves can also be used in logic devices aswas reported by Kostylev et al.38 The properties ofspin wave interference are used in a Mach�Zehnder-type current-controlled spin wave interferometer.Di®erent kinds of spin wave gates have beendemonstrated by using spin wave amplitude andphase.41

The concept of spin waves in a periodic mediawas ¯rst introduced by Bloch.42 Spin waves arebasically collective dynamics of spins in a continu-ous or patterned magnetic media. Magnons are theparticle counterpart of the spin waves. The periodicarrays of nanomagnets may be considered as anarti¯cial crystals where spin waves or magnonsare the carriers, and are therefore called as the\magnonic crystal" analogous to the \photoniccrystal" where the light or photons are the carriers.Following this, a newly born and evolving researcharea called \magnonics" is developing.16,43�45 Theperiodic modulation of the magnetic parameterstailors the magnonic band structure with the cre-ation of magnonic band gaps. The aim of the modernresearch in magnonics is to control and manipulatethe magnonic band structure46�48 by varying variousphysical and material parameters of the magnoniccrystals as well as the external magnetic ¯elds.Ordered arrays of magnetostatically coupled mag-netic nanodots, nanostripes have the potential toserve the purpose of the magnonic crystals. The dotsor nanostripes can be made of a single component(element), bi-component, and multilayer structurealso. Magnetic antidot lattices, which are basicallythe array of holes on the magnetic thin ¯lms are also

showing various promises as an exchange coupledmagnonic crystal, which show larger spin wavepropagation velocity (steeper dispersion) than mag-netic dot lattices.

The successful applications of spin waves in futuretechnology strongly rely upon their generation, con-trolled propagation and detection mechanisms. Thechallenge is to generate controlled spin waves atde¯ned frequencies. The spin waves can be generatedby picosecond magnetic ¯eld pulses either launchedfrom a photoconductive switch49 or generated by apulsed electric ¯eld.50 Harmonic51 or pulsed49 mag-netic ¯elds can be used in this case. One can also useultrashort laser pulses52,53 to generate the spin waves.The STT, a recently discovered physical phenom-enon, being used to excite spin waves with the help ofspin polarized current.54 The spin waves are generallyguided by one-dimensional magnetic stripes.55�57

and detected by probing optically,58 electrically59,60

or by inductive method.61,62 The future technologydemands various spin wave based nanoscale micro-wave components and devices. Consequently, devicessuch as spin wave ampli¯er,63 spin wave phase shif-ter64 and frequency ¯lter and interferometer39 arealso subject of intense research. Di®erent propertiesof spin waves like re°ection and refraction,65�68

interference and di®ractions,69�72 tunneling73,74 andfocusing and self-focusing75�78 have been studiedtheoretically and experimentally in last few decades.

Beyond the applications in spintronics and mag-nonics, nanomagnets have opened new windows forbiomedical and biotechnological applications. Mag-netic nanoparticles show new promising applicationsin improved contrast enhanced magnetic resonanceimaging,79,80 targeted drug delivery,81 manipulationof cell membrane,82 and hyperthermic treatmentfor malignant cells.83,84 Since ferromagnetic nano-particles could be harmful for the biological cells,superparamagnetic nanoparticles are used for bio-medical treatments as they do not retain any netmagnetization in absence of magnetic ¯eld.

For most of the potential applications mentionedabove, the nanomagnets should be arranged in anordered array such that they can be addressedeasily. Therefore, in recent years fabrication and thestudy of magnetization dynamics of ordered arraysof nanomagnets have attracted much attentionof the nanomagnetism community.85�95 Moderndata storage and memory technologies also demandhigher operational speed along with the higherstorage density. The conventional storage technology

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depends upon the slow magnetic domain wall dy-namics, which limits the data writing and readingspeed in the nanosecond regime. The faster speeddemands the operation of nanomagnets far beyondthe gigahertz regime. The picoseconds precessionalmagnetization dynamics of nanomagnets may ful¯llthe above requirement. The ultrafast precessionalswitching96�98 where the magnetization reversal istriggered by a large angle precessional dynamics canbe used for high speed data storage and memorydevices. However, a thorough understanding andcontrol of the precessional dynamics including theprecession frequency and the damping coe±cient ofthe nanomagnets are required for its application intechnology.

The quasi-static and ultrafast magnetizationdynamics of con¯ned magnetic structures are di®er-ent from their bulk counterparts and continuous thin¯lms. Magnetization dynamics of nanomagnetsstrongly depends upon their static magnetizationstates, which depend not only on their intrinsic ma-terial parameters such as exchange sti®ness constant,saturation magnetization and magnetocrystallineanisotropy, but also on their physical structuresincluding the shape, size and aspect ratio99�102 aswell as the external parameters like the strength anddirection of the bias magnetic ¯eld.103 For an array ofnanomagnets, the static magnetization state alsodepends upon the physical parameters of the arrayssuch as the lattice constant and the lattice symmetry.The quasi-static magnetization reversal of nano-magnets can be of various types. For an isolatedsingle or quasi-single domain nanomagnet the rever-sal may occur through the coherent rotation ofmagnetization or through the formation of curling,buckling, fanning, vortex, onion, °ower, leaf, C- andS-states depending upon the shape, size and aspectratio of the nanomagnet.89,104�109 On the other hand,multidomain nanomagnets reverse through domainwall dynamics.109 In contrary, the reversal for clus-ters and arrays of dipolar or exchange couplednanomagnets may occur through the formation ofdi®erent collective modes, where the °ux °owsthrough a number of nanomagnets to form theminimum energy con¯guration and various reversalmodes as above may also be observed.108,110�112

While the shape, size and aspect ratio of the indi-vidual nanomagnets control the nonuniform demag-netizing ¯eld within them, lattice periodicity andlattice constant and the external magnetic ¯eld con-trol the strength and type of the interactions among

the nanomagnets. The static magnetization states ofnonellipsoidal nanomagnets are generally nonuniformdue to the nonuniform nature of the demagnetizing¯eld. The nonuniform static magnetization statesmay trap the spin waves locally or quantizethem.90,113�119 Magnetic vortices are very interestingnonuniform ground state of magnetization, which arefound in micro- and nano-disks when the aspect ratiode¯ned by thickness/diameter is much less than oneand both thickness and diameter exceed the exchangelength in that material. The vortex dynamics isdominated by gyration of the core and radial andazimuthal spin wave modes of the chiral spins, whichare much di®erent from the dynamics of a singledomain nanomagnet. Magnetic vortices are the po-tential candidates for future nonvolatile memory andlogic devices.120

The key challenges to study the physical phe-nomenon mentioned above are the synthesis orfabrication of high quality magnetic nanostructuresand their characterizations. Technology demandsfabrication of nanomagnets with narrow size dis-persion and in an ordered array over a macroscopiclength scale. These led to the development of anumber of \top-down" and \bottom-up" approachesin nanofabrication methods and more recently acombination of the two. The solution phase colloidalchemistry is applied in \bottom-up" approach tosynthesize monodisperse nanocrystals of uniformshapes and sizes.121 There are many colloidal chem-istry-based methods like reduction, nonhydrolyticsol-gel and thermal decomposition processes. Inreduction method, various kinds of reductants areused to synthesize metallic nanoparticles, whereasnonhydrolytic sol-gel process is mainly used tosynthesize metal oxide nanoparticles. In thermaldecomposition methods, the decomposition of orga-nometallic compounds is performed in hot surfactantsolutions to synthesize nanoparticles of variousmaterials. Reverse micelle methods are also verye±cient to synthesize various kinds of nanocrystals.However, the physical methods are utilized in \top-down" approach, which includes di®erent kinds oflithographic techniques such as photolithography,122

electron beam lithography (EBL),123 deep ultra-violet (DUV) lithography,124 X-ray lithography,125

interference or holographic lithography (IL),126

nanoimprint lithography (NIL),127 and ion beamlithography (IBL).128 Scanning probe lithography,129

step growth methods,130 shadow masks,131 laser orion irradiation132 are also very promising techniques.

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Other di®erent techniques for preparing orderednanostructures are self-assembly mechanism likeheterogeneous nucleation of magnetic atoms on met-allic surfaces,133 seeded growth,134 and nanotemplatefabrication technique using templates such as diblockcopolymers,135 anodized alumina membranes136 andnuclear-track etched membranes.137

To study the quasi-static and ultrafast dynamicproperties of nanomagnets di®erent kinds of sensi-tive characterization techniques have been develo-ped in last few decades.58 Magnetic force microscopy(MFM)138,139 and Lorentz force microscopy140 arenow commonly used microscopies, which are verye±cient to map the stray magnetic ¯eld and thesample magnetization with a spatial resolutionbetter than 10 nm. The contrast of the images comefrom magnetic force between the scanning magnetictip and the sample stray ¯eld or magnetization incase of MFM and the de°ection of the acceleratedelectrons by Lorentz force after transmittingthrough the sample surface in case of Lorentz mi-croscopy. The main drawback for both techniques isthe di±culty of extracting quantitative informationdirectly from the images. Electron holography141 isanother electron transmission method based on theelectron interference. The amplitude as well as thephase information of the magnetic domains andstray magnetic ¯eld can be mapped with a very highspatial resolution close to 2 nm. Magneto-opticalKerr microscopy (MOKE)142 is widely used to mapthe sample magnetization with a di®raction limitedspatial resolution of about few hundreds of nan-ometer. The photoemission electron microscopy(PEEM),143 which is basically X-ray microscopy isused with better resolution than visible light imag-ing. Spin polarized low energy electron microscopy(SPLEEM),144 scanning electron microscopy withpolarization analysis (SEMPA),145 spin polarized-scanning tunneling microscopy (SP-STM)146 andballistic electron magnetic microscopy (BEMM)147

are other imaging techniques, which give spatialresolution close to 10 nm or better. All these tech-niques use the spin dependent transmission, scat-tering, tunneling for image contrast. The imagingmethods mentioned above are used to map staticmagnetization state with tremendous spatial reso-lution but with very poor temporal resolution. Toinvestigate fast magnetization dynamics of nano-magnets di®erent kinds of techniques have emerged.The conventional ferromagnetic resonance (FMR)148

or vector network analyzer�ferromagnetic resonance

(VNA�FMR)149 techniques are very e±cient tomeasure the high frequency magnetic response fromMHz to GHz regime with a very good frequencyresolution. The poor spatial resolution of FMRtechniques pushed to develop spatially resolvedFMR.150 Pulsed inductive microwave magne-tometry (PIMM)151 is another technique to measurethe dynamics in time-domain with tens of pico-seconds time resolution. The Brillouin light scat-tering (BLS) is a very e±cient technique to studythe spin waves in the wave-vector domain.152 Thefrequency dispersion of the spin waves with theirwave-vector can be measured directly by usingthis technique. Recently space-resolved and timeresolved BLS techniques have been developed toobtain sub-�m spatial resolution and few ns time-resolution.153 The best spatio-temporal resolution isobtained from time-resolved magneto-optical Kerre®ect (TRMOKE) microscopes.99 They are used toprobe the ultrafast magnetization dynamics in time-domain with sub-hundred femtosecond time resol-ution limited by the pulse width of the laser. Themagnetoresistive methods154 and X-ray micro-scopy155 also have the potential to achieve very goodspatio-temporal resolution like TRMOKE. Time-resolved scanning Kerr microscopy (TRSKM) isanother variant of TRMOKE, which is used toimage the time evolution of magnetization excitedby a time-dependent magnetic ¯eld.156�158

Here, we will review the collective precessionalmagnetization dynamics of arrays of magneticnanodots by an all-optical TRMOKE microscope.The time resolution of the set-up is � 100 fs alongwith a spatial resolution of � 800 nm. The reviewis organized as follows. In Sec. 2, we will describethe details of sample fabrication. In Sec. 3, we willdescribe the details of the TRMOKE techniques.In Sec. 4, we will describe the time-resolved mag-netization dynamics of magnetic nanodot arrays.This section contains a brief review of the collectivedynamics of magnetic nanodot arrays and singlemagnetic nanodots studied by di®erent experimen-tal techniques followed by some recent results by theauthors on the collective magnetization dynamicsof arrays of permalloy nanodots. This is aided byanalysis of the experimental results with the aid ofmicromagnetic simulations using OOMMF159 andDotmag softwares.48,160 We will discuss the methodto achieve very high measurement sensitivity in theTRMOKE set-up with a two-color microfocusedcollinear pump-probe geometry, which enabled the

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detection of the picoseconds magnetization dynam-ics of 50 nm nanodot arrays down to the singlenanodot regime. At the end, we will discuss aboutthe future perspectives and the scopes on collectivedynamics of magnetic nanodot arrays.

2. Fabrication Methods

2.1. Introduction

Nanofabrication is at the heart of nanoscience andnanotechnology. To implement the nanomaterialsin di®erent kinds of advanced nanodevices, it is verycrucial to fabricate good quality nanostructureswith uniform shapes and sizes in an ordered arrayover macroscopic length scales. The physical prop-erties of them have to be well studied prior to thedesired applications. For di®erent experimentalstudies, the nanostructures should be physicallyuniform with excellent surface quality and highchemical purity with uniform chemical composition.Another challenge is the development of cost e®ec-tive fabrication techniques with a high yield. Anumber of di®erent nanofabrication methods havebeen developed over the years, out of which thelithographic methods are the most precise and e±-cient methods for fabrication of large ordered arrayof nanomagnets. These methods are not very coste®ective, particularly for very large area nano-fabrication. However, they are the best option forfabrication of high quality samples suitable for sci-enti¯c research, where sample quality may playimportant roles over the intrinsic properties of thematerials. Below, we will describe some selectivelithographic fabrication methods used for our study.

2.2. Optical lithography

Ultraviolet (UV) photolithography was used tofabricate 10� 10 �m2 unpatterned permalloy ¯lm.First of all the Si substrate (100) was ultrasonicatedin acetone for 20min to remove the organic ma-terials from the top surface of the substrate. Anitrogen gun was used for drying the substrate aftercleaning. A uniform layer of positive photoresistdissolved in some organic liquid was then coated onthe substrate using a spin coater. The typicalthickness of photoresist layer was about one microndepending upon the viscosity of the resist and thespinning speed. A photo mask (chrome on glass)containing the square patterns to be fabricated was

placed on top of the substrate. A convex lens wasused in between source and the mask for uniformillumination of the source on the mask. The polymerchains in the resist were broken at the exposed areaafter the exposure. The exposed area was etched outby wet chemical process. The substrate was ¯rst putinto distilled water at room temperature, rinsed for60 s and dried using nitrogen gun. The substrate wasthen put into MIBK:IPA (1:3) solution and rinsedfor 30 s. Finally, the substrate was put into acetone,rinsed for 60 s and dried with nitrogen gun. There-fore a two-dimensional pattern was developed ontothe resist. After the etching process a 20 nm thick¯lm was deposited at a deposition rate of 0.3 nm/sby electron-beam evaporation at a base pressure of2:0� 10�8 Torr. A thin capping layer of alumina(Al2O3Þ of 10 nm thickness was deposited by elec-tron-beam evaporation on top of the ¯lm to protectit from oxidization or any kind of external damage.Finally, the ¯lm deposited on the resist was lifted o®along with the resist leaving only the depositedmetal onto the surface.

2.3. Electron beam lithography

As electron beams have very short wavelength atlarge enough accelerating voltage, they are used forpatterning smaller features with a very high spatialresolution of few nanometers. The square arraysof square permalloy dots with di®erent sizes andseparations were fabricated by electron-beam litho-graphy. The schematic diagram of the process isshown in Fig. 1. First of all the substrate wascleaned with water, acetone and IPA (2-propanol)respectively by using an ultrasonicator. PMMAresist was spin coated onto a cleaned Si substrate(100) for 40 s at a speed of 5000 rpm to make auniform layer of about 1.5�m thickness. Anotherlayer of 1�m thick MMA was made on the PMMAlayer at a rotation speed of 1500 rpm for 90 s. In thisway, the Si substrate was coated with the bilayer(PMMA/MMA) resist. The scanning electronmicroscope (Elionix ELS-7700H) was used to exposethe resist with focused electrons. At ¯rst, the desiredstructures were designed in a PC using commerciallyavailable design software (Auto-CAD). The design¯les were then converted into the positions de¯ningCELL ¯les and were uploaded to the PC attachedto the lithography system. Ultimately, the designpatterns were written onto the resist layers by

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accelerated e-beam controlled by the PC. The beamcurrent and dose time were optimized at 100 pA and0.9�1.0�s. The chemical etching process, alreadydiscussed in optical lithography technique was usedto etch out the exposed material from the substrate.A 20 nm thick permalloy ¯lm was post-depositedby electron-beam evaporation with an optimizeddeposition rate 0.3 nm/s at a base pressure of 2:0�10�8 Torr. A 10 nm thick capping layer of alumina(Al2O2Þ was deposited by electron-beam evapora-tion on top of the pattern dots and resist to protectthe sample from further oxidization or degradationduring the exposure of the laser beam duringmeasurements. The lift-o® technique was used toremove the unexposed resist along with the ¯lmdeposited on it. Ultimately the substrate was dippedinto acetone for 12 h to remove residual resists.

3. Time-Resolved Magneto-OpticalKerr E®ect Measurement

MOKE has been an e±cient tool for measuring themagnetic hysteresis loops and imaging magnetiza-tion con¯gurations including domain imaging for along time since its discovery in 1877.161 Since then, ithas been used in various con¯gurations in static anddynamic regimes.162,163 The ¯rst TRMOKE measure-ment was performed by Freeman et al. in 1992.164

Since then di®erent variants of time-resolved Kerre®ect measurements have been reported. Some im-portant developments have been the introduction of

time-resolved scanning Kerr microscope (TRSKM),156

all-optical TRMOKE for measurement of ultrafastdemagnetization165 and coherent spin waves,166 ap-plication of time-resolved Kerr measurements tosamples fabricated on opaque substrate,167,168 time-resolved cavity enhancedMOKEmeasurements99 andmore recently a benchtop time-resolved Kerr mag-netometer.169,170 Since then time-resolved magneto-optical Kerr microscope has emerged as a powerfultechnique for a noninvasive probing of ultrafastmagnetization dynamics of arrays of magneticnanostructures and single magnetic nanodots.

3.1. Magneto-optical Kerr e®ect(MOKE) and various MOKEgeometries

When a linearly polarized optical beam is re°ectedfrom or transmitted through a magnetic material, itis converted into an elliptically polarized light wherethe major axis is rotated from the original polari-zation direction. The re°ection geometry corres-ponds to the magneto-optical Kerr e®ect, while thetransmission geometry corresponds to the magneto-optical Faraday e®ect. For a very small rotation(�mdeg), i.e., in the linear regime the angle ofrotation or Kerr rotation is linearly proportional tothe magnetization of the system. There are threekinds of MOKE geometries, namely the longitudi-nal, transverse and polar geometries dependingupon the orientation of magnetization vector w.r.t.the sample surface and plane of incidence of light.170

In the longitudinal geometry, the magnetization(M) lies parallel to the sample surface and the planeof incidence [Fig. 2(a)]. For polar geometry M liesperpendicular to the sample surface and parallel tothe plane of incidence [Fig. 2(c)]. In both cases, theincident linearly polarized light (p- or s-polarized)is converted to elliptically polarized light with therotation of the major axis w.r.t. the plane of polar-ization of incident light. The Kerr rotation (�kÞ isshown schematically in Fig. 2(d). Another param-eter called Kerr ellipticity ("k) measures the °atnessof the ellipse. The Kerr rotation (�k) and ellipticity("k) are related to each other by a relation: �k þi"k ¼ k=r, in the limit k � r.171 The third MOKEgeometry is called the transverse geometry whereM lies in plane of the sample, but perpendicular tothe plane of the incident of light [Fig. 2(b)]. Onlyp-polarized light shows transverse Kerr e®ect. In this

Fig. 1. Schematic diagram of fabrication of magnetic dotsarray by electron-beam lithography method (color online).

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case, the re°ected beam remains linearly polarizedwithout any Kerr rotation, but the amplitude ischanged as magnetization vector changes sign fromþM to �M.

3.2. Field pumped time-resolvedscanning Kerr e®ect microscope(TRSKM )

TRSKM is used to measure the precessional mag-netization dynamics in the time domain. Thedynamics is pumped by a pulsed magnetic ¯eld172

generated by optical triggering of a photoconductiveswitch, popularly known as an Auston switch or bya picosecond pulse generator.169,170,173 In a ¯eldpump geometry using an Auston switch, the femto-second pulsed laser beam is divided into two parts, astronger part known as the pump beam and aweaker part known as the probe beam.174 The pumpbeam goes through a variable delay line and isincident on an Auston type photoconductive (PC)switch (usually Au on intrinsic or doped GaAs) fortriggering the PC switch. The PC switch is generallyconnected with a microwave transmission line (TL)or a coplanar waveguide (CPW) structure and a DC

bias voltage is applied across the circuit as shown inFig. 3(a). The optical triggering by a femtosecondlaser pulse generates a fast photocurrent with rise-time (typically 10�100 ps) determined by the laserpulsewidth and carrier parameters of the GaAssubstrate and the decay time (typically 100 ps�1 ns)is determined by the carrier recombination time ofthe GaAs. The photocurrent °ows through the TLor the CPW and generates a pulsed magnetic ¯eld.The samples are either grown directly on the TL/CPW structure or on a transparent substrates andthe substrate is overlaid onto the TL/CPW struc-ture. The pulsed magnetic ¯eld excites the dynamicsof the sample, while the probe is used to probe theKerr rotation as a function of the time-delay bet-ween the pump and probe beams. In the TRSKM,the probe is focused to a sub-micron spot on thesample surface and the sample is scanned underthe focused probe spot to build the scanning Kerrimages at a ¯xed time-delay. The measurement of theKerr rotation is done either by a crossed polarizerarrangement set at extinction, by using a photo-elastic modulator or by an optical bridge detector(OBD).170,175 The optical bridge detector consists ofa polarized beam splitter (PBS) set at 45� to there°ected light and two photodiodes. The di®erence inthe signal between the two photodiodes is pro-portional to the Kerr rotation, while the sum of thetwo photodiodes gives the total re°ectivity.

3.3. All-optical time-resolvedmagneto-optical Kerr e®ect(TRMOKE ) microscope

The magnetization dynamics of ferromagnetic thin¯lms and nanostructures can also be excited all-optically by femtosecond laser pulses. Figure 3(b)shows the schematic diagram of all-optical exci-tation and detection technique. All the experimentaltime-resolved data presented in this review havebeen measured by an all-optical TRMOKE micro-scope based upon a two-color collinear pump-probegeometry. The schematic diagram of the TRMOKEset up is shown in Fig. 3(c). A mode-locked Ti-sapphire laser pumped by a diode pump solid statelaser is used in this experiment. The output pulseshave maximum power of 2W at a repetition rate80MHz (25 nJ/pulse) with �70 fs pulsewidth andthe wavelength can be tuned between 700 nm and1100 nm. The output is generally ¯xed to 800 nm for

(a) (b)

(c) (d)

Fig. 2. Schematic diagrams of (a) longitudinal, (b) transverseand (c) polar MOKE con¯gurations are shown. Here, r is theFresnel re°ection coe±cient, k is the Kerr coe±cient, andM is the magnetization of the sample. (d) Geometry of the Kerrrotation (�k) and Kerr ellipticity ("k) is demonstrated (coloronline).

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stable operation and for increased sensitivity of theSi-based detectors. The output beam is divided intotwo parts (80:20 or 90:10). The stronger part isfrequency doubled (� ¼ 400 nm) by passing intoa second harmonic generator (SHG) with pulsewidth �100 fs. The second harmonic is used topump the magnetization dynamics whereas the lin-early polarized fundamental beam is used for prob-ing. A spectral ¯lter is placed outside the SHG to¯lter out the residual fundamental beam, becausea trace amount of the fundamental beam mixedwith the pump beam can produce a very noisy sig-nal. The pump beam travels through a ¯xed pathand is combined with the aid of a beam combinerwith the probe beam, which comes through a vari-able delay line. Both the beams are made collinearfrom the point at which they meet at the beamcombiner with the help of steering mirrors. Theprobe beam is focused to a di®raction limited spotsize of � 800 nm at the sample surface with the helpof the microscope objective (N:A: ¼ 0:65) and an

x�y�z piezoelectric scanning stage. The pump beamis spatially overlapped with the probe beam afterpassing through the same microscope objective in acollinear geometry (probe beam is placed at thecenter of the pump beam). At the sample plane,which is also the focal plane of the probe beam, thepump beam is slightly defocused with spot size of�1�m. The optical arrangement is shown in Fig. 4.

Both pump and probe beams are re°ected fromthe sample surface, a small fraction of which are sentto a CCD camera for viewing the pump and probebeams as well as the sample. A white light source isused to illuminate the sample surface for a betterview. The major parts of the pump and probe beamsare then sent through a spectral ¯lter to eliminatethe pump beam before sending to the OBD. Thedynamics is measured by measuring the polar Kerrrotation by means of the OBD. In absence of anyKerr rotation (pump beam blocked) a linearlypolarized beam will produce identical intensity intwo orthogonal components of polarization and

(a) (b)

(c)

Fig. 3. The schematic diagrams of (a) magnetic ¯eld-pumped excitation and (b) all-optical excitation are shown. (c) Schematicdiagram of a two-color all-optical TRMOKE microscope (color online).

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consequently identical signals in two photodiodes(A and B) and the OBD is said to be balanced(A� B ¼ 0). When the pump is incident on thesample, the magnetization state of the sample willchange, causing a Kerr rotation. The Kerr rotationmodi¯es the intensities in the two orthogonal com-ponents of polarization and gives rise to a ¯niteelectronic signal at the output of the optical bridgedetector (A� B 6¼ 0). Therefore A� B gives theinformation about the magnetization dynamicsincluding ultrafast demagnetization, remagnetiza-tion and generation and decay of spin waves. Onthe other hand, the sum signals of the two photo-diodes (Aþ B) gives the total re°ectivity signaland gives information about the dynamics of theelectronic state of the sample as well as the acousticmodes. The Kerr rotation and re°ectivity signalsare measured by two lock-in ampli¯ers in a phasesensitive manner. The pump beam is modulatedby a chopper at a frequency of 1�2 kHz. The fre-quency of the chopper is used as the reference sig-nal for the lock-in detection. The average pumppower used in these measurements vary between7mW and 15mW (7.7�15.4mJ/cm2), while theprobe power is much weaker and is about 1�2mW(1.4�2.8mJ/cm2).

A large enough magnetic ¯eld (well above satu-ration) is applied at an angle (�15�) to the sampleplane to saturate its magnetization. The magnetic¯eld strength is then reduced to the bias ¯eld value(Hb ¼ component of bias ¯eld along x-direction),which ensures that the magnetization remainssaturated along the bias ¯eld direction. The bias¯eld is tilted 15� out of the plane of the sample tohave a ¯nite demagnetizing ¯eld along the directionof the pump pulse, which is eventually modi¯ed bythe pump pulse to induce precessional magnetiza-tion dynamics within the dots.

A very good spatio-temporal resolution alongwith a high measurement sensitivity (�� deg) isobtained by the collinear pump-probe geometry.Figure 5 shows a schematic representation of adirect comparison of temporal resolution and thesize of the individual dot measurement sensitivityamong di®erent techniques. Electron microscopyhas very good spatial resolution but poor temporalresolution. Brillouin light scattering, X-ray micro-scopy and GMR-based technique has moderate timeand spatial resolution. Cavity enhanced magneto-optical Kerr e®ect (CE-MOKE) microscope hasshown the best spatio-temporal sensitivity so far.Using the collinear microfocused pump-probe tech-niques we have successfully measured the dynamicsof 50 nm dots down to the single nanomagnet regime

Fig. 4. Schematic diagram showing the geometry of the bias¯eld and the pump and probe beams focused onto the sample bya microscope objective (color online).

Fig. 5. A schematic representation of the spatio-temporal¯gure of merit of di®erent techniques for the measurements ofthe time-resolved magnetization dynamics of single (isolated)nanomagnets (color online).

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with about 100 fs temporal resolution, which trans-lates into an overall improvement by a factor of3 than the previously published results.

4. Time-Resolved MagnetizationDynamics of Magnetic NanodotsArrays and Single Nanodots

4.1. Background

Ordered arrays of magnetic nanodots are potentialcandidates for various applications including mag-netic data storage. The interest in studying mag-netization dynamics in magnetic nanodots stemsfrom exploring the con¯ned spin wave modes inmagnetic nanodots and its size dependence.90,116,176

However, the magnetostatic interaction among thenanodots in a closely packed array may signi¯-cantly a®ect the intrinsic dynamic of the individualelements. E®orts have been made later to probe theintrinsic dynamics of single magnetic nanoelements asdescribed later in this article. The magnetostaticallycoupled nanodot arrays, shows a collective dynamics,where all the elements maintains a constant ampli-tude and phase relationships. This opened a newwindow of opportunity for applications in magnoniccrystals,16,45,46,177 spin wave logic devices11,13 andSTNOs14,15 in addition to fundamental interests.Consequently, collective magnetization dynamics inarrays of spatially con¯ned magnetic nanoelementsespecially in dot arrays have been studied by variousgroups in the last decade.

The collective dynamics of dot arrays have beenstudied by a variety of techniques in the frequency,time and wave-vector domains. All the techniqueshave their own advantages and limitations. In 2002,Jung et al.88 reported collective magnetizationdynamics of square arrays of circular permalloydots with varying sizes and separations in sub-micron range by using FMR. Two additional res-onant peaks were observed on both sides of themain resonant peak corresponding to the uniformmode. A four-fold anisotropy in the frequency of theresonant peaks was observed when the azimuthalangle of the bias ¯eld was varied re°ecting thesymmetry of the arrangement. The peak with fre-quency below the uniform mode was identi¯ed asexchange spin wave mode whereas the peak aboveuniform mode was identi¯ed as dipole-exchangespin wave mode with the help of micromagneticsimulation.

BLS measurements on arrays of circular permal-loy dots with varying radius in sub-micron range178

reveal dispersionless quantized dipole-exchangeDamon�Esbach (DE) and backward volume mag-netostatic (BWVMS) modes. It was found that theBWVMS modes for small dots (< 200 nm) can alsobe detected in DE geometry of the measurement,because the lateral con¯nement of smaller dotsbreaks the translational symmetry. Zero- and ¯rst-order Bessel solutions were used to reproduce themeasured frequencies of the modes. In anotherreport, the collective dynamics of arrays of cylin-drical dots with ¯xed diameter and with varyinginterelement separations shows the existence of fourfundamental modes namely, the edge mode, back-ward volume like mode, DE mode and mixedmodes.119 The existence of the normal modes withineach isolated element in the array and the splittingof the fundamental mode into three collective modeswere shown experimentally as well as with the aidof micromagnetic simulations. The dispersion pro-perties of propagating collective modes in squarearray of magnetic dots were studied in Ref. 95. Aconsiderable group velocity was found for the pro-pagating collective spin waves in the DE geometryand negligible group velocity was found for the spinwaves in the backward volume geometry. The ani-sotropy in static demagnetizing ¯eld and dynamicstray ¯eld are responsible for this anisotropic beha-vior of propagating collective modes. In a di®erentstudy,179 the collective spin excitation of chains ofrectangular rounded-corner dots put side-by-sidealong either their major or minor axis were studied.The frequency dispersion of the spin-wave exci-tations was measured as a function of the wave-vector along the chains of dots and for an externalmagnetic ¯eld applied perpendicularly to thetransferred wave-vector in the dots plane. The pro-nounced frequency dispersion was found for funda-mental modes among di®erent kinds of resonantmodes correspond to di®erent branches due to thesigni¯cant stray ¯eld. The larger oscillation ampli-tude for the fundamental modes with the chainsarranged along the hard axis was reported as arequirement for spin wave-based devices.

TRSKMs have been thoroughly used to study thepicosecond collective magnetization dynamics in arraysof magnetic nanodots.103,117,180�182 The dynamics ofarrays of sub-�m square magnetic dots with varyingsizes shows the importance of micromagnetic statesdue to the nonuniformity of the demagnetizing ¯eld

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as the element size and aspect ratio (width/height)is decreased. A crossover from center mode domi-nated to edge mode dominated regime is observed asthe dot size decreases below 220 nm. The micro-magnetic simulations revealed that the ground statebecomes more nonuniform with the decrease of thedot sizes due to the increment of the demagnetizededge regions. Therefore, the edge mode localizedat the demagnetized edge region becomes dominantfor smaller dots. In continuation, the authors180

show that with the increasing thickness of magneticdots the ground states become more nonuniform,leading to some additional modes in their collectivedynamics. The edge modes are very sensitive to theexchange interaction and dipolar interactions fromthe surrounded dots. The edge modes for smallerdots are strongly a®ected by the edge roughness,defects and rounded corners of dots. Later, thedynamic con¯gurational anisotropy of arrays ofnanomagnets was also demonstrated.103 A four-foldvariation in the frequency of the resonant modes wasobserved in thin square magnetic dots. The micro-magnetic simulation revealed that four-fold aniso-tropy could not be explained from the variation ofthe static e®ective magnetic ¯eld, and instead, thevariation of the dynamic e®ective magnetic ¯eldtakes the dominant role for very thin magnetic dots.In 2008, an extensive study of collective dynamics ofsquare ferromagnetic nanoelements of di®erent sizesand for a range of bias ¯elds was reported.181 It wasshown that the dynamic behavior strongly dependsupon the ground magnetization state of nanodots.The coexistence of two branches of modes (centermode and edge mode) above a cross-over ¯eld andvanishing of higher branch were observed. Micro-magnetic simulations show the spatial nature of theexcited modes and the magnetic ground states. Thecollective behavior of the edge mode of stadiumshaped magnetic dots was reported by TRSKM.182

It was shown that as the bias ¯eld decreased, thebroad single peak splits into three narrower peaks ata cross-over ¯eld. Micromagnetic simulations showthe nature of the modes as backward volume likemode, uniform mode and DE mode.

Recently frequency-resolved magneto-optical Kerre®ect was used to study the spin dynamics and modestructure in permalloy nanomagnets with diametersranging from 50�200 nm.94 The intrinsic Gilbertdamping parameter is found to be largely una®ectedby the nanopatterning despite a large linewidthdependence on the size of the nanomagnets. Both the

end (edge) and center modes are observed in largernanomagnets and their linewidth di®er consider-ably, possibly due to the sensitivity of the end modeto small variations and imperfection of the shapeand edge materials. The authors proposed that thise®ect can be exploited as a means to separatelycharacterize the magnetic properties of the nano-magnets as well as the size and shape variationswithin the array.

Few theoretical and simulation reports are alsofound on the collective dynamics of dot arrays. In2006, Kakazei et al.148 studied the origin of in-planeanisotropy in square array of circular dots. It wasfound that when the interdot distance becomescomparable to the dot diameter, the e®ect of four-fold anisotropy becomes more prominent. This ani-sotropy comes from the nonuniform magnetizationin the dot due to the dipolar forces in the patternedmagnetic structure under strong enough applied¯elds. This was explained by an iterative solutionof a continuous variational procedure. In 2009, asystematic micromagnetic study on the static anddynamic behavior of thin circular and square mag-netic elements were reported by Barman et al.93 Theprecessional frequency and damping were studied asa function of the width to separation ratio in arraysof square and circular magnetic elements as a com-parison with the single nanomagnet dynamics. Asigni¯cant variation in the resonant frequency anddamping of the arrays with the variation of inter-element separation was observed due to the vari-ation in the dipolar magnetic ¯eld in the array.Generally, resonant frequency and damping increa-ses with the decrease in the interelement separationand single resonant frequency split into multiplemodes due to collective e®ect when width to sepa-ration ratio becomes greater than 1. The dampingincreases due to the mutual dynamic dephasing. Itwas also shown that, for nanomagnets with largerwidth and thickness, the ground state becomes morenonuniform, which leads to nonuniform dynamicsfor single nanomagnets. In this case, the arrayincreases the incoherence of precession. In 2010,another report on the dispersion relations of collec-tive dynamics of array of magnetic dots with per-pendicular magnetization was published.183 Fromthe dispersion relation it was found that there is acritical bias ¯eld, below which the chessboard anti-ferromagnetic state is stable and above that ferro-magnetic state is stable. A nonanalytic behavior ofthe mode frequency was found for the wave-vector

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close to zero value. It was shown that, for ferro-magnetic state, the center of the Brillouin zonecorresponds to a nonparabolic minimum, while forantiferromagnetic state, the saddle point with non-analytic dependence of the components of the wavevector is located at small values of the wave vector.

In the meantime, intrinsic magnetization dy-namics of single nanomagnets have been reportedby various authors. The intrinsic magnetizationdynamics of single Ni dots of varying dot diameters(aspect ratios) down to 125 nm were reported inRef. 99 and 184.99,184 The measurements were doneby a cavity enhanced all-optical TRMOKE setup.To enhance the Kerr signal from the nanodots, withdimensions well below the di®raction limited probelaser spot, they were coated with SiN layer (Fabry-P�erot cavity) along with an antire°ection coating onthe Si substrate to minimize the nonmagneticbackground. A sharp increase in the resonant modefrequency and a transition with the decrease in thedot diameter was observed. The damping alsoshowed a signi¯cant variation with a transition withthe variation in dot diameter. The observationswere interpreted with the aid of a macrospin modelafter considering the micromagnetic ground states.Micromagnetic simulations revealed that the groundmagnetization state varied from a multidomainstate with in-plane magnetization to single domainstate with out-of-plane magnetization via a vortexstate with the decrease in dot diameter. The incre-ment of the out-of-plane component of magnetiza-tion and surface anisotropy with decrease in dotdiameter were found responsible for increase in fre-quency. A sudden jump in the volume anisotropy atthe sub-nanometer size explained the sudden jumpin the resonant frequency in this regime. The vari-ation in damping was interpreted due to the mutualdephasing of two closely spaced modes, which sep-arate out further with the decrease in dot diameter.Consequently, when the dephasing reduced thedamping was also reduced. In another study, themagnetization dynamics of 160 nm magnetic dotswere studied by TRSKM in Ref. 185.185 The tran-sition between vortex state and quasi-uniform statewas demonstrated by sweeping the bias ¯eld and thefrequencies of the fundamental modes of the nano-disk show a hysteresis behavior as a function of theapplied bias ¯eld. The main outcome of this studywas to demonstrate that by changing the shape ofthe disk the annihilation ¯eld of the vortex can becontrolled over a wide range keeping the nucleation

¯eld almost unchanged. Liu et al. in 2011186 repor-ted the measurement of magnetization dynamics ofa smaller magnetic dot (150 nm in diameter) in theenvironment of a larger magnetic dot with knownresonant frequency by all-optical TRMOKE and thee®ect of the magnetostatic interaction on the dy-namics of the smaller dot. Another recent TRSKMstudy on single nanomagnet187 showed the sup-pression of edge mode of single nanomagnet by largeamplitude magnetization dynamics. The resultswere supported by micromagnetic simulation. Thisobservation is useful for application of single nano-magnets as STT oscillator where a large amplitudemagnetic excitation is required.

In the following sections, we will describe somerecent results on the collective magnetization dy-namics in square arrays of square magnetic nano-dots. The dynamics will be studied as function ofstrength and in-plane orientation of the bias ¯eld,as a function of the areal density and size of thedots. We will show how the collective dynamics varysystematically with the areal density and undergoa transition from a strongly collective regime to anisolated regime through various weakly collectiveregimes. We will demonstrate the measurement ofthe dynamics of 50 nm magnetic dots down to thesingle nanodot regime and the e®ects of the higher-order magnetostatic interactions and dynamic de-phasing in the variation of precessional modes andthe damping behaviors of such dots as a function ofthe areal density. We will also show anisotropy inthe collective magnetization dynamics and a tran-sition from a strongly collective dynamics to com-pletely isolated dynamics in the same array onlywith the variation of the direction of the bias mag-netic ¯eld.

4.2. Collective dynamics of 200nmdot arrays as a functionof the areal density

We will investigate here the e®ect of areal density ofmagnetic nanodots on their collective magnetizationdynamics. Speci¯cally, 10� 10�m2 square arraysof permalloy dots with 200 nm width (W ), 20 nmthickness (t), and interdot (edge to edge) separation(S ) varying from 50 nm to 400 nm were measured.The scanning electron micrographs (SEMs) of thosedot arrays are shown in Fig. 6(a). A representativeatomic force microscope (AFM) image of dot arraywith 100nm interdot separation is shown in Fig. 6(b).

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The three-dimensional view of the image clearlyshows a smooth ¯lm surface as well as the depthpro¯le. The samples were prepared by a combi-nation of electron beam evaporation and electron-beam lithography as discussed in Sec. 2.3. The SEMand the AFM images show that the arrays werewell fabricated with a slightly rounded corner andwith a small (< 5%) deviation from their nominaldimensions.

To determine the material parameters of theunpatterned sample, a 10� 10�m2 permalloy ¯lmwith 20 nm thickness was also prepared at the same

deposition conditions. Figure 7(a) shows the AFMimage of unpatterned permalloy ¯lm. Figures 7(b)and 7(c) show the re°ectivity and Kerr signal as afunction of the time delay between pump and probemeasured from the unpatterned ¯lm at a bias ¯eldof 1.4 kOe. The bias ¯eld was applied parallel tothe edge of the ¯lm as shown on top of Fig. 7(a). There°ectivity signal [Fig. 7(b)] shows a sharp increaseat zero delay followed by a bi-exponential decay.The Kerr signal shows four distinct regions markedas 1, 2, 3 and 4 in Fig. 7(c). In region 1 (�ve delay),probe reaches before pump and no variation in

(a) (b) (c)

Fig. 7. (a) AFM image of a 10� 10�m2 permalloy ¯lm is shown along with the geometry of the bias magnetic ¯eld. (b) Measuredre°ectivity signal from same permalloy ¯lm is shown as a function of the time delay between pump and probe. (c) The Kerr rotationsignal corresponds to uniform precession of magnetization at bias ¯eld of 1.4 kOe shows four di®erent region of interest. In region 1,probe reaches before pump (�ve delay) and no signi¯cant change in the magnetization is observed. In region 2, the magnetizationdrops sharply to a minimum value within� 500 fs. This corresponds to demagnetization of the sample. In region 3, the magnetizationis recovered very sharply along with the emission of a terahertz signal. After 10�15 ps, a slow recovery of magnetization is observedalong with the precession, which is marked as region 4 (color online).

(a) (b)

Fig. 6. (a) Scanning electron micrographs of arrays of square permalloy dots with width ðWÞ ¼ 200 nm, thickness ðtÞ ¼ 20 nm andwith varying edge to edge separations (S ) from 50 nm to 400 nm. (b) AFM image of array of 200 nm square dots with S ¼ 100 nm(color online).

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magnetization is observed. This gives the saturationmagnetization state at the applied bias ¯eld. Inregion 2, the probe beam reaches at the same time(zero delay) or just after the pump. The magneti-zation drops sharply to a minimum value within�500 fs. This corresponds to the ultrafast demag-netization of the sample.165 In region 3, magnetiza-tion is recovered very sharply with a time constantof <10 ps accompanied with the emission of a ter-ahertz signal. After 10 ps, the slow recovery ofmagnetization is observed with a time constantof about 500 ps along with the precession of mag-netization which is marked as region 4. The pre-cessional part (marked as 4) is separated out anda bi-exponential background is subtracted beforeperforming the fast Fourier transform (FFT) to ¯ndout the corresponding power spectra. The frequencyresolution of the FFT spectra depends upon thetotal measurement time.

The left panel of Fig. 8(a) shows the backgroundsubtracted Kerr signals (solid circles) measuredfrom a 10� 10�m2 permalloy ¯lm at di®erent biasmagnetic ¯eld magnitudes. Right panel shows cor-responding FFT power spectra. The single preces-sional mode corresponds to the uniform precession ofmagnetization. Frequencies of uniform precessionalmodes are plotted as a function of bias magnetic¯eld (solid circles) and are ¯tted with the Kittelformula188 (solid line) for uniform precession ofmagnetization [Fig. 8(b)].

f ¼ �

2�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðHb þHkÞðHb þHk þ 4�MsÞ

p; ð1Þ

where � is gyromagnetic ratio, Hb is the bias ¯eld,Hk is magnetocrystalline anisotropy and Ms issaturation magnetization for permalloy. From the¯tting, the magnetic parameters were obtained as� ¼ 18:5MHz/Oe, anisotropy ¯eld HK � 0, andsaturation magnetization Ms ¼ 860 emu/cm3.

To ¯nd out the Gilbert damping constant thetime-resolved background subtracted Kerr rotationdata at di®erent bias ¯elds were ¯tted with adamped sine curve

MðtÞ ¼ Mð0Þe�t� sinð2�ft� �Þ: ð2Þ

The relaxation time � is related to the Gilbertdamping coe±cient � by the relation � ¼ 1=ð2�f�Þ,where f is the frequency of the uniform mode, and� is the initial phase of the oscillation.184 The ¯tteddata are shown by solid lines in Fig. 8(a). Theextracted values of the damping coe±cient (�) for

the permalloy ¯lm is 0:017� 0:001 for all values ofthe bias ¯eld.

Figures 9(a) and 9(b) show the backgroundsubtracted time-resolved Kerr rotation data andthe corresponding FFT spectra from the arrays ofpermalloy dots with W ¼ 200 nm and with varyingS at a ¯xed bias magnetic ¯eld value Hb ¼ 1:25 kOe.The geometry of the bias ¯eld is shown on the top ofFig. 6(a). The FFT spectra show a clear variation inthe precession frequencies with the increase in S.189

A single resonant mode is observed for S ¼ 50 nmand 75 nm, but the amplitude of precession for

(a)

(b) (c)

Fig. 8. (a) Left panel (solid circles) shows the backgroundsubtracted time-resolved Kerr rotation signals measured froma 10� 10�m2 permalloy ¯lm as function of the bias magnetic¯eld. Solid black lines are the ¯ttings with an exponentiallydecaying sinusoidal function. Right panel shows the powerspectra corresponding to the FFT of the time-resolved data.(b) Frequencies of the uniform precessional mode as a functionof the bias ¯eld (solid circles) and ¯t with the Kittel's formula(solid line) are shown. (c) SEM of 10� 10�m2 square permalloy¯lm (color online).

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S ¼ 75 nm decays faster than that for S ¼ 50 nmdue to the inhomogeneous line broadening. ForS ¼ 100 nm, a lower frequency mode is appearedalong with the partial splitting of the intense res-onant mode. For S ¼ 150 nm, the three distinctmodes are observed and the situation remains sametill S < 400 nm although the splitting amplitudeincreases with the increase in S. For S ¼ 400 nm,another drastic change is observed and only twodistinct well separated modes are observed.

We have performed ¯nite di®erence methodbased micromagnetic simulations by using OOMMFsoftware159 to understand the details of the observedmodes. For the simulation, 7� 7 square arrays ofpermalloy dots with W ¼ 200 nm, t ¼ 20 nm andvarying separations (S ) from 50 nm to 400 nm wereconsidered. It was found that the minimum 7� 7array is required to take into account the e®ect oflong range magnetostatic interaction correctly.190

The observed deviation in the dot size and theinterdot separation from the nominal dimensionsfrom the SEM images are incorporated in the samplegeometry although the precise pro¯le of the edgeroughness could not be incorporated in the ¯nitedi®erence method based micromagnetic simulationsused here. The extracted materials parameters (�,Hk and Ms) from the dynamics of 10� 10�m2

permalloy ¯lm were used in the micromagnetic

simulations for array. The exchange sti®ness con-stant A was obtained from the literature.191 Thesamples were discretized into a number of cuboidalcells of size 5� 5� 20 nm3. The \x" and \y"dimensions were set lower than the exchange lengthfor permalloy (�5:3 nm) to incorporate the role ofexchange interaction on the dynamics. The \z"dimensions were set same as the thickness which ismuch larger than the exchange length. This isbecause we were not interested in studying theperpendicular standing spin wave (PSSW) modes,instead we are intended to study the collectivemodes in the x�y (sample) plane. Further testsimulations with reduced cell sizes do not show anysigni¯cant changes in the static and dynamicbehaviors and also in the mode pro¯les. The groundstates of magnetization were carefully prepared inthe same way as was done in the experiment. Firsta large enough magnetic ¯eld was applied alongparallel to the edge of the array (10�15� tilted fromthe sample plane) [Fig. 6(a)] and then the magni-tude of the magnetic ¯eld was reduced to the bias¯eld value with a su±cient time for relaxationso that the maximum torque, m�H wherem ¼ M=Ms, goes well below 10�6 A/m. This is veryimportant because the ground state stronglydepends upon the history of the applied magnetic¯eld. In this case, a high value of � ¼ 0:9 is used so

Fig. 9. (a) Experimental time-resolved Kerr rotations and (b) corresponding FFT spectra are shown for arrays of permalloy squaredots with width ¼ 200 nm, thickness ¼ 20 nm, and varying interelement separations S at Hb ¼ 1:25 kOe. (c) FFT spectra of simu-lated time-domain magnetization for 7� 7 array of dots with same speci¯cations as the experiment (color online).

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that magnetization can reach to the equilibriumstate quickly. To trigger the precession, a pulsedmagnetic ¯eld of amplitude 30Oe and rise time of40 ps were applied perpendicular to the sampleplane keeping ground state as the initial magneti-zation state. This time � is set at 0.008, the typicalvalue for permalloy. The dynamic simulations wererun for a time duration of 4 ns at time steps of 5 ps.The convergence criterion for the dynamic simu-lation is set on the simulation time steps. The ex-perimentally observed modes and the key featuresare qualitatively reproduced by micromagneticsimulation as seen from Fig. 9(c). In the simulation,appearance of a single resonance mode at S ¼ 50,presence of a small lower frequency mode at S ¼75nm, broadening and splitting of the resonantmode at S ¼ 100 nm and 150 nm, and the appear-ance of two well de¯ned modes for S ¼ 400 nm areobserved. The dynamic behavior can be dividedinto three distinct regimes: a completely uniform(strongly) collective regime (S � 75 nm), a non-uniform (weakly) collective regime (100 nm � S �300 nm) and a noncollective (isolated) regime (S >300 nm). With the decrease in the areal density thetransition from uniform collective regime to non-collective regime is observed from experiment aswell as from simulated results.

In the above, we varied the areal density of thearray, which e®ectively varied the magnetostaticinteraction among the dots. In order to get an ideahow this interaction changes as a function of theareal density, we have simulated magnetostatic ¯elddistributions for 7� 7 arrays of dots with di®erentseparations (S ). Figure 10(a) shows contour plotsof the simulated magnetostatic ¯elds for 3� 3 dotstaken from the center of 7� 7 array. The magne-tostatic ¯eld pro¯les have been shown for fourdi®erent separations (S ¼ 50 nm, 100 nm, 200 nmand 400 nm). For larger separations (S 400 nm),the interaction among the dots is negligible and theyare magnetostatically isolated. As the separationdecreases, the interactions among the dots increase,which modify the magnetic ground states and thecorresponding dynamics. For S � 75 nm, the dotsare strongly coupled to each other, giving rise to astrongly collective dynamics. To get the quantita-tive value of the stray ¯elds, we have also shown thelinescans of the e®ective ¯eld for arrays of 200 nmdots with separations ðS ¼ 50 nm, 100 nm, 200 nmand 400 nm) in Fig. 10(b). The linescans were donealong the dotted lines (parallel to x-axis) as shownin Fig. 10(a). The linescans show that the straymagnetic ¯eld is very high (� 2 kOe at the centerof the gaps between the elements) for S ¼ 50 nm

(a) (b)

Fig. 10. (a) Pro¯les of the simulated magnetostatic ¯eld distribution (x-component) for 3� 3 array of dots with W ¼ 200 nmand with varying S values are shown for Hb ¼ 1:25 kOe. The magnetostatic ¯elds were calculated for array of 7� 7 elements, out ofwhich 3� 3 elements from the center have been shown. The arrows inside the dots represent the magnetization states of the dots.The strength of the stray ¯eld is presented with a color bar as shown at the bottom right corner of the ¯gure. (b) Linescans of themagnetostatic ¯elds from arrays of 200 nm dots with S ¼ 50 nm, 100 nm, 200 nm and 400 nm are shown. The linescans were takenalong the dotted lines (parallel to x-axis) as shown in Fig. 10(a) (color online).

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and decreases steeply with an increase in S andbecomes <200Oe for S ¼ 400 nm.

To understand the nature of the observed modeswe have further simulated the pro¯les of powerand phases correspond to each mode.48 The spatialpro¯les of the power and phase for various resonantmodes are obtained by ¯xing one of the spatial co-ordinates in the space and time dependent magne-tization and by performing a discrete Fouriertransform with respect to the time domain. At¯rst we simulated the mode pro¯les for a singledot with W ¼ 200 nm and 50 nm and that for anunpatterned ¯lm with W ¼ 10�m. The simulatedmode spectra for 10� 10�m2 ¯lm and the singledots with W ¼ 200 nm and 50 nm are shown inFig. 11 (¯rst column) along with their static mag-netization states (second column). The mode pro-¯les are shown in the third, fourth and ¯fth columnsfor Hb ¼ 1:25 kOe applied parallel to the horizontaledge of the elements. The 10�m ¯lm shows a singlemode, whereas 200 nm dot shows three modes and50 nm dot shows a single mode with a much lowerfrequency than the 10�m ¯lm. The ground state for10�m ¯lm is almost uniform with negligible edgedemagnetized regions, which leads to the uniformprecession of the majority spins and hence a singleuniform mode. However, for 200 nm dot the groundstate becomes nonuniform with a prominentdemagnetized region near the edge and three dis-tinct modes are observed. Mode 1 is the center mode(CM) and mode 3 is the edge mode (EM). Mode 2

is a mixture of edge mode and a DE (EM�DE)mode. As the dot size becomes smaller, as seen for50 nm dot, the volume fraction of the demagnetizedregion increases enormously. Consequently, the EMachieves su±ciently large power to suppress othermodes and only a single mode is observed for the50 nm dot.

Power and phase pro¯les of the resonant modesfor arrays of 200 nm dots with varying separationsare shown in Fig. 12 for 3� 3 elements extractedfrom the center of the arrays. For S ¼ 50 nm, auniform collective precession (in-phase) of all theelements are observed due to the strong magneto-static interaction among the elements. This mode iscompletely di®erent from the CM observed in single200 nm dot as the mode occupies a major volume ofthe elements including regions originally occupiedby the EM for a single dot. For S ¼ 100 nm,the highest frequency mode (mode 1) corresponds tothe collective precession of the elements where themode pro¯les of individual elements look similar tothe CM of a single element. In mode 3, the powerpro¯le of individual element looks identical to themixed EM�DE mode of single 200 nm element.Mode 2 is BWVMS-like mode. The mode pro¯lesfor S ¼ 200 nm and S ¼ 300 nm (not shown) lookidentical to the modes observed for S ¼ 100 nm,with slight modi¯cations due to the change in thestrength of magnetostatic stray ¯elds. For S ¼400 nm, the mode pro¯les of individual elementslook similar to the modes observed for single 200 nm

Fig. 11. Simulated mode spectra for a 10� 10�m2 ¯lm and single dots with W ¼ 200 nm and 50 nm are shown (¯rst column) alongwith their static magnetization states (second column) and mode pro¯les (third, fourth and ¯fth column) for Hb ¼ 1:25 kOe appliedparallel to the horizontal edge of the elements. The color scale is shown in the ¯rst row (color online).

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element. For this separation, the dots are magne-tostatically isolated and collective behavior is notobserved.

In the above, we have described the collectivebehavior of the higher frequency modes, as observedin the experiment. The lower frequency EM was nottaken into account. In Fig. 13(a), the EMs from thearrays are shown along with their spatial pro¯les,which look identical to the EM of the individual200 nm element. Surprisingly, this mode remains

una®ected in the array both in the frequency spectraand in the mode pro¯le. This EM was not consis-tently observed in the experimental data. This isprobably because the defects and edge roughness ofthe lithographically fabricated nanodots stronglya®ect these modes leading towards very smallpower of these modes. Further, inhomogeneous linebroadening and smaller experimental time windowin the TRMOKE measurements make it di±cultto resolve this mode for the 200 nm element.181 Thetransition from uniform collective mode to nonuni-form collective mode is shown in Fig. 13(b).

4.3. Damping as a functionof the areal density

We have calculated the e®ective damping coe±cient(�eff) of the collective precession of the array as afunction of the interdot separation. This e®ectivedamping coe±cient (�eff), also known as the appar-ent damping is di®erent from the intrinsic Gilbertdamping �. The experimental time-resolved Kerrsignals were ¯tted with a damped sinusoidal curveusing Eq. (2), where the frequency f is the averagefrequency of the observed modes. The values of the�eff were extracted by the method as described inSec. 4.2 and are plotted along with the error bars asa function of S (Fig. 14). The plot shows that �eff �0.017 for array of 200 nm dots with S ¼ 50 nm,identical to that obtained for the 10� 10�m2

unpatterned thin ¯lm of same thickness. The �eff

increases sharply with increase in S and saturatesat about 0.025 at S ¼ 100 nm. For S ¼ 50 nm, thecoherent precession of all the elements in array and

(a) (b)

Fig. 13. (a) Edge modes of the arrays (marked as *) fordi®erent values of S and corresponding mode pro¯les are shownfor arrays of 200 nm permalloy dots. (b) FFT spectra of thesimulated magnetization dynamics from arrays of 200 nmpermalloy dots with 50 nm � S � 100 nm are shown withsmaller steps of S to reveal the precise value of S, at which themode splitting occurs (color online).

Fig. 14. Variation of e®ective damping coe±cient �eff isplotted with error bars as a function of interdot separation S.The dotted line shows the damping of the uniform mode of a10� 10�m2 unpatterned ¯lm (color online).

Fig. 12. Power and phase distributions of the simulated modesfor arrays of 200 nm dots with varying separations S. The colorscales are shown in the top row (color online).

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suppression of the edge modes observed in an iso-lated element leads to a lower �eff same as theunpatterned thin ¯lm. As the S increases, theinterdot coupling decreases and the coherence ofthe precession also decreases. For S > 100 nm amode splitting and occurrence of a third mode leadto the mutual dephasing among the modes andcause a higher �eff value 0.025.

4.4. Collective dynamics as a functionof bias ¯eld

The bias magnetic ¯eld strongly a®ects the staticmagnetic con¯gurations of an individual nanodot aswell as that of an array. The internal ¯elds withinthe individual elements become more nonuniformwith the decrease in the bias ¯eld resulting in astrong variation in the interdot magnetostatic in-teraction. Consequently, the dynamic behaviors ofthe dot array also changes signi¯cantly with the bias¯eld magnitude (Hb). Figure 15(a) shows the staticmagnetic con¯gurations at di®erent Hb values forthe array of permalloy dots with W ¼ 200 nm andwith separation S ¼ 50 nm. At Hb ¼ 1:25 kOe, thestatic magnetic con¯gurations of all the elements arein °ower (F) states. However as Hb is reduced to0.6 kOe a distribution of \C" and \S" states areobserved. The geometry of \C" and \S" states areslightly changed with the ¯eld magnitude, but theoverall distribution looks similar below 0.6 kOe.

Left panel of Fig. 15(b) shows the time-resolvedKerr rotation data at various Hb values while theright panel shows the corresponding FFT spectra.For 1.25 kOe a single uniform mode is observed asshown before. The mode is broadened at 0.8 kOedue to the inhomogeneous line broadening and asplitting occur for Hb � 0:6 kOe. Consequently, thestrongly collective dynamics of the array becomesweakly collective. The frequency of the modes alsodecreases monotonically with the decrease in Hb.The experimental and simulated resonant modes areplotted in Fig. 15(c). From the experimental datatwo branches of modes are observed, both of whichvary systematically with the bias magnetic ¯eld. Onthe other hand the simulation shows three branches.The highest frequency branch obtained from thesimulation agree with the experimental data andthis mode ¯ts well with the Kittel's formula with theparameters obtained from the 10� 10�m2 permal-loy thin ¯lm. This clearly shows that this modecorresponds to the strongly collective precession of

all elements in the 10� 10�m2 array and theelements lose their individual characters in thismode. In the case of simulated data two additionalmodes appear for Hb < 0:6 kOe. Out of those the

(a)

(b)

(c)

Fig. 15. (a) Simulated static magnetization states for array of200nm square permalloy dots with S ¼ 50 nm are shown atdi®erent bias ¯eldmagnitudes. The color scale (red�white�blue orblack�grey�white) shows the y-component of magnetization.\F", \S" and \C" denote °ower-like, S-like and C-like states.(b) Experimental time-resolved Kerr rotations at di®erent bias¯elds (left panel) and the corresponding FFT spectra (right panel)are shown for the same array. (c) Plot of the experimentallyobserved resonant peaks and simulated resonant peaks for di®erentbias¯eldmagnitudes.Theplottedmodesare¯ttedwith theuniformKittel mode with the parameters obtained from the 10� 10�m2

permalloy ¯lm (color online).

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higher frequency mode has a frequency very closeto the Kittel mode, which was not resolved in theexperiment, probably due to the lack of requiredspectral resolution. The lowest frequency modeobtained in the simulation varies systematically withHb and agree qualitatively with the lowest frequencybranch observed from experiment. The lack ofquantitative agreement may be attributed to thephysical di®erences between the experimental andsimulated samples, in particular the edge roughnessand deformation during the lift-o® process.

4.5. Time-resolved magnetizationdynamics of arrays of 50nmmagnetic dots

Figure 11 shows the simulated dynamic mode of asingle 50 nm magnetic dot. The mode spectrum fora 50 nm dot is remarkably di®erent from that of a200 nm dot. Only a single mode is observed in the50 nm dot as opposed to three modes for a 200 nmdot, with a much lower frequency than that of thehighest frequency mode for a 200 nm dot at the samebias ¯eld. It would be interesting to see whether thismode can be detected at the single nanodot regimeusing TRMOKE experiment. Since we are using asub-�m laser spot without any cavity enhancementof MOKE as used earlier,99 we have decided to workwith arrays of 50 nm dots down to the limit wherethe dots are completely magnetostatically isolated.We will also study how the resonant mode frequencyand damping in the 50 nm dot get a®ected by theinterdot separation in a densely packed array. InSec. 4.1, we discussed that magnetization dynamicsof an isolated nanomagnets with lateral dimensionsdown to 125 nm have been reported by time-resolvedcavity enhanced magneto-optical technique. How-ever, the picosecond magnetization dynamics of50 nm magnetic dots down to single nanomagnetregime has not been reported in the literature. So, itis also technically challenging to detect the pico-second magnetization dynamics of a magnetostati-cally isolated 50 nm dot.192

4.6. Precessional frequency of arraysof 50nm permalloy dots

10� 10�m2 arrays of square permalloy dots withwidth (WÞ ¼ 50 nm, thickness ðtÞ ¼ 20 nm andedge-to-edge separation (S) varying from 50 nm to200 nm were prepared by electron-beam lithography

and electron beam evaporation. The SEMs of dotarrays are shown in Fig. 16(a). A representativeAFM image of an array of 50 nm dots withS ¼ 100 nm is also shown in Fig. 16(b). In AFMimage, the width to separation ratio is apparentlydeviated from the actual values due to the ¯nite tipradius (�10nm) of AFM. The images show somedeviations in the dimensions and shapes of thedots from their nominal dimensions. The cornersare rounded due to the limitations of electron-beam lithography. In spite of these deviations, thegeneral features are maintained. The magnetizationdynamics were measured by the all-optical pump-probe set up. Extreme care was taken to increase thesensitivity of the measurements and the Kerrrotation signals from a thin ¯lm and a largerpermalloy dot were enhanced to the largest possible

(a)

(b)

Fig. 16. (a) Scanning electron micrographs of 50 nm squarepermalloy dots with edge-to-edge separations (S) varying from50 nm to 200 nm are shown. The direction of the applied bias¯eld is also shown on top of the image for S ¼ 150 nm. (b) AFMimage of array of 50 nm square dots with S ¼ 100 nm (coloronline).

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values by optimizing the optical alignments. Thetime-resolved data was recorded for 1 ns time win-dow, which was found to be su±cient to collect allthe important features of dynamics and at the sametime total measurement time does not a®ect theoptical alignment signi¯cantly. The backgroundsubtracted time-resolved data (solid circles) for dotarrays with various separations and atHb ¼ 2:5 kOeare shown in Fig. 17(a). We obtained su±cientlygood quality data for all arrays with S up to 200 nmso that the precession frequency and damping can beextracted for all arrays. The corresponding FFTspectra are shown in Fig. 17(b), which shows a singleresonant mode at around 9.04GHz for S ¼ 200 nm.As S decreases, the frequency of the single modeincreases. For S ¼ 150 nm, the mode frequencyslightly decreases, but the larger error bars con¯rmthe general trend of the increase in the frequencywith the decrease in S. However, for S ¼ 50nm thesingle resonant mode splits into two closely spacedmodes, out of which the higher frequency mode stillfollows the increase in frequency with the decrease inS, while a new lower frequency mode shows up.

The experimental results were reproduced quali-tatively by micromagnetic simulations. Figure 18(a)shows the simulated FFT spectra of the time-resolved magnetization obtained from arrays of7� 7 dots. The rounded corners of the elements and

slight variation in the dimensions in the experi-mental samples were taken into account in thesimulations, but it was not possible to take intoaccount of the edge roughness and other defor-mation of the samples precisely in the ¯nite

(a) (b)

Fig. 17. (a) Background subtracted time-resolved Kerr ro-tation data (solid circles) measured from arrays of 50 nmpermalloy dots with varying interdot separation (S ) are pre-sented. The solid black lines are the ¯t with exponentiallydamped sinusoidal function. (b) FFT spectra of the time-resolved data (color online).

Fig. 18. (a) FFT spectra of the simulated time-resolvedmagnetization for arrays of permalloy dots with 50 nm width,20 nm thickness, and interdot separation S varies from 50 nmto 200 nm. Dotted line shows the position of the resonant modefor a single 50 nm permalloy dot with 20 nm thickness. (b) Theexperimental (solid circles) and simulated (un¯lled circles)precession frequencies are plotted along with the error bars as afunction of w=a, where w is the width of dots and a is the center-to-center distance among the dots. Solid curves correspond tothe ¯t to Eq. (3). (c) The damping coe±cient � is plotted as afunction of the interdot separations S. The solid circles corre-spond to the experimental data with error bars and the solid lineis the linear ¯t. The dashed line shows the damping coe±cientfor a 10� 10�m2 permalloy ¯lm. (d) Contour plots of thesimulated magnetostatic ¯eld distributions (x-component) forarrays of magnetic dots with size ðwÞ ¼ 50 nm and separationsðSÞ ¼ 50; 75; 100; 150 and 200 nm. The strength of the magne-tostatic ¯elds are presented by the color scale as shown in thebottom right corner of the ¯gure (color online).

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di®erence method based micromagnetic simulationswhere the samples are divided into a number ofcuboidal cells of equal size. In our simulations, thearrays were divided into the cells of dimensions2:5� 2:5� 20 nm3 to take into account the roundedshape of the dots. The material parameters wereused as extracted from the 10� 10�m2 squareunpatterned thin ¯lm. The simulation method isalready explained in Sec. 4.2. The bias ¯eld valueused for the simulations was 2.5 kOe, same as theexperimental value. The simulations were run for4 ns in order to resolve all modes with high precision.Consequently, the linewidth of the simulated modesare narrower than the experimental resonant modes.The simulated FFT spectra qualitatively reproduceall the key features observed in the experiment.A single dominant resonant peak is observed forS ¼ 200 nm. The systematic increment of the res-onant frequency with the decrease in the interdotseparation is observed along with a mode splitting atS ¼ 50 nm [Fig. 18(a)]. In contrary, two importantdiscrepancies are observed between the experimen-tal and simulated results. In the experiment, theincrease in the resonant frequency with a decrease inS is steeper than the simulated results. The devi-ation is found to be larger for smaller separations.The reason behind that is increment of the non-idealities in the structure of the dots at lowerseparations. Also, the relative intensities of themodes observed in the experiment for S ¼ 50 nm arenot reproduced by the simulations. We will showlater in this article that the lower frequency mode isthe propagating spin wave mode. The experimentalarray is much larger (upto 100� 100 elements) ascompared to the simulated array (7� 7 elements).The smaller dimension of the array used in simu-lation causes a faster decay of the propagating modeas opposed to the experiment, and hence the relativeintensities of the modes in experiment and simu-lation di®er.

The frequencies of the experimental (solid circles)and simulated (un¯lled circles) resonant modes areplotted [Fig. 18(b)] along with the error bars as afunction of the width (w) to center-to-center (a)ratio (w=a), where ðwþ SÞ ¼ a. The experimentaland simulated data show that frequency remainsalmost constant upto w=a ¼ 0:25 (i.e., S ¼ 150 nm)and then increases sharply as w=a increases i.e.,S decreases. The variation of the frequency withthe interdot separation may be expressed as Eq. (3),where the ¯rst term corresponds to the frequency

of the individual elements, whereas the second andthird terms arise from the dipolar and quadrupolarinteraction.193

f ¼ f0 �Aw

a

� �3 þB

w

a

� �5: ð3Þ

Here A and B are the strengths of the dipolar andquadrupolar interactions. The experimental andsimulated data were ¯tted with Eq. (3), as shown bythe solid curves in Fig. 18(b). The simulated data¯ts well with Eq. (3). Though a large deviation inthe data points for the arrays with S ¼ 75 nm and150 nm is found for experiment, the ¯tted curvepasses through the error bars of all data points andthe ¯tting is reasonably good for the experimentaldata. In general, the contribution of the quadrupolarinteraction is dominant over the dipolar interaction,which is also evident from the sharp increase in thefrequency for w=a > 0:25. It has been observed fromthe ¯tting parameters that the dipolar contributionsare almost equal for experimental and simulatedresults, whereas the quadrupolar contribution in theexperimental data is found to be 30% greater thanthat of the simulated data. This is because the exactphysical shapes of the experimental dots were notpossible to be considered in the simulation.

4.7. Damping of arrays of 50nmpermalloy dots

We have extracted the value of the damping coe±-cient � from the background subtracted exper-imental Kerr rotation data [Fig. 17(a)]. Theextracted values of � are plotted as a function of S[Fig. 18(c)] along with the error bars. The dottedhorizontal line is the value of � measured for10� 10�m2 permalloy ¯lm of thickness ðtÞ ¼ 20 nm.

To explain the variation of resonant frequencyand damping with interdot separation ðSÞ, we cal-culated the distribution of magnetostatic ¯eld. Thecontour plots of the ¯eld distribution for 3� 3 dotstaken from the center of 7� 7 array are shown inFig. 18(d). The ¯gure shows at larger separations(S ¼ 200 nm), the stray ¯elds are mainly con¯nedclose to the dots. The dots are magnetostaticallyisolated with a negligible interaction. As the inter-dot separation decreases, the stray ¯elds of theneighboring dots start to overlap with each other,which increases the e®ective ¯eld within the dots.Therefore the frequency of the resonant modesincreases. For smaller separations (S ¼ 50 nm), the

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coupling increases to the extent, where a collectivebehavior of the whole array is also observed.

Figure 18(c) shows that the array with S ¼200 nm has a lowest value of damping coe±cient (�)of 0.023 where the dots are magnetostatically iso-lated. This value is comparable with the apparentdamping coe±cient (0.024) extracted for the dotarray with W ¼ 200 nm and S ¼ 400 nm, in whichcase the 200 nm dots were magnetostatically iso-lated. The value of � for array with S ¼ 200 nm,is slightly higher than the damping coe±cient(0.017) of the permalloy ¯lm. Since the dots weremagnetostatically isolated, the possibility of mutualdynamic dephasing of the 50 nm permalloy dots canbe ruled out for S ¼ 200 nm. The dominant singleresonant edge mode also rules out the possibility ofdephasing of more than one mode within individualdots. Therefore, we believe that the defects intro-duced in the nanodots during the nanofabricationare responsible for the enhanced damping. As Sdecreases, the magnetostatic coupling among thedots increases [Fig. 18(d)] causing increased mutualdynamic dephasing of slightly out-of-phase mag-netization precession of the dots in the array.Consequently, � increases monotonically with thedecrease in S down to 75 nm. A di®erent situationarises at S ¼ 50 nm, where the single mode is splitonto two closely spaced modes. The dephasingdue to the out-of-phase precession of two modesincreases the value of the apparent damping dras-tically from 0.032 to 0.066. To extract the value of �for uniform resonant mode (mode 1) only, we haveisolated the time-resolved Kerr signal of uniformresonant mode from the lower frequency mode byusing fast Fourier ¯ltering. The extracted value of �for mode 1 comes out to be 0.033, which is consistentwith the linear increment of � as shown by solidstraight line in Fig. 18(c).

4.8. Mode pro¯les of arrays of 50nmpermalloy dots

The mode pro¯les were studied by calculatingthe spatial distributions of power and phase corre-sponding to the resonant modes of the arrays. InFig. 19(a), the mode pro¯les for arrays with S ¼50 nm and 200 nm are shown along with that of asingle 50 nm dot for comparison. For S ¼ 50 nm,mode 1 (higher frequency mode) is the uniformmode, where the precession of almost all the dots arein-phase except for few dots near the edges of the

array due to the di®erent magnetostatic environ-ment. The power is maximum at the center of thearray and decreases towards the edges. The phasepro¯le for mode 2 shows that the dots in the con-secutive columns precess out-of-phase, while thedots in the alternative columns precess in-phase.The power varies from a higher value to a lowervalue as we go from the edges to the center. Mode 2is similar to a propagating mode in a continuousthin ¯lm with wave-vector parallel to the biasmagnetic ¯eld (Hb) i.e., BWVMS mode. The largeinterdot magnetostatic interaction in the array isresponsible for formation of this long wavelengthcollective mode in the array.

For S ¼ 200 nm, the dots are magnetostaticallyisolated. Therefore the single resonant mode (mode 1)corresponds to the precession of the individual dots,which is clear from their identical power and phasepro¯les. We have also calculated the power and phasepro¯les for the single resonant mode of a single 50 nmdot with two di®erent cell sizes 1� 1� 20 nm3 and2:5� 2:5� 20 nm3. The corresponding power andphase pro¯les are shown in Figs. 19(b) and 19(c).The power pro¯le shows that the power of singlemode is localized at the demagnetized edge region ofthe dot. This is the EM of the element. The pro¯le ofthis mode remains independent of the chosen cellsize. A magni¯ed view to the central dot of the array

Fig. 19. (a) The spatial maps of power and phase corres-ponding to the resonant modes for the arrays of 50 nm permalloydots with S ¼ 50 nm and 200 nm. We have also simulated thepower and phase maps for a single 50 nm permalloy dot with20nm thickness with di®erent cell sizes as (b) 1� 1� 20 nm3

and (c) 2:5� 2:5� 20 nm3. The color maps for the power andphase are shown at right bottom corner (color online).

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with S ¼ 200 nm shows [Fig. 19(a)] that the powerand phase pro¯les are identical to those of a singledot. It can be seen that the dynamics is dominatedby EMs of constituent dots in this array. Therefore,we conclude that we have successfully measured thedynamics of 50 nm magnetic dots in single nano-magnet (magnetostatically isolated) regime.

4.9. Collective dynamics as a functionof dot size

Size of nanomagnets can a®ect the intrinsic dy-namics behavior or collective behavior signi¯cantly.In the all-optical TRMOKE experiment, we haveprobed the dynamics of square nanodots in twodi®erent regimes. For 200 nm dots the intrinsicdynamics are governed by CM and EM, whereas for50 nm dots the EM dominates and the CM is com-pletely suppressed. Consequently, their collectivedynamics also vary signi¯cantly. However, it is im-portant to understand how this transition with thedot width occurs. We study below the magnetiza-tion dynamics of single dots and arrays of dots as afunction of dot size by micromagnetic simulationsand investigate the transition.

4.10. Static magnetic con¯gurationsof nanodots with varying dot sizes

Figure 20 shows the simulated static magneticcon¯gurations for single square permalloy dots with

varying sizes (W ) with the applied bias magnetic¯eld Hb ¼ 1:25 kOe parallel to the horizontal edge.The color scale shows the z-component (out-ofplane) of magnetization. From the images it is clearthat for con¯ned magnetic nanostructures, the edgeregions perpendicular to the direction of the bias¯eld becomes demagnetized to minimize the totalmagnetic energy of the system. With the decrease inthe size the volume percentage of the demagnetizedregion becomes larger. This variation in the staticmagnetization state with dot size is responsible forthe variation in the observed dynamic behavior.

4.11. Intrinsic dynamics of single dotsas a function of dot size

The simulated resonant modes of single dots areplotted as a function of the dot width (W) in Fig. 21.The thickness (t) of the dots is 20 nm while a bias¯elds Hb ¼ 1:25 kOe was applied to the dots. Thespatial distributions of the powers correspondingto all resonant modes are shown in the same ¯gure.As already shown in Fig. 11, for 200 nm dot, threedi®erent modes are observed. Mode 1 is the CM,mode 2 is the mixed EM�DE mode, while mode 3 isthe EM. For 150 nm dot three resonant modes aresimilar to those for the 200 nm dot with slightlymodi¯ed spatial nature such as the EM widthincreases for this dot. The frequencies of the modescorresponding to each branch increase and thefrequency gap between modes 1 and 2 decreases.

Fig. 20. Simulated static magnetization states for singlesquare permalloy dots with varying widths (W ). The color mapshow the z-component (out-of-plane) of the magnetization. The¯gures are not in the same length scale (color online).

Fig. 21. Frequencies of the simulated dynamic resonantmodes of single permalloy square dots with varying sizes(W ¼ 50; 75; 100; 150 and 200 nm) are plotted as a function ofthe dot size. The mode (power) pro¯les for all modes are shown.The three di®erent background colors show three di®erentregions of interest. Di®erent branches of frequencies areassigned with di®erent colored symbols. The color scale is shownat top of the ¯gure (color online).

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This region is marked as region 1. In region 2 (S ¼100 nm and 75 nm), mode 2 (EM�DE) disappearsand mode 3 (EM) becomes more prominent withmore extended spatial pro¯les as opposed to thesame mode for the dots in region 1. The frequency ofmode 3 increases very sharply with the decrease inW in this region. For 75 nm dot, mode 1 is the CM,occupying a smaller volume fraction of the elementsas opposed to the larger dots, while mode 2 (EM)has extended signi¯cantly. In region 3 (W < 75 nm),mode 1 (CM) completely disappears and we observea single mode (EM), which now occupies the majorvolume fraction of the element.

4.12. Collective dynamics of arrayof dots with varying sizes

To observe the collective precessional modes of dotarrays with varying dot size, we further simulatedthe resonant modes of 7� 7 array of permalloy dotswith constant edge-to-edge separation ðSÞ ¼ 50 nm,thickness ðtÞ ¼ 20 nm, and with size (W) varyingfrom 50 nm to 200 nm. This is supposed to be acollective regime for arrays of all dot sizes, where thedynamic behaviors of the dots change signi¯cantlyfrom the intrinsic behavior of the individual dots.The resonant modes are plotted as a function of dotsize [Fig. 22(a)]. The graph shows two clear fre-quency branches for dot sizes (W ) varying from200 nm to 100 nm. The frequency of each branchincreases with the decrease in the dot size. For thearray with 75 nm dot, three modes are observedas the lower frequency mode is now split into twoclosely spaced modes. The general trend of increas-ing the frequency with the decrease in dot size ismaintained. A cross-over from the higher frequencybranch to the lower branch occurs for the arraywith 50 nm dot size. The higher frequency modecompletely disappears at this point, while the twoclosely spaced lower frequency modes are observedlike the array of 75 nm dots. The increase in fre-quency with the dot size is steeper for W < 75nm.

The resonant mode pro¯les for the arrays arenumerically calculated as shown in Fig. 22(b).Power and phase maps of 3� 3 dots are shown fromthe center of 7� 7 array. The whole frequency rangeis split into three regions according to the dynamicbehavior. The color map for power, as shown inregion 1, is same for all the regions. The phase colormap for 150 nm and 200 nm dot arrays are shownin region 1, while the phase color map for 100 nm,

75 nm and 50 nm dot arrays are shown in region 3.For the array of 200 nm dots, the higher frequencymode (mode 1) is the uniform collective mode, wherethe power is uniformly distributed throughout thearray and the precession of all the elements are inphase. In this case, mode 2 of the single 200 nm dot issuppressed as discussed previously. Mode 3 is thecoherent precession of the EM of the individualelements. For the array of 150 nm dots, mode 1 is thecoherent precession of the CM of the constituentelements. This mode looks similar to the CM ofsingle 150 nm dot with a modi¯cation of the spatial

(a)

(b)

Fig. 22. (a) Simulated resonant mode frequencies for arrays ofpermalloy dots with S ¼ 50 nm and thickness ðtÞ ¼ 20 nm areplotted as a function of dot size. (b) The power and phase mapsfor the resonant modes are shown. The whole range is split intothree regions. The color map for power is shown in region 1. Thecolor map for phase for ¯rst two rows in region 1 is shown withinthat region, while the phase color map for last row in regions 1, 2and 3 are shown in region 3 (color online).

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pro¯le due to the collective e®ect. Mode 3 is coher-ent precession of the EM of the individual elements.Mode 2 of single 150 nm dot is suppressed similar tothat of the 200 nm dot. Mode 1 for the array of100 nm dots is the coherent precession of the modi-¯ed CM of the individual dots. Mode 2 is absent.Mode 3 is the coherent precession of the BWVMSlike standing wave modes of the individual elementswith two nodes inside the elements. For the array of75 nm dots, mode 1 is again the coherent precessionof the modi¯ed CM of the dots. Modes 2 and 3 arevery closely spaced in the frequency domain andboth are coherent precession of BWVMS likestanding wave modes of the elements of the array.While in mode 2 there are two nodes inside theelement, for mode 3 there is a single node. The onlydominant EM of a single 50 nm dot is split into twocollective modes (modes 2 and 3). Mode 2 is theuniform mode, where the precessions of all theelements are is phase, but a small variation inthe power is observed from center to the edges.Mode 3 is the BWVMS of the array, where the dotsin the consecutive columns precess out-of-phase,while the dots in the alternative columns precessin-phase as discussed earlier.

4.13. Role of con¯gurational anisotropyon collective magnetizationdynamics

The collective dynamics of an array of elements isdetermined by the interplay between the static anddynamic stray ¯elds. With the variation of therelative orientation of the bias ¯eld both the internal¯elds within the individual elements and the stray¯elds are expected to vary and result in a variationin the collective mode structures as opposed to themodes of the individual elements. However, to thisend very few attempts have been made to study thedependence of collective modes on the relativeorientation of the applied bias ¯eld and the sym-metry of the array.103 We have studied the preces-sional dynamics of square arrays of square magneticelements, where the individual elements possessnonuniform static magnetic con¯gurations. Conse-quently a number of resonant modes are observed asthe bias ¯eld direction is varied from the edge to thediagonal of the individual elements. We investigatehow the anisotropy in the spin wave manifold isa®ected by the strength of the magnetostatic cou-pling as the arrays undergo transition from a uniform

collective mode to a number of nonuniform collec-tive modes.194

We investigated square arrays of square permal-loy dots of 200 nm width, 20 nm thickness and withS of 50 nm and 100 nm in this particular work.The SEMs [Fig. 23(a)] show the dots are of verygood quality. The geometry of applied bias ¯eld isshown on top of the SEM image. To get an idea onhow the internal magnetization state of individualelements and the magnetostatic ¯elds of the arrayvary with the azimuthal angle (�) of the bias ¯eld,we simulated the static magnetic con¯gurations atdi®erent values of � of the bias ¯eld for a single200 nm dot and also for arrays of 200 nm dots. TheSEM [Fig. 23(b)] shows that the static magneticcon¯gurations change from \°ower" state to \leaf"state via \S" state as the bias ¯eld is rotated from� ¼ 0� to 45�. Magnetostatic ¯eld distributionswere calculated for 5� 5 arrays of dots and the ¯elddistributions around the central dot at variousvalues of � are shown in Fig. 24. It is observed thatthe stray ¯eld and the internal ¯eld pro¯les changesigni¯cantly with �. A four-fold anisotropy may beexpected to be observed with the variation of � inthe dot arrays as a re°ection of the combined e®ectsof the four-fold symmetry in the shape of individualelements and the lattice symmetry of the array.

(a)

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Fig. 23. (a) SEM for array of square permalloy dots with sizeðWÞ ¼ 200 nm and separation ðSÞ ¼ 50 nm. The geometry ofthe applied bias ¯eld is shown on top of the SEM image.(b) Simulated static magnetization states are shown for a single200 nm square dot of thickness ðtÞ ¼ 20 nm at a bias ¯eld of1.15 kOe applied at di®erent azimuthal angles (�) (color online).

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4.14. Anisotropy in the dynamicalmodes

Figure 25(a) shows the background subtracted ex-perimental Kerr rotation at Hb ¼ 1:15 kOe appliedat di®erent values of �ð0� � � � 180�, at steps of15�) and Fig. 25(b) shows the corresponding FFTpower spectra. A single uniform collective modeof the elements is observed for � ¼ 0� but at � ¼15�, the single mode is split into two modes. As �increases, the frequency gap between two modes alsoincreases along with the increment in the frequencyof each mode. This trend is observed upto � ¼ 45�.For � > 45�, the behavior is reversed upto � ¼ 90�and at this angle a single mode is observed withidentical frequency as that for � ¼ 0�. For � > 90�,this trend is repeated upto � ¼ 180�. In this way, asingle uniform mode is observed for � ¼ 0�, 90� and

(a) (b)

(c)

Fig. 25. (a) Experimentally measured Kerr rotations at1.15 kOe bias ¯eld applied at di®erent azimuthal angles (�) and(b) the corresponding FFTs are shown. (c) FFT spectra of thesimulated time-domain magnetization obtained from a 7� 7array of 200 nm magnetic dots with 50 nm edge-to-edge sepa-ration are shown for di®erent values of � (color online).

Fig. 24. (a) Contour plots of magnetostatic ¯elds at di®erentazimuthal angles (�) of the in-plane bias ¯eld of magnitude1.15 kOe for arrays of 200 nm permalloy dots with interdotseparations (S ) of (a) 50 nm and (b) 100 nm. The arrows insidethe dots show the magnetization states of the dots and the colormap as shown in between (a) and (b) represent the x-componentof magnetization (color online).

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180�, while two modes are observed for all otherangles. In addition, clear four-fold symmetries in thefrequency spectra are observed for both frequencybranches with minima at � ¼ 0�, 90� and 180� andmaxima at � ¼ 45� and 135�. In Fig. 25(c), FFTpower spectra of simulated time-domain magneti-zation obtained from a 7� 7 array of 200 nmpermalloy dots with S ¼ 50 nm are shown. Thesimulated spectra reproduce the experimentalresults qualitatively along with all the key featuresas described above.

The experimental and simulated resonant modesare plotted as a function of � in Figs. 26(a) and 26(b),respectively. Two well separated branches are ob-served with a four-fold anisotropy of the frequencyvariation with �. This four-fold anisotropy re°ectsthe lattice symmetry of individual dots as well asthe four-fold symmetry of the array with the easyaxes along � ¼ 45� and 135� and hard axes along 0�,90� and 180�. The upper branch is observed for allvalues of �, whereas the lower branch is missing

along three symmetry axes (� ¼ 0�, 90� and 180�).The angular variation of mode frequencies is ¯ttedwith Kittel's formula after taking into account of afour-fold anisotropy constant (K4) and assumingzero magnetocrystalline anisotropy for permalloy.

f ¼ �

2�

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiH � 4K4

Mcos 4�

� �H þ 4�M �K4

Mð3þ cos 4�Þ

� �s

ð4Þwhere � is the gyromagnetic ratio and M is thee®ective value of magnetization within the regionswhere the modes are localized. The ¯tted curves tothe experimental and simulated results for bothbranches are shown as solid curves in Figs. 26(a)and 26(b). The deviation of the simulated resultsfrom the ¯t to Eq. (4) stems from the fact that theequation is based upon macrospin approximationof LLG equation whereas the simulation includesthe full micromagnetic con¯guration of the sample.From the experimental data, the extracted valuesof four-fold anisotropy ¯elds (K4=M) for the upperand lower branches are about 22Oe and 14Oe, re-spectively. On the otherhand, from the simulatedresults K4=M for the upper and lower branches areabout 25Oe and 13Oe, respectively, which arein good agreement with the experimental results.The extracted values of M for the lower branchesare well below than those for the upper branches.This is due to the fact that the lower branch isrelated to the collective dynamics of the edge modesof the elements, which are localized within thedemagnetized regions with much lower e®ectivemagnetization than those for the central regions ofthe elements.

To understand the role of con¯guration ani-sotropy on the collective dynamics of the array, weshould ¯rst understand the intrinsic behavior ofa single 200 nm dot. We therefore simulated themagnetization dynamics of a single 200 nm squaredot for various azimuthal angles � of the bias mag-netic ¯eld. The simulated mode frequencies areplotted as a function of � for a constant bias ¯eldHb ¼ 1:15 kOe in Fig. 27(a). Four distinct branchesare observed. The frequency branches 2 and 4 showclear four-fold anisotropy with � with anisotropy¯elds (K4=M) of 16Oe and 8Oe, respectively.However, the other two branches do not show anyclear symmetry. At � ¼ 0�, 90� and 180� two well

(a)

(b)

Fig. 26. (a) The experimental and (b) simulated resonantmodes for array of 200 nm magnetic dots with S ¼ 50 nm areplotted as a function of the azimuthal angle (�) of the appliedbias ¯eld. The solid lines correspond to Eq. (4) (color online).

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separated modes are observed, out of which thelower frequency mode is clearly suppressed in thearray with S ¼ 50 nm. At 45� and 135� two modesare also observed. At intermediate values of �, weobserve four modes for the single elements but in thearrays two of those modes completely disappear,while two other modes are modi¯ed.

We have further simulated the power and phasepro¯les of each resonant mode for both the single200 nm element [Fig. 27(b)] and for the array withS ¼ 50 nm (Fig. 28). At � ¼ 0� the two modes arethe center and edge modes of the single elementand clearly the edge mode is suppressed in this arraydue to the strong magnetostatic ¯elds from the

neighboring elements. When the bias ¯eld is appliedalong directions between two edges of the elements amixture of the BWVMS and DE modes with dif-ferent mode numbers (mixed modes) are observed.At the intermediate angles (15� and 30�Þ, modes 1and 4 are completely suppressed in the array, whilemodes 2 and 3 are modi¯ed due to the interdot in-teraction. However for � ¼ 45�, both the modesremain intact only with slight modi¯cations in theirfrequencies. Hence, the collective dynamical beha-vior is almost nonexistent for � ¼ 45� although weobserve a strongly collective dynamics for � ¼ 0�.For 0� < � < 45�, the strength of collective dy-namics gradually decreases.

We have also simulated the angular variation ofthe resonant modes for array with S ¼ 100 nm.Figure 29(a) shows the plot of mode frequencies as afunction of �. Two branches of frequencies showingclear four-fold anisotropy are observed. Few othermodes are also observed in between two branchesbut, those modes do not show any clear four-foldsymmetry. We have ¯tted branches 1 and 2 withEq. (4). From the ¯ttings, the four-fold anisotropy¯eld (K4=M) values come out to be 22 Oe and 25Oe. Figure 29(b) shows the power and phase pro¯lesof the simulated modes shown in Fig. 29(a). At� ¼ 0�, three modes are observed. Mode 1 is thecenter mode, mode 2 is the BWVMS-like mode, andmode 3 is the mixed mode (EM�DE) of individual

Fig. 28. Power and phase distributions for the resonant modesfor an array of 200 nm square dots with 50 nm interdot sepa-ration are shown (extreme left and extreme right column) alongwith the FFT spectra (middle column) for four di®erent valuesof �. The scales of the color maps are shown at the left topcorner of the ¯gure (color online).

(a)

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Fig. 27. (a) Simulated resonant frequencies are plotted as afunction of the azimuthal angle (�) of the applied bias ¯eldfor a single 200 nm square permalloy dot of 20 nm thickness.The solid curves ¯t with Eq. (4). (b) The power and phasedistributions for the resonant modes are shown. The scales ofthe color maps are shown at the right top corner of the ¯gure(color online).

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dots. However, at � ¼ 15� two mixed modes areobserved. The power pro¯les of modes 1 and 2 of thearray looks identical to the power pro¯les of modes2 and 4 of the single element. For � ¼ 30� threemixed modes are observed, while for � ¼ 45� againtwo mixed modes (identical to the single element)are observed. The EM is absent for � ¼ 15�, 30�and 45�. Although, there is a common observationof noncollective dynamics for � ¼ 45� for both S ¼50nm and 100 nm, in the former case the transitionfrom strongly collective regime (� ¼ 0�) to non-collective regime (� ¼ 45�) occurs gradually, whilein the latter case no clear trend is observed.

4.15. Summary

In summary, the collective magnetization dynamicsof arrays of magnetic nanodots of various shapes

and sizes have been investigated in last decade byusing a variety of experimental techniques in thefrequency, time and wave-vector domains and alsoby analytical methods and micromagnetic simu-lations. In these studies di®erent kinds of localized,quantized and propagating spin wave modes havebeen observed along with the Kittel mode insome cases. The BLS studies reveal that the fre-quency dispersions of di®erent kinds of quantizedand propagating spin wave modes with wave vectorstrongly depend upon interdot separation, shape ofthe dots and also angle of the applied bias ¯elddirection. For smaller dots (<200 nm), the lateralcon¯nement breaks the translational symmetryand hence the BWVMS modes can also be observedin the DE geometry. The intrinsic magnetizationdynamics of single nanodots are modi¯ed to collec-tive magnetization dynamics when the dot sepa-ration becomes comparable to the dimension of dots.The anisotropy in static demagnetizing ¯eld anddynamic stray ¯elds which come from shape ani-sotropy of nanodots and lattice geometry of arrayare responsible for anisotropic behavior of collectivemodes. With the decrease in dot size, the groundstate of magnetization becomes more nonuniformdue to the increase of volume percentage ofdemagnetized region and edge mode becomes moreprominent. A crossover from upper frequencybranch of resonant modes to lower frequency branchis observed as the dot size decreases to sub-micrometer length scale. The pro¯le, frequency anddamping of collective modes vary signi¯cantly withthe variation of interdot separation i.e., the mag-netostatic interaction among the dots in an array.The intrinsic magnetization dynamics of perpen-dicularly magnetized cylindrical magnetic nanodotsof diameter down to 125 nm have been measuredby cavity enhanced TRMOKE measurement. Thefrequency of intrinsic modes increases with thedecrease of dot size due to the increment of the out-of-plane component of magnetization and surfaceanisotropy with the decrease in dot diameter. Thedamping of precession of magnetization increaseswith the increase in dot size due to the increment ofmutual dynamic dephasing of two closely spacedmodes. The edge mode of single nanodot can besuppressed by large angle magnetization dynamics.

Recent studies of the collective magnetizationdynamics of square arrays of square magneticnanodots by all-optical TRMOKE microscope withcollinear pump-probe geometry show a systematic

(a)

(b)

Fig. 29. (a) Simulated resonant modes for an array of 200 nmsquare dots with 100 nm interdot separations are plotted as afunction of the azimuthal angle (�) of the bias ¯eld. (b) Powerand phase distributions of the simulated modes are shown forfour di®erent values of �. The color scales are shown at the topright corner of the ¯gure (color online).

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transition from uniform collective regime to non-collective regime via weakly collective regime as theinterdot separation (S ) of 200 nm dots increasesfrom 50 nm to 400 nm. The intrinsic EM�DE modeof 200 nm dots is suppressed by uniform collectivemode when the dots are closely packed with S ¼50 nm. For S ¼ 50nm, the coherent precession of allthe elements in the array leads to a lower value ofthe e®ective damping (�eff) similar to that of anunpatterned thin ¯lm. However, as S increases, theincrement of the incoherence of precession due to thedecrease of interdot coupling and mode splittingincreases the �eff value. The reason behind that isthe variation of magnetostatic coupling among thenanodots with the interdot separation. The collec-tive dynamics is found to be anisotropic with thevariation in the azimuthal angle of in-plane bias¯eld. As the azimuthal angle of the bias ¯eld ischanged from 0� to 45� w.r.t the symmetric axis asystematic transition from a uniform collectivedynamics to noncollective dynamics is observed instrongly collective regime (S ¼ 50 nm), whereas noclear trend is observed in the weakly collectiveregime (S ¼ 100 nm). As the dot size decreases from200 nm down to 50 nm, a crossover from upperbranch of resonance frequency to lower branch isobserved for intrinsic dynamics and also for collec-tive dynamics. The picoseconds magnetizationdynamics of 50 nm magnetic dots reveals that inthe single nanomagnet regime, the dynamics 50 nmdots are governed by the edge mode of the dot.As the interdot separation decreases, the resonantfrequency increases due to increment of interdotcoupling and damping of precession increases due toincrease of mutual dynamic dephasing. For 50 nmseparation the single resonant mode splits intotwo closely spaced collective modes of the array andthe apparent damping increases drastically. Fromthe analysis of the experimental data for the arrayof 50 nm dots the main contribution to the magne-tostatic interaction are found to be quadrupolarin nature.

5. Future Directions

The collective magnetization dynamics of arrays ofmagnetic nanodots is an important problem withfuture prospects of applications in various nano-magnetic devices. For example, applications inmagnonic crystals and spin wave logic devices

demand the detailed understanding of the spin wavespectra. Mode splitting, band formation, tunabilityof bandgaps, control on propagation velocity, ani-sotropy in the spin wave propagation w.r.t. therelative orientation of the bias ¯eld with the latticesymmetry and variation in Gilbert damping andextrinsic contributions to the damping due todynamic dephasing are some of the important issuesin magnonics and spin wave logic. Hence, moredetailed understanding of the spin wave dynamics inmagnetic nanodot arrays and possible control ofabove properties would open up exciting new pros-pects in these ¯elds.

Luckily, the collective magnetization dynamicsof magnetic nanodot arrays is a multidimensionalproblem with many control parameters. The dy-namics of magnetic nanodots strongly depends upontheir static magnetic con¯gurations. For thin mag-netic ¯lms the magnetization is uniform over the¯lm and the dynamics is mainly determined byintrinsic material parameters like saturation mag-netization, magnetic anisotropy, exchange sti®nessconstant and external parameters like magnitudeand direction of the applied magnetic ¯elds. Themagnetization of con¯ned magnetic elements devi-ates from the bias magnetic ¯eld even in presenceof a large enough magnetic ¯elds thus creatingdemagnetized regions. These demagnetized regionsplay important roles in determining the magneticground states, quasi-static magnetization reversaland the precessional dynamics of magnetization.They o®er di®erent potentials to the propagatingspin waves and also spatially localize them withinthe demagnetized regions. In addition, the interdotmagnetostatic interaction in ordered arrays ofnanomagnets gets a®ected by the shape of theelements. This is because the pro¯le of the straymagnetic ¯eld depends on the shapes of the bound-ary of the elements as well as the internal magnetic¯eld and one may tailor the stray magnetic ¯eld toany complicated nature beyond the dipolar andhigher order multipolar terms.

The periodicity of the lattice determines theBrillouin zone and the allowed wave-vectors forthe magnonic modes. In addition to the periodicityof the lattice, the interdot magnetostatic inter-actions in the lattices should play a very importantrole in determining its collective magnonic modesand further tailoring of this interaction ¯eld o®ersexciting prospects in magnonics. The variation oflattice symmetry is an attractive option, by which

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the magnetostatic ¯elds of highly complex naturecan potentially be introduced but it has not yet beentried in the context of the spin wave dynamicsof magnonic crystals. Investigation of magnonicspectra in nanodot lattices with variable latticesymmetry and the e®ects of reduction of latticesymmetry, symmetry breaking and quasi-symmetricand defective crystals will provide a wealth of in-formation on the propagating and localized spinwaves.

In addition, advents of new techniques forstudying magnetization dynamics with very highspatio-temporal resolution will enable the study ofthe intrinsic dynamics of smaller single nano-magnets as well as their dynamics under the mag-netostatic interaction of the neighboring elementswhen placed in arrays of various lattice constantsand lattice symmetries.

Acknowledgment

The authors gratefully acknowledge the ¯nancialsupports from the Department of Science andTechnology under the Grant Nos. SR/NM/NS-09/2007, INT/EC/CMS (24/233552) and INT/JP/JST/P-23/09, and the Department of InformationTechnology, Government of India under Grant No.1(7)/2010/M&C. The authors acknowledge thesupports from Y. Otani, Y. Fukuma and S. Sugi-moto for assistances in sample preparation and foruseful discussions. The authors also acknowledgeassistances of D. Kumar and S. Barman in micro-magnetic simulations and S. Pal, S. Saha andR. Mandal in some time-resolved magneto-opticalmeasurements.

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