kirilyuk_kimel_rasing_2010_reviews of modern physics_ultrafast optical manipulation of magnetic...

Ultrafast optical manipulation of magnetic order

Andrei Kirilyuk,* Alexey V. Kimel, and Theo Rasing

Radboud University Nijmegen, Institute for Molecules and Materials, Heyendaalseweg135, 6525 AJ Nijmegen, The Netherlands

Published 22 September 2010

The interaction of subpicosecond laser pulses with magnetically ordered materials has developed intoa fascinating research topic in modern magnetism. From the discovery of subpicoseconddemagnetization over a decade ago to the recent demonstration of magnetization reversal by a single40 fs laser pulse, the manipulation of magnetic order by ultrashort laser pulses has become afundamentally challenging topic with a potentially high impact for future spintronics, data storage andmanipulation, and quantum computation. Understanding the underlying mechanisms impliesunderstanding the interaction of photons with charges, spins, and lattice, and the angular momentumtransfer between them. This paper will review the progress in this field of laser manipulation ofmagnetic order in a systematic way. Starting with a historical introduction, the interaction of light withmagnetically ordered matter is discussed. By investigating metals, semiconductors, and dielectrics, theroles of nearly free electrons, charge redistributions, and spin-orbit and spin-lattice interactions canpartly be separated, and effects due to heating can be distinguished from those that are not. It will beshown that there is a fundamental distinction between processes that involve the actual absorption ofphotons and those that do not. It turns out that for the latter, the polarization of light plays an essentialrole in the manipulation of the magnetic moments at the femtosecond time scale. Thus, circularly andlinearly polarized pulses are shown to act as strong transient magnetic field pulses originating from thenonabsorptive inverse Faraday and inverse Cotton-Mouton effects, respectively. The recent progressin the understanding of magneto-optical effects on the femtosecond time scale together with thementioned inverse, optomagnetic effects promises a bright future for this field of ultrafast opticalmanipulation of magnetic order or femtomagnetism.

DOI: 10.1103/RevModPhys.82.2731 PACS numbers: 75.78.Jp, 85.75.d, 78.47.jh, 75.60.d

CONTENTS

I. Introduction 2732

A. Issues and challenges in magnetization dynamics 2732

B. The problem of ultrafast laser control of magnetic

order 2733

II. Theoretical Considerations 2734

A. Dynamics of magnetic moments:Landau-Lifshitz-Gilbert equation 2734

B. Finite temperature: Landau-Lifshitz-Bloch equation 2735C. Interaction of photons and spins 2735

III. Experimental Techniques 2737A. Pump-and-probe method 2737B. Optical probe 2737C. Ultraviolet probe and spin-polarized electrons 2738D. Far-infrared probe 2739E. X-ray probe 2739

IV. Thermal Effects of Laser Excitation 2739A. Ultrafast demagnetization of metallic ferromagnets 2739

1. Experimental observation of ultrafastdemagnetization 2740

2. Phenomenological three-temperature model 27423. Spin-flip and angular momentum transfer in

metals 27434. Interaction among charge, lattice, and spin

subsystems 2744

a. Time-resolved electron photoemission

experiments 2744

b. Time-resolved x-ray experiments 2745

5. Microscopic models of ultrafast

demagnetization 2746

B. Demagnetization of magnetic semiconductors 2747

C. Demagnetization of magnetic dielectrics 2748

D. Demagnetization of magnetic half metals 2749

E. Laser-induced coherent magnetic precession 2750

1. Precession in exchange-biased bilayers 2750

2. Precession in nanostructures 2751

3. Precession in III,MnAs ferromagnetic

semiconductors 2752

4. Precession in ferrimagnetic materials 2753

F. Laser-induced phase transitions between twomagnetic states 2755

1. Spin reorientation in TmFeO3 27552. Spin reorientation in Ga,MnAs 27563. AM-to-FM phase transition in FeRh 2756

G. Magnetization reversal 2757V. Nonthermal Photomagnetic Effects 2758A. Photomagnetic modification of magnetocrystalline

anisotropy 27581. Laser-induced precession in magnetic garnet

films 2758a. Experimental observations 2758b. Phenomenological model 2759c. Microscopic mechanism 2760

2. Laser-induced magnetic anisotropy inantiferromagnetic NiO 2761*[email protected]

REVIEWS OF MODERN PHYSICS, VOLUME 82, JULY–SEPTEMBER 2010

0034-6861/2010/823/273154 ©2010 The American Physical Society2731

http://dx.doi.org/10.1103/RevModPhys.82.2731

3. Photomagnetic excitation of spin precessionin Ga,MnAs 2761

B. Light-enhanced magnetization in III,MnAssemiconductors 2762

VI. Nonthermal Optomagnetic Effects 2763A. Inverse magneto-optical excitation of magnetization

dynamics: Theory 27631. Formal theory of inverse optomagnetic

effects 27632. Example I: Cubic ferromagnet 27653. Example II: Two-sublattice antiferromagnet 2766

B. Excitation of precessional magnetization dynamicswith circularly polarized light 2766

1. Optical excitation of magnetic precession ingarnets 2766

2. Optical excitation of antiferromagneticresonance in DyFeO3 2767

3. Optical excitation of precession in GdFeCo 2768C. All-optical control and switching 2769

1. Double-pump coherent control of magneticprecession 2769

2. Femtosecond switching in magnetic garnets 27713. Inertia-driven switching in antiferromagnets 27714. All-optical magnetization reversal 27725. Reversal mechanism via a nonequilibrium

state 2773D. Excitation of the magnetization dynamics with

linearly polarized light 27741. Detection of the FMR mode via magnetic

linear birefringence in FeBO3 27752. Excitation of coherent magnons by linearly

and circularly polarized pump pulses 27763. ISRS as the mechanism of coherent

magnon excitation 27764. Effective light-induced field approach 2778

VII. Conclusions and Outlook 2778Acknowledgments 2779References 2779

I. INTRODUCTION

A. Issues and challenges in magnetization dynamics

The time scale for magnetization dynamics is ex-tremely long and varies from the billions of years con-nected to geological events such as the reversal of themagnetic poles down to the femtosecond regime con-nected with the exchange interaction between spins.From a more practical point of view, the demands forthe ever-increasing speed of storage of information inmagnetic media plus the intrinsic limitations that areconnected with the generation of magnetic field pulsesby current have triggered intense searches for ways tocontrol magnetization by means other than magneticfields. Since the demonstration of subpicosecond demag-netization by a 60 fs laser pulse by Beaurepaire et al.1996, manipulating and controlling magnetizationwith ultrashort laser pulses has become a challenge.Femtosecond laser pulses offer the intriguing possi-bility to probe a magnetic system on a time scale that

corresponds to the equilibrium exchange interaction,responsible for the existence of magnetic order, whilebeing much faster than the time scale of spin-orbit inter-action 1–10 ps or magnetic precession 100–1000 ps;see Fig. 1. Because the latter is considered to set thelimiting time scale for magnetization reversal, the optionof femtosecond optical excitation immediately leads tothe question whether it would be possible to reversemagnetization faster than within half a precessional pe-riod. As magnetism is intimately connected to angularmomentum, this question can be rephrased in terms ofthe more fundamental issues of conservation and trans-fer of angular momentum: How fast and between whichreservoirs can angular momentum be exchanged and isthis even possible on time scales shorter than that of thespin-orbit interaction?

While such questions are not relevant at longer timesand for equilibrium states, they become increasingly im-portant as times become shorter and, one by one, thevarious reservoirs of a magnetic system, such as themagnetically ordered spins, the electron system, and thelattice, become dynamically isolated. The field of ul-trafast magnetization dynamics is therefore concernedwith the investigation of the changes in a magnetic sys-tem as energy and angular momentum are exchangedbetween the thermodynamic reservoirs of the systemStöhr and Siegmann, 2006.

Although deeply fundamental in nature, such studiesare also highly relevant for technological applications.Indeed, whereas electronic industry is successfully enter-ing the nanoworld following Moore’s law, the speedof manipulating and storing data lags behind, creatinga so-called ultrafast technology gap. This is also evidentin modern PCs that already have a clock speed of afew gigahertz while the storage on a magnetic hard diskrequires a few nanoseconds. A similar problem is expe-rienced by the emerging field of spintronics as in, forexample, magnetic random access memory devices.Therefore, the study of the fundamental and practicallimits on the speed of manipulation of the magnetiza-tion direction is obviously also of great importance for

Laser

1 ns

1 ps

1 fs

100 ps

10 ps

100 fs

10 fs

Magneticfield

~ 0.01 ps – 0.1 psExchangeinteraction

Spin-orbit (LS)interaction

L S

Spinprecession~ 1 ps - 1 ns

~ 0.1 ps-1 ps

FIG. 1. Color online Time scales in magnetism as comparedto magnetic field and laser pulses. The short duration of thelaser pulses makes them an attractive alternative to manipulatethe magnetization.

2732 Kirilyuk, Kimel, and Rasing: Ultrafast optical manipulation of magnetic order

Rev. Mod. Phys., Vol. 82, No. 3, July–September 2010

magnetic recording and information processing tech-nologies.

In magnetic memory devices, logical bits “ones” and“zeros” are stored by setting the magnetization vectorof individual magnetic domains either “up” or “down.”The conventional way to record a magnetic bit is to re-verse the magnetization by applying a magnetic fieldLandau and Lifshitz, 1984; Hillebrands and Ounadjela,2002. Intuitively, one would expect that switching couldbe infinitely fast, limited only by the attainable strengthand shortness of the magnetic field pulse. However, re-cent experiments on magnetization reversal usinguniquely short and strong magnetic field pulses gener-ated by relativistic electrons from the Stanford LinearAccelerator Tudosa et al., 2004 suggest that there is aspeed limit on such a switching. It was shown that deter-ministic magnetization reversal does not take place ifthe magnetic field pulse is shorter than 2 ps. Couldoptical pulses be an alternative?

B. The problem of ultrafast laser control of magnetic order

The discovery of ultrafast demagnetization of a Nifilm by a 60 fs optical laser pulse Beaurepaire et al.,1996 triggered the new and booming field of ultrafastlaser manipulation of magnetization. Subsequent experi-ments not only confirmed these findings Hohlfeld et al.,1997; Scholl et al., 1997; Güdde et al., 1999; Ju et al.,1999; Koopmans et al., 2000; Bigot, 2001; Hicken, 2003;Rhie et al., 2003; Bigot et al., 2004; Ogasawara et al.,2005 but also demonstrated the possibility to opticallygenerate coherent magnetic precession Ju, Nurmikko,et al., 1998; van Kampen et al., 2002, laser-induced spinreorientation Kimel, Kirilyuk, et al., 2004; Bigot et al.,2005, or even modification of the magnetic structure Juet al., 2004; Thiele et al., 2004 and this all on a time scaleof 1 ps or less. However, despite all these exciting ex-perimental results, the physics of ultrafast optical ma-nipulation of magnetism is still poorly understood.

A closer look at this problem reveals that excitationwith a femtosecond laser pulse puts a magnetic mediumin a highly nonequilibrium state, where the conventionalmacrospin approximation fails and a description of mag-netic phenomena in terms of thermodynamics is nolonger valid. In the subpicosecond time domain, typicaltimes are comparable to or shorter than the characteris-tic time of spin-orbit interaction, and the magnetic an-isotropy becomes a time-dependent parameter. Notethat, although the spin-orbit coupling is an importantingredient of the magnetic anisotropy mechanism, thelatter is the result of a balance between different crystal-field-split states. Therefore, the typical anisotropy en-ergy is considerably lower than that of spin-orbit cou-pling, which is also translated into the correspondingresponse times. At shorter time scales even the ex-change interaction should be considered as time depen-dent. All these issues seriously complicate a theoreticalanalysis of this problem. In addition, experimental stud-ies of ultrafast magnetization dynamics are often ham-

pered by artifacts, and interpretations of the same dataare often the subject of heated debates.

What are the roles of spin-orbit, spin-lattice, andelectron-lattice interactions in the ultrafast optical con-trol of magnetism? How does the electronic band struc-ture affect the speed of the laser-induced magneticchanges? A systematic study of the laser-induced phe-nomena in a broad class of materials may answer thesequestions, as optical control of magnetic order has beendemonstrated in metals, semiconductors, and dielectrics.So far several attempts to summarize and systematizethese studies have been dedicated to metals Bigot,2001; Zhang, Hübner, et al., 2002; Hicken, 2003; Benne-mann, 2004; Bovensiepen, 2007, ferromagnetic semi-conductors Wang et al., 2006, or dielectrics Kirilyuk etal., 2006; Kimel et al., 2007.

This review aims to introduce and summarize the ex-perimental and theoretical studies of ultrafast opticalmanipulation of spins in all the classes of both ferromag-netically and antiferromagnetically ordered solids stud-ied so far, including metals, semiconductors, and di-electrics. We present an overview of the different experi-mental and theoretical approaches to the problem anddistinguish effects of light on the net magnetization,magnetic anisotropy, and magnetic structure. As a result,important conclusions can be drawn about the role ofdifferent reservoirs of angular momentum free elec-trons, orbital motion, and lattice and their mutual ex-change for the ultrafast optical manipulation of magne-tism.

The effects of a pump laser pulse on a magnetic sys-tem could be classified as belonging to one of the follow-ing classes:

1 Thermal effects: Because of the absorption of pho-tons, energy is pumped into the medium. Thechange in the magnetization corresponds to that ofspin temperature: M=MTs. Since in the electric di-pole approximation spin-flip transitions are forbid-den, the direct pumping of energy from light to spinsis not effective. Instead, light pumps the energy intothe electron and phonon system. The time scale ofthe subsequent magnetization change is determinedby internal equilibration processes such as electron-electron, electron-phonon, and electron-spin inter-actions, which for itinerant ferromagnets can bevery short, down to 50 fs. For dielectric magnets, incontrast, this time is of the order of a nanoseconddue to the absence of direct electron-spin processes.The lifetime of such thermal effects is given by ex-ternal parameters such as thermal conductivity of asubstrate as well as the geometry of the sample.

2 Nonthermal photomagnetic effects involving theabsorption of pump photons Kabychenkov, 1991:In this case the photons are absorbed via certainelectronic states that have a direct influence on mag-netic parameters, such as, for example, the magne-tocrystalline anisotropy. The change is instantaneouse.g., the rise time of the pump pulse. These param-eters, in turn, cause a motion of the magnetic mo-

2733Kirilyuk, Kimel, and Rasing: Ultrafast optical manipulation of magnetic order


ments that obeys the usual precessional behavior.The lifetime of this effect is given by the lifetime ofthe corresponding electronic states. Note, however,that if this time is much shorter than the precessionperiod, the effect will be difficult to detect.

3 And finally, there are nonthermal optomagnetic ef-fects that do not require the absorption of pumpphotons but are based on an optically coherentstimulated Raman scattering mechanism Kaby-chenkov, 1991. The action of this mechanism can beconsidered as instantaneous and is limited by thespin-orbit coupling, which is the driving force be-hind the change in the magnetization in this case20 fs for a typical 50 meV value of spin-orbit cou-pling. The lifetime of the effect coincides with thatof optical coherence 100–200 fs. Note that in prac-tice thermal effects are always present to someextent.

II. THEORETICAL CONSIDERATIONS

A. Dynamics of magnetic moments: Landau-Lifshitz-Gilbertequation

The interactions of magnetic moments with magneticfields are basic to the understanding of all magnetic phe-nomena and may be applied in many ways. Homoge-neously magnetized solids exhibit a magnetic moment,which for a volume V is given by m=VM, where M isthe magnetization. If V is the atomic volume, then m isthe magnetic moment per atom; if V is the volume of themagnetic solid, m is the total magnetic moment of thebody. The latter case is often called the “macrospin ap-proximation.” Also, for the inhomogeneous case, themagnetic solid can often be subdivided into small re-gions in which the magnetization can be assumed homo-geneous. These regions are large enough that the motionof the magnetization can in most cases be described clas-sically.

The precessional motion of a magnetic moment in theabsence of damping is described by the torque equation.According to quantum theory, the angular momentumassociated with a magnetic moment m is

L = m/ , 1

where is the gyromagnetic ratio. The torque on themagnetic moment m exerted by a magnetic field H is

T = m H . 2

The change in angular momentum with time equals thetorque:

dLdt

=d

dt

m

= m H . 3

If the spins not only experience the action of the exter-nal magnetic field but are also affected by the magneto-crystalline anisotropy, shape anisotropy, magnetic dipoleinteraction, etc., the situation becomes more compli-

cated. All these interactions will contribute to the ther-modynamical potential , and the combined action ofall these contributions can be considered as an effectivemagnetic field

Heff = − /M . 4

Thus the motion of the magnetization vector can bewritten as the following equation, named after Landauand Lifshitz Landau and Lifshitz, 1935:

dm/dt = m Heff, 5

which describes the precession of the magnetic momentaround the effective field Heff. As mentioned, Heff con-tains many contributions:

Heff = Hext + Hani + Hdem + ¯ , 6

where Hext is the external applied field, Hani is the an-isotropy field, and Hdem is the demagnetization field. Ex-cept for Hext, all other contributions will be material de-pendent. Consequently, optical excitation of a magneticmaterial may result, via optically induced changes in thematerial-related fields, in a change in Heff, giving rise tooptically induced magnetization dynamics.

At equilibrium, the change in angular momentumwith time is zero, and thus the torque is zero. A viscousdamping term can be included to describe the motion ofa precessing magnetic moment toward equilibrium. Adissipative term proportional to the generalized velocity−m /t is then added to the effective field. This dissi-pative term slows down the motion of the magnetic mo-ment and eventually aligns m parallel to Heff. This givesthe Landau-Lifshitz-Gilbert LLG equation of motionGilbert, 1955:

mt

= m Heff +

mm

mt

, 7

where is the dimensionless phenomenological Gilbertdamping constant.

Equation 7 may be used to study the switching dy-namics of small magnetic particles. If the particles aresufficiently small, the magnetization may be assumed toremain uniform during this reversal process, and theonly contributions to the effective field are the aniso-tropy field, the demagnetizing field, and the applied ex-ternal field. For larger samples, and in the case of inho-mogeneous dynamics, such as spin waves with k0, themagnetic moment becomes a function of spatial coordi-nates: m=mr. The effective magnetic field in this casealso acquires a contribution from the exchange interac-tion. In this case, nonhomogeneous elementary excita-tions of the magnetic medium may exist, first proposedby Bloch in 1930 Bloch, 1930. These excitations arecalled spin waves and involve many lattice sites. Moredetails on these aspects can be found in Hillebrands andOunadjela 2002.

The LLG equation can also be used in the atomisticlimit to calculate the evolution of the spin system usingLangevin dynamics, which has proved to be a powerful



approach to modeling ultrafast magnetization processesKazantseva, Nowak, et al., 2008; Atxitia et al., 2009a;see also the next section.

Some limitation of the LLG equation may come fromthe fact that, on a time scale shorter than the spin-orbitcoupling of the order of 20 fs, the description with asingle gyromagnetic ratio fails and spin and orbital con-tributions must be considered separately.

B. Finite temperature: Landau-Lifshitz-Bloch equation

The Landau-Lifshitz-Gilbert equation considersmagnetization as a vector of fixed length and ignores itslongitudinal relaxation. Such an approach is obviouslyunsatisfactory at elevated temperatures since magnetiza-tion entering the LLG equation is an average over somedistribution function and its magnitude can change. Onthe other hand, the atomistic approach described in theprevious section is prohibitively time consuming in mac-roscopic systems.

A macroscopic equation of motion for the magnetiza-tion of a ferromagnet at elevated temperatures shouldthus contain both transverse and longitudinal relaxationterms and should interpolate between the Landau-Lifshitz equation at low temperatures and the Blochequation Bloch, 1946 at high temperatures.

We start with an atomistic approach where a magneticatom is described as a classical spin vector s of unitlength. The magnetic moment of the atom is given by=0s. In the case of a weak coupling with the bath, thedynamics of the vector s can be described with the helpof the stochastic Landau-Lifshitz equation,

dsdt

= s H + − †s s H‡ , 8

with 0, 1, where correlators of the ,=x ,y ,zcomponents of the Langevin field t are given by

tt =2T

0t − t , 9

with 0 the atomic magnetic moment and the param-eter describing the coupling to the bath system. The co-efficient in front of the delta function in Eq. 9 is deter-mined by the fluctuation-dissipation theorem, whichmay limit the applicability of this approach on a timescale comparable to the correlation time of electron sys-tem, about 10 fs for metals Stöhr and Siegmann, 2006.

The basis of Eq. 9 is thus the separation of timescales, assuming that the heat bath phonon or electronsystem acts much faster than the spin system. In thiscase, the bath degrees of freedom can be averaged outand replaced by a stochastic field with white noise cor-relation functions. The ensemble-averaged spin polar-ization gives the magnetization of the material: m=0s. Then, Eq. 8 results in the following Landau-Lifshitz-Bloch equation Garanin, 1991, 1997:

dmdt

= m Heff −

m · Heffmm2

+

†m m Heff‡m2 , 10

where and are dimensionless longitudinal andtransverse damping parameters given by

= 2T

3TcMF, = 1 −

T

3TcMF 11

for TTcMF and the same with ⇒ for T Tc

MF,where Tc

MF is the mean-field Curie temperature.Obviously, the damping parameters and depend onthe Langevin field. The correlation between the spec-trum of the Langevin field and the damping parametershas been recently investigated by Atxitia et al., 2009.

The LLB equation 10 was shown to capture thecomplex physics revealed by the atomistic model, in par-ticular, the variation in the magnetization magnitudeduring reversal and the increase in damping with tem-perature Chubykalo-Fesenko et al., 2006. The longitu-dinal and transverse relaxation times calculated from theLLB equation also agree well with those calculated fromthe atomistic model. The LLB equation could thereforeserve in the future as a basis for an improved micromag-netics at elevated temperatures. Particularly interestingwould be its application in the area of multiscale simu-lations where atomistic simulations of nanometer-sizeareas are linked to micromagnetic regions to extendthose calculations to macroscopic length scales.

C. Interaction of photons and spins

We now discuss the possibility of a direct interactionbetween photons and spins using energy considerationsPershan, 1963. It can be shown that the thermodynami-cal potential of an isotropic, nonabsorbing, magneti-cally ordered medium with static magnetization M0 ina monochromatic light field E includes a term

= ijkEiEj*Mk0 , 12

where ijk is the magneto-optical susceptibility Pita-evskii, 1961; van der Ziel et al., 1965; Pershan et al., 1966;Landau and Lifshitz, 1984; Kabychenkov, 1991. The ex-istence of this potential arises from the absence of dissi-pation; moreover, in the case of an isothermal processconsidered here, this is equivalent to the time-averagedfree energy of the system; see Pershan, 1963; Nye, 2004for details.

In the electric dipole approximation the linear opticalresponse of a medium to a field E is defined by theoptical polarization P= /E. From Eq. 12one can easily see that the optical polarization Pshould have a contribution Pm proportional to the mag-netization M,



Pim = ijkEjMk0 . 13

From this equation one can determine that, when lin-early polarized light is transmitted through a magnetizedmedium, the polarization plane of the light gradually ro-tates over an angle F given by

F =ijkMk0L

cn, 14

where c is the speed of light in vacuum, n is the refrac-tion coefficient of the medium, and L is the propagationdistance of the light in the medium Zvezdin and Kotov,1997. Equation 14 describes the so-called magneto-optical Faraday effect discovered by M. Faraday in 1846Faraday, 1846. This polarization rotation is the resultof spin-orbit interaction which creates an asymmetry ofthe electronic wave functions, leading to a rotation ofthe excited dipolar currents Argyres, 1955.

The magneto-optical Faraday effect demonstrates thata magnetically ordered medium can indeed affect pho-tons and change the polarization of light Zvezdin andKotov, 1997. Is the inverse phenomenon feasible, thatis, is it possible that polarized photons affect the magne-tization? From Eq. 12 one can find that an electric fieldof light at frequency will act on the magnetization asan effective magnetic field Heff directed along the wavevector of the light k:

Hk = −

Mk= ijkEiEj*. 15

In isotropic media, ijk is a fully antisymmetric tensorwith a single independent element . Therefore, Eq. 15can be rewritten as

H = E E* . 16

From this it becomes obvious that right- and left-handedcircularly polarized waves should act as magnetic fieldsof opposite sign Pitaevskii, 1961; van der Ziel et al.,1965; Pershan et al., 1966; Landau and Lifshitz, 1984.Therefore, it is seen from Eq. 16 that, in addition tothe well-known magneto-optical Faraday effect wherethe polarization of light is affected by the magnetizationM, the same susceptibility also determines the inverseoptomagnetic phenomenon: circularly polarized light af-fects the magnetization via the inverse Faraday effect.Thus in a thermodynamical approach, the effect of lighton spins in a magnetically ordered material can be de-scribed by the Landau-Lifshitz equation, where the mag-netic field is generated by light via the inverse Faradayeffect.

We note, however, that Eq. 16 has been derived for amonochromatic optical excitation in the approximationof thermal equilibrium. It is therefore interesting to con-sider the situation when the product EE*

changes much faster than the fundamental time scales ina magnetically ordered material, given by the spin pre-cession period and the spin-lattice relaxation time Per-shan et al., 1966. Consider the excitation of spins by alaser pulse with duration t=100 fs. Using Fourier trans-

formation, one can see that such a laser pulse, beingultrashort in the time domain, is spectrally broad 5 THz. Such laser excitation of spins is convenientlydescribed in the frequency domain see Fig. 2. Initiallythe electron is in the ground state |1 and its spin is up. Ifthe state is nondegenerate, being an orbital singlet, thespin-orbit coupling for the electron in this state can beneglected. If we act on this electron with a photon, dur-ing the optical transition the wave function of the elec-tron becomes a superposition of several eigenstates.This will effectively increase the orbital momentum ofthe electron, leading to an increase in the interactionbetween spins and orbital moments and thus resulting inan increase in the probability of a spin-flip process. If theenergy of the photon is smaller than the gap betweenthe ground state |1 and the nearest excited state |2, thephoton will not excite any real electronic transition butjust result in a spin flip of the electron in the groundstate. In other words, the spin flip in the ground state isdue to the fact that circularly polarized light mixes afraction of the excited-state wave function into theground state Pershan et al., 1966. This process will beaccompanied by the coherent reemission of a photon ofenergy 2=1−m. In magnetically ordered materi-als, m corresponds to the energy of a magnon. More-over, such a laser-induced spin-flip process can be coher-ently stimulated if both frequencies 1 and 2 arepresent in the laser pulse see Fig. 2. The time of thespin-flip process sf is given by the energy of the spin-orbit interaction in the perturbed ground state ESO. Formaterials with a large magneto-optical susceptibility, theenergy of the spin-orbit coupling may exceed 50 meVKahn et al., 1969, and thus the spin-flip process can beas fast as sf /ESO20 fs. This microscopic pictureshows that the generation of effective magnetic fields bycircularly polarized light should also work on the subpi-cosecond time scale.

Note that the spin-flip process as described above isallowed in the electric dipole approximation Shen andBloembergen, 1966. This mechanism is much more ef-

1>|

Ωm

Ωm

1ω2ω

2ω

1ω

Ωm1ω −

ωI( )

ω2>|

FIG. 2. Color online Illustration of stimulated coherent Ra-man scattering, a possible microscopic mechanism responsiblefor an ultrafast optically generated magnetic field. Two fre-quency components of the electromagnetic radiation from thespectrally broad laser pulse take part in the process. The fre-quency 1 causes a transition into a virtual state with strongspin-orbit coupling. Radiation at the frequency 2 stimulatesthe relaxation back to the ground state together with the cre-ation of a magnon.



fective than magnetic dipole transitions and does notrequire annihilation of a photon. It means that the en-ergy transfer from photons to spins magnons is real-ized via an inelastic scattering process. While some pho-tons lose a small part of their energy, the total number ofphotons remains unchanged. It is important to note thatin a spin flip via stimulated Raman scattering, as de-scribed above, the stimulating and reemitted photonshave identical polarization, implying that such a photon-induced spin flip is not accompanied by a loss of theangular momentum of the photons. It has been argued,on the other hand, that even in the absence of spin-lattice interaction, the angular momentum can be sup-plied via the change in the direction of the pump light,i.e., via the linear momentum of photons Woodford,2009. This, however, still needs to be confirmed in anexperiment.

III. EXPERIMENTAL TECHNIQUES

A. Pump-and-probe method

The investigation of ultrafast spin dynamics triggeredby a picosecond or subpicosecond laser pulse requiresmethods that allow detection of the changes in the mag-netization in a medium with subpicosecond temporalresolution. Practically all the studies of ultrafast laser-induced magnetization dynamics employ some kind ofpump-and-probe method, where the magnetization dy-namics is triggered by a short laser pulse in the opticalrange pump, while detection probe of the laser-induced magnetic changes is done with the help of asecond pulse of electromagnetic radiation Shah, 1996.Analysis of the properties of the probe pulse after inter-action with the medium for different time separationsbetween pump-and-probe pulses allows reconstructionof the ultrafast laser-induced magnetic changes actuallytaking place in the medium. The duration of the probepulse defines the temporal resolution, while the spec-trum of the probe is crucial for the sensitivity of suchmeasurements. Choosing the probe pulse in the far-infrared, optical, ultraviolet, or x-ray spectral ranges,one defines the electronic transitions responsible for theinteraction of the probe pulse with the medium. As aresult, all these spectroscopic methods are characterizedby different sensitivity to spin and orbital degrees offreedom. Therefore measurements in the far-infrared,optical, ultraviolet, or x-ray spectral ranges allow one toobtain four different views on the same phenomenonand thus obtain the most complete information.

B. Optical probe

The interaction of a medium with a probe pulse in thevisible range or its vicinity 0.4–10 m can be describedin the electric dipole approximation. Using energy con-siderations, one can analyze such an interaction by con-sidering the thermodynamical potential . For an isotro-pic nondissipating, magnetically ordered medium withstatic magnetization M0 or antiferromagnetic vector

l0 in a monochromatic light field E, neglectingterms of order higher than 3 in E, can be written as

= ijlEi*Ej + ijk

l Ei*EjM0k

+ ijkl Ei*Ejl0k + ijk

nlEi2*EjEk

+ ijklnlEi2*EjEkM0l

+ ijklnlEi2*EjEkl0l, 17

where ijl, ijk

l , ijkl , ijk

nl, ijklnl, and ijkl

nl are tensors thatdefine the optical properties of the medium, and the su-perscripts l nl indicate the linear nonlinear response,respectively Pershan, 1963. Equation 17 is a generali-zation of Eq. 12 from which the Faraday effect wasderived.

Similarly, because of the polarization Pm, one canobtain polarization rotation of light upon reflection froma magnetized medium; this phenomenon is called themagneto-optical Kerr effect MOKE. Therefore, linearmagneto-optical effects can be used as a probe of themagnetization of a medium and, using modulation tech-niques, can be sensitive to even submonolayer mag-netic films Qiu and Bader, 1999.

Considering in Eq. 17 the terms linear in the antifer-romagnetic vector l0, it is easy to see that linear opticscan also serve as a probe of magnetic order in geom-etries where no Faraday or Kerr effect is present or inmaterials with no net magnetization, such as antiferro-magnets or ferrimagnets at their compensation pointChaudhari et al., 1973. It is important to note that alllinear magneto-optical phenomena are sensitive to cer-tain projections of the magnetic vectors M and l andcan be observed in all media, irrespective of their crystalsymmetry or crystallographic orientation Smolenski etal., 1975; Ferré and Gehring, 1984.

Although the magneto-optical Faraday effect i.e.,magnetic circular birefringence and the magnetic linearbirefringence at a frequency do not require absorptionat this frequency, causality does not allow complete ne-glect of absorption. Because of the Kramers-Kronig re-lations, an observation of circular or linear birefringencein a certain spectral range should be accompanied bysimilar effects of polarization-dependent absorption inanother spectral domain. Such effects are known asmagnetic circular dichroism and magnetic linear dichro-ism and are often employed as efficient probes of mag-netization as well Zvezdin and Kotov, 1997; Kottler etal., 1998.

In the nonlinear optical approximation of Eq. 17,one should take into account the terms of third orderwith respect to the electric field of light. In that approxi-mation, the optical field E is able to induce a polar-ization in the medium at the double frequency P2.The appearance of this polarization results in the phe-nomenon of second-harmonic generation when the me-dium excited by the optical field E generates light atthe double frequency 2. The intensity of the second-harmonic light in ferromagnets or antiferromagnets canbe found to be



I2 P2i2

= ijknlEjEk + ijkl

nlEjEkM0l2

or ijknlEjEk + ijkl

nlEjEkl0l2,

respectively.From this one can conclude that the second-harmonic

generation can also be a measure of the magnetizationM0 or antiferromagnetic vector l0 in the medium.However, in contrast to linear magneto-optical effects,the second harmonic in the electric dipole approxima-tion is generated only in media in which the crystallo-graphic symmetry does not contain a center of inversion.In centrosymmetric media the second-harmonic genera-tion is thus only allowed at surfaces and interfaces wherethe bulk symmetry is broken. Consequently, the mag-netic second-harmonic generation MSHG techniquecan serve as a unique tool for probing surface and inter-face magnetism Kirilyuk and Rasing, 2005.

Since the optical response of media in the visible spec-tral range is dominated by electric dipole transitions andthe selection rules for such transitions do not allow aspin flip, the sensitivity of light to magnetic order in caseof both linear and nonlinear magneto-optical effects re-quires a strong spin-orbit interaction. A strong couplingbetween spins and orbitals results in substantial valuesof the and components in Eq. 17 so that one canuse the magneto-optical effects to obtain informationabout the magnetic state of a medium.

Interpretation of the results of magneto-optical stud-ies of solids is hampered by the fact that optical transi-tions in the visible spectral range are relatively broad.The corresponding spectral lines often overlap even fordielectric materials Lever, 1984. For an analysis of op-tical excitations of metals, one has therefore to considera continuum of transitions, making a theoretical descrip-tion of the magneto-optical spectra an extremely diffi-cult problem. Moreover, to apply them for ultrafast stud-ies one has to take into account a possiblenonequilibrium behavior of the system that also stronglyaffects the magneto-optical response Oppeneer andLiebsch, 2004.

In addition, magneto-optical effects are only propor-tional to M and l and are thus only indirect probes ofspin ordering. This results in a number of uncertaintiesin the interpretation of time-resolved magneto-opticalmeasurements. Laser excitation can change populationsof the excited states this process is called “bleaching”and modify the symmetry of the ground state, and thusmay result in changes in the magneto-optical response ofa medium even if its magnetic order and magnetizationvector have not been affected Koopmans et al., 2000;Regensburger et al., 2000; Oppeneer and Liebsch, 2004.In terms of a thermodynamical description of themagneto-optical effects, such a situation corresponds tolaser-induced changes in the tensors ijk

l , ijkl , ijkl

nl, andijkl

nl. However, we stress here that, although a thermo-dynamical description is suitable to demonstrate the factthat optical phenomena can serve as a probe of magneticorder, an explanation of femtosecond laser-induced ef-

fects in magnets in terms of time-dependent tensors can-not be expected to fully describe the phenomena thatactually take place at the ultrafast time scale. Laser-induced magneto-optical Kerr or Faraday effects inpump-probe experiments should be described in termsof a nonlinear polarization of third order, as done bySham 1999 and Kimel et al. 2001. Such a theory canbe easily developed for a simple two- or three-level sys-tem. However, a proper theoretical framework that al-lows an adequate description of the time-resolvedpump-probe magneto-optical experiments in metals,magnetic semiconductors, and even dielectrics remainschallenging, and the number of approaches is limitedOppeneer and Liebsch, 2004; Vernes and Weinberger,2005; Zhang et al., 2009.

Recently Zhang et al. claimed the emergence of a new“paradigm of the time-resolved magneto-optical Kerreffect for femtosecond magnetism” Zhang et al., 2009.Despite the timeliness and relevance of this result, itshould be noted that this article a does not considerpump-and-probe experiments at all since there is onlyone pulse in the model; b considers only the Mxy com-ponent of the magnetization tensor, while the relationbetween the calculated off-diagonal components of themagnetization tensor Mxy and the net magnetization ofthe sample Mzz remains unclear. The work of Zhang etal. is no doubt a step toward a better understanding ofthe results of time-resolved magneto-optical experi-ments on magnetic metals. However, as the study in-volves quite some simplifications and leaves many ques-tions unanswered, a new paradigm might be a slightoverstatement.

C. Ultraviolet probe and spin-polarized electrons

The excitation of a medium with photons with an en-ergy larger than the work function, the difference be-tween the Fermi level and the vacuum energy, may re-sult in an emission of electrons from the surface.Photoelectron spectroscopies facilitate access to thestatic electronic structure of condensed matter as it en-ables one to obtain information about both the energyand momentum distribution of the electrons Hüfner,1995. With spin-sensitive detectors one can detect theaverage spin polarization of the emitted electrons, pro-viding thus direct access to the spin distribution of theelectron gas in a metal. This is in contrast to themagneto-optical methods, which are indirect. For thisreason the interpretation of time-resolved photoemis-sion experiments is much more straightforward than thatof time-resolved magneto-optical measurements Vater-laus, Beutler, and Meier, 1990, 1991; Scholl et al., 1997;Lisowski et al., 2005. The disadvantages of photoemis-sion are the strong requirements on the conductivity andthe quality of the surface. Consequently, the techniquecan be applied to a limited set of compounds, mainly,metals or half metals.

In principle, important insights into the dynamics andrelaxation processes of excited electrons can be obtainedwith the help of two-photon photoemission 2PPE that



is also spin, energy, and time resolved Aeschlimann etal., 1997. This is a two-step process, where an ultrashortpump laser pulse excites electrons from the Fermi level,and the transient population of the intermediate excita-tion is probed by a second pulse which promotes the stillexcited electrons to the vacuum where their energy isanalyzed.

The nonlinearity of the two-photon process leads toan increase in the 2PPE yield when the pulses are spa-tially and temporarily superimposed. As long as the twolaser pulses temporarily overlap it is obvious that anelectron can be emitted by absorbing just one photonfrom each pulse. However, if the pulses are temporarilyseparated, then an excited electron from the first pulse isable to absorb a photon from the second pulse only aslong as the inelastic lifetime of the intermediate stateexceeds the delay or if the normally unoccupied elec-tronic state is refilled by a secondary electron. Becauseof the precise measurement of the time delay betweenthe two pulses, this technique allows analysis of relax-ation times which are considerably shorter than the laserpulse duration; see, e.g., Schmidt et al. 2005.

D. Far-infrared probe

The energy of far-infrared photons falls in the rangewhere metals have an intense response of their free elec-trons. This Drude response of free electrons to electricfield transients allows one to obtain information aboutthe conductivity of the material. Therefore, time-resolved far-infrared spectroscopy has become a power-ful tool for the investigation of phase transitions instrongly correlated systems, such as superconductor-conductor and metal-dielectric transitions Averitt et al.,2001. If the conductance of the free-electron gas is sen-sitive to the magnetic order, it can be used as an indirectprobe of magnetic phase transitions as well. It is inter-esting to note that the frequencies of magnetic reso-nance of most antiferromagnets lie in the range from100 GHz to 3 THz. Moreover, several rare-earth ionshave spin-flip transitions in the range between 100 GHzand 10 THz. Thus terahertz spectroscopy can potentiallydeliver a direct access to the spin system and thus be aunique probe of transient laser-induced magneticchanges complementary to optical or ultraviolet meth-ods Hilton et al., 2006.

E. X-ray probe

The development of synchrotron x-ray sources has en-abled the generation of linearly and circularly polarizedradiation with intensities sufficient for the measure-ments of magnetic circular and magnetic linear dichro-ism in the x-ray domain XMCD and XMLD, respec-tively van der Laan et al., 1986; Schütz et al., 1987;Schneider et al., 1993; Stöhr et al., 1993. Similar tomagneto-optical phenomena in the visible spectralrange, XMCD and XMLD can be employed in order todeduce the orientation of the magnetization and antifer-romagnetic vector, respectively. However, in contrast to

the optical spectral range, in the range of x-rays mag-netic materials are characterized by relatively narrowtransitions at well-defined energies, so that x-ray absorp-tion techniques are characterized by elemental and evenchemical specificity. Moreover, because spectral lines inthe x-ray regime are narrow and well separated, one canuse so-called sum rules to obtain quantitative informa-tion about spin and orbital moments of the ground stateCarra et al., 1993.

Because of the advantages of x-ray techniques, time-resolved XMCD measurement with subpicosecond tem-poral resolution would provide novel and unique in-sights into ultrafast laser-induced magnetic changes. Thedevelopment of such a method has become one maingoal in the physics of magnetism over the last ten years.The first XMCD results with 100 fs temporal resolutionhave recently been reported Stamm et al., 2007. Itshould be noted, however, that ultrafast measurementsin the x-ray spectral range are still technically challeng-ing, requiring much effort. Also, theory for the interpre-tation of ultrafast XMCD measurements still needs tobe developed Carva et al., 2009.

IV. THERMAL EFFECTS OF LASER EXCITATION

A. Ultrafast demagnetization of metallic ferromagnets

Metallic ferromagnets happen to be complex systemsto understand because of the itinerant character of theirmagnetism. At the same time, metallic magnets are usedin numerous applications, from power transformers andsensors to data storage and spintronics. It is thereforenot surprising that the experimental studies of subpico-second magnetization dynamics has started with such“simple” systems as Fe Kampfrath et al., 2002; Carpeneet al., 2008, Ni Beaurepaire et al., 1996, or Gd samples.Despite some progress, the results are still being de-bated, both theoretically and experimentally.

The first ultrafast time-resolved studies of the impactof laser pulses on the magnetization were done on Niand Fe using picosecond laser pulses. However, thesewere not successful in observing any magnetic effects upto the melting point of the samples Agranat et al., 1984;Vaterlaus, Beutler, and Meier, 1990. Later, using time-resolved spin-polarized photoemission as a probe of themagnetization Vaterlaus et al. succeeded in estimatingthe spin-lattice relaxation time in Gd films to be100±80 ps Vaterlaus, Beutler, and Meier, 1991. Thedifficulties in the measurements were related to the factthat the available laser pulses were of the same durationor even longer 60 ps to 10 ns than the relevant timescales. Only later was it realized that, with such longexcitations, the various parts of the system were alwaysin equilibrium with each other so that the system fol-lowed the excitation profile. To study the intrinsic timescales mentioned earlier, much shorter stimuli are re-quired.



1. Experimental observation of ultrafast demagnetization

Beaurepaire et al. were the first to use 60 fs laserpulses to measure both the transient transmittivity andthe linear MOKE of 22 nm Ni films Beaurepaire et al.,1996, see Fig. 3. It was estimated from the transientreflectivity that the electron thermalization time is about260 fs and the electron temperature decay constant is1 ps. In contrast, the spin temperature deduced from thetime dependence of hysteresis loops reached its maxi-mum around 2 ps only. Thus, different electron and spindynamics were postulated. On the other hand, in thefollowing MSHG experiments by Hohlfeld et al., a simi-lar electron thermalization time was obtained 280 fs,but no delay between electron excitation and the loss ofmagnetization was seen Hohlfeld et al., 1997. In bothcases, the observed dynamics was much faster than whatcould be expected on the basis of the spin-lattice relax-ation time. In fact, ignoring a possible small differencebetween electron and spin temperature, these resultsshowed that the excitation by an ultrashort laser pulseleads to ultrafast heating of the electronic system. As aresult, the magnetization will decrease equally fast fol-lowing its normal temperature dependence M=MTe,where Te is the electron temperature. In the two-photonphotoemission of Scholl et al., two different demagneti-zation processes were observed, a fast one in less than300 fs, and a very slow one around 500 ps which wasascribed to the excitation of spin waves Scholl et al.,1997. Femtosecond demagnetization was also demon-strated by Stamm et al. using XMCD Stamm et al.,2007.

While it has been generally accepted that the magne-tization follows the electron temperature with a possibledelay between the electron excitation and the magneticbreakdown of no more than 50 fs, questions arose about

the interpretation of the experimental results. For ex-ample, is the magneto-optical response from a nonequi-librium system indeed proportional to the magnetiza-tion? A study on Cu/Ni/Cu wedges Koopmans et al.,2000 demonstrated that during the first hundreds offemtoseconds the dynamical evolution of the Kerr ellip-ticity and rotation do not coincide see Fig. 4, breakingdown the proportionality between the magnetizationand the Voigt vector that is the basis of magneto-optics.Such breakdown is indeed a consequence of the out-of-equilibrium character of the electron system immedi-ately after the femtosecond excitation. Only after theelectronic equilibration process can one reliably deducethe change in the magnetization from changes in Fara-day or Kerr rotation. Such arguments were used to castdoubt on ultrafast instantaneous light-induced changesin the magnetization Koopmans et al., 2000.

Nevertheless, using laser pulses as short as 20 fs dura-tion and carefully separating the dynamics of the diago-nal and the nondiagonal elements of the time-dependentdielectric tensor, it has been shown that a significant de-magnetization can be obtained at a sub-100 fs time scale,for example, in CoPt3 Guidoni et al., 2002 see Fig. 5.Moreover, it has been shown by time-resolved photo-emission that the exchange splitting between majorityand minority spin bands is affected at a similar timescale Rhie et al., 2003.

The breakdown between the magneto-optical re-sponse and magnetization in ultrafast pump-probe ex-periments has been also supported by ab initio calcula-tion of the magneto-optical Kerr effect in Ni Oppeneerand Liebsch, 2004. By evaluating the complex conduc-tivity tensor of Ni for nonequilibrium electron distribu-tions, Oppeneer and Liebsch considered dichroicbleaching and state-blocking effects. It was shown thatthe conductivity tensor and therefore the complex Kerrangle can be substantially modified so that the Kerr ro-tation and ellipticity are no longer proportional to themagnetization of the sample and the Kerr response atultrashort times can therefore not be taken as a measureof demagnetization. For CoPt3, the role of dichroicbleaching and state-blocking effects in ultrafastmagneto-optical pump-probe experiments has been veri-

0.9

0.8

1.0

0.5

0.7

0.6

0 15105

t (ps)

No

rma

lize

dre

ma

ne

nce

H (Oe)

Kerr

sig

nal

no pump

t = 2.3 ps

-50 500

FIG. 3. Remanent MO contrast measured for a nickel thin filmas a function of time after exciting by a 60 fs laser pulse. Theresults demonstrate ultrafast loss of the magnetic order of theferromagnetic material within a picosecond after laser excita-tion. From Beaurepaire et al., 1996.

FIG. 4. Comparison of induced MO ellipticity open circlesand rotation filled diamonds for a Cu111 /Ni/Cu epitaxialfilm and Ni thickness and pulse energy as indicated. The insetdepicts the polar configuration with pump 1 and probe 2beams. From Koopmans et al., 2000.



fied experimentally, employing measurements where thewavelength of the probe was varied in the broad spectralrange between 500 and 700 nm Bigot et al., 2004. Ifbleaching and state-blocking effects are indeed present,one would expect to observe a spectral dependence inthe magneto-optical response, associated with the popu-lation bleaching. However, the spectral response wasfound to be rather flat, and it was concluded that themagneto-optical signal predominantly reflects the spindynamics in this ferromagnet. We mention that rela-tively small magneto-optical signals from Ni did not al-low the performance of similar measurements in thismetal. Therefore, strictly speaking the conclusions ofBigot et al. 2004 cannot be expanded to other metals.Moreover, a saturation of ultrafast laser-induced demag-netization at high excitation densities has recently beenobserved in Ni and has been explained in terms of bandfilling effects Cheskis et al., 2005.

Another signature of ultrafast laser-induced demagne-tization is emission in the terahertz spectral range. If themagnetization of a ferromagnetic film is changed on asubpicosecond time scale, an electromagnetic wave willbe generated according to Maxwell’s equations. This ra-diated wave thus contains information about the intrin-sic spin dynamics. Similar ideas were used to investigatecarrier dynamics in semiconductors and Cooper-pairbreaking in superconductors. Beaurepaire et al. werefirst to apply this technique on thin Ni films in order toobtain an extra proof of the subpicosecond demagneti-zation Beaurepaire et al., 2004. A rather standardelectro-optic sampling technique in a 0.5-mm-thickZnTe crystal was used for a review see Ferguson andZhang 2002.

The electric field thus measured upon ultrafast laser-induced demagnetization of the thin Ni film is shown inFig. 6a. If one assumes that pump pulses coherentlyexcite the film, producing a time-varying magnetization,the electric field in the far field polarized orthogonal tothe magnetization vector Mx, propagating in the z di-rection is

Eyt =0

42r

2Mx

t2 t − r/c , 18

where r is the distance to the dipole. Thus the measuredtransient electric field emitted by the sample is given bythe second derivative of the magnetization.

With this in mind, and assuming the magnetizationvariation as shown in Fig. 6b, the solid curve of Fig.6a is obtained, showing a perfect correspondence withthe experimentally observed terahertz radiation. It hasalso been verified that the outgoing terahertz wave isindeed polarized along the y axis and that the effect isindependent of the incoming polarization of the visiblepump light.

Similar experiments have also been performed on thinFe films Hilton et al., 2004. There the terahertz emis-sion was found to have two contributions. One part ofthe terahertz signal depended on the polarization of thepump beam and was attributed to the nonlinear effect ofoptical rectification. Another part did not depend on thepump polarization and likely originated from the ul-trafast demagnetization.

Finally, we mention that ultrafast laser-induced de-magnetization of metals can be accompanied by achange in magnetization reversal behavior Weber et al.,2004; Wilks et al., 2005; Roth et al., 2008; Xu et al., 2008.

50 100 2001500-50

-2

-8

-6

-4

0

-10x10-3

time delay (fs)

time delay (fs)

q

/q

/

50 100 2001500-50

-2

-8

-6

-4

0

-10x10-3

F F

F F

(a)

(b)

0 100fs

FIG. 5. Ultrafast laser-induced dynamics in CoPt3. a Time-resolved Faraday MO signals, rotation open circles, and el-lipticity filled circles. The pump-probe cross correlationdashed line is displayed for reference. b Short-delay relativevariations of real and imaginary dielectric tensor componentsretrieved from the data in a. For t100 fs, the real andimaginary parts follow different dynamics. From Guidoni et al.,2002.

0 420-2-4

-1.0

0.0

-0.5

0.0

0.5

1.0

Time (ps)

TH

zam

plit

ude

or

magnetization

(arb

.units)

(a)

(b)

FIG. 6. Laser-induced generation of TH2 radiation. a TheTHz pulse generated upon ultrafast laser heating of 42-Å-thickin-plane magnetized Ni film. The smooth line is a simulationassuming the time-dependent magnetization of b convolutedwith a Gaussian instrument response function full width athalf maximum 540 fs. From Beaurepaire et al., 2004.



2. Phenomenological three-temperature model

Already, since the first laser-induced magnetizationdynamics experiments, it has become clear that the re-sulting demagnetization is a complicated process, wherevarious components of the system are participating thatinvolve different relaxation processes.

The interactions in such a system can be qualitativelydescribed with a help of a single model that containsthree separate but interacting reservoirs, namely, theelectrons, lattice, and spins see Fig. 7. These three res-ervoirs are connected by interactions of different originand efficiency. A particular effective temperature can beassigned to each of these reservoirs. Note, however, thatthis assignment is possible only if a certain equilibrium isassumed within the considered subsystem. Given theshort time scales, this assumption is not always valid,imposing a limit on the validity of this model.

The three-temperature model describes the temporalevolution of the system by three coupled differentialequations,

CedTe/dt = − GelTe − Tl − GesTe − Ts + Pt ,

CsdTs/dt = − GesTs − Te − GslTs − Tl , 19

CldTl/dt = − GelTl − Te − GslTl − Ts ,

where Gij represents the coupling between the ith andjth baths, Ci describes their heat capacity, Ti is the tem-perature of the corresponding system, and Pt is theoptical input. The Gij coefficients are phenomenologicalparameters that once determined tell us how strong aparticular link is but at the same time tell us nothingabout the nature of the interaction. Depending on theheat capacities Cj, effective temperature differences canbe very large. For example, since the electron heat ca-pacity is typically one to two orders of magnitudesmaller than that of the lattice, Te may reach severalthousand Kelvin within the first tens of femtosecondsafter excitation, while the lattice remains relatively cold

Fatti et al., 2000 even after the equilibration process.Schematically the process of the equilibration betweendifferent reservoirs following the excitation with a fem-tosecond laser pulse is shown in Fig. 8.

A typical scenario of the processes leading to a laser-induced demagnetization is the following. At optical fre-quencies, only electrons are able to respond to an elec-tromagnetic excitation, absorbing or scattering thephotons practically instantaneously. Thus, in the firststep i the laser beam hits the sample and createselectron-hole pairs hot electrons on a time scale of1 fs; ii the electronic system equilibrates at elevatedtemperatures Te by electron-electron interactions within50–500 fs, depending on the system; and iii the equili-brated electronic excitations decay via phonon cascadeson a time scale given by the electron-phonon interactiontime within 100 fs–1 ps for metals and heat up the lat-tice, increasing Tl. Thus, at the end of a picosecond,electron and lattice systems are in thermal equilibriumwith each other. What happens with the spins?

As the essence of magnetization is angular momen-tum, apart from the energy, angular momentum conser-vation needs to be considered in magnetic systemsZhang and George, 2008. Also, in the process of de-magnetization, a certain amount of angular momentummust be taken away from the spin system. It is thereforecrucial to disclose the possible channels for this angularmomentum transfer when considering the various pro-cesses involved in ultrafast demagnetization.

Generally speaking, both the electron system and thelattice should be able to absorb this angular momentum,even if only temporarily. The spin-lattice interaction isusually considered to be of the same order of magnitudeas the magnetocrystalline anisotropy. The latter is ratherweak 100 eV in transition metals because of thecrystal-field-induced quenching of the electron orbits.Therefore the corresponding interaction time is ex-pected to be quite long, e.g., 300 ps in Ni Hübner andZhang, 1998. However, after femtosecond demagnetiza-tion had been demonstrated Beaurepaire et al., 1996, itbecame clear that in metals there is a much strongercoupling between the spins and the two other reservoirs.Mechanisms of an effective electron-spin or phonon-

electrons lattice

spins

Gel

Ges Gsl

FIG. 7. Color online Interacting reservoirs carriers, spins,and lattice in the three-temperature model as suggested byAgranat et al. 1984. Some of the possible interaction chan-nels, such as spin lattice and electron-phonon interactions, areconsidered to be well understood. The direct electron-spincoupling is a subject of current research and active debates.

time

tem

per

atu

re

Te

Tl

0 ~1ps

Ts

Ts

metal

diel

FIG. 8. Color online Temporal behavior of electron, lattice,and spin temperatures following excitation with a short laserpulse. The behavior of the spin temperature is shown for bothstrong coupling metals and weak coupling dielectrics cases.



spin coupling responsible for a demagnetization fasterthan 100 fs are the subject of current research and activedebates.

Recently Bigot et al. suggested that the laser-induceddemagnetization of Ni could be related to a direct cou-pling between photons and spins Bigot et al., 2009. Inparticular, a novel relativistic quantum mechanicalmechanism was claimed to be responsible for the ul-trafast change in magnetization. However, in this workthe only direct, i.e., polarization-dependent, effect wasobserved during the overlap of pump and probe pulses.This is similar to earlier work by Dalla Longa et al.2007, where an analogous effect was attributed to theeffects of optical coherence. Indeed, if the pump pulseleaves the system in the same magnetic state regardlessthe polarization, what then is the result of that “directcoupling”? It should also be mentioned that a relativisticcoupling of spins to a light field has been consideredearlier for the description of magneto-optical effects inuranium compounds by Kraft et al. 1995, but theirHamiltonian is different from the one suggested byBigot et al. In this respect, it is worth noting that it hasrecently been demonstrated, using polarization pulseshaping, for the case of an elementary two-level systemthat such instantaneous laser-induced magneto-opticaleffect has nothing to do with spin excitations but origi-nates from off-resonant pumping of excited-state popu-lation, obviously present only during the action of thelight pulse Versluis et al., 2009. Clearly, on the topic ofdirect coupling between photons and spins, the last wordhas not yet been said.

3. Spin-flip and angular momentum transfer in metals

The interaction that is responsible for the angular mo-mentum flow into the electron orbits is the spin-orbitcoupling, about 50 meV in ferromagnetic metals. Thiswould correspond to a minimum relaxation time ofsome 20 fs, i.e., fast enough to explain the observed ul-trafast effects. There are three main mechanismsthrough which an excited electron can flip its spin in aferromagnetic metal. These are i a Stoner excitation,effective at relatively high energies Stöhr and Sieg-mann, 2006; ii an inelastic electron-spin-wave scatter-ing event, effective in the cases of relatively low excita-tion photon energy Edwards and Hertz, 1973; Hertzand Edwards, 1973; Zhukov and Chulkov, 2009, and iiia single-particle-like spin-flip scattering with impuritiesor phonons, called the Elliott-Yafet mechanism Elliott,1954; Yafet, 1963. These three mechanisms are charac-terized by short time scales and can lead, in principle, tothe observed subpicosecond changes of magnetization inmetals after excitation with a laser pulse, as far as theenergy relaxation is concerned. The relaxation of theenergy into the lattice occurs in parallel to this and canbe considered only as a factor limiting the demagnetiza-tion magnitude because of too fast energy dissipation.

In metals, the spin-flip probability is strongly affectedby the band structure. The electron-electron interactionsin itinerant-electron systems lead to a separation of the

electronic bands for spin-up and spin-down electrons.When the Fermi level lies between these two bands, thelower-energy spin-up or majority states are occupied,while the higher-energy spin-down or minority statesare not. It is this difference that leads to a net spin po-larization of the conduction electrons. Moreover, bandsthat are exchange-split across the Fermi level alsopresent the possibility of a unique single-particle mag-netic excitation. Such excitation is nevertheless many-body-like in the sense of a Coulomb interaction. Anelectron may be removed from an occupied majoritystate, undergo spin reversal, and be placed in a previ-ously unoccupied minority state. The resulting electron-hole pair with opposite spins and net vector q is called aStoner excitation. Note that for each combination of qand spin, a distribution of Stoner excitations is possibleMohn, 2003.

Since Stoner excitations involve a spin reversal, theseare the fundamental single-particle magnetic excitationsin itinerant-electron magnets. At low energy and wavevector, the configuration of the electronic bands thelarge value of exchange splitting does not allow indi-vidual Stoner excitations. In this case, a coherent super-position of virtual excitations of wave vector q can pro-duce a collective magnetic excitation at q called spinwave at low energy. In the regions of energy-momentum space where individual Stoner excitationsare almost allowed, the spin waves and Stoner modeswill be coupled so that the former become heavilydamped.

From this picture it is clear that in the low-energyregion spin waves will dominate over Stoner excitations.This is true in the case of most though not all laser-induced magnetization dynamics experiments that usenear-infrared wavelengths around 800 nm, correspond-ing to a photon energy of 1.5 eV. Note, however, thatno finite q vector is required for on-site spin flips, whilenevertheless a low-energy excitation becomes possibledue to many-body effects.

In general, a possible mechanism for a flip of onesingle spin in a metal was studied more than 50 yearsago and called Elliott-Yafet scattering Elliott, 1954;Yafet, 1963. In this theory it is shown that a scatteringevent of an excited electron with a phonon changes theprobability to find that electron in one of the spin states↑ or ↓ , thus delivering angular momentum from theelectronic system to the lattice. Because of the spin-orbitcoupling, an electronic state in a solid is always a mix-ture of the two spin states, e.g., a dominant spin-up con-tribution and a small spin-down contribution, definingthe total crystal wave function Steiauf and Fähnle,2009,

k,↑ = akr↑ + bkr↓ , 20

where k is the wave vector. The potential describing theinteraction of an electron with a phonon of wave vectorq consists of two parts



Uq = Uq0 + Uq

spin-orbit, 21

where Uq0 is the spin-independent part and Uq

spin-orbit re-fers to the spin-dependent part of the scattering poten-tial. The matrix element which is of interest for demag-netization is the spin-flip matrix element between aspin-up state k,↑ and a spin-down state k+q,↓:

mk+q,↓;k,↑ = k+q,↓Uq0 + Uq

spin-orbitk,↑ . 22

It was the important observation of Elliott 1954 thatthe first part of this matrix element, originating from Uq

0,is already nonzero because it connects the large andsmall components of two spinors. It was shown to beessential, however, that both parts of the potential aretaken into account Yafet, 1963.

Steiauf and Fähnle recently calculated the spin-mixingparameters in several transition metals using ab initiodensity functional theory Steiauf and Fähnle, 2009.The calculations showed that in Co and Fe as well as Nithe spin-mixing parameter is about a factor of 25 largerthan the value commonly used for Cu. Although theseresults support the idea that angular momentum transferfrom spins to the lattice via the Elliott-Yafet-like mecha-nism may play an important role in the femtoseconddemagnetization, one should remember that this is asingle-electron model, where exchange interaction is nottaken into account and hence the spin-flip events areassumed to be independent. Therefore a straightforwardapplication of these conclusions to magnetically orderedmaterials would result in an oversimplified model whichneglects fundamental features of magnetism.

An attempt to predict the time scale of ultrafast laser-induced demagnetization of metals has been recentlyperformed under the assumption that the speed of thedemagnetization is defined by the speed of spin-flip pro-cesses via the Elliot-Yafet mechanism Koopmans,Kicken, et al., 2005; Koopmans, Ruigrok, et al., 2005.Using the Landau-Lifshitz-Gilbert equation see Sec.II.A, an analytical expression was derived that connectsthe demagnetization time constant with the Gilbertdamping parameter via the Curie temperature. Basi-cally this was an attempt to connect the characteristictimes of magnon scattering at the center and at the edgeof the Brillouin zone. However, in strong contrast to abinitio calculations Gómez-Abal et al. 2004 unavoidablyneglected the electronic band structure of the materialin light-matter interactions. In contrast to atomistic andmicromagnetic calculations, the spin-wave band struc-ture was neglected as well Djordjevic and Münzenberg,2007. Such crude approximations have raised doubtsabout the adequateness of the approach Walowski et al.,2008; Radu et al., 2009. It has recently been demon-strated that the model fails to describe the experimentalresults on demagnetization of the rare-earth-doped met-als even qualitatively Radu et al., 2009. In the follow upto this work Koopmans et al., 2010 the spin-scatteringmechanisms are described by a single dimensionlessspin-flip parameter asf that can be derived directly fromexperimental data. This allows the macroscopic demag-netization process to be related to microscopic spin-flip

processes. It will be interesting to see how this param-eter asf compares with the one extracted from spin-transport experiments; see, e.g., Moreau et al. 2007.

4. Interaction among charge, lattice, and spinsubsystems

a. Time-resolved electron photoemission experiments

Valuable information on the microscopic processesthat accompany the ultrafast laser-induced demagnetiza-tion can be obtained by studying magnetic surfaces withsurface-sensitive techniques. Such a low-dimensionalstructure reduces the phase space available for relax-ation processes. As a consequence, relaxation rates aredecreased compared to those in the bulk, facilitating amore detailed analysis. This advantage of surfaces hasled in the last decade to a comprehensive understandingof the principles governing the decay of electronic exci-tations in metals Echenique et al., 2004.

The Gd0001 surface was found to be a favorablemodel system to study electron-phonon, electron-magnon, and phonon-magnon interactions and their re-spective dynamics by femtosecond time-resolved laserspectroscopy Melnikov et al., 2003, 2008; Lisowski et al.,2005; Bovensiepen, 2007. The surface presents anexchange-split 5dz2 surface state with the majority spincomponent being occupied and the minority componentunoccupied, respectively. Time-resolved photoemissionTRPE was used to investigate the pump-induced varia-tion in this exchange-split surface state. The top panel ofFig. 9 shows a spectrum around EF at 300 fs with the PEyield measured over five orders of magnitude. The occu-pied surface state S↑ is readily visible, while the unoccu-

10-3

10-2

10-1

100

101

Inte

nsity

1.5

1.5

1.0

1.0

0.5

0.5

0.0

0.0

-0.5

-0.5

E - EF (eV)

10-3

10-2

10-1

100

101

Inte

nsity

40

30

20

10

0

t = 300 fs

-2 ps0 fs50 fs100 fs200 fs300 fs400 fs500 fs600 fs700 fs1000 fs

Ekin

EF

Evach1

h2

f(E)

-e

S

S

S

S

bulk surface

e-TOF

ex

FIG. 9. Color online Time-resolved electron photoemission.Top: Normalized photoelectron spectrum at 300 fs circleswith fit solid line and distribution function fE ; t dashedline on a logarithmic scale and divided by f triangles on alinear scale at right. The inset sketches the TRPE experiment.Bottom: Photoelectron spectra at different delays are dis-played in color fits in black and are offset vertically. FromLisowski et al., 2005.



pied component S↓ shows up as a shoulder at 0.4 eVabove EF. To fit the spectra two Lorentzians for S↑↓, aconstant bulk density of states, and a distribution func-tion fE ; t are used. Convolution with a Gaussian of45 meV width accounts for broadening of the laser pulseand the spectrometer resolution. As shown by Greber etal. 1997 and Rhie et al. 2003, unoccupied states can beanalyzed with photoemission by normalizing the spec-trum with a Fermi-Dirac distribution. On ultrafast timescales, nonthermalized electrons have to be includedLisowski et al., 2004, which leads to small correctionsof the Fermi distribution for Gd. After normalization tof, the surface contributions S↑ and S↓ are clearly discern-ible. Figure 9 shows that the optical excitation leads toan instantaneous population of S↓ that decays withinabout 1 ps.

To develop a microscopic understanding of the ob-served ultrafast spin dynamics, two contributions shouldbe considered: i secondary electrons in combinationwith transport processes and ii spin-flip scattering ofhot electrons among spin-mixed states. The pump pulseexcites S↑ electrons from the occupied surface state tounoccupied bulk states, minority electrons from bulk tounoccupied surface states, and transitions between bulkstates. Transitions involving the surface state redistrib-ute spin polarization from the surface to the bulk be-cause majority electrons are excited into bulk states orminority electrons are excited to the surface state. Thisredistribution is a consequence of transport effects sincethe minority holes or majority electrons, which are ex-cited in the vicinity of the surface, propagate into thebulk.

b. Time-resolved x-ray experiments

In an attempt to identify the angular momentum res-ervoir responsible for the disappearance of the spin or-der, Bartelt et al. performed studies of the laser-induceddemagnetization of Fe/Gd multilayers using time-resolved measurements of x-ray magnetic circular di-chroism with 2 ps temporal resolution Bartelt et al.,2007. Later Stamm et al. performed similar studies ofultrafast laser-induced demagnetization of Ni but withfemtosecond temporal resolution and using femtosecondx-ray pulses see Fig. 10 Stamm et al., 2007. In bothcases by separating spin and orbital contributions to theXMCD, it was unambiguously demonstrated that theelectron orbitals were not responsible for this fast de-magnetization. Therefore the only possible reservoirshould be the lattice. Because of the above-mentionedweak spin-lattice interaction, they argued that this canonly be possible if some extra interaction, for example,via light-induced virtual states, enhances the spin-latticeinteraction.

This finding seems to favor the Elliot-Yafet scatteringas the most likely interaction mechanism. Indeed,electron-spin-wave scattering would keep the angularmomentum inside the electron and spin systems, andonly phonon- or impurity-induced scattering can trans-fer it to the lattice. Combining spin-, energy-, and time-

resolved two-photon photoemission with the time-resolved MOKE, Cinchetti et al. claimed to have foundexperimental evidence for the relevance of the Elliott-Yafet spin-flip processes for the ultrafast demagnetiza-tion on a time scale around 300 fs Cinchetti et al., 2006.

How do these findings correlate with the generally ac-cepted weak spin-lattice interaction? To answer thisquestion, we recall the underlying physics. In general,spins couple to the electron orbital motion via spin-orbitcoupling Uso=S ·L, which is of the order of 50 meV.When placed in a crystal lattice, however, the orbits be-come distorted, so that the final states of the system arelinear combinations of the original spherical harmonics.Such combinations are best viewed as standing waves ofcylindrical symmetry with a periodic perturbation, andcontain harmonics with opposite values of angular mo-mentum. This interaction is in fact very strong, with typi-cal values of the state splitting of the order of 1 eV. Theexact value and sign of the splitting is given by the par-ticular crystal symmetry. As a result of this interaction,the average orbital moment in each of the final states iszero or close to it quenching of the orbital moments intransition metals. This leads to a strong reduction in theaveraged value of the spin-orbit coupling in transition-metal magnets which also leads to a weak static magne-tocrystalline anisotropy.

Dynamically, however, the energy and angular mo-mentum transfer from spins to lattice can be treated as atwo-step process. In the first step, exchange of the en-ergy and momentum between spins and orbitals occurs;in the second, it is the one between orbitals and lattice.At first it seems that such picture contradicts the obser-vations by Stamm et al., who have not found any in-crease in the orbital angular momentum during the de-magnetization process. Note, however, that in such two-step processes, the second step happens much fasterthan the first one so that any accumulation of orbitalmomentum is not to be expected. The transfer rate isthus determined by the slowest part, namely, the cou-pling between spins and the original atomiclike orbitals,

FIG. 10. Color online Time-resolved XMCD signal with cir-cularly polarized x rays incident at 60° relative to the samplesurface vs pump-probe time delay symbols measured at theL3 edge maximum. As the measured value is proportional toS+ 3

2L, overall decrease in the signal excludes orbital momentsfrom gaining any significant value at the cost of spins. FromStamm et al., 2007.



around 50 meV, and thus occurs at a time scale of tensof femtoseconds.

Additional aspects of the role of angular momentumwill be treated in Sec. VI.D.

5. Microscopic models of ultrafast demagnetization

Ab initio modeling has revealed possible mechanismsof ultrafast demagnetization without having the latticeinvolved. To achieve a control on demagnetization andeventually magnetic switching, the electromagnetic fieldof the laser pulse was invoked as an active lever to drivethe process Zhang and Hübner, 2000. Also a model forultrafast demagnetization of materials with discrete elec-tronic structure was suggested Gómez-Abal and Hüb-ner, 2002 and all-optical subpicosecond spin switchingwas observed in ab initio simulations performed for NiOGómez-Abal et al., 2004. However, currently there isno experimental evidence that the mechanisms of thedemagnetization and spin switching in Ni and NiO sug-gested by the ab initio calculations are really takingplace in these materials. Nevertheless, these works con-tained a number of fruitful ideas related to ultrafast cou-pling of light and matter. For instance, it was noted thatswitching by optical femtosecond pulses may follow atotally different trajectory, such as via a Raman-like two-photon excitation process.

An alternative approach to describe ultrafast demag-netization is based on an analysis of magnon-magnonand phonon-magnon interactions, while it neglects theelectronic structure of a material. The classical micro-magnetic modeling is performed for magnetic elementslarger than 0.5 nm, assuming a fixed length of the mag-netization vector macrospin and using the Landau-Lifshitz-Gilbert equation see Sec. II.A. Often an as-sumption of zero temperature is made; however, astochastic variant of micromagnetic simulations allowsone to take finite temperature into account Berkov andGorn, 1998; Fidler and Schrefl, 2000; Nowak et al., 2005.A modeling using the LLG equation cannot describe theprocesses in a magnetic system close to the Curie tem-perature, in general, and the phenomenon of ultrafastdemagnetization, in particular. In this case the dampingis enhanced when approaching the Curie temperatureand the magnetization magnitude is not constant in timeNowak et al., 2005. However, it is expected that anultrafast laser excitation of free electrons in metals isfollowed by an electron-phonon interaction at the timescale of about 1 ps. Because of this interaction, the elec-tron temperature decreases, bringing the magnetic sys-tem well below the Curie point. It is therefore reason-able to expect that micromagnetic simulations mayprovide an adequate modeling of the processes that oc-cur in a spin system on a time scale longer than 1 psafter laser excitation. An attempt to describe the pro-cesses following ultrafast laser excitation of a magneticmetal using such micromagnetic simulations was per-formed by Djordjevic and Münzenberg 2007 Fig. 11.The model did not regard the details of the demagneti-zation and the quenching was assumed to be an instan-

taneous process; it was described by orienting the mag-netization vectors of 0.5 nm elements by a random angleEilers et al., 2006. From the micromagnetic simulationsone sees that the instantaneous demagnetization isequivalent to the generation of a large number of mag-nons at the edge of the Brillion zone. In the followingtime domain the magnetization slowly recovers, and thespeed of the recovery is defined by the speed of theenergy transfer from the short-wavelength magnons tolong-wavelength spin excitations via a spin-wave relax-ation chain i.e., magnon-magnon interaction. The re-sults of the micromagnetic simulations based on theLLG equation were found in good agreement with time-resolved magneto-optical measurements of the laser-induced magnetization dynamics but only at time scaleslonger than 1 ps. For an adequate description of fasterdynamics, a development of micromagnetic theory thatis able to deal with elevated temperatures and mac-rospin dynamics beyond the LLG equation is necessary.

Another approach to modeling ultrafast laser-inducedspin dynamics in a metallic magnet is to use the LLGequation on the atomic level Chubykalo-Fesenko et al.,2006; Kazantseva, Nowak, et al., 2008. Essentially suchmodeling consists of the use of the Heisenberg modelfor the exchange coupling and the Langevin dynamicapproach the LLG equation augmented by a randomfield to include the effects of temperature to describethe evolution of the ensemble of coupled spins. Ab initiocalculations are used to provide information on the localproperties such as the spin and exchange integral. Atypical amount of modeled spins at the moment does notexceed 106, corresponding to a maximum size of 202020 nm3. Nevertheless, the method appeared to bequite powerful for modeling magnetic systems in the vi-cinity of phase transitions Chubykalo-Fesenko et al.,2006. For instance, it was able to reproduce an increasein the macroscopic transverse damping when the Curietemperature was approached. Similarly to micromag-

0.0 0.1 0.2 0.3 0.4 0.510-12

10-7

10-2

103

108

1013

t=16.7 ps

t=0.7 ps

t=125 ps

t=6.7 ps

PowerFFT(Mx,y,z/M

0)/PowerFFT(250ps)

kz/2π [nm-1]

t=0 ps

0 10 20 30 40 50

-8

-6

-4

-2

0

2

t=16.7 ps

t=0.7 ps

t=125 ps

t=6.7 ps

Mx,y,z(z)/M0

z [nm]

MxMyMz

t=0 ps

x5

FIG. 11. Color online Cut through micromagnetic simulationfor the Ni film after a partial demagnetization. Left: The evo-lution of spin-wave emission from the excited area shown inreal space. Right: The corresponding Fourier transform shownas a function of spatial frequency. From Djordjevic and Mün-zenberg, 2007.



netic simulations based on the LLG equation, the atom-istic modeling was able to reproduce the recovery of themagnetization at the time scale of a few picosecondsKazantseva, Nowak, et al., 2008. Most importantly, theresults of the simulations with the stochastic LLG equa-tion on an atomistic level were found in excellent agree-ment with the experimental observations of a subpico-second time scale for laser-induced demagnetization of aferromagnet. Moreover, it was found that, even if energypumped into the spin system at the subpicosecond timescale corresponds to heating above the Curie point, thesystem does not necessarily demagnetizes fully. Such afinding shows that for a spin system far from equilibriumthe concept of spin temperature, often used for the de-scription of ultrafast demagnetization Beaurepaire etal., 1996, is not valid. It should be mentioned that mi-crospin simulations based on the Landau-Lifshitz-Blochequation seemed to capture the physics revealed by theatomistic model Chubykalo-Fesenko et al., 2006 seeSec. II.B. Finally, we note that it is becoming more ob-vious that for a solution of the problem of ultrafast de-magnetization a multiscale approach, combining ab ini-tio, atomistic, and micromagnetic simulations, should bedeveloped.

B. Demagnetization of magnetic semiconductors

The separation of spins and electrons into two sepa-rate baths see Sec. IV.A.2 is a crude approach for treat-ing itinerant ferromagnets, where no sharp separationbetween s and d bands is present. In contrast, such anapproach is ideally suited for materials, where most ofthe macroscopic magnetization comes from the localizedd or f shell spins, while the exchange interaction be-tween the spins is mediated via itinerant s carriers Jung-wirth et al., 2006. It is generally accepted that such amechanism of exchange interaction is typical for rare-earth metals such as Gd. At the same time, a similarmechanism of exchange is considered to be responsiblefor ferromagnetism in diluted magnetic semiconductorssuch as EuO, Ga1−xMnxAs, and In1−xMnxAs.

In contrast to EuO and EuS, fabrication of the novelIII,MnV ferromagnetic semiconductors only becamepossible not long ago. The first successful doping of Mnions into InAs exceeding the heavy doping regime wasreported by Munekata and co-workers in 1989 Mu-nekata et al., 1989. The subsequent discovery of ferro-magnetism in p-type InMnAs films in 1991 triggeredmuch interest in these materials Ohno et al., 1992.

Ferromagnetism of III,MnV semiconductors in-duced by relatively low concentrations of free carriersmakes these materials unique model systems for the in-vestigation of laser control of magnetic order. Unliketransition metals, in ferromagnetic semiconductors sepa-ration of the spins and electrons into two distinct sub-systems is theoretically substantiated. And, unlike inrare-earth metals, the concentration of the free carriersresponsible for the exchange between localized spins isrelatively low and can be substantially controlled withthe help of optically injected photocarriers.

The first experiments on excitation of GaMnAs withsubpicosecond laser pulses the energy of the photonsused was about 3.1 eV revealed the feasibility of opticalcontrol of magnetism in these materials Kojima et al.,2003. A photoinduced demagnetization process was ob-served at a time scale of 1 ns, and no demagnetization ata subpicosecond time scale has been detected.

Later, however, experiments using time-resolvedmagneto-optical Kerr spectroscopy of ferromagneticGaMnAs with mid-infrared pump pulses have allowedtwo distinct demagnetization processes to be revealed, afast 1 ps and a slow 100 ps one; see Fig. 12Wang et al., 2006, 2008. While the slow componentwith a characteristic time of about 100 ps is interpretedas a spin temperature increase via spin-lattice relaxation,an original concept was suggested for the interpretationof the time of the fast demagnetization processes1 ps Cywinski and Sham, 2007; Wang et al., 2008.

Since the energy of the photons used in the expe-riment 0.61 eV was insufficient for excitation of anelectron from the valence to the conduction band, itwas concluded that the observed demagnetization ofGaMnAs resulted from the excitation of holes and asubsequent increase in their kinetic energy tempera-ture Wang et al., 2006, 2008 Fig. 13. It was shownthat the probability of a spin-flip process for Mn ions isenhanced with increasing effective hole temperature; thephenomenon was called the “inverse Overhauser effect”Cywinski and Sham, 2007. It was noticed, however,that, if the demagnetization of the ferromagnet is notcomplete, the speed at which the magnetization dropsmight have no relation to the characteristic interactiontime of the spins with other subsystems electrons andphonons, see Fig. 13. Instead, the demagnetization timecan be given by the characteristic time of the electrontemperature drop. Since the probability for spin flip ofMn spins in the inverse Overhauser effect is propor-tional to the temperature of the free carriers, the fastcooling of the electron subsystem below a certain level

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

∆θK

(mra

d)

86420Time Delay (ps)

-170 mT

170 mT

0 mT

T = 4.1 K

(a)

∆θK

(1m

rad

per

div.

)


6 K

60 K

50 K

45 K

40 K

35 K

25 K

15 K

(b)

FIG. 12. Transient magneto-optical Kerr rotation inGa0.974Mn0.026As triggered by a 140 fs laser pulse with a pho-ton energy of 0.61 eV. a Magnetic-field dependence of theKerr rotation. b Temperature dependence of the Kerr rota-tion between 6 and 60 K. The signal decreases quickly withincreasing temperature and is absent above the Curie tempera-ture TC=50 K. From Wang et al., 2006, 2008.



will effectively stop the demagnetization. Thus, the sub-picosecond time scale of the demagnetization processmay simply be the characteristic time of carrier-phononinteraction. It is important that there is again no elec-tronic many-body interaction necessary. The connectionbetween demagnetization time and energy relaxation ofthe excited carriers has been experimentally demon-strated in the study of the correlations between thesetwo processes in GaMnAs Wang et al., 2006, 2008.

We note, however, that an efficient angular momen-tum transfer from Mn spins to a hot hole gas is notenough to result in demagnetization since the p-d inter-action conserves the total angular momentum. In addi-tion to the redistribution of the angular momentum be-tween the Mn spins and holes, an effective sink of theangular momentum from the holes to the lattice is re-quired. Calculations show that short hole-spin relaxa-ion times of the order of 10 fs are required for the ex-planation of ultrafast laser-induced demagnetization inGaMnAs and InMnAs Cywinski and Sham, 2007; Mo-randi et al., 2009.

Finally, we note that the demagnetization of Ga,MnAs can also be seen in an ultrafast change in coer-civity Hall et al., 2008 or terahertz emission Héroux etal., 2006. Terahertz emission induced by ultrashort laserexcitation has also been observed in InMnAs. However,this was not associated with ultrafast demagnetizationsince the signal was also observed above the Curie pointZhan et al., 2007.

C. Demagnetization of magnetic dielectrics

It is instructive to compare the ultrafast demagnetiza-tion of ferromagnetic metals with a similar process inmagnetic dielectrics. In the latter case, electrons arestrongly localized and spins are located at the magneticions. Optical excitations in such a system rapidly relaxvia phonon cascades without affecting the magnetiza-tion, leading to an increase in the lattice temperature ona subpicosecond time scale. Only after this can the spinsystem be heated via phonon-magnon interaction. Thecollision integral between these quasiparticles is rela-tively small and is predominantly related to the spin-orbit coupling in the magnetic ions, in the limited spec-tral range near the center of the Brillouin zone. It is thusno wonder that the typical times are of the order of ananosecond.

To illustrate this, we consider here the laser-induceddemagnetization of iron borate FeBO3 Kimel et al.,2002. The detailed magnetic structure of this material isnot important and will be considered below; see Sec.VI.D.1. In short, FeBO3 is a weak ferromagnet, i.e., theantiferromagnetically coupled spins are slightly tilted,leading to a net magnetization M that can be probed viathe Faraday effects in a pump-probe experiment.

The intrinsic Faraday effect in FeBO3 is shown inFig. 14 as a function of temperature. Since the Faradayrotation is proportional to the order parameter, its tem-perature dependence is generally given by Landau andLifshitz, 1984

FT = 01 − T/TN, 23

where TN is the Néel temperature, is the critical expo-nent, and T is the temperature of the spin system, whichdrives the order parameter. A fitting of Eq. 23 to thecorresponding measurements in Fig. 14 results in =0.364±0.008 and TN=347.0±0.1 K. These values are in

1.0

0.5

−∆θ K

/θK

121086420Fluence (mJ/cm2)

2 ps20 ps80 ps

(b)

∆θK/ θK(1

per

div.)


0.33 mJ/cm2

0.84 mJ/cm2

1.67 mJ/cm2

3.34 mJ/cm2

6.68 mJ/cm2

10.0 mJ/cm2

11.7 mJ/cm2

13.4 mJ/cm2

(a) 170 mT, 20 K

FIG. 13. Normalized photoinduced magneto-optical Kerr ro-tation in In0.87Mn0.13As/GaSb heterostructure with Curie tem-perature 60 K a vs time for different pump fluences and bvs pump fluence for different time delays. At high fluences, thesignal saturates to 1, suggesting a complete quenching of fer-romagnetic order. From Wang et al., 2005.

F(0

ps)

,

F(-

20

ps)

(mra

d)

F(-

20

ps)

-(m

rad

)

F(5

00

ps)

Bias temperature (K)

2.4

2.0

1.6

1.2

0.8

0.4

0

12

10

8

6

4

2

0

335 340 345 350330

T

(t),

KS

1.50

0.75

0.00

0 1 2Time (ns)

F(0 ps)

F(-20 ps)

FIG. 14. The Faraday rotation at negative solid circles andzero open circles time delay as a function of the bias tem-perature with the fit to Eq. 23 solid and dotted lines, respec-tively. The difference between the intrinsic magneto-opticalsignal and that at 500 ps is shown by diamonds, together withthe calculation based on the fitted parameters dashed line.The inset shows the transient component of the magnon tem-perature as a function of the time delay. The solid line is the fitaccording to Eq. 24. From Kimel et al., 2002.



good agreement with =0.354 and TN=348.35 K as re-ported before Wijn, 1994.

Figure 14 also shows the difference between the in-trinsic magneto-optical signal and that at a time delay of500 ps. This difference increases drastically before drop-ping to zero at the Néel point. This implies that thepump-induced relaxation of the magneto-optical signalis related to an increase in the magnon temperature. Ata temperature of T=346.5 K and a delay time of 500 ps,the Néel point is reached and magnetic order is de-stroyed.

Thus the magnetization dynamics is not directly af-fected by the optical excitation itself but instead is de-layed considerably. In order to derive information aboutthe dynamics from the measured transient Faraday rota-tion, the latter is converted into transient magnon tem-peratures Tst by means of Eq. 23 for all data belowTN. Decomposing these temperatures into a static tem-perature T and an optically induced transient compo-nent Tst, we found all Tst to be identical withinthe experimental error. Their average is shown in theinset of Fig. 14 and is characterized by a monotonic in-crease that was fitted by

Tst = Ts01 − exp− t/sl , 24

where Ts0 is the amplitude of the dynamical spin tem-

perature and sl is the phonon-magnon interaction time.All variables were set as fitting parameters, and the re-sult of the fit for Ts

0=1.4 K and sl=700 ps is shown inthe inset by a solid line.

Using the deduced parameters Ts0, sl, and 0, the dif-

ference between the intrinsic Faraday rotation and thatat a time delay of 500 ps as a function of temperaturedashed line in Fig. 14 is directly derived. The excellentagreement with the experimental data supports the va-lidity of such description.

Microscopically these results can be understood as fol-lows. Because of the 61

+→ 44+ excitation, the electron

potential energy increases by only 1.4 eV, while the ex-cess of the photon energy is transferred either to thelattice or to the magnetic system. Generally, magnon-assisted transitions are less intense than phonon-assistedones Tanabe and Aoyagi, 1982. Consequently, after theoptical excitation the temperature of the phonons ishigher than that of the magnons: Tl Ts. This differencegradually vanishes and the magnon temperature in-creases with a time constant sl=700 ps determined bythe phonon-magnon interaction, which is predominantlyrelated to the spin-orbit coupling in the magnetic ionsKittel, 1958; Akhiezer et al., 1968.

A comparative study of different types of magneticmaterials in the work of Ogasawara et al. 2005 showedthat the relaxation time constant sl is strongly materialdependent and scales with the magnetocrystalline aniso-tropy, indicating that the spin-orbit coupling is a domi-nant interaction for this process. It can also be under-stood in terms of the magnetic spin-wave frequencyscaling with the anisotropy constant and thus accelerat-ing all relevant processes.

Note that, although iron borate is an antiferromagnet,the presence of the weak ferromagnetic moment greatlyfacilitates the observation of the demagnetization dy-namics. Similar experiments in pure antiferromagnetsCr2O3 Satoh et al., 2007 are unavoidably connectedwith interpretation difficulties.

D. Demagnetization of magnetic half metals

A half metal is characterized by an electronic bandstructure that shows metallic behavior for one spin com-ponent, whereas it is a semiconductor having a bandgap for the other. Experiments on laser-induced demag-netization of half-metallic materials therefore representanother intriguing opportunity to reveal a connectionbetween electronic band structure and the dynamics oflaser-induced demagnetization.

Laser-induced demagnetization was studied in thehalf-metallic ferromagnet Sr2FeMoO6 Kise et al., 2000.An extremely slow relaxation of the spins on a timescale of about 500 ps was observed. It was claimed thatthis slow dynamics was due to the half-metallic nature ofthe material, where the spins are thermally insulatedfrom the electron and lattice systems. In another halfmetal, CrO2, a similarly slow laser-induced demagnetiza-tion was observed and likewise explained by a vanish-ingly small interaction between electrons and spins typi-cal for a half-metallic material Zhang et al., 2006.

For ferromagnetic Ga,MnAs, the observed slow spindynamics upon laser-induced demagnetization was alsoexplained as a consequence of the half-metallic bandstructure of this material Kojima et al., 2003. After arapid nonradiative relaxation, the subsequent carrierscattering occurs mainly near the Fermi surface. There-fore the demagnetization rate of carriers strongly de-pends on their spin polarization near the Fermi level. Inparticular, if the density of states for one spin direction ismuch smaller than that of the other direction, spin flipsare inefficient, and spins and charges become thermallyisolated, thus excluding the possibility of ultrafast laser-induced demagnetization. Later ultrafast demagnetiza-tion was demonstrated in both InMnAs Fig. 13 andGaMnAs Fig. 12, which seriously challenged this de-scription in terms of half metallicity. The correlation be-tween demagnetization time and spin polarization at theFermi level has recently been studied systematicallyMüller et al., 2009.

It should be mentioned, however, that the absence ofultrafast laser-induced demagnetization can have a mucheasier explanation. It has been noted in Sec. IV.C thatinsulating materials do not show ultrafast laser-induceddemagnetization either. Because of the fundamental dif-ferences in the magnetic and transport properties ofmetals and insulators, the effect of a femtosecond pumppulse on these two types of magnetic material is differ-ent. In insulators, the effect of laser excitation is latticeheating, while in materials with wider energy bands, la-ser excitation leads to an increase in the kinetic energyof the itinerant electrons. As shown in Sec. IV.A, anincrease in electron temperature can in principle lead to



a rapid increase in spin temperature and consequentlyultrafast demagnetization, while owing to the phonon-magnon bottleneck, a heating of the lattice is followedby a much slower demagnetization at the subnanosec-ond time scale.

E. Laser-induced coherent magnetic precession

Ultrafast laser-induced changes of the magnetizationcan be exploited to provide an easy experimentalmethod to study magnetic precession. For this purpose,an external field is applied in the near-hard-axis direc-tions, in order to create a noncollinear geometry of theanisotropy, applied, and possibly also demagnetizingshape anisotropy, Hd=−4M fields see Fig. 15a. Arapid laser-induced change in the temperature and themagnetization value results in a change in the magneto-crystalline and/or shape anisotropies, and consequentlyin a change in the total effective field direction Fig.15b. If this change is faster than the correspondingprecession period, the magnetization vector will not beable to follow the net field and will start to precessaround its new equilibrium Fig. 15c. Thus a noncol-linear initial configuration of the various components ofthe net magnetic field is important so that the change inthe value of the one component modifies the direction ofthe sum.

Time-domain measurements on the excited precessiongive quantitative information on the anisotropy, switch-ing, and damping phenomena Ju, Nurmikko, et al.,1998; van Kampen et al., 2002; Zhang, Nurmikko, et al.,2002; Vomir et al., 2005; Kimel et al., 2006; Rzhevsky etal., 2007; Langner et al., 2009. This technique can beused for a local probe of the magnetization dynamics insmall structures. In addition, because of the nonuniformcharacter of the excitation, standing spin waves can beexcited, allowing for the investigation of spin-wave dis-persion. Moreover, this technique is also well suited tostudy lateral standing waves, such as occur in nanostruc-tures Kruglyak, Barman, et al., 2005; Keatley et al.,2008.

1. Precession in exchange-biased bilayers

As a specific test system, we consider the work of Ju etal., where an exchange-coupled NiFe/NiO ferromag-netic FM/antiferromagnetic AF bilayer was studiedJu, Nurmikko, et al., 1998; Ju et al., 1999, 2000. Thisbilayer is characterized by a distinct unidirectional mag-netic anisotropy Meiklejohn and Bean, 1956. The mag-netic characteristics of such coupled FM/AF systemsshow both an effective exchange bias field Hex shiftedhysteresis loop and an increased coercivity Hc. Theidea is to create spin excitations by laser excitation atthe FM/AF interface by abruptly reducing the exchangecoupling. When the magnetization of the FM layer isinitially biased antiparallel to the external applied fieldH, the optically induced “unpinning” of the exchangewould result in an ultrafast switching of the internal fieldHex and provide the driving force for the subsequentcoherent magnetization dynamics. The low blockingtemperature Tb220 °C of the NiO/NiFe system isconvenient for breaking the exchange bias thermally.

Figure 16a shows “snapshots” of the easy-axis tran-sient Kerr hysteresis loops acquired from a100 Å/400 Å NiFe/NiO bilayer sample at probe delaysof t1 and 200 ps, following the pulsed laser excitation.The results are typical of several different samples stud-ied and should be compared with the static unper-turbed Kerr loop for the sample dashed lines in the top

Hext

mH

a 4 m

Hext

m* H* ma 4 *

Hext

m Ha 4 m

(a) (b) (c)

FIG. 15. Color online All-optical excitation of coherent pre-cession. a t0, the magnetization points in the equilibriumdirection, given by the sum of the anisotropy, demagnetizing,and external fields; b 0t1 ps, the magnitude of Mand/or the anisotropy change due to heating, thereby alteringthe equilibrium orientation; c the magnetization precessesaround the new equilibrium direction that is slowly restoreddue to the heat diffusing away.

FIG. 16. Ultrafast laser-induced dynamics in NiFe/NiOexchange-biased bilayer: a Easy-axis transient Kerr loops forthe photoexcited sample at t=1 and 200 ps following pulsephotoexcitation at t=0. The open circles inserted in the bottomtrace show expected behavior in the absence of coherent mag-netization rotation. b Illustration of the trajectory of the mag-netization vector in the typical LLG calculation. The initialdirection of M is directed along the z axis. M is subject to adamped rotation with a large y component and a finite out-of-plane x component indicated schematically. The transientmodulation of the z component, measured by experiment, isprojected out explicitly in c. From Ju et al., 1999, 2000.



panel. Within about 1 ps the transient Kerr loops showa significant reduction in the Hex. Some “softening” ofthe loop is seen, demonstrating direct electronic accessto both the exchange coupling and the spins in the NiFelayer. On the other hand, the coercivity remains nearlyunaffected.

With increasing time delay, the photoexcited, coupledFM/AF bilayer system relaxes via electron, spin, and lat-tice interactions. Over several tens of picoseconds, thetransient Kerr loop, while reduced in amplitude due tothese relaxation processes, begins to display a pro-nounced change in its shape. An asymmetric distortionsets in, the effect being concentrated in the lower rightcorner of the loop, which becomes quite dramatic at t200 ps. Note that the detected signal is a measure ofthe pump-induced modulation of the magnetization andcontains a contribution that is proportional to thechanges in magnetization component in the plane of in-cidence, Mz. The large “negative Kerr loop anomaly”implies that a coherent magnetization rotation is trig-gered upon photoexcitation of the bilayer. When cali-brated against the static Kerr rotation, this magnetiza-tion modulation induced by each pump laser pulse inFig. 16a, corresponds to Mz /Ms0.4 under these ex-perimental conditions, or an average magnetization ro-tation of about 53°.

Figure 16b shows a pictorial view of the typical LLGsimulation of the dynamical magnetization rotation pro-cess. The initial direction of the magnetization vector Mis taken along the z axis in the plane of the thin film.Following the photoinduced change in the exchange biasfield, M is subject to a damped rotation with a large ycomponent and a finite out-of-plane x component indi-cated schematically in Fig. 16b. The transient modula-tion of the z component is projected out explicitly in Fig.16c.

2. Precession in nanostructures

Another example of light-induced magnetic preces-sion concerns magnetic nanodots, where pump-probemeasurements represent a real-time alternative to thespectroscopic tool such as Brillouin light scatteringDemokritov and Hillebrands, 2002.

The ground-state magnetization of a continuous thinfilm is virtually uniform. Consequently, the precessionfrequency is fully determined by the magnetic param-eters of a sample and the value of the applied magneticfield Gurevich and Melkov, 1996. In contrast, the inter-nal magnetic field in finite-sized nonellipsoidal magneticelements such as used in practice is nonuniform, whichintroduces additional complexity into the character ofthe observed magnetization dynamics. For example, thenonuniform demagnetizing field may lead to a spatialconfinement and quantization of spin-wave modes onthe nanometer length scale Demokritov et al., 2001.

Here we illustrate this on an example of magnetic pre-cession measured in a time-resolved Kerr-effect schemefrom arrays of square Permalloy elements of differentsizes in the nanometer range.

Fast Fourier transform FFT spectra calculated fromthe measured and simulated time-resolved signals areshown in Fig. 17 for two values of the bias field a 1 andb 150 Oe. At the bias field value of 1 kOe, the depen-dence of the experimental FFT spectra upon the size ofthe elements closely resembles that observed by Atxitiaet al. 2009. As the element size is reduced from637 to 236 nm, the frequency of the dominant mode ini-tially increases. At an element size of 124 nm, at leasttwo modes are clearly seen in both the experimental andsimulated data. The “higher-frequency mode” continuesthe trend of increasing frequency as the element size isreduced; however, in the 124 nm element, its spectralpower is reduced. The “lower-frequency mode” is sepa-rated from the higher-frequency mode by about 5 GHzin the experimental spectra and has slightly higher spec-tral power. The frequency of the lower-frequency modethen increases as the element size is reduced further.The trend is reproduced by the simulated spectra. Thespectra show a crossover in mode intensity for an ele-ment size of about 124 nm.

Comparison of the measured spectra with simulationsenables derivation of the spatial character of the ob-served precessional modes, as shown in the insets of Fig.17. Modes can be classified according to their spatialcharacter as center, edge, or detached edge modes. In

FIG. 17. Color online The dependence of the mode fre-quency upon the element size is shown for bias field values ofa 1 kOe and b 150 Oe. The shaded spectra were obtainedexperimentally while simulated spectra are shown with a solidcurve. For each element size, the spatial distribution of theFFT magnitude of the modes in the simulated spectra is shownin the insets. From Keatley et al., 2008.



addition, because of the small 200 nm separation,interaction between the elements affects the dynamicalbehavior as well. A similar experimental and micromag-netic study of ultrafast laser-induced precession fornanostructured Ni films has been performed by Mülleret al. 2008. Magnetic and elastic dynamics in periodicarrays of Permalloy nanodots induced by an ultrashortlaser pulse has been investigated by Comin et al. 2006.

When the size of magnetic nanostructures is so smallthat their anisotropy energy KV becomes comparable tothe thermal energy kT, their magnetization will fluctuatebetween the two unidirectional states Akharoni, 2000.This so-called superparamagnetic behavior results in azero magnetization when it is averaged over a time scalelarger than the typical time of these fluctuations. Thisfluctuation time generally varies over a very broadtime scale, depending on the size of the particles. Al-though superparamagnetism was discussed by Néel in1949 Néel, 1949, it has recently become a real criticalissue for magnetic storage where the bit sizes are becom-ing smaller and smaller.

Using 4 nm cobalt particles embedded in an Al2O3matrix, Andrade et al. succeeded in visualizing the“Brownian” magnetization trajectories of superpara-magnetic particles for the first time Andrade et al.,2006. Obviously, for the investigation of such a motionthe temporal resolution of the experiment has to bemuch better than the characteristic times investigatedtypically a few picoseconds. Time-resolved magneto-optical measurements with subpicosecond temporal res-olution is an elegant way to solve this problem. The par-ticles were magnetized by an external field and thenheated by a femtosecond laser pulse. The subsequentdynamics of the magnetization shown in Fig. 18 displaysan oscillatory behavior characteristic of a stronglydamped motion of precession. It is clearly present in theprojection of the magnetization trajectory displayed inthe longitudinal-polar plane up to 1 ns see Fig. 18a.Figure 18b shows the corresponding time-dependentdifferential polar and longitudinal components of themagnetization displayed only up to 125 ps. The overallmagnetization dynamics results from the ultrafast riseand subsequent decrease in the electron-spin tempera-ture induced by the laser pump pulse, two processes

which occur at time scales much faster than the motionof precession. In other words, the ultrafast change intemperature acts as a function excitation of the mag-netization which induces a change in both its modulusand orientation. The reorientation of the magnetizationvector is due to a dynamical change in the effective fieldrelated to the time-dependent anisotropy and exchangeinteractions, as shown recently in the case of thin cobaltfilms Bigot et al., 2005. The precession frequency variesfrom 14 to 25 GHz when H varies from 2.1 to 3.3 kOe.It is seen that the maximum of the demagnetization at400 fs coincides with the thermalization of the electrons.Next, a partial remagnetization occurs within 1 ps whenthe electrons cool down to the lattice via the electron-phonon interaction. Simultaneously, the orientation ofthe magnetization changes and starts precessing.

The damping was found to be much larger in thenanoparticles than in the bulk and, in addition, it consis-tently increased when the particle size was decreased.The exact mechanisms of this enhanced damping ob-served in nanoparticles are still unknown but these re-sults suggest that the metal-dielectric interface plays animportant role to damp the precessional motion. It con-firms previous studies of magnetization damping in-Fe2O3 or cobalt nanoparticles. Using ac susceptibilityand Mössbauer spectroscopy measurements, Dormannet al. reported a value of close to unity for -Fe2O3particles Dormann et al., 1996. Respaud et al. studiedthe ferromagnetic resonance of cobalt particles and re-ported values of of 0.3 and 0.55 for particles contain-ing, 310 and 150 cobalt atoms, respectively Respaudet al., 1999. They attributed these large damping valuesto surface spin disorder.

3. Precession in (III,Mn)As ferromagnetic semiconductors

The recently fabricated novel ferromagnetic semicon-ducting alloys III,MnAs are materials the physicalproperties of which are poorly understood Munekata etal., 1989; Ohno et al., 1992. During the last ten years,these materials have been intensively studied using vari-ous experimental techniques and methods. Several havereported time-resolved studies of laser-induced magneticphenomena in Ga,MnAs semiconductors, demonstrat-ing that excitation of these semiconductors with a subpi-cosecond laser pulse may trigger precession of the Mnspins at the frequency of the ferromagnetic resonanceWang, Ren, et al., 2007; Hashimoto and Munekata,2008; Hashimoto et al., 2008; Rozkotová et al., 2008a,2008b; Kobayashi et al., 2009; Qi et al., 2009; Zhu et al.,2009. On the one hand, these observations have raisedquestions about the mechanism of such an all-opticalexcitation of the spin precession. On the other hand, thepossibility of all-optical excitation of ferromagnetic reso-nance creates an intriguing opportunity to obtain extrainformation about the magnetic properties of the semi-conducting alloys and their heterostructures and nano-structures.

It has been concluded by several research groups thatthe mechanism responsible for all-optical excitation of

0.0

-0.5

-1.0

-1.5

-0.2

-0.1

0.0

Pol/Pol

Long/Long

P

ol/P

ol( x

10

)

Long/Long (x10)1005000-1

a) b)t = 0

t = 1 ns

t = 400 fs

Time (ps)-2

FIG. 18. Magnetization dynamics of 4 nm superparamagneticcobalt particles in Al2O3. a Magnetization trajectory in thepolar and longitudinal plane. b Dynamical polar and longitu-dinal Kerr signals Andrade et al., 2006.



ferromagnetic resonance in Ga,MnAs semiconductorsprimarily relies on the laser-induced heating of the ma-terials Wang, Ren, et al., 2007; Rozkotová et al., 2008a;Qi et al., 2009. Because of absorption, an excitation ofthe semiconductor results in an effective energy transferfrom light to the lattice. Because of the temperature de-pendence of the magnetic anisotropy of the materials,the heating changes the balance between different con-tributions of the magnetic anisotropy or affects the in-terplay between the magnetic anisotropy energy and themagnetostatic energy. Consequently, laser-induced heat-ing changes the equilibrium orientation of the magneti-zation and if this change occurs much faster than theperiod of ferromagnetic resonance, it triggers the pre-cession of the Mn spins.

It was observed that the polarization of the pumppulse hardly influenced the amplitude and phase of thespin oscillations Wang, Ren, et al., 2007; Hashimoto etal., 2008; Qi et al., 2009. Moreover, the amplitude andfrequency of the precession were found to be functionsof the sample temperature Rozkotová et al., 2008a;Kobayashi et al., 2009; Qi et al., 2009; Zhu et al., 2009.Both these observations are in agreement with the hy-pothesis that spin precession is excited via laser-inducedheating. It is interesting to analyze the Gilbert dampingof laser-induced spin precession in the annealed and as-grown samples Qi et al., 2009. The Gilbert dampingwas seen to hardly depend on sample temperature orpump intensity in annealed samples. However, in the as-grown samples the damping increased with an increasein the pump intensity. This observation indicated a con-nection between the damping and Mn-induced magneticdefects in GaMnAs. However, a theoretical analysis ofthe problem has not been performed yet.

A systematic analysis of the transient magneto-opticalsignals from GaMnAs ferromagnetic alloys was per-formed by Qi et al. 2009. It was shown that, if the pho-ton energy of the pump pulse is below the band gap ofGaMnAs Eg=1.53 eV at 10 K, the photoinducedmagneto-optical signals can be entirely attributed to fer-romagnetically coupled spins. However, an above-band-gap excitation results in both Mn spin precession andpumping of electrons into the conduction band. The lat-ter result in an additional contribution to the photoin-duced magneto-optical Kerr effect, which, however, isweakly sensitive to temperature and insensitive to anexternal magnetic field. It was concluded that the elec-tron spins are decoupled from the Mn spins Qi et al.,2009.

A comprehensive analysis of the spin dynamics andmagnetic eigenmodes excited by femtosecond laserpulses in Ga,MnAs slabs was performed by Wang,Ren, et al. 2007 yielding accurate values of the ex-change, bulk, and surface anisotropy constants for thisnovel compound see Fig. 19.

4. Precession in ferrimagnetic materials

Laser-induced coherent spin precession in systemswhere both magnetic moment and angular momentum

can be continuously tuned, for example, by changingtemperature, is a unique playground in which to studythe role of the angular momentum in optical control ofmagnetism.

As one example of such a system, rare-earth–3dtransition-metal RE-TM ferrimagnetic compoundswere once widely used materials for magneto-opticalMO recording. Depending on their composition,RE-TM ferrimagnets can exhibit a magnetization com-pensation temperature TM where the magnetizationsof the RE and TM sublattices cancel each other and,similarly, an angular momentum compensation tempera-ture TA where the net angular momentum of the sublat-tices vanishes. The theory of ferrimagnetic resonanceWangsness, 1953, 1954 predicts a strong temperaturedependence of the dynamic behavior in such systems. Inparticular, the frequency of the homogeneous spin pre-cession as well as the Gilbert damping parameter areexpected to diverge at the temperature TA Kobayashi etal., 2005. Such combination of a high frequency andlarge damping of the spin precession would provide ameans for ultrafast and ringing-free magnetization re-versal via precessional motion Schumacher et al., 2003.

In a ferrimagnetic system, the Landau-Lifshitz-Gilbertequation should be written for each ith sublattice

0 200 400-5

0

5

10 12 14

0 z

FOURIERTRANSFORM

FREQUENCY (GHz)

δθ( µradians)

TIME DELAY (ps)

S0S2

0

S2

S0S2

S1

S0

FIG. 19. Color online Light-induced magnetic precession inGa,MnAs slabs. The top panel shows the transient magneto-optical Kerr signal measured in the Voigt geometry opencircles at H=0.17 T. The applied field is parallel to 100. Theblack curve is the linear prediction fit, which gives two modesof period of 99.7 and 83.9 ps. Their contributions to the fittedsignal are shown by the curves. The bottom panel shows theFourier transform of the fit and, in the inset, the calculated Mzcomponent for the three lowest eigenmodes. From Wang, Ren,et al., 2007.



i=RE,TM. These equations are coupled by the pres-ence of the exchange field HRE,TM

ex =−exMTM,RE andgive rise to two modes. Here ex is a parameter thatcharacterizes the strength of the exchange field. The fer-romagnetic mode with FMR=effHeff can be describedby a single LLG equation but now employing an effec-tive gyromagnetic ratio eff Kittel, 1949; Wangsness,1953,

effT =MRET − MTMT

MRET/RE − MTMT/TM=

MTAT

,

25

and an effective Gilbert damping parameter eff Man-suripur, 1995,

effT =RE/RE2 + TM/TM2

MRET/RE − MTMT/TM=

A0

AT,

26

where MT and AT are the net magnetic moment andnet angular momentum, respectively; TM and RE arethe Landau-Lifshitz damping parameters for the rare-earth and transition-metal sublattices, respectively; A0 isa constant under the assumption that the Landau-Lifshitz damping parameters are independent of tem-perature Kobayashi et al., 2005. The validity of thisassumption was confirmed over a wide temperature in-terval by FMR measurements in 3d-TM Bhagat andLubitz, 1974.

In addition to the ferromagnetic mode FMR, spins ina ferrimagnetic system may oscillate with the exchangeresonance frequency Kaplan and Kittel, 1953

ex = exTMMRE − REMTM

= exRETMAT . 27

Equations 25 and 26 indicate a divergence of both theprecession frequency and the Gilbert damping param-eter of the FMR mode at the temperature TA. More-over, from Eq. 25, one notices that at the temperatureTM the FMR frequency becomes zero. In contrast, Eq.27 indicates that the exchange resonance branch soft-ens at the angular momentum compensation tempera-ture TA Wangsness, 1955, where the FMR mode di-verges.

Laser-induced spin precession in the ferrimagneticGdFeCo alloy was investigated using a pump-and-probemethod see Secs. II.A and II.B Stanciu et al., 2006.The sample used in these experiments was a 20-nm-thickGdFeCo layer, a ferrimagnetic amorphous alloy withperpendicular magnetic anisotropy. Strong coupling be-tween the RE and TM subsystems leads to a commonCurie temperature TC which for the given alloy com-position Gd22Fe74.6Co3.4 is about 500 K.

Figure 20 shows the temperature dependence of athe magnetization precession frequency and b the Gil-bert damping parameter. At T=220 K a significant in-crease is observed in both the precession frequency andthe damping parameter. As expected from Eqs. 25 and

26, the fact that both eff and FMR peak at the sametemperature indicates the existence of angular momen-tum compensation near this temperature of 220 K. Thestrong temperature dependence of eff demonstrates thenonequivalent character of the gyromagnetic ratios ofthe two magnetic sublattices in GdFeCo. This inequiva-lence also leads to the difference between the point ofmagnetization compensation TM and angular momen-tum compensation TA. In addition to the peak near TA,an enhancement of eff has also been observed as thetemperature is increased toward TC. Again Eq. 26 pre-dicts this enhancement under the assumption oftemperature-independent Landau-Lifshitz damping pa-rameters RE and TM. This enhancement is consistentwith earlier data Li and Baberschke, 1992; Silva et al.,2004. These measurements demonstrate the consistencyof the theoretical prediction of Eq. 26 with the tem-perature dependence of eff in RE-TM alloys such asGdFeCo. The observed strong increase in the precessionfrequency and the Gilbert damping when the tempera-ture approaches TA is ideal for ultrafast ringing-free pre-cessional switching in magnetic and magneto-optical re-cording.

In the temperature region just above TA, Fourieranalysis of the measured time dependencies reveals twofrequencies, one decreasing and the other increasingwith temperature. While the former can be attributed tothe FMR mode, the temperature behavior allows clearidentification of the latter as the exchange mode see

ω

Hext= 0.29 T

hω= 1.54 eV

Heff

MTM

dMTM/dt

dMRE/dt

MRE

ωex

FMR

αeff

T TAM

(a)

(b)

=

0 100 200 300 400

0.08

0.12

0.16

0.20

Gilbertdampingparameter

Temperature (K)

0

20

Frequency(GHz)

θ 600 Tc= 500K

TA=220K

TM=160K

ext

FIG. 20. Color online Temperature dependence of a themagnetization precession frequencies FMR and ex. As tem-perature decreases from 310 K toward TA, the exchange reso-nance mode ex open circles softens and mix with the ordi-nary FMR resonance FMR closed circles. The solid lines area qualitative representation of the expected trend of the tworesonance branches as indicated by Eqs. 25 and 27. b Tem-perature dependence of the Gilbert damping parameter eff.The lines are guides to the eye. From Stanciu et al., 2006.



Eq. 27, the frequency of which can be low around TAbut is usually very high everywhere else.

F. Laser-induced phase transitions between two magneticstates

Ultrafast laser-induced heating can also be used totrigger magnetic phase transitions on a subpicosecondtime scale. Soon after the discovery of ultrafast demag-netization in Ni, Agranat et al. investigated the ultrafastdynamics of first- and second-order magnetic phase tran-sitions in a TbFeCo film Agranat et al., 1998. At roomtemperature the studied compound was in an amor-phous ferrimagnetic phase with the easy axis of mag-netic anisotropy perpendicular to the film. However,heating led to a drastic change in the magnetic andstructural properties of the film. At a temperature TC1420 K, the material undergoes a second-order phasetransition into an amorphous paramagnetic state. AtTac570 K a first-order and a second-order phase tran-sition occur, causing both a crystallization and the for-mation of a magnetic state with the magnetization in theplane of the sample. This TbFeCo film was excited by anintense picosecond laser pulse. It was found that a laser-induced heating up to temperatures TC1TTac re-sulted in a demagnetization within 5–10 ps. A similardemagnetization was observed if the laser brought thematerial up to T TC2, where TC2720 K is the Curietemperature of the crystalline phase. Surprisingly, if thelaser intensity was tuned such that the film was heatedup to TacTTC2, the crystallization and spin reorien-tation occurred within 1 ps directly, i.e., without the for-mation of the amorphous paramagnetic phase. Althoughit is clear that demagnetization of TbFeCo alloys maytake place at a much shorter time scale Kim et al.,2009, these experiments raise fundamental questionsabout the mechanisms and ultimate speed of phase tran-sitions between two magnetic states.

1. Spin reorientation in TmFeO3

In antiferromagnetic TmFeO3, as the temperature islowered a spontaneous spin reorientation occurs as aresult of a strong temperature dependence of the mag-netic anisotropy Belov et al., 1969; White, 1969; Wijn,1994. In this process, the antiferromagnetic vector lturns continuously from its position along the x axis at atemperature T2 to a position along the z axis at a tem-perature T1.

The temperature-dependent anisotropy energy inTmFeO3 has the form Horner and Varma, 1968; Sha-piro et al., 1974

FT = F0 + K2Tsin2 + K4 sin4 , 28

where is the angle in the x-z plane between the x axisand the antiferromagnetic AM moment l and K2 andK4 are the anisotropy constants of second and fourthorders, respectively. Application of equilibrium condi-tions to Eq. 28 yields three temperature regions corre-sponding to different spin orientations:

4GxFz: = 0, T T2,

2GzFx: =12, T T1, 29

24: sin2 = −K2T2K4

, T1 T T2,

where T1 and T2 are determined by the conditionsK2T1=−2K4 and K2T2=0. Here s indicate the rep-resentations of the respective symmetry groups Kosterand Statz, 1963. Therefore, depending on the anisot-ropy constants, a spin reorientation may be expectedthat shows two second-order phase transitions at T1 andT2. The temperature dependence of in the phase 24 isdetermined by K2T, which varies roughly linearly withtemperature Belov et al., 1969.

The laser-induced transition between the two spinconfigurations in this antiferromagnet can be monitoredwith the help of linear optical birefringence see Sec.III.B. The resulting changes in the birefringence aresummarized in Fig. 21 Kimel, Kirilyuk, et al., 2004. Inthe time domain, the relaxation process can be dividedin three distinct regions. First, the excitation of hot elec-trons decays via phonon cascades and the phonon sys-tem thermalizes in a very short time process 1 in Fig. 21,with a time constant of around 0.3 ps. The phonon-

1 2 3

10 K

15 K

25 K

35 K45 K

55 K

65 K

70 K

74 K

76 K

84 K

86 K

88 K90 K

105 K

125 K

78 K

80 K

82 K

160 K

0 1 2 3 4 5 6 7 8 10 20 30 40 50 60 70

Time delay (ps)

Photo

-induce

dbir

efri

ngen

ce

spin-reorientation spin-oscillations

FIG. 21. Excitation and relaxation of the AFM vector momentmeasured via changes in the magnetic birefringence. Here onecan distinguish three processes: 1 electron-phonon thermali-zation with 0.3 ps relaxation time, 2 rotation of the AFMvector with 5 ps response time, and 3 oscillations of the AFMvector around its equilibrium direction with an approximate10 ps period. From Kimel, Kirilyuk, et al., 2004.



phonon interaction sets a new lattice temperature sothat the equilibrium anisotropy axis is changed. Undersuch conditions, the magnetization vector in a FM ma-terial would precess around its new equilibrium direc-tion, approaching it due to damping Landau and Lif-shitz, 1935. In an antiferromagnet, the exchange-coupled spins start to precess in opposite directions, thuscreating a strong exchange torque Tex that opposes thisprecession. The resulting motion of the spins to the newequilibrium should then occur in the plane spanned bythe anisotropy field HA and spins S. This process ismarked 2 in the time dependencies of Fig. 21 and has acharacteristic time of about 4 ps. This relaxation timecorresponds to an AM resonance frequency of 80 GHz.The amplitude of this spin reorientation reaches 30°.

After the initial relaxation, the antiferromagnetic vec-tor oscillates around its new equilibrium process 3. Par-ticularly strong oscillations are observable in the rangeof 80–90 K, i.e., in the region of the reorientationaltransition. In fact, the derived frequencies closely re-semble these of the spin waves with k=0 Shapiro et al.,1974; White et al., 1982. Such spin waves are equivalentto a homogeneous precession of the magnetization ob-served at such conditions in ferromagnets van Kampenet al., 2002.

The experimental results show that the AM spins inTmFeO3 are reoriented by several tens of degreeswithin only a few picoseconds. For comparison, in a fer-romagnet with an anisotropy energy similar to that inTmFeO3 104 J /m3 Wijn, 1994 the magnetization pre-cesses with a period of several hundred picosecondsGerrits et al., 2002; Schumacher et al., 2003.

The dynamics of a similar laser-induced phase transi-tion has been investigated in Fe/Gd multilayers with thehelp of element-specific XMCD with temporal reso-lution of about 80 ps Eimüller et al., 2007. Surprisingly,the magnetic relaxation was found to be noticeablyslower than the cooling by heat diffusion. It was specu-lated that this slow relaxation can be attributed to theweak magnetic anisotropy.

2. Spin reorientation in (Ga,Mn)As

The ferromagnetic semiconductor Ga,MnAs has abiaxial magnetic anisotropy Jungwirth et al., 2006 andas a result possesses two equivalent easy axes. Thisproperty modifies the hysteresis loop such that a switch-ing of the magnetization between two different pairs ofstates can be observed. It has been demonstrated thatthe switching can be triggered by laser-induced heating.Owing to the temperature dependence of the fourfoldmagnetic anisotropy of this magnetic alloy, excitation ofthe material with a single subpicosecond light pulseleads to spin rotation over 90° in the plane of the sampleAstakhov et al., 2005. To monitor such a reorientationthe phenomenon of magnetic linear birefringence wasused see Sec. III.B for details. In particular, the angleof incidence was around 20° to the structure growth di-rection and the incoming polarization was set at 45° withrespect to the 100 axis. A magnetic field was applied in

the plane of the sample at an angle of 42° with respect tothe 100 axis. Amplified 100 fs pulses from a Ti:sapphirelaser at a repetition rate of 1 kHz at a wavelength805 nm were used for the ultrafast laser excitation of thesample. A mechanical shutter was used to select onesingle pulse. This pulse was focused on the sample to aspot size of 100 m. The pump fluence was up to150 mJ/cm2. Excitation of the sample with such a laserpulse led to the spin reorientation over 90°. In the ge-ometry of the experiment such an reorientation wasseen as a reversal of the magnetic hysteresis loop seeFig. 22. This observation demonstrates that ultrafast la-ser excitation is able to induce far more drastic changesthan just a switch in the orientation of the magnetiza-tion. In the present experiment, the hysteresis loop as awhole is completely reversed. It is also demonstratedthat by applying ultrashort optical pulses we can ma-nipulate the magnetization in a Ga,MnAs layer be-tween its four metastable states, which can be used forapplication in a magneto-optical memory.

3. AM-to-FM phase transition in FeRh

Recently two groups independently showed the possi-bility of generating ferromagnetic order at a subpicosec-ond time scale in FeRh thin films Ju et al., 2004; Thieleet al., 2004. FeRh is a particularly interesting systemthat displays an antiferromagnetic phase below and aferromagnetic phase above a phase transition tempera-ture around 370 K. Laser heating of such a film, drivingit through its AM-to-FM phase transition, showed a firstonset of ferromagnetism within the first picosecond,which implied that the phase transition was of electronicorigin. On a longer time scale a continued growth of theferromagnetic order was witnessed, possibly due to do-main expansion. Not only does FeRh provide a uniquemodel system for improving our theoretical understand-ing of subpicosecond magnetic processes, it also re-ceives considerable interest for its potential applicationin heat-assisted magnetic recording Thiele et al., 2003.

Time-resolved MOKE measurements were made onFe45Rh55 thin film samples thickness 100 nm, made

-30 -20 -10 0 10 20 30 40

-0.5

0.0

0.5

T=10K

direction of HAfter pulse

Before pulse

Magnetic field (mT)

Pola

riza

tion

rota

tion

(mra

d)

M

M

FIG. 22. Color online Minor hysteresis loops of theGa0.98Mn0.02As sample, recorded before open symbols andafter solid symbols excitation by single optical pulse with aduration of 100 fs, shown together with corresponding magne-tization orientations insets. From Astakhov et al., 2005.



with dc-magnetron sputtering on a MgO substrate. InFig. 23b the Kerr rotation during the first 700 ps afterpump excitation of a 100 nm FeRh film is presented forincreasing pump powers. A clear threshold behavior isobserved. A minimum fluence of 6 mJ/cm2 is found tobe required to pass the transition temperature, and onlyfor higher powers is a finite laser-induced Kerr rotationobserved. We stress that the observation of a thresholdpower is considered as an essential fingerprint of thelaser-induced phase transition. As a second signature ofdriving the film into the ferromagnetic phase, the dura-tion over which the induced Kerr signal lasts increases asa function of pump fluence up to nanoseconds at thehighest fluences, because it takes increasingly longerbefore the magnetic film is cooled down below the phasetransitions by diffusive cooling.

Time-resolved MOKE and transient reflectivity datawere compared with simulations that included magneti-zation growth and alignment processes Bergman et al.,2006. The initial rapid growth of magnetic momentwithin about 10 ps was followed by a gradual alignmentprocess where local magnetic moments are aligned rela-tive to each other in about 50 ps. The observedmagneto-optical transients demonstrated an interplayamong the local magnetic moment, nonlocal domaingrowth or alignment, and magnetization precessionwhich was launched by the varying demagnetizing fields.

G. Magnetization reversal

Laser-induced heating of a magnetic medium in thepresence of an external field is used in thermomagneticwriting to record a bit of information via magnetizationreversal. Revealing the ultimate speed at which such arecording event can be realized is a fundamentally inter-esting issue with possible consequences for the futuredevelopment of magnetic recording and informationprocessing. Fast magnetization reversal induced by fem-tosecond laser pulses was investigated in GdFeCO alloysHohlfeld et al., 2001. In the experiment an externalmagnetic field directed antiparallel to the magnetizationand a strength smaller than the coercive field was ap-plied. Femtosecond laser excitation of the sample led toa total demagnetization of the material and a slow re-covery of the magnetic order with the magnetization re-versed. Although the demagnetization occurred at thesubpicosecond time scale, the medium remained in thedemagnetized state for a few picoseconds Kazantseva,Nowak, et al., 2008 and then exhibited slow recovery ofthe magnetization, which was completed in about750 ps. Moreover, comparison of temperature-induceddynamics and of field-induced magnetization reversal todata obtained for the same high pump fluence in rema-nence demonstrated that the dynamics of remanentmagnetization cannot be interpreted by temperature dy-namics only, and the dynamics of magnetic domainsshould be taken into account. The important role playedby the magnetic domain wall dynamics in the process ofthe laser-induced magnetization reversal has recentlybeen demonstrated using time-resolved magneto-opticalimaging of laser-excited TbFeCo thin films Ogasawaraet al., 2009.

As shown above, in a ferrimagnet close to the com-pensation point both the frequency of the magnetizationprecession Stanciu et al., 2006 and the domain wall ve-locity Weng and Kryder, 1993; Randoshkin et al., 2003strongly increase due to the divergence of the gyromag-netic ratio see Sec. IV.E.4. Therefore one might expectthat in such a ferrimagnetic material application of amagnetic field would instantaneously flip the magnetiza-tion. In other words, at TA, magnetization can be re-garded as a mechanical system with no inertia which canbe moved by the slightest torque. However, to verify thisultrafast switching, instantaneous application of a mag-netic field is required which in real experiments is notfeasible. Instead, a dc magnetic field can be applied tothe ferrimagnet parallel to the original magnetization di-rection at a temperature TTM. When the temperatureof the ferrimagnet increases above both TM and TA, itsspin structure will reverse Aeschlimann et al., 1990,1991. Thus a femtosecond laser pulse heating of thesample might effectively act as an instantaneously ap-plied magnetic field, allowing the investigation of themagnetization reversal speed near TA.

The question arises: How fast is this switching pro-cess? To answer this question, hysteresis loops weremeasured at different time delays after the pump pulsesheated the sample Stanciu, Tsukamoto, et al., 2007.

0 100 200 300 400-0.10.00.10.20.30.40.50.60.70.80.91.0

17.7

11.1

7.16.24.4

∆θk

Time (ps)0 100 200 300 400

0

2

4

6

8

10

12

14

16

18

6.27.111.1

17.7

4.4

Time (ps)

∆

0 100 200 300 400-0.10.00.10.20.30.40.50.60.70.80.91.0

17.7

11.1

7.16.24.4

∆θk(arb.units)

Time (ps)0 100 200 300 400

0

2

4

6

8

10

12

14

16

18

6.27.111.1

17.7

4.4

Time (ps)

∆R(arb.units)

?

10-13 sec

?

10-13 sec

?

10-13 sec

?

10-13 sec(a)

(b) (c)

FIG. 23. Ultrafast laser-induced dynamics in RhFe: a Sche-matic of the ultrafast generation of ferromagnetic order byinducing an AM-FM transformation in FeRh when excitedwith femtosecond optical pulses. Time evolution of b thetransient Kerr effect Kt and c the transient reflectivityRt as a function of pump fluences labeled for each curve inmJ/cm2. The curves are vertically displaced for clarity. FromJu et al., 2004.



The results are shown in Fig. 24. One can observe that inthis field range the measured Faraday signal changessign after about 700 fs. The sign change reflects thechange in the FeCo sublattice direction toward the ap-plied magnetic field, as the spin temperature of the fer-rimagnetic system increases over TM in the probed area.This observation unambiguously demonstrates that themagnetization reversal takes place on a subpicosecondtime scale. Note that this reversal time is considerablyfaster than that found in GdFeCo at temperatures abovethe compensation points Hohlfeld et al., 2001. On theother hand, the growth of the reversed domain to its full100% is determined by the cooling rate and takes placeat a much longer time scale nanosecond.

What are the implications of the observed subpicosec-ond magnetization reversal? Although it is well knownthat ultrafast laser excitation of itinerant ferromagnetsas Co, Ni, or Fe leads to a demagnetization on the fem-tosecond time scale see Sec. IV.A, little is known abouthow fast the magnetic moments in metals as Gd can beexcited with photon energies in the visible range. This isbecause in Gd the optically excitable electrons of the5d-6s bands carry only 9% of the total moment, whilethe localized 4f electrons dominate the magnetic spinmoment. The localization of the magnetic moment to-gether with the weak spin-lattice coupling that can bededuced from the small value of the magnetocrystallineanisotropy, characteristic for Gd, suggests that the trans-fer of the photon energy to the localized states should bea slow process of the order of 100 ps Vaterlaus, Beutler,and Meier, 1991. In this context it was recently claimedthat laser excitation of a CoGd sample resulted in anindependent excitation of the Co sublattice only Binderet al., 2006. In contrast to this, the subpicosecond mag-netization reversal over the compensation points dem-onstrated here implies that both the Gd and FeCo sub-lattice magnetizations are considerably reduced on a

subpicosecond time scale. In particular, an increase inthe spin temperature above TM requires a partial de-magnetization of the FeCo sublattice of 25%. Conse-quently, the magnetization of the Gd sublattice, whichbelow TM is larger than that of FeCo, must be reducedby more than 25%. For Gd this represents a reduction inthe magnetic moment far larger than that contributed bythe itinerant 5d-6s electrons. A subpicosecond access tothe localized 4f spin moments is therefore required, thusrevealing the important role played by the Gd 4f elec-trons in this fast reversal process. Such excitation timescale is indeed allowed in Gd by the strong exchangecoupling between 5d-4f, responsible for its ferromag-netic order Bovensiepen, 2006. The strength of thisintra-atomic exchange is 100 meV, corresponding to10 fs. From this it follows that indeed the localized 4fspin magnetic moment in Gd can be optically excited ona time scale comparable with that observed for the itin-erant ferromagnets.

V. NONTHERMAL PHOTOMAGNETIC EFFECTS

The laser-induced magnetization dynamics discussedabove was caused by heating of electrons induced by alaser pulse that could be of any polarization. It is a de-pendence on the pump pulse polarization, however, thatis the fingerprint of a nonthermal effect. In the followingwe discuss various nonthermal laser-induced effects thatcan lead to changes in the magnetic anisotropy, induceprecessional dynamics, and even switching of the mag-netization. Although all these effects do depend on thepolarization of the exciting laser pulse, two differenttypes of nonthermal effect should be distinguished. Thefirst, so-called photomagnetic effects, do depend on theabsorption of photons, resulting in an effective excita-tion of the system. The second group of effects are saidto be optomagnetic as they do not require the absorptionof photons but instead are related to, but the inverse of,magneto-optical effects such as Faraday or Cotton-Mouton effects.

A. Photomagnetic modification of magnetocrystallineanisotropy

1. Laser-induced precession in magnetic garnet films

First we discuss nonthermal photomagnetic modifica-tion of the magnetocrystalline anisotropy, which wasdemonstrated in thin garnet films Hansteen et al., 2005,2006.

a. Experimental observations

Applying an external magnetic field Hext in the planeof a magnetic garnet sample so that M is in plane, =90°, see below, and pumping with linearly polarizedlaser pulses, optically induced precession of the magne-tization M was observed; see Fig. 25a. In the opticaltransmittivity of the sample, a sudden drop was seenwhich did not relax significantly within 3 ns. The ampli-tude and phase of the precession in Fig. 25a was found

-1 0 1 2 400 800 1200

-1

-0.5

0

0.5

1

1.5

-0.6

0

0.6

-0.4

0

0.4

-0.1

0

0.1

-0.1

0

0.1

-0.2 0 0.2-1

0

1

Field (T)

Delay time∆t (ps)

∆Mz/M

-0.2 0 0.2

-0.2 0 0.2

-0.2 0 0.2

∆M/M

Z

-0.2 0 0.2

∆t = - 1ps

0.7 ps

500 ps

0.5 ps 200 ps

∆M/M

Z

∆M/M

Z

∆M/M

Z

∆M/M

Z

Field (T)

Field (T)

Field (T)Field (T)

FIG. 24. Color online Transient magnetization reversal dy-namics measured for a pump fluence of 6.29 mJ/cm2. Insetsshow hysteresis loops measured at distinct pump-probe delays.The loops demonstrate the magnetization reversal after about700 fs. From Stanciu, Tsukamoto, et al., 2007.



to depend on the plane of polarization of the pumppulses as shown in Fig. 25b. Negative values of theamplitude indicate precession of M with opposite phase.Maxima of the precessional amplitude were observedfor every 90° rotation of the polarization. From this de-pendence on pump polarization it is evident that theunderlying effect must be nonthermal: heating wouldonly reduce the magnitude of the magnetization and theanisotropy field, independent of the pump polarization.

It is also interesting to note that M always starts itsprecessional motion by moving normal to the film planealong the ±z direction. This follows from the initialphase of the measured signal in Fig. 25a, which alwaysstarts from the inflection point where Mz is changingmost rapidly. From the Landau-Lifshitz equation Eq.5, it can be inferred that immediately after the photo-excitation both M and Heff are in the film plane but notparallel to each other. Consequently, the observed mag-netization dynamics must be due to an ultrafast changein either the magnetization M or the anisotropy fieldHa or a combination of the two that effectively createsan in-plane angular displacement =M ,Heff be-tween M and Heff. These two possibilities can be distin-guished by analyzing the precession amplitude as

function of the applied field. If triggered by an ultrafastrotation of the magnetization M→M+M, the ampli-tude of the subsequent precession should be indepen-dent of the strength of the applied magnetic field asM ,Heff does not depend on Hext. However, if theprecession is caused by a photoinduced anisotropy fieldHa, the precession amplitude is expected to decreasewith increasing applied magnetic field as

= Heff,Heff + Ha 1

Hext + Ha, 30

which is valid for small-amplitude precessions, and forHaHeff=Hext+Ha. As shown in Fig. 26 solid line,the measurements exhibit exactly this behavior. Agraphical illustration of the excitation process and thesubsequent precession is shown in Fig. 27.

There appeared to be a linear relation between theprecession amplitude and the pump power up to pulseenergies of almost 10 J, whereafter the effect saturatescompletely. Based on the absorption coefficient the esti-mated density of absorbed photons is about one perhundred unit cells in the illuminated crystal volume.Saturation effects are therefore not expected unless theyare caused by the presence of a low concentration ofimpurities. This will be discussed in more detail below,where the microscopic basis of the photomagnetic effectis considered.

b. Phenomenological model

In this section we give a macroscopic phenomenologi-cal description of the observed photoinduced magneticanisotropy. The model is not concerned with the micro-scopic mechanism of the effect, but gives some insightinto its symmetry properties.1

The creation of a static magnetic field Ha0 in thegarnet sample can be described as a combination of thenonlinear process of optical rectification Shen, 1984and a linear magnetoelectric effect O’Dell, 1970:

1Strictly speaking, such an approach is only applicable to ef-fects that do not involve absorption, i.e., when the result His only present simultaneously with the stimulus E. Withthis restriction in mind, it is applied here to describe the sym-metry properties of the effect. Time integration should actuallybe used on the right-hand side of Eq. 31 and below, to ac-count for this difference. We omit this for simplicity.

∆

0 2 40

1

External field (kOe)

Mz

/M(%

)

FIG. 26. Color online Dependence of the precessional am-plitude on the applied in-plane magnetic field Hext. Round andsquare symbols represent amplitudes extracted from measure-ments at ±Hext. From Hansteen et al., 2006.

t < 0 t = 0 t > 0

FIG. 27. Color online Graphical illustration of the process ofphotoinduced magnetic anisotropy caused by linearly polar-ized laser excitation and the subsequent precessional dynam-ics. From Hansteen et al., 2006.

∆

θ = 100

80

60

40

20

0 1 2 3

0

2

4

Time delay (ns)

Mz/M

(%)

(a)

∆

θ (deg.)0 180 360

-1

0

1

Mz/M

(%)

(b)o

o

o

o

o

FIG. 25. Color online Coherent precession of the magnetiza-tion triggered by linearly polarized laser pulses. a Time de-pendence of the precession for different planes of pump polar-ization , with an applied field of Hext=350 Oe in the plane ofthe sample. Circles represent measurements and solid linessimulations based on the Landau-Lifshitz equation. b Preces-sional amplitude as a function of the plane of pump polariza-tion. Round and square symbols represent amplitudes ex-tracted from measurements at ±Hext. The solid line is a best fit.From Hansteen et al., 2006.



Hia0 = ijklEjEkMl0 . 31

Here E is the electric field component of light and Mis the magnetization of the garnet film. The fourth-rankpolar tensor ijkl has nonzero components for crystals ofany symmetry Birss, 1966.

When the experimental geometry and the symmetryof ijkl for the 4 mm point group of these samples aretaken into account, only four independent nonzero com-ponents of the tensor ijkl remain,

A = xxxx = yyyy,

B = xyxy = xxyy = yxyx = yyxx,32

C = xyyx = yxxy,

D = zxxz = zyyz,

and the vector components of the photoinduced aniso-tropy field are then given by

Hxa E0

2Ms sin A + Ccos

+ A − Ccos 2 cos + 2B sin 2 sin , 33

Hya E0

2Ms sin A + Csin

− A − Ccos 2 sin + 2B sin 2 cos , 34

Hza E0

2MsD cos . 35

Here Hia is the photoinduced field along the i direction,

i= x ,y ,z refers to the crystal axes of the sample, de-notes the azimuthal angle between the sample x axis andthe projection of the magnetization vector on the filmplane, and is the angle between the film normal andthe magnetization.

The fact that there is no amplitude offset in the curveshown in Fig. 25b requires that A=−C so that the firstterm in Eqs. 33 and 34 vanishes. Furthermore, thesinusoidal shape of the curve implies that A=B andleaves us with only two independent components of thetensor ijkl,

A = xxxx = yyyy = − xyyx = − yxxy

= xyxy = xxyy = yxyx = yyxx,36

D = zxxz = zyyz.

These additional equalities indicate that the ijkl tensorhas a higher symmetry than the garnet crystal. The op-tically induced anisotropy field can now be written as

Hxa AE0

2Ms sin sin 2 sin + cos 2 cos , 37

Hya AE0

2Ms sin sin 2 cos − cos 2 sin , 38

Hza DE0

2Ms cos . 39

For the in-plane field geometry cos =0 this describesa vector of constant length and with a direction depend-ing on the angle of the magnetization with respect to

the x axis and the plane of polarization of the pumppulses.

Computer simulations based on this simple model andthe numerical integration of Landau-Lifshitz equationexhibit good agreement with our experimental results,for the in-plane Hext geometry shown in Fig. 25a,where the results of the simulation are shown by solidlines.

c. Microscopic mechanism

Photomagnetic effects were shown to exist in garnetscontaining certain dopants Teale and Temple, 1967; Dil-lon et al., 1969, in particular Si and Co Chizhik et al.,1998; Stupakiewicz et al., 2001. Optically induced elec-tron transfer between ions on nonequivalent sites in thecrystal is believed to cause a change in the magnetocrys-talline anisotropy due to a redistribution of ions Albenet al., 1972. This effect is strong in crystals doped withelements that can assume different valence states andwhere their contribution to the anisotropy is different.However, it has also been observed in undoped garnetsamples containing Pb impurities Veselago et al., 1994,which we believe is the case here.

The linear dependence of Ha for low power and thesaturation at high pump power suggest that linear opti-cal absorption by Pb impurities is the dominating ab-sorption process. Divalent Pb2+ ions substitute trivalentLu3+ ions on dodecahedral sites in the crystal and act aselectron acceptors. This is a p-type doping which createsholes that are assumed to be located on iron ions intetrahedral sites Metselaar et al., 1975; Paoletti, 1978.To maintain overall charge neutrality in the crystal,some tetrahedrally coordinated trivalent iron ionschange their valency to 4+. Photoexcitation can thus in-duce a charge transfer between these Fe4+ ions and Fe3+

magnetic ions on octahedral sites, thus effectively “mov-ing” the Fe4+ ions to sites with different symmetry seeFig. 28 and thereby causing a change in the magneticanisotropy.

The low concentration of Pb impurities creates a lim-ited number of photoactive ions and the photomagnetic

c

[a]

(d)4+Fe

e−Fe3+

2+Pb

FIG. 28. Color online Illustration of the photoexcitation ofelectrons between iron ions in different crystallographic sites.A laser pulse induces electron transfer from a Fe3+ ion in theoctahedral site denoted by a to a Fe4+ ion in the tetrahedralsite denoted by d. The dodecahedral site with the divalentlead impurity is denoted by c.



effect can therefore be expected to saturate under in-tense illumination. An estimation shows that the illumi-nated volume of garnet film contains about 1012 Pb ions.An optical pulse of 20 J delivers 1014 photons fromwhich about 1% is expected to be absorbed Paoletti,1978. This allows, in principle, for all of the photoactiveions to be excited and it is thus not surprising that satu-ration can occur at these pump intensities. The pump-induced change in transmittivity is also believed to berelated to the photoexcitation of impurities Eremenkoet al., 2000.

2. Laser-induced magnetic anisotropy in antiferromagneticNiO

The antiferromagnetic dielectric NiO has a high Néeltemperature, allowing it to be used in magnetic sensors.The magnetic anisotropy in this compound is deter-mined by dipolar and quadrupolar interactions betweenthe Ni2+ spins. It is therefore easily modified by the shiftof the 3d orbital wave functions accompanying the exci-tation of d-d transitions by an optical pump pulse. Thismay lead to a change in the easy direction of the single-

ion magnetic anisotropy from 112 to 111. Duong et al.probed the magnetic changes following the photoexcita-tion of NiO by optical second-harmonic generationSHG Duong et al., 2004; Satoh et al., 2006. The mea-surements revealed oscillations of the nonlinear opticalsignal with two frequencies of about 54 and 108 GHzsee Fig. 29. It is remarkable that, in contrast to theexperiments in magnetic garnets, these frequencies donot correspond to those of magnetic resonance but areclose to the magnetic anisotropy energy of 0.11 meV in-

stead. The observed oscillations of the second harmoniccould be tentatively explained by quantum beating,

which arises from the fact that the 112 ground stateand the 111 excited state interfere coherently. It wasclaimed, furthermore, that such a modification of theanisotropy is able to trigger an ultrafast 90° switching ofthe antiferromagnetic vector Fiebig et al., 2008.

Although Fiebig et al. supported their model by anumber of arguments, one cannot ignore the fact that itraises some questions as well. The most important is:Why does SHG show these quantum beats but does notreveal spin oscillations at the frequency of the antiferro-magnetic resonance? An instantaneous spin flip betweentwo metastable states should trigger spin oscillations atthe frequency of antiferromagnetic resonance and not atthe frequency separating these two states. Unfortu-nately, theoretical investigations of the femtosecond re-sponse of NiO have not been able to address this ques-tion so far Ney et al., 2002; Gómez-Abal et al., 2004;Lefkidis and Hübner, 2007; Lefkidis et al., 2009 and fur-ther studies of the nonlinear optical properties of thiscompound are required.

The experimental difficulty in this case consists in thepractical impossibility of distinguishing between the dy-namics of the magnetic order parameter and that of thenonmagnetic electronic contributions. Indeed, themagnetization-sensitive elements of the nonlinear opti-cal tensor represent a not very well-understood convo-lution of magnetic and structural components, separa-tion of which constitutes a major difficulty even in staticsKirilyuk et al., 1998. Thus the oscillations in Fig. 29might also be due to some other excitations, for ex-ample, acoustic phonons, the frequencies of which areclose in the present geometry Mazurenko and vanLoosdrecht, 2007. Also, no magnetic excitations such asspin waves were observed in these experiments Fiebiget al., 2008. Thus the study of ultrafast dynamics inpurely antiferromagnetic materials remains an interest-ing experimental challenge for the future.

3. Photomagnetic excitation of spin precession in (Ga,Mn)As

Another type of nonthermal laser-induced effect,based on the absorption of photons, can be observed inferromagnetic semiconductors, such as GaMnAs. Forinstance, Wang et al. reported on spin precession inGaMnAs induced by 100 fs laser pulses Wang, Ren, etal., 2007. These results were interpreted in terms of amodel based on the temperature dependence of themagnetic anisotropy. In particular, the temperature risethat follows the absorption of the laser pulse wasclaimed to be responsible for a rapid change in magneticanisotropy and consequently for the change in the equi-librium orientation of the magnetization. Such a changeon a time scale much faster than the period of magneticresonance in GaMnAs led to the excitation of a homo-geneous spin precession. The problem, however, is thatsuch a mechanism would require either a modulation ofthe magnetic shape anisotropy due to an ultrafast de-magnetization of GaMnAs, as usually observed in met-

0.0 0.1 0.2 0.30

1

2

-15

-10

-5

0

0 20 40 60 80 100

0 20 40 60 80 100

1.5

1.0

0.5

0.0

0.5

Delay (ps)

(b)

Chan

geof

intensity

(%)

Delay (ps)Frequency (meV/h)

Amplitu

de(arb.u

nits)

~SHG

THG ~ lattice

spins

(a)

THG

-

-

-

(c)

Chan

geof

intensity

(%)

FIG. 29. Temporal behavior of second-harmonic generationSHG and third-harmonic generation THG measured in re-flection from single crystal NiO 111 at 6 K. b Fourier trans-form of the SHG data after subtraction of the steplike de-crease at t=0. Dashed and straight lines: fitted spectralcontributions and envelope. c THG signal from a. FromDuong et al., 2004.



als van Kampen et al., 2002, or a modulation of themagnetocrystalline anisotropy via an increase in the lat-tice temperature; see Kimel, Kirilyuk, et al. 2004 andBigot et al. 2005. Both effects normally require pumpfluences of about 1 mJ/cm2, while the phenomenon re-ported by Wang et al. was triggered by a substantiallylower pump fluence of 0.2 J /cm2 photon energy of1.5 eV. The effect was found to be insensitive to thepolarization of the pump. Soon a similar surprisinglyeffective laser excitation of a homogeneous spin preces-sion in GaMnAs was reported in experiments witha photon energy of 1.569 eV and a pump fluence of3.4 J /cm2 Hashimoto et al., 2008; see Fig. 30. The lat-ter was again much too small for a substantial demagne-tization, required to excite precession via a change inshape anisotropy.

To solve this problem, trajectories of the laser-inducedspin motion were analyzed. Simulations based on theLandau-Lifshitz-Gilbert equation showed that one candescribe the oscillations if one assumes that the laserpulse introduces an impulsive change in the effectivemagnetic field due to a change in the magnetic anisot-ropy. The amplitude of the impulse was estimated to beabout 0.2 T. The impulse was characterized by a risetime of 50 ps and a decay time of 500 ps. It was noted

that such a pulse shape suggests the presence of a post-photocarrier process. Hashimoto et al. speculated thatsuch photoinduced anisotropy change may be due to achange in the number of holes near the Fermi level. Theafter effect is associated with the time required for thelaser-generated holes to cool down via emission ofacoustic phonons.

Chovan and Perakis analyzed the problem of lasercontrol of magnetization in magnetic semiconductorsusing density matrix equations of motion in a mean-field and single-band approximation Chovan et al.,2006; Chovan and Perakis, 2008. They showed that inIII,MnV semiconductors one may expect a modifica-tion of the magnetic anisotropy on the time scale of theoptical pulse duration, so that a laser pulse acts on theMn spin as an effective magnetic field. Unfortunately,qualitative estimates of the strength of the effectivemagnetic field and pump fluences required for this effecthave not been made. Recently Wang et al. reported thatan excitation of GaMnAs with a pump fluence of10 J /cm2 and photon energy of 3.1 eV allowed to de-tect a magnetization rotation during the action of thelaser pulse Wang et al., 2009. A comparison of the am-plitude of the transient magneto-optical signal and staticmagneto-optical measurements allowed the estimationthat during the action of the laser pulse 200 fs the mag-netization in GaMnAs deviated over 0.5° from equilib-rium. Taking the gyromagnetic ratio for GaMnAs fromLiu et al. 2003 and Wang, Ren, et al. 2007, one mayestimate that such a deviation of the magnetization fromequilibrium can be observed if the laser pulse acts as aneffective magnetic field with an amplitude of 0.1 T. Thisis comparable with the value reported by Hashimoto etal. 2008. However, Wang et al. 2009 also stated thatthe pumping of GaMnAs with a photon energy of 3.1 eVwas crucial for the observation of the femtosecond mag-netization rotation. Unfortunately, the calculations ofChovan and Perakis were performed in a single-bandapproximation and thus did not allow for the analysis ofthe spectral dependence of the photoinduced magneticanisotropy or the revelation of the origin of the differ-ences between the findings reported by these three ex-perimental groups Wang, Ren, et al., 2007; Hashimotoet al., 2008; Wang et al., 2009.

B. Light-enhanced magnetization in (III,Mn)As semiconductors

Surprising and intriguing observations were obtainedwhen a III,MnAs semiconductor sample was irradiatedby low-power continuous radiation. In In,MnAs suchan excitation with unpolarized light fluence of the laserequivalent to about 50 photons/cm2 per 100 fs led to anenhancement of the magnetization Koshihara et al.,1997. For a long time, time-resolved experiments aim-ing to reveal the relevant time scales of such an en-hancement remained a challenge. Only recently an ul-trafast enhancement of magnetic order in GaMnAs as aresult of excitation with 120 fs linearly polarized pulseswith a photon energy of 3.1 eV and peak fluences10 J /cm2 was reported by Wang, Cotoros, et al.

6420θ

[mrad.]

θ(a)

840(

10-3)

10005000Time delay [ps]

My/MsMz/Ms

(c)

0.4

0.2

0.0

-0.2

My/Ms[%]

1.00.80.60.40.20.0Mz/Ms [%]

(d)63 ps

113 ps

158 ps

210 ps

(Ms-Mx)/Ms

Mn/Ms

0.040.020.00

(10-3 ) (Ms-Mx)/Ms(b)

FIG. 30. Color online Laser-induced magnetic anisotropyand subsequent spin precession in Ga0.8Mn0.2As. a Temporalprofiles of the angle of tilt of the effective magnetic field Heffexperienced by the spins. The time dependence originatesfrom transient photoinduced magnetic anisotropy in the mate-rial, b temporal evolution of the x component of magnetiza-tion, and c y and z as function of time. All profiles are ex-tracted on the basis of experimental curves. d Precessionalmotion of magnetization in My−Mz parameter space Hash-imoto et al., 2008.



2007; see Fig. 31. No effect of the polarization of lighton this optical control of magnetic order was found.However, both the initial subpicosecond demagnetiza-tion and a distinct magnetization rise on a 100 ps timescale were observed. The observed ultrafast enhance-ment of magnetization was attributed to laser-inducedphotoinjection of holes and subsequent enhancement ofcollective ordering of Mn spins, leading to a transientincrease in the Curie temperature over 0.5–1.1 K. Ac-cording to the Zener model Dietl et al., 2000 this valueis in reasonable agreement with the ratio between back-ground and photoexcited holes.

An excitation of Ga,MnAs with circularly polarizedcw light was claimed to lead to a large helicity-dependent change in magnetization, reaching up to 15%of the saturation magnetization Oiwa et al., 2002; seeFig. 32. Note that III,MnAs semiconductors possessingferromagnetic order have a relatively high concentrationof holes of the order of 1020 cm3. The laser-inducedchanges by low-power excitation 4105 photons/cm2

per 100 fs in combination with the fast hole-spin relax-ation 10 fs is surprising. These experiments have im-mediately raised questions about the mechanisms andrelevant time scales of such an extremely efficient opti-cal control of magnetism. Two attempts to investigatethese questions with the help of time-resolved magneto-optical measurements have been done Kimel, Asta-khov, et al., 2004; Mitsumori et al., 2004. Although dif-ferent interpretations of these experimental results were

suggested, none of these dynamic measurements wasable to show that 100 fs laser pulses induce large valuesof photoinduced magnetization, as reported for cw exci-tation Oiwa et al., 2002. The mechanism that wouldallow efficient control of magnetization in GaMnAs isstill an open question that is a subject of intense re-search interest Aoyama et al., 2008; Astakhov et al.,2009.

VI. NONTHERMAL OPTOMAGNETIC EFFECTS

A. Inverse magneto-optical excitation of magnetizationdynamics: Theory

We now discuss nonthermal laser-induced effects thatdo not depend on the absorption of photons. In Sec. II itwas shown both phenomenologically and microscopi-cally that circularly polarized light can directly influencespins in magnetically ordered materials on a femtosec-ond time scale. The suggested mechanism for laser con-trol of spins does not require annihilation of photonsand the light-induced spin flip does not require the lossof the angular momentum of the photons. Instead, thephotons stimulate an efficient angular momentum trans-fer from spins by generating magnons via orbits to thelattice.

1. Formal theory of inverse optomagnetic effects

A formal mathematical theory is considered here for atransparent magnetic dielectric Gridnev, 2008.2 Takinginto account the small wave vectors k0 of light, only

2Following the original reference Gridnev, 2008, the Gauss-ian CGS system of units is used in this section.

8

4

0

-4

∆θk(µrad)

8006004002000T ime Delay (ps )

GaMnAs, T = 70K, pump 10 µJ /cm2

E xpFit

(b)

Mn, d-like

Valenceband

3.1 eV120 fs

E F

Mn, d-like

Beforeexternal stimuli

M

pTc ∆∆

After external stimuli

E nhanced FM

T

(a)

(c)

Holes , p-like

FIG. 31. Color online Laser-induced magnetism in GaMnAs:a Illustration of hole-density-tuning effects via externalstimuli in III-V FMSs seen in the static experiments Koshi-hara et al., 1997. FM: ferromagnetism. 4p is hole densitychange. b Schematic diagram of the spin-dependent densityof states in GaMnAs. Femtosecond pump pulses create a tran-sient population of holes in the valence band. c Time-resolved magneto-optical Kerr effect dynamics at 70 K and un-der 1.0 T field. Transient enhancement of magnetization, with100 ps rise time, is clearly seen after initial fast demagneti-zation. From Wang, Cotoros, et al., 2007.

5

0

-5

6.5

6.0

5.5

4003002001000

Angle of λ /4 wave plate (degree)

-190

-170

-150

-130

-0.020

0.02

RH

all/R

shee

t

-0.4 0 0.4H (T)

(a)

(b)

H =0 T, 4.2 K

H⊥plane

Light on

σ- σ-σ+ σ+

RH

all(

Ω)

Rsh

eet(

kΩ)

RH

all /R(10

)sheet

-3

FIG. 32. Changes in Hall resistance and sheet resistance at4.2 K for a GaMnAs/GaAs sample under cw light irradiation,the polarization state of which is controlled by a quarter wave-plate. The inset shows the magnetization curve extracted fromthe anomalous Hall effect measurements, indicating in-planemagnetic anisotropy of the current sample. From Oiwa et al.,2002.



spin-wave modes with k=0 are considered, that is, thetemporary behavior of M is homogeneous across thesample.

The deviation of Mt from equilibrium is given by thecomponents Mx and My, the z axis directed along theequilibrium magnetic moment M0. For the followingconsideration, it is convenient to introduce the canonicalvariables b through the linearized Holstein-Primakofftransformation:

b = 2M−1/2Mx + iMy , 40

where M= M is the magnitude of the sublattice magne-tization and is the gyromagnetic ratio. In the harmonicapproximation, the Hamiltonian H0 of the spin system isa quadratic function of the variables b ,b.

The electric field of a pump pulse exciting the spinsystem can be represented in the form Et=Re Eteit,where is the central frequency of the pulse and Et isthe time-dependent amplitude.

Phenomenologically, the interaction between mag-nons or other low-energy collective excitations and thepump pulse in a transparent medium is given by Lan-dau and Lifshitz, 1984

V = −ij

16Ei

*tEkt , 41

where ij is the modulation of the optical dielectric per-mittivity by magnons. V can be expressed as a powerseries in b and b. When considering spin-wave excita-tions, one should take into account only terms in ijthat are linear in b and b. For a transparent medium,the tensor ij is Hermitian: ij=ji

.In the impulsive limit, when the optical pulse width

−1, where is the frequency of the induced spinwave, one may use the formal representation

Ei*tEjt =

4I0

ncei

*ejt , 42

where n is the refractive index, c is the speed of light, I0is the integrated pulse intensity, and ei is a component ofthe unit vector defining the polarization of light. Thedelta function t accounts for the impulsive character−1 of the pump light. It is convenient to decom-pose the product ei

ej into its symmetric and antisymmet-ric parts,

ei*ej = Sij +

i

2aijkhk, 43

where Sij=12 ei

ej+ejei, aijk is the antisymmetric unit ten-

sor, and hk is a component of the vector h= iee. his a measure of the light helicity and varies in the rangefrom −1 to 1.

Under the action of the light pulse, excluding absorp-tion, b varies with time according to the Hamilton equa-tion Zakharov and Kuznetsov, 1984,

db

dt= − i

b*H0 + V . 44

The advantage of the classical approach used here isthat there is no need to specify exactly the optical tran-sitions caused by the photoexcitation. The optical prop-erties of a medium relevant for the generation of coher-ent excitations are contained in the modulation of theoptical dielectric permittivity. This is a consequence ofthe fact that all optical transitions are virtual for a trans-parent medium. When optical absorption is significant,this approach does not work and a microscopic consid-eration is necessary Stevens et al., 2002.

Equations 41 and 44 are rather general and can beapplied to pump pulses of arbitrary duration includingcw light as a limiting case, provided that the electric fieldentering the interaction potential Eq. 41 is the totalelectric field, i.e., is the sum of the incident EI and scat-tered ES electric fields. Despite the smallness of the scat-tering field in comparison with the incident one, ES inEq. 41 may not always be neglected. When consideringstimulated Raman scattering RS of cw light, it is theterm containing the product Ei

IEjS that is responsible

for the scattering and for spin-wave generation be-cause EI and ES oscillate at different frequencies. Thecase of impulsive stimulated RS ISRS Yan et al., 1985is quite a different situation. In this case, the spectralwidth of the excitation pulses exceeds the spin-wave fre-quency. For this reason, in the impulsive stimulated scat-tering process, higher-frequency photons from a pumppulse are coherently scattered into lower-frequency pho-tons within the pump pulse bandwidth and propagatedin the same direction but with a slightly smaller wavevector magnitude. Thus, ISRS is a forward-scatteringprocess which is stimulated because the Stokes fre-quency is contained within the bandwidth of the incom-ing pulse. The pump pulse is asymmetrically redshiftedthrough ISRS when it leaves the sample. Since the in-tensity of the scattered light is small compared to theincident one, the scattered electric field can be neglectedin the interaction potential.

Equation 44 can be applied to any instant of time,including times within the pump pulse, t. However,experimentally, light-induced spin polarization can bereliably measured only at times after the pump pulse,t . For this reason, it makes sense to calculate theinduced magnetization at those times. In that case, it isallowed to take the limit of zero pump pulse duration,→0.

Integrating Eq. 44 over the pulse width and account-ing for Eq. 42, we obtain b just after the photoexcita-tion t=0+,

b0+ = iI0

4ncij

ij

b*ei

*ej. 45

Thus, in general, immediately after the photoexcitationthe spins are rotated away from the equilibrium direc-tion. This is simply a consequence of the fact that theelectron spins straightforwardly participate in the opticaltransitions induced by the light pulse.

Equations 44 and 45 are applicable to all coherentcollective excitations, which are characterized by non-



zero values of the canonical variables. Note that this isnot the case for coherent two-magnon excitations Zhaoet al., 2004. For magnons, ij depends, in general, onboth Re bMx and Im bMy. Consequently, both realand imaginary parts of b0+ are finite at t=0+, i.e., eachspin acquires a finite rotation during a pump pulse.

In order to study the temporal evolution of the spinsystem with N sublattices, it is convenient to transformbr, where r=1, . . . ,N is a site index, to the normal coor-dinates Q,

Q = r

urbr + vrbr* , 46

where =1, . . . ,N is a mode index and ur and vr arematrices, satisfying the following equations that guaran-tee that the transformation Eq. 46 is canonical:

r

urur* − vrvr

* = ,

47

r

urvr* − vru

r* = 0.

After the transformation to normal coordinates, theHamiltonian of the spin system has the simple form

H0 =

QQ* , 48

where is the frequency of the spin-wave mode .Being canonical variables, the normal coordinates alsoobey the Hamilton equations

dQ

dt+ iQ = − i

V

Q* . 49

The modulation of the dielectric permittivity by thespin waves is given by

ij =

PijQ + Rij

Q* , 50

where

Pij = ij/Q, Rij

= ij/Q* 51

are scattering matrices for the mode . Solving Eq. 49for Q and taking into account Eqs. 41, 42, and 50,we obtain

Q = iI0

4ncij

Rijeiej

*e−it. 52

These laser-induced magnetic excitations, in turn, leadto changes in the dielectric permittivity of the medium.For these, using Eqs. 50 and 52 we obtain

ijind = i

I0

4nc nm

PijRnm

e−it − Pnm Rij

eitenem* . 53

Thus, Eq. 53 explicitly shows the oscillatory responsein the pump-probe experiments. The particular phase ofthe measured oscillations will depend on the propertiesof the given medium, such as the magnetic symmetry aswell as the magnetocrystalline anisotropy that determine

the properties of the scattering matrices Pij and Rij

. Thelatter contain all information about the dependence ofthe spin-wave generation on the light polarization, crys-tal orientation, and magnetic structure. In order to cal-culate these matrices, one needs to know the modulationof the dielectric permittivity ij induced by the devia-tion of the spin system from equilibrium. ij can be ex-panded in powers of the sublattice magnetizations Cot-tam and Lockwood, 1986. To reveal the distinctivefeatures of the magnon generation by an optical pulse inmore detail, we perform the explicit calculations of thelight-induced spin precession for a simple cubic ferro-magnet and a rutile-type structure antiferromagnet.

2. Example I: Cubic ferromagnet

For a simple cubic ferromagnet N=1 themagnetization-dependent part of ij is given by Cottamand Lockwood, 1986

ij = iKk

eijkMk + kl

GijklMkMl, 54

where the first and second terms on the right-hand sidedescribe the Faraday and Cotton-Mouton Voigt ef-fects, respectively.

For simplicity, let the external magnetic field H andthe equilibrium magnetization M0 point along the z axis.Then the change in the permittivity as a function of thespin-wave amplitudes Mxt and Myt is

ij = k=x,y

iKeijkMk + 2GijkzM0Mk . 55

Our ferromagnet possesses only one spin-wave eigen-mode with the frequency =H and the normal coordi-nate

Q = 2M0−1/2Mx + iMy . 56

Using Eqs. 55 and 56 we obtain the scattering matri-ces

Pij =M0

2Kieijx + eijy + 2M0Gijxz − iGijyz ,

57

Rij =M0

2Kieijx − eijy + 2M0Gijxz + iGijyz ,

Then, using Eqs. 52 and 56 the light-induced dynamiccomponents of the magnetization are derived:

Mxt = − c1 cos t + c2 sin t ,58

Myt = c2 cos t + c1 sin t ,

where

c1 = Khy + GM0Syz ,59

c2 = Khx + GM0Sxz ,

with G=Gxzxz=Gyzyz and =M0I0 /4nc. Although in acubic ferromagnet the tensor Gijkl has three independentcomponents, only one of them relates to the one-



magnon RS when M0 001 Cottam and Lockwood,1986. According to Eqs. 58 and 59, the initial changein magnetization M= Mx0+ ,My0+= −c1 ,c2 con-sists of two contributions, M=MK+MG, propor-tional to the Faraday and Cotton-Mouton magneto-optical constants, respectively. MK vanishes for linearlypolarized light h=0, while MG can be nonzero forany light polarization.

Note that the magneto-optical constants K and Gdefine the Faraday rotation and magnetic linear bire-fringence, respectively, and can be determined by mea-surements of these effects. In magnetic dielectrics,KM00.001 and GM0 /K usually lies in the range of0.1–1. Because the magnetic linear birefringence is rela-tively strong in magnetically ordered media Ferré andGehring, 1984, it may happen that linearly and circu-larly polarized light can induce spin waves of compa-rable amplitude see Sec. VI.D.2 below.

3. Example II: Two-sublattice antiferromagnet

Here we consider a two-sublattice N=2 antiferro-magnet with a tetragonal symmetry D4h, with spins ori-ented along the fourfold z axis. The spin-wave spectrumof such an antiferromagnet consists of two branches thatare degenerate in zero magnetic field, with a frequency

= HAHA + 2HE , 60

where HE is the exchange field and HA is the anisotropyfield. In this case, the coefficients ur and vr defining thenormal coordinates are given by Fleury and Loudon,1968; Kittel, 1987;

u11 = u, v21 = v, u21 = v11 = 0,

61

u22 = u, v12 = v, u12 = v22 = 0,

where

u,v =HE + HA ±

2. 62

From these, the normal coordinates for the two modesare

Q1 = ub1 + Vb2*, Q2 = ub2 + vb1

*, 63

where b1,2 are defined by Eq. 40.To describe the light-induced spin dynamics of the an-

tiferromagnet in more detail, it is convenient to intro-duce the ferromagnetic and antiferromagnetic vectorsM=M1+M2 and L=M1−M2. The components Mx andMy characterize mainly the relative orientation of thesublattice magnetizations M1 and M2, while Lx and Lydescribe their deviations away from the z axis. To calcu-late these components, we start with the calculation ofthe normal coordinates Q1,2t from Eq. 52.

The interaction of the nonequilibrium spin system ofthe antiferromagnet with light is given by Cottam, 1975;Cottam and Lockwood, 1986

V = K+Mxhx + Myhy + K−Lxhy + Lyhx

+ G+LxSxz + LySyz + G−MxSxz + MySyz , 64

where K± and G± are magneto-optical constants. Next,using Eqs. 40, 47, and 52, one obtains

Mx = − m1 cos t + l2u − v2 sin t ,

My = m2 cos t + l1u − v2 sin t ,65

Lx = − l1 cos t + m2u + v2 sin t ,

Ly = l2 cos t + m1u + v2 sin t ,

where

m1 = K−hx + G+Syz ,

m2 = K−hy + G+Sxz ,66

l1 = K+hy + G−Syz ,

l2 = K+hx + G−Sxz .

Equations 65 show that, as in a ferromagnet, the spinsin the antiferromagnet are rotated during the light pulse,and this rotation is independent of HE and HA. More-over, the parameters m1 and m2 determine the anglebetween the sublattice magnetizations just after thepump pulse. These parameters depend on the polariza-tion of the pump pulse as well as on the magneto-opticalconstants K and G. The initial noncollinearity leads toan enhancement of the spin rotation from the z axis inthe subsequent motion.

B. Excitation of precessional magnetization dynamics withcircularly polarized light

1. Optical excitation of magnetic precession in garnets

Garnets show a minimum of absorption at the excita-tion wavelength of 800 nm. The key factor, however, indistinguishing the nonthermal effects described below istheir dependence on the pump pulse polarization. Thus,left- and right-handed circularly polarized laser pulsesincident along the z direction were used to excite a mag-netic garnet film exposed to an in-plane applied mag-netic field Hext. Precession of M with opposite phase wastriggered by pulses of helicity + and −; see Fig. 33. Theinitial phase of the signal reveals that M initially movesalong the ±z direction and therefore both M and Heffare parallel to the film plane immediately after the pho-toexcitation.

These experimental observations can be understood ifduring the presence of the laser pulse a strong magneticfield along the k vector of light is created. Such an axialmagnetic field HF can be generated by intense circularlypolarized light through what is known as the inverse Far-aday effect Pitaevskii, 1961; van der Ziel et al., 1965;Pershan et al., 1966; see Sec. II.C. These optically gen-erated field pulses are much stronger than both the an-



isotropy field Ha and the applied field Hext and thereforecompletely dominate during the t=100 fs presence ofthe laser pulse. The magnetization will respond by pre-cessing in the plane of the film normal to HF to a newin-plane orientation. After the pulse is gone, the magne-tization will precess in the effective in-plane field Heff=Hext+Ha+Ha, as shown in Fig. 33, where Ha is theoptically induced change in the anisotropy field.

The strength of the photoinduced field HF can be es-timated from the precession amplitude :

HF / /tpulse, 67

where is the precession frequency, is the gyromag-netic ratio, and tpulse is the duration of the opticalpulse. We find that laser pulses of energy 20 J createtransient magnetic field pulses of about 0.6 T in the gar-net films.

2. Optical excitation of antiferromagnetic resonance inDyFeO3

The optomagnetic inverse Faraday effect can be usedto excite magnetization dynamics in a situation whenany other method is difficult or impossible to apply. Inthis section we describe optical excitation of antiferro-magnetic resonance modes in DyFeO3, in the hundredsof GHz frequency range.

Details of the structure of this antiferromagnethave been given by Wijn 1994. Because of theDzyaloshinskii-Moriya interaction, the antiferromag-netically coupled Fe spins are slightly canted, giving riseto a spontaneous magnetization Ms8 G. Despite thesmall magnetization, this material exhibits a giant Fara-day rotation of about 3000° cm−1 owing to its strong

spin-orbit interaction Zvezdin and Kotov, 1997. Suchweak ferromagnets represent an intriguing combinationof the static properties of a ferromagnet with the dynam-ics of an antiferromagnet.

For the detection of the optically induced magnetiza-tion the direct magneto-optical Faraday effect was used,which was possible due to the presence of a weak ferro-magnetic moment. Figure 34 shows the temporal evolu-tion of the Faraday rotation in a z-cut DyFeO3 samplefor two circularly polarized pump pulses of opposite he-licities. On the scale of 60 ps one can distinguish twodifferent processes that start after excitation with apump pulse. At zero time delay instantaneous changesin the Faraday rotation are observed that result from theexcitation of virtual and real transitions in the Fe3+ ionsfrom the high-spin ground state S=5/2. The instanta-neous changes of the Faraday rotation are followed byoscillations with a frequency of about 200 GHz that canbe assigned to oscillations of the magnetization. It isseen from Fig. 34 that the helicity of the pump lightcontrols the sign of the photoinduced magnetization.This observation unambiguously indicates that the cou-pling between spins and photons in DyFeO3 is directbecause the phase of the spin oscillations is given by thesign of the helicity of the exciting photon.

The amplitude of the oscillations corresponds toa photoinduced change in the magnetization M0.06Ms, where Ms is the saturation magnetization.

From Fig. 34 one can distinguish not only oscillationsbut also an exponential decay of the equilibrium level ona time scale of about 30 ps. This can be explained by aphotoinduced change in the equilibrium orientation ofthe magnetization and a subsequent decay of this equi-librium orientation to the initial state.

σ−

σ+∆

-2

0

2M

z/M

(%)

0 1 2 3Time delay (ns)

FIG. 33. Color online Precession following excitation withcircularly polarized light. The two helicities + and − give riseto precession with an opposite phase and a different ampli-tude. During the 100 fs presence of the laser pulse the magne-tization precesses in the dominating axial magnetic field HFcreated by the circularly pump pulse. Subsequent precessiontakes place in the effective magnetic field Heff , possibly modi-fied by the photomagnetic effects, see Sec. V.A. From Hans-teen et al., 2005, 2006.

DyFeO

T = 95 K3

+

Fara

day

rota

tion

(deg.)

H

0 15 30 45 60

0.0

0.1

0.2

Time delay (ps)

H

FIG. 34. Color online Magnetic excitations in DyFeO3probed by the magneto-optical Faraday effect. Two processescan be distinguished: 1 instantaneous changes of the Faradayeffect due to the photoexcitation of Fe ions and relaxationback to the high spin ground state S=5/2 and 2 oscillations ofthe Fe spins around their equilibrium direction with an ap-proximately 5 ps period. The circularly polarized pump pulsesof opposite helicities excite oscillations of opposite phase. In-set shows the geometry of the experiment. Vectors H+ andH− represent the effective magnetic fields induced by right-handed + and left-handed + circularly polarized pumppulses, respectively. From Kimel et al., 2005.



Although in principle the effect of optically inducedmagnetization does not require the absorption of pho-tons, the laser control of the spontaneous magnetizationand the excitation of coherent spin oscillations is equiva-lent to photoexcitation of magnons and thus requiressome energy. Figure 35 shows the amplitude of the pho-toexcited spin oscillations as a function of the pump in-tensity. The linearity of this dependence is in perfectagreement with the expression for the inverse Faradayeffect Eq. 16. Note that extrapolation of the intensitydependence shows that the photoinduced effect on themagnetization would reach the saturation value of Ms ata pump fluence of about 500 mJ/cm2. The effect of sucha 100 fs laser pulse on the magnetic system would beequivalent to the application of a magnetic field pulse ofabout 5 T. According to our measurements, the absorp-tion in DyFeO3 in the near infrared spectral range is onthe order of 100–200 cm−1. Given this low value of theabsorption, a photoexcitation of 500 mJ/cm2 is still be-low the damage threshold of DyFeO3 and thus quitefeasible, provided the sample is of high optical quality.

3. Optical excitation of precession in GdFeCo

In the two examples above, the inverse Faraday effectwas shown to act as an effective magnetic field on thespins in oxidic materials. The observed effects were non-thermal and any additional laser-induced thermal effectswere small in these transparent crystals. But what aboutmetals? Most magnetic materials of practical interest areferromagnetic metals. Would a circularly polarized laserpulse also act as a magnetic field pulse in this case?

Many attempts to detect an effect of the light helicityon ferromagnetic metals such as Ni and other transitionmetals remained unsuccessful Ju, Vertikov, et al., 1998;Wilks et al., 2004; Kruglyak, Hicken, et al., 2005. Heat-driven effects of light were found to dominate in allthese experiments. In order to diminish the unavoidableeffects due to laser-induced heating, we used the ap-proach of turning the sample into a multidomain state,in this way minimizing the usually dominating heat-

driven effects of light on the magnetization, as explainedbelow.

The material chosen for this study was the ferri-magnetic alloy GdFeCo, discussed in Sec. IV.G. TheGdFeCo 20-nm-thick layer is amorphous and is charac-terized by a strong perpendicular anisotropy and a Curietemperature of about 500 K.

Initially, the sample was in a single-domain state. Theapplication of an out-of-plane opposite field with the si-multaneous presence of an in-plane external magneticfield results in the nucleation of a large number of smalldomains Wu, 1997; Kisielewski et al., 2003, with sizesmuch smaller than the probe area. Because of the coer-civity, these magnetic domains are sufficiently stable intime and thus suitable for stroboscopic pump-probe ex-periments. The balance among magnetocrystalline an-isotropy field Ha, shape anisotropy field Hs, and externalfield He creates an initial state for these experimentssuch that the magnetization in each of the domains willbe tilted see Fig. 36a. Laser excitation of such asample induces locally a change in the magnetocrystal-line and shape anisotropy due to heating. This leads to achange in the equilibrium orientation for the magnetiza-tion and triggers a precession of the magnetic momentsof the domains. These precessions proceed in such a waythat the z components of the magnetization in oppo-sitely oriented domains like M1 and M2 in Fig. 36balways oscillate out of phase. Since the probe beam isaveraged over a large number of such oppositely ori-ented domains, the heat-driven effect of light on themagnetization is effectively averaged out in the Faradaysignal by achieving M1

i +M2j 0, where the sums

are taken over all domains within the probed area. Ex-perimentally, the optimum cancellation is achieved byadjusting the external magnetic field He. On the otherhand, an optically induced effective magnetic field willapply a torque on all the various magnetic moments inthe same direction and consequently lead to a nonzerochange in Mz.

0 25 50 750.0

0.5

1.0

Amplitude(arb.units)

Pulse fluence (mJ/cm2)

FIG. 35. The amplitude of the spin oscillations as a function ofpump fluence. From Kimel et al., 2005.

H - H M

M

M

a s H - Ha s

1

1

2

''

'

HeHe

M2'

Heat-driven

excitation

Initial multidomain

magnetic state

∆t < 0 ∆t > 0

a) b)

FIG. 36. Color online Suppression of heat-driven oscillationsof the net magnetization in a multidomain sample: a Att0, the magnetizations M1 and M2 of oppositely magne-tized domains are tilted due to the balance among magneto-crystalline anisotropy field Ha, shape anisotropy field Hs, andexternal field He. b After photoexcitation, t 0, heating in-duces out-of-phase oscillations of the z components of M1 andM2.



The results of these time-resolved measurements ofthe magneto-optical Faraday effect for right-handed +

and left-handed − circularly polarized pump pulses atdifferent He are shown in Fig. 37. The variation in themagneto-optical signal F, representing the oscillatorybehavior of Mz, was plotted relative to the total Faradayrotation of the sample F. On the scale of 500 ps one candistinguish two different processes. At zero time delay,instantaneous changes in the Faraday rotation are ob-served that result partly from ultrafast demagnetizationand partly from changes in the magnetocrystalline aniso-tropy Vomir et al., 2005. These instantaneous changesof the Faraday rotation are followed by oscillations witha frequency of about 7 GHz, which can be ascribed toprecession of the magnetization Stanciu et al., 2006.The observed strong damping can be understood as aresult of the averaging over the many domains. Figure37 shows that the difference between the magneto-optical Faraday signal for right- and left-handed excita-tion strongly depends on the external applied field He.At He=0.172 T the + and − pump pulses excite pre-cession of opposite phases. Because the phase of thespin precession is given by the polarization of the excit-ing photons, this observation provides a first indicationof an optomagnetic effect in the metallic ferrimagnetGdFeCo.

Equation 16 shows that the strength of the effectivemagnetic field HF depends on the laser fluence. Al-though in these experiments a relatively high laser flu-ence for a metallic magnetic material 2 mJ/cm2 wasused, this is still low compared with the experiments onnonabsorbing dielectrics e.g., 60 mJ/cm2 for garnetsHansteen et al., 2006 where a strong optomagnetic ef-fect was observed. Therefore the usual experimentalconditions for metallic magnets of low laser fluence inaddition to a strong heat-driven effect require a special

technique for the detection of ultrafast optomagneticeffects, as shown here. The observation of the inverseFaraday effect in a metallic magnet together with thepreviously discussed observation of this effect in ortho-ferrites and garnets indicate that this mechanism doesnot rely on specific material properties but is a generalphenomenon see also Hertel 2006. Its strength, aswell as the experimental possibility of observing it, de-pend on the material parameters. On the one hand, it isthe value of the magneto-optical response that deter-mines the nonthermal interaction. On the other hand,there are competing thermal effects that are usuallydominating in metals. To make a classification, the roleof the bandwidth may be invoked: narrowband materi-als, such as insulators, oxides, and to some extent also fmetals can be excited much more selectively than thebroadband transition metals. Similarly, the former ex-hibit more of a helicity dependence than the latter. Thisexplains, for example, the absence of helicity-dependenteffects in the experiments on Ni Longa et al., 2007.

C. All-optical control and switching

1. Double-pump coherent control of magnetic precession

The nonthermal effects of light on the magnetizationallow for ultrafast coherent control of spin precession.Such control can be achieved using multiple laser pulsesin rapid succession.

In Fig. 38 the results of such coherent control experi-ments are shown for a magnetic garnet. Initially, fort0 the spins are aligned along the total effective mag-netic field Heff that is of the order of 0.05 T. A pumppulse of helicity + arriving at t=0 acts as a strong pulseof magnetic field HFHeff and thus it triggers preces-sion of the magnetization, as explained in the previoussections. A second pump pulse of helicity − arrivingafter an odd number of half-precessional periods rotatesthe magnetization further away from Heff, causing thesubsequent precession to have almost twice the ampli-tude upper graph. If, however, this second pump pulse

0

2

4

0 100 200 300 400 500

-4

-2

0

-1

0

1

σ(+)

σ(-)

He=0.189T

σ(-)

He=0.208T

He=0.172T

He=0.115T

σ(+)

σ(+)

σ(-)

σ(+)

σ(-)

Time delay ∆t (ps)

∆θ F/(%)

0

2θ F

FIG. 37. Color online Precession of the magnetization ex-cited by circularly polarized pump pulses in GdFeCo, probedvia the magneto-optical Faraday effect. For certain values ofthe external field He, the two helicities + and − give rise toprecession with different phase. The lines are guides to theeye.

∆

0 0.5 1 1.5

0

1

2

Time delay (ns)

Mz

/M(%

)

σ+ σ−

FIG. 38. Double-pump experiment in magnetic garnet withcircularly polarized laser pulses of opposite helicity. Shown ishow amplification and complete stopping of the magnetizationprecession can be achieved depending on the phase of the pre-cession when the second laser pulse arrives. The time delaybetween the two pump pulses is fixed at approximately 0.6 ns,and the precession frequency is controlled by varying the ex-ternal field. From Hansteen et al., 2006.



arrives after an integer number of full periods, the mag-netization is rotated back into its original equilibriumorientation along Heff and no further precession takesplace lower graph. Figure 39 gives a pictorial illustra-tion of these two situations. Similarly, circularly polar-ized light can control the precession of antiferromag-netic spins in the terahertz domain see Fig. 40.

These experiments demonstrate that femtosecondoptical pulses can be used to directly and coherentlycontrol spin motions. Depending on the phase of theprecession when the second pulse arrives, energy istransferred either from the laser pulse to the magnetic

system amplification of the precession or from themagnetic excitation to the optical pulse stopping of theprecession. In view of the low intrinsic damping in theorthoferrites and garnets and therefore the long lifetimeof their magnetic excitations, it is remarkable how ul-trashort laser pulses can completely and instantaneouslystop the long-period coherent precession of spins. Thisprocess of transferring the energy back into the opticalpulse can also be viewed as coherent laser cooling ofmagnons.

The complex spin oscillations in orthoferrites trig-gered by a train of laser pulses was recently studiedtheoretically using nonlinear Landau-Lifshitz-Gilbertequations. It was demonstrated that such a periodicalexcitation of spins results in various patterns of spin os-cillations, which depend on the intensity and periodicityof the laser pulses Perroni and Liebsch, 2006.

It should be pointed out that the present double-pump experiments, which demonstrate control of themagnetization in ferrimagnetic garnets and antiferro-magnetic orthoferrites, are considerably different fromthose previously reported in diamagnetic and paramag-netic materials. During the past two decades many pub-lications have been devoted to the photoexcitation of anonequilibrium spin polarization in direct-band-gapsemiconductors through the phenomenon of optical ori-entation Meier and Zakharchenya, 1984; Awschalom etal., 1987; Žutic et al., 2004. In these materials, absorp-tion of circularly polarized photons may lead to a non-equilibrium population of spin-polarized electrons andholes in the conduction and valence bands, respectively.In paramagnetic semiconductors these spin-polarizedcarriers can cause partial alignment of the momentsof magnetic ions due to an sp-d exchange interactionand thereby also affect their precession in a magneticfield Furdyna and Kossut, 1988. Using this phenom-enon of optical orientation, Akimoto et al. 1998 dem-onstrated control of the precession of Mn2+ moments inCdTe/Cd1−xMnxTe quantum wells.

A nonabsorptive mechanism for manipulation of spinsin Zn1−xCdxSe quantum well structures was reported byGupta et al. 2001, who used below-band-gap opticalpulses to control the spin precession of photoexcitedelectrons in the conduction band via the optical Starkeffect. However, these experiments were performed onparamagnetic materials, where coupling between thespins of magnetic ions is small and the spins oscillateindependently. Therefore, in a double-pump experimentwith paramagnets, the first and second laser pulses cansimply excite different spins so that the integrated signalwill show either amplification or quenching of the oscil-lations. However, the amplification and quenching of theoscillations in such an experiment would only mean thatthe spins excited by the first or second pump pulses os-cillate in phase or out of phase, respectively. In magneti-cally ordered materials, discussed in this review, spinsare strongly coupled by the exchange interaction andspin excitations are delocalized. Therefore, in contrast toparamagnets, laser control of spins in magnetically or-

100fs < t < t2 t2 < t < t2+100fs t > t2+100fs

(a)

100fs < t < t2 t2 < t < t2+100fs t > t2+100fs

(b)

FIG. 39. Color online Illustration of the double-pump experi-ment for circularly polarized pump pulses of opposite helicityarriving at an a odd number of half precessional periods andb an integer number of full precessional periods. The magne-tization is either rotated further away from the effective fielddirection causing subsequent precession to take place with al-most twice the original amplitude or the magnetization is ro-tated back into the effective field direction and no further pre-cession takes place. From Hansteen et al., 2006.

FIG. 40. Color online Coherent control of spins in DyFeO3with two circularly polarized laser pulses. a Precession trig-gered by the first laser pulse, b amplification of spin preces-sion by the second laser pulse that comes after an even numberof full periods, and c stopping of the spin oscillations by thesecond pump that comes after an odd number of half periods.



dered materials indeed means control of the collectivemotion of spins.

2. Femtosecond switching in magnetic garnets

A proper combination of the inverse Faraday effectand the photoinduced anisotropy Sec. V.A allows foran interesting demonstration of photomagnetic switch-ing on a femtosecond time scale Hansteen et al., 2005.When the laser pulse is circularly polarized, the direc-tion of Ha depends only on the initial angle of themagnetization with respect to the crystal axes. There-fore, it can be tuned by rotating the sample with respectto the applied field. Alternatively, since the initial equi-librium of M is along Heff, which is determined by thebalance between the magnetocrystalline anisotropy fieldHa and the externally applied field Hext, it can also betuned simply by varying the strength of the applied field.

In Fig. 41 the coherent precession of the magnetiza-tion following excitation with pulses of helicity − and+ is shown for different values of Hext. The amplitudeof precession is consistently larger in the case of +, asduring 0 t100 fs, M precesses away from the newequilibrium created by Ha. For pulses of helicity −,this precession is toward the new equilibrium, leading tosmaller precessional amplitude in the time after thepulse. With an applied field of Hext150 Oe, no pre-cession is triggered owing to a perfect balance of twoeffects: the in-plane precession of the magnetization dur-ing the 100 fs magnetic field pulse HF brings the mag-netization exactly to its new equilibrium orientation cre-ated by the simultaneously optically modified anisotropyfield. It remains stable in this orientation until the aniso-

tropy field relaxes back to its original state, i.e., for sev-eral nanoseconds. An illustration of this switching pro-cess is shown in Fig. 42.

Note also that for the − helicity at weak appliedfields the precession has an opposite phase compared tothe precession in stronger applied fields and that thisphase is the same as for the precession triggered by the+ pulses. At weak fields the direction of the photoin-duced Ha is such that the precession of M by HF duringthe optical pulse is not sufficient to bring it into the di-rection of Heff . At stronger fields, however, Ha is in adifferent direction producing an Heff that is less inclinedwith respect to the original effective field. During thepresence of HF the magnetization now precesses pastthe direction of Heff and therefore with the oppositephase in the time directly after the laser pulse.

3. Inertia-driven switching in antiferromagnets

The dynamics of spins in a ferromagnet is describedby the LLG equation that does not contain inertialterms such as acceleration. Therefore, a switching be-tween two minima requires that the system is draggedover the potential barrier. However, if an inertial motionof the magnetic moments, similar to that of a finite-massobject was possible, one could “kick” the system so thatthe following motion will bring the system over the bar-rier and into the new equilibrium see Fig. 43. Such amechanism would allow the use of pulses much shorterthan the precessional periods.

The inertial motion can actually be realized in antifer-romagnets Kimel et al., 2009. The dynamics of an anti-ferromagnet with two sublattices M1 and M2 is describedin terms of the motion of the antiferromagnetic unit vec-tor l= M1−M2 / M1−M2. In angular variables, theequation of motion can be written as Kimel et al., 2009

d2

dt2 + 2d

dt+ 0

2dw d

−2HD

sin 0Htcos = 0. 68

Here the presence of the acceleration d2 /dt2 describesthe appearance of inertia, similar to that in the Newtonequation of motion for unit mass; the second term pre-

∆

0 1 2

0

2

4

6

8

10

12

14

Time delay (ns)

Mz

/M(%

)

-352 Oe

-221 Oe

-151 Oe

-77 Oe

65 Oe

135 Oe

206 Oe

348 Oe

0 1 2Time delay (ns)

σ− σ+

FIG. 41. Color online Precession of the magnetization trig-gered by left- and right-handed circularly polarized laser pulsesat different values of the in-plane applied magnetic field. Forthe − helicity, at an applied field of ±150 Oe, no precessionis observed due to a perfect balance of the two photomagneticeffects Ha and HF. From Hansteen et al., 2005.

Initial statet < 0 0 < t < 100fs t > 100fs

Final state

FIG. 42. Color online Illustration of the switching process.Initially at t0 the magnetization is along Heff. During thepresence of the laser pulse 0 t100 fs photoinduced modifi-cation of the anisotropy fields leads to a new long-lived equi-librium along Heff . Simultaneously, the strong optomagneti-cally generated field HF causes the magnetization to precessinto the new state. After t 100 fs the optical pulse is gone andthe approximately 0.6° switching of M is complete. From Han-steen et al., 2005.



sents a viscous force with damping coefficient ; termsdw /d and HD represent the restoring and drivingforce, respectively; w is a dimensionless function pro-portional to the magnetic anisotropy energy; 0 is thefrequency of the lower antiferromagnetic mode.

A short magnetic field pulse should trigger an inertialspin reorientation between two magnetic phases. To ob-serve this, however, the pulse must be shorter than thecharacteristic time of the magnetic eigenmodes, a fewpicoseconds in the case of an antiferromagnet. Suchshort pulses can be provided only by the inverse Faradayeffect.

To observe such inertial motion of spins in HoFeO3,pump-probe experiments were performed with 100 fscircularly polarized laser pulses. The pump-induced spindynamics was monitored by detecting the Mz compo-nent of the magnetization via the magneto-optical Fara-day effect in the probe pulse, measured as a function ofdelay between the pump and probe pulses. The resultsof such time-resolved measurements were in excellentagreement with the results of simulations and thus dem-onstrated the inertial motion of antiferromagneticallycoupled spins Kimel et al., 2009.

4. All-optical magnetization reversal

In Sec. IV.G we showed how laser-induced heating ofGdFeCo in an external magnetic field, induced a subse-quent ultrafast reversal switching of the magnetization.As is obvious from the previous sections, circularly po-larized femtosecond laser pulses act as equally shortmagnetic field pulses via the inverse Faraday effect. Thisnaturally leads to the question: Can such optically in-duced field pulses be used to completely reverse themagnetization of a magnetic domain?

The experiments were performed by placing a sampleof the GdFeCo magnetic alloy under a polarizing micro-scope, where domains with magnetization “up” and“down” could be observed as white and black regions,

respectively. To excite the material, amplified pulsesfrom a Ti:sapphire laser were used at a wavelength of=800 nm, repetition rate of 1 kHz, and a pulse widthof 40 fs. The laser pulses were incident normal to thesample surface, so that an effective optically generatedmagnetic field would be directed along the magnetiza-tion, similar to a conventional recording scheme.

In order to unambiguously determine whether excita-tion by a single 40 fs laser pulse is sufficient to reversethe magnetization, the laser beam was swept at highspeed across the sample so that each pulse landed at adifferent spot; see Fig. 44. One can see that each of the+ pulses reversed the magnetization in the black do-main but did not affect the magnetization of the whitedomain. The opposite situation is observed when thesample is exposed to − pulses. Thus, during the pres-ence of a single 40 fs laser pulse, information about thephotons’ angular momentum is transferred to the mag-netic medium and subsequently recording occurs. Theseexperiments unambiguously demonstrate that all-opticalmagnetization reversal can be achieved by single 40 fscircularly polarized laser pulses without the aid of anexternal magnetic field.

As a simple illustration of optomagnetic recording it isshown in Fig. 45 how optically written bits can be over-lapped and made much smaller than the beam waist by

Free motion

NO-INERTIA

INERTIA

t

t

t

t

Stimulus

Free motionStimulus

FIG. 43. Color online Noninertial and inertial scenarios totransfer a point mass over a potential. The noninertial mecha-nism above requires a continuous driving force that pulls themass over the potential barrier. A similar scenario is realizedin magnetization reversal via precessional motion in ferromag-nets. In contrast, in the inertial mechanism below, during theaction of the driving force the coordinate of the particle ishardly changed, but the particle acquires enough momentumto overcome the barrier afterwards. From Kimel et al., 2009.

σ+

-σ

µm150

FIG. 44. The effect of single 40 fs circular polarized laserpulses on the magnetic domains in Gd22Fe74.6Co3.4. Thedomain pattern was obtained by sweeping at high-speed50 mm/s circularly polarized beams across the surface sothat every single laser pulse landed at a different spot. Thelaser fluence was about 2.9 mJ/cm2. The small size variation inthe written domains is caused by the pulse-to-pulse fluctuationof the laser intensity. From Stanciu, Hansteen, et al., 2007.

µm300

FIG. 45. Color online Demonstration of compact all-opticalrecording of magnetic bits. This was achieved by scanning acircularly polarized laser beam across the sample and simulta-neously modulating the polarization of the beam between leftand right circular. From Stanciu, Hansteen, et al., 2007.



modulating the polarization between + and − as thelaser beam is swept across the sample. High-density re-cording may be achieved in principle by employing es-pecially designed near-field antenna structures Challe-ner et al., 2003 such as those currently being developedfor heat-assisted magnetic recording HAMR.

5. Reversal mechanism via a nonequilibrium state

Although the experiments in Sec. VI.C.4 showed theintriguing possibility of triggering magnetization reversalwith a femtosecond stimulus, the relevant time scalesand mechanism of such an optically induced magnetiza-tion reversal are unclear, since a precessional switchingwithin 40 fs would require enormous effective magneticfields above 102 T and unrealistically strong damping. Toclarify the dynamics of the switching process, femtosec-ond single-shot time-resolved optical imaging of mag-netic domains was used together with multiscale model-ing beyond the macrospin approximation Vahaplar etal., 2009.

GdFeCo samples were excited by a single circularlypolarized laser pulse. A single linearly polarized probepulse delayed with respect to the pump was used forultrafast imaging of the magnetic domain structure viathe magneto-optical Faraday effect. Magnetic domainswith magnetization parallel up or antiparallel downto the sample normal are seen as white or black regions,respectively, in an image on a charge-coupled devicecamera. After each write-read event, the initial magneticstate was restored by applying a magnetic field pulse.Taking images of the magnetic structure for differentdelays between the pump and probe pulses, we wereable to visualize the ultrafast dynamics of the laser-induced magnetic changes in the material.

Figure 46 shows images of magnetic domains at differ-ent delays after excitation by right-handed + or left-handed − circularly polarized pulses. In the first fewhundreds of femtoseconds, pump pulses of both helici-ties bring the originally magnetized medium into a

strongly nonequilibrium state with no measurable netmagnetization, seen as a gray area in the second columnof Fig. 46, the size of which is given by the laser beamdiameter and the intensity profile. In the following fewtens of picoseconds either the medium relaxes back tothe initial state or a small 5 m domain with a re-versed magnetization is formed. It is thus obvious thati switching proceeds via a strongly nonequilibrium de-magnetized state, clearly not following the conventionalroute of precessional motion and ii the final state isdefined by the helicity of the 100 fs pump pulse lastcolumn of Fig. 46.

The switching time is in fact surprising because in con-trast to heat-assisted magnetic recording Hohlfeld et al.,2001, the reversal time is much longer than the effectivelight-induced magnetic field pulse Heff. The duration ofthe latter teff is still an open question but can be differ-ent from the full width at half maximum FWHM of theoptical pulse. However, teff can be estimated from thespectrum of terahertz radiation generated by an Fe filmexcited by a subpicosecond visible laser pulse. Based ona half-period oscillation with the lowest frequency in thespectrum reported by Bartelt et al. 2007, the maximumteff is about 3 ps. The pulse amplitude Heff, for a typicalpump fluence of 2.5 J /m2 and the magneto-optical con-stant of GdFeCo 3105 deg/cm, reaches 20 T.

To understand this reversal process the Landau-Lifshitz-Bloch equation was solved. This macrospin ap-proach encapsulates well the response of a set ofcoupled atomic spins subjected to rapidly varying tem-perature changes, including the reduction in the magni-tude of M; see Sec. II.B. The temperature-dependentparameters for the LLB equation, i.e., the longitudinaland transverse susceptibilities and the temperaturevariation in the magnetization, were calculated atomisti-cally using Langevin dynamics combined with a Landau-Lifshitz-Gilbert equation for each spin Kazantseva,Hinzke, et al., 2008. The laser-induced increase in thekinetic energy temperature of the electrons was simu-lated using a two-temperature model Kazantseva,Nowak, et al., 2008, the parameters for which weretaken to be typical for a metal Zhang, Hübner, et al.,2002 electron heat capacity Ce=1.8106 J /m3 K atroom temperature and electron-phonon coupling Gel-ph=1.71018 J /K s. The simulations showed that in thefirst 100 fs the electron temperature Tel increases from300 K up to Tel

* and relaxes with a time constant of0.5 ps down to the vicinity of TC. Simultaneously thespins experience a short pulse of effective magnetic fieldwith amplitude Heff=20 T and duration teff. The possi-bility of magnetization reversal under these circum-stances has been analyzed numerically for a volume of30 nm3. The results of the simulations are plotted inFig. 47a as a phase diagram, defining the combinationsof Tel

* and teff for which switching occurs for the givenHeff. As can be seen from the diagram, a field durationas short as teff=250 fs can reverse the magnetization.For better insight into the reversal process, we simulatedthe latter for teff=250 fs and Tel

* =1130 K. The result is

Negativedelay 1 ps 12 ps 58 ps

Timedelay91 ps Final state

...

...

...

...

“up”

“down”

FIG. 46. Color online The magnetization evolution inGd24Fe66.5Co9.5 after the excitation with right- + and left-handed + circularly polarized pulses at room temperature.The domain is initially magnetized “up” white domain and“down” black domain. The last column shows the final stateof the domains after a few seconds. The circles show areasactually affected by pump pulses. From Vahaplar et al., 2009.



plotted in Fig. 47b, showing that already after 250 fsthe effective fields of two different polarities bring themedium into two different states, while the magnetiza-tion is nearly quenched within less than 0.5 ps. This isfollowed by relaxation either to the initial state or to thestate with reversed magnetization, achieved alreadywithin 10 ps. The considered pulse duration teff of250 fs is only 2.5 times larger than the FWHM of theoptical pulse in the experiments and well within the es-timated lifetime of a medium excitation responsible forHeff. An important result is that in simulations teff wasfound to be sensitive to the parameters of the two-temperature model. In particular, an increase in Gel-phleads to a reduction in the minimum field pulse duration.This shows that the suggested mechanism may, in prin-ciple, explain the experimentally observed laser-inducedmagnetization reversal. This magnetization reversaldoes not involve precession; instead, it occurs via a lin-ear reversal mechanism, where the magnetization firstvanishes and then reappears in the direction of Heff,avoiding any transverse magnetization components, asshown in Fig. 46. Exactly as in the experiments, the ini-tial 250 fs effective magnetic field pulse drives the rever-sal process, which takes one to two orders of magnitudelonger.

The state of magnetization after the pulse is criticallydependent on the peak temperature Tel

* and the pulseduration. For ultrafast linear reversal by a 250 fs fieldpulse it is necessary that, within this time, Tel

* reaches thevicinity of TC. If, however, this temperature is too highand persists above TC for too long, the reversed magne-tization is destroyed and the effect of the helicity is lost.This leads to a phase diagram Fig. 47a, showing that

the magnetization reversal may occur in a certain rangeof Tel

* . Such a theoretically predicted reversal window ofelectron temperature can be easily verified in the experi-ment when one changes the intensity of the laser pulse.Figure 47c shows the switchability, i.e., the differencebetween the final states of magnetization achieved in theexperiment with +- and −-polarized pulses, as a func-tion of Tel

* , calculated from the laser pulse intensity. It isseen that, indeed, switching occurs within a fairly narrowlaser intensity range. For intensities below this windowno laser-induced magnetization reversal occurs, while ifthe intensity exceeds a certain level both helicities resultin magnetization reversal since the laser pulse destroysthe magnetic order completely, which is then recon-structed by stray fields Ogasawara et al., 2009. Despitethis qualitative agreement between simulations and ex-periments, the experimentally observed reversal time isseveral times larger than the calculated 10 ps. The latter,however, is calculated for a 30 nm domain whereas inour experiments the magnetization in a 5 m spot is ma-nipulated.

D. Excitation of the magnetization dynamics with linearlypolarized light

As shown above in many examples, the phase of theoptically excited spin precession is unambiguously de-fined by the helicity of the laser pulses. This fact actuallycreated certain controversies in the description of thephenomena, namely, about the role of the angular mo-mentum of the photons in the process. Thus, a completeunderstanding of the processes taking place duringand after the optical excitation by a subpicosecond laser

Theory

Experiment

Theory(a) (b)

(c)

FIG. 47. Color online Magnetization reversal triggered by a combined action of a short magnetic field pulse Heff and ultrafastheating of electrons to a temperature of Tel

. a Phase diagram showing the magnetic state of the 30 nm3 volume achieved within10 ps after the action of the optomagnetic pulse with parameters Heff=20 T, teff, and Tel

. b The averaged z component of themagnetization vs delay time as calculated for 250 fs magnetic field pulses Heff=20 T and Tel

=1130 K. c Switchability vs the pumpintensity for Gd22Fe68.2Co9.8 at room temperature. Peak electron temperature Tel

is calculated using Ce. Note that in this range ofintensities the amplitude of the effective light-induced magnetic field varies within 19.2–20.8 T. From Vahaplar et al., 2009.



pulse is of particular importance. Here we expand thediscussion and describe the nonthermal optomagneticeffects with linearly polarized pump pulses Kalashni-kova et al., 2007, 2008, measured in iron borateFeBO3 that possesses a strong magneto-optical re-sponse, both linear Faraday effect and quadraticMLD in the magnetization.

1. Detection of the FMR mode via magnetic linearbirefringence in FeBO3

Iron borate FeBO3 crystallizes in the rhombohedral

calcite-type crystallographic structure space group R3cwith two formula units per unit cell Bernal et al., 1963;Diehl, 1975; Wijn, 1994. Below the Néel temperatureTN=348 K, the magnetic moments M1 and M2 of thetwo Fe3+ sublattices are coupled antiferromagnetically.It is common to describe such magnetic structure interms of the ferromagnetic vector M=M1+M2 and theantiferromagnetic vector L=M1−M2. The magnetocrys-talline anisotropy of FeBO3 corresponds to an “easy-plane” type of spin alignment, where spins of both sub-lattices are perpendicular to the crystallographic z axis.The effective magnetocrystalline anisotropy field is HA1.7 kOe Schober, 1976. The presence of the antisym-metric exchange interactions Dzyaloshinskii, 1957;Moriya, 1960 leads to a canting of the spins in thex-y plane. The effective Dzyaloshinskii field is HD61.9 kOe Schober, 1976, corresponding to a cantingangle of HD /HE1° and a net magnetic moment of4Ms=115 G at T=300 K Kurtzig et al., 1969 238 Gat T→0 K. The magnetocrystalline anisotropy in thex-y plane is as weak as HA0.26 Oe Schober, 1976.

The optically excited spin precession in FeBO3 wasstudied by means of a magneto-optical pump-probetechnique see Sec. III.A at photon energy of E0=1.54 eV. Two experimental geometries were used: themagnetic field was always applied along the x axis, whilethe pump pulses were propagating along either the z orthe y axis.

The spin precession induced by the pump pulses leadsto a perturbation of the dielectric permittivity tensorsee Sec. III.B which, in turn, leads to a change in thepolarization of the probe beam via either the Faradayeffect or magnetic linear birefringence.

The rotation of the probe polarization as a function oftime delay between pump propagating along the z axisand probe pulses is shown in Fig. 48a for different val-ues of the applied magnetic field. Clear field-dependentoscillations are observed in these data. In Fig. 48b thedependence of the oscillation frequency on the magneticfield is plotted. This dependence is in good agreementwith the behavior of the FMR mode in FeBO3 Jantz etal., 1976. Also the dependence of the frequency of theoscillations on temperature Fig. 48d is consistent withthe temperature behavior that one would expect for theFMR mode. Thus these data show that the laser pulsespropagating along the z axis excite the FMR mode ofcoherent spin precession in FeBO3.

The FMR mode supposes oscillations of the lx, my,and mz components of lt and mt. There are variousmagneto-optical effects that can serve as a probe of sucha precession. That is, in the experimental geometryshown in Fig. 48a, the spin precession may lead to atransient rotation of the probe polarization via both theFaraday effect and the magnetic linear birefringenceMLB. For instance, mzt can be detected using theFaraday effect with the probe polarization rotation tgiven by

Ft = 0d!xy

a tn

= 0dKmzt

n, 69

where 0 is the pulse central frequency, d is the samplethickness, and n is the refractive index. In turn, lxt os-cillations cause MLB, which also leads to a rotation ofthe probe polarization,

pump

probe

L

MH

x

φyz

(a)

(b)

(c) (d)

FIG. 48. Color online Excitation and detection of the ferro-magnetic mode of spin precession by linearly polarized pumppulses. a Experimental geometry. b Pump-induced rotationof the probe polarization as a function of time delay betweenlinearly polarized pump and probe pulses for different valuesof the applied magnetic field. c The frequency 0 of the ob-served oscillations as a function of the applied field strength H.d The frequency 0 of the oscillations as a function of tem-perature T. The results were obtained for a pump intensity of10 mJ/cm2. Pump pulses were linearly polarized with azi-muthal angle =45°. Results in b and c are obtained at atemperature T=10 K. From Kalashnikova et al., 2008.



MLBt = 0d!xy

s tn

0dGLy

0lxtcos2"n

, 70

where G=b1−b2 is the magneto-optical coefficient and "is the incoming polarization of the probe pulse. In thecase of the Faraday effect, the measured signal twould not depend on the incoming polarization. Notethat, in the given geometry, it is the symmetric part ofthe off-diagonal tensor components which is most af-fected by the FMR mode of precession linear with re-spect to lxt. The change in the diagonal components isquadratic with respect to lxt and can be neglected. Ex-perimentally, a clear 180° dependence of the signal on "was observed. Therefore the measured signal originatesfrom the transient MLB Eq. 70 and reveals an in-plane motion of the antiferromagnetic vector L. The factthat MLB dominates over the Faraday effect is causedby the strong ellipticity of the FMR mode of spin pre-cession: since the magneto-optical constants K and Gare comparable for the photon energy E=1.54 eV, theratio between the transient Faraday effect and MLB ismainly defined by the ratio of the dynamic componentsof magnetic vectors mz / lx that is as small as 0.01 becauseof strong easy-plane-type anisotropy.

2. Excitation of coherent magnons by linearly and circularlypolarized pump pulses

Figure 49a shows the spin precession excited by lin-early polarized pump pulses for various azimuthal orien-tations of the pump polarization , as shown in Fig.

49b. The spin precession amplitude depends on Fig.49c.

The effect of circularly polarized pulses appeared todepend on the mutual orientation of the pump propaga-tion direction and antiferromagnetic vector Ly

0. As canbe seen from Fig. 50a, the circularly polarized pumppulses propagating along the z axis do excite spin pre-cession but a change in their helicity affects neither theamplitude nor the phase of the oscillations

+−

−

=0. In contrast, the spin precession excited by circularlypolarized pump pulses propagating along the y axischanges phase by 180° when the pump helicity of thelight is reversed Fig. 50b. We previously explainedhow circularly polarized pulses act on the spins as effec-tive magnetic field pulses with a direction depending onthe helicity. The phase of the excited precession, there-fore, should be defined by the helicity of the pumppulses. This inverse Faraday effect is determined by thesame magneto-optical susceptibility that also accountsfor the Faraday effect and is expected to be allowed inmedia of any symmetry. Therefore, the absence of thiseffect in the results shown in Fig. 50a is, at first glance,puzzling considering the fact that the Faraday effect inFeBO3 is one of the strongest among the iron oxidesKurtzig et al., 1969. Note that the incompleteness ofsuch an interpretation was pointed out by Woodford,2007.

Below we show that this together with the observedpolarization dependence of the excitation can be ex-plained by taking into account the strongly ellipticalcharacter of the spin precession modes in FeBO3. Thatis, it is not only the strength of the generated effectivefield that plays a role but also susceptibility of the sys-tem to that field.

3. ISRS as the mechanism of coherent magnon excitation

We first consider the excitation of spin precession bythe linearly polarized pump pulses propagating along

(a)

(b) (c)θ

H

x

y

M

L

pump

z

FIG. 49. Color online Laser-induced magneto-optical re-sponse from FeBO3: a Oscillations of the probe polarizationas a function of the time delay between linearly polarizedpump and probe pulses for different orientations of the pumppolarization. The pump pulses propagate along the z axisshown in b. The probe pulse is not shown in b for simplicity.c The amplitude of oscillations as a function of the pumppolarization azimuthal angle symbols and its fit using Eq. 71line. The results are obtained at T=10 K, H=1.75 kOe, andI=10 mJ/cm2. From Kalashnikova et al., 2008.

(a) (b)

H

xM

L

H

y

M

Lz

H=1.75 kG H=2 kG

pump

pump

y

x

FIG. 50. Color online Spin precession excited by circularlypolarized pump pulses propagating along a the z axis andalong b the y axis. +−− is the difference between the spinprecession amplitude excited by right- and left-handed circu-larly polarized pump pulses. In both cases the magnetic field isapplied along the x axis and the probe is at 10° from the pumppropagation direction. The results are obtained at T=10 K andI=10 mJ/cm2. From Kalashnikova et al., 2008.



the z axis Fig. 48a. Combining Eqs. 65, 66, and70 see Sec. VI.A.3 for the transient rotation of theprobe polarization caused by magnons excited via ISRS,we obtain

lint = AI0

4ncGLy

02ax2 sin 2 sin 0t , 71

where A=0d /n, is the azimuthal angle of the pumppolarization, 0 is the FMR frequency, G=b1−b2 is themagneto-optical coefficient, I0 is the integrated pumppulse intensity, 0 is the pump pulse central frequency,and n is the refraction index at 0.

Comparison of Eq. 71 with the experimental resultsshows good agreement. In particular, the experimentallyobtained dependence of the oscillation amplitude on theapplied magnetic field H Fig. 51a is described bylinHax

21/01/H, following Eq. 71. Thepump-induced oscillations of the probe polarizationshould, according to Eq. 71, possess a sinelike behaviorin the time domain, which is, indeed, observed in theexperiment Figs. 51b and 51d for the magneticallysaturated sample. The theoretically predicted depen-dence of the oscillation amplitude on the polarization ofthe pump pulses linsin 2 shows good agreementwith our experimental data Fig. 49c.

In the case of circularly polarized pump pulses, thetransient rotation of the probe polarization excited bypulses propagating along the z and y axes, respectively,can be expressed as

±zt = ± AGLy

0ax I0

4ncKbz cos 0t bzax,

72a

±yt = ± AK2 I0

4ncK1by + K2axax sin 0t ax

2.

72b

Now, keeping in mind that the FMR precession pos-sesses strong ellipticity, i.e., bzax, we obtain that

±z±ylinz. Our experimental data Fig. 50

show that, indeed, circularly polarized laser pulses effec-tively excite a helicity-dependent spin precession onlywhen propagating along the y axis. It is worth notingthat these results are in perfect agreement with the ob-servation of spontaneous Raman scattering in FeBO3reported by Jantz et al. 1976. It was shown there that,because of spin precession ellipticity, the scattering oflight propagating along the z axis is defined mainly bythe second-order magneto-optical constant i.e., by G inEq. 71, while the scattering of light propagating alongthe y axis is defined by a first-order magneto-optical con-stant i.e., by K1 and K2 in Eq. 72b. A similar effect ofspin precession ellipticity on the Raman scattering inten-sity was reported by White et al. 1982 for orthoferrites.

Here we comment on the distinction between theseresults and those discussed in Sec. V.A.1. There theexcitation of coherent spin precession by linearly polar-ized laser pulses in ferrimagnetic garnet films was dueto a quasistationary photoinduced change in magneticanisotropy. The oscillations of lxt and, consequently,t excited via such a process should obey the1−cos 0t-like dependence on the time delay betweenpump and probe pulses. Here t lxtsin 0t seeFigs. 48a, 49a, and 51b. In addition, the depen-dence of the amplitude of the excited precession on thepump pulse intensity Fig. 51d is also different fromthe one due to the photomagnetic effects Hansteen etal., 2006, where the photoinduced change is related tothe absorption by impurity centers. As their concentra-tion is limited, the dependence of the excited spin pre-cession amplitude on the pump intensity indeed showedsaturation. However, no saturation of the spin preces-sion amplitude with pump intensity was observed in thepresent case Fig. 51b.

Thus, the experimentally observed excitation of co-herent magnons in FeBO3 can be unambiguously de-scribed in terms of impulsive stimulated Raman scatter-ing. The efficiency of the excitation by the pump pulseswith certain polarization is defined by the ellipticity ofthe magnon mode.

It is therefore interesting and important to clarify theapproach of the ultrafast inverse Faraday effect usedpreviously to describe the light-induced spin precessionin orthoferrites, garnets, and metallic alloys Kimel et al.,2007 and to compare it with this ISRS-based interpre-tation.

(a)

(c)

(b)

(d)

FIG. 51. Color online Experimental dependencies of the os-cillations amplitude 0 dots on a the magnetic field H andb pump intensity I0. Solid lines in a and b represent thedependencies described by Eq. 71. c Initial phase of thepump-induced oscillations as a function of the magnetic fieldH when described by t=0 sin0t+. A typical expe-rimental curve t is shown circles in d. For reference,the curves t=0 sin0t+0 solid line and t=0 sin0t+ /2 dashed line are shown. The experimentalresults shown in a–d are obtained for the pump polarization=45° and temperature T=10 K. From Kalashnikova et al.,2008.



4. Effective light-induced field approach

The excitation of coherent spin precession by shortlaser pulses has been described using the approach inwhich light acts on spins as an effective light-inducedmagnetic field Heff=KEE*. Strictly speaking, thiswas originally defined for a cubic paramagnetic mediumPershan et al., 1966. To describe the impulsive action oflight on the spins of a multisublattice medium, we intro-duce the light-induced effective fields

Hieff = − Hi,int/Mi, 73

where Mi is the sublattice magnetization and Hi,int is theHamiltonian describing the interaction of light with theith sublattice. For the case of a one-sublattice ferromag-netic medium, the further description of the light-matterinteraction is trivial because one ferromagnetic vector Mis sufficient to describe the collective response of themagnetic system to the light action see Sec. VI.A.2. Amultisublattice magnetic medium, however, is describedby several magnetic vectors M and Lj, where j=1, . . . ,n−1 and n is the number of magnetic sublattices. In par-ticular, FeBO3 has two magnetic sublattices with magne-tizations M1 and M2. The interaction of light with eachsublattice can be described by the effective field H1

eff

=−H1,int /M1 and H2eff=−H2,int /M2. Making a transi-

tion from the sublattice magnetizations to the ferromag-netic M=M1+M2 and antiferromagnetic L=M1−M2vectors, one obtains two effective fields

Heff = − Hint/M , 74a

heff = − Hint/L , 74b

which can also be understood as Heff=H1eff+H2

eff andheff=H1

eff−H2eff. The latter field accounts for the non-

equivalent responses of the Fe3+ ions at different crys-tallographic positions to the action of light. It can beshown that it is this field heff that induces the spin pre-cession in a weak ferromagnet, such as FeBO3. Notethat, in general, for a medium with n magnetic sublat-tices there are n−1 fields hj

eff=−Hint /Lj.A detailed analysis of the response of the FeBO3 spin

system reveals that the efficiency of the excitation byeither linearly or circularly polarized optical pulses isdetermined by the details of the magnetic structures. Inparticular, the effect of a circularly polarized pulsepropagating along the y axis resembles that of a linearlypolarized one propagating along the z axis. Namely, theeffective field induced by the former and latter pulses ishx

eff and the spins move out of the x-y plane during thepulse. In analogy to the inverse Faraday effect, the ap-pearance of an effective fields Heff and heff generated bylinearly polarized laser pulses can be seen as a manifes-tation of the inverse Cotton-Mouton effect Zon andKupershmidt, 1983.

Finally, we note that the approaches based on theequation of motion for magnon normal coordinatesSecs. VI.A and VI.D.3 and on the LL equations areequivalent for the treatment of the experimental results

presented in this section. However, the latter approachwill be more convenient for the description of large de-viations or even switching Stanciu, Hansteen, et al.,2007 of spins caused by strong laser pulses, when theequation of motion Eq. 49 becomes highly nonlinear.

VII. CONCLUSIONS AND OUTLOOK

Optical manipulation of magnetic order by femtosec-ond laser pulses has developed into an exciting and stillexpanding research field that keeps being fueled by acontinuous stream of new and sometimes counterintui-tive results. The discovery of ultrafast demagnetizationby a 60 fs laser pulse, that is, demagnetization on a timescale much faster than the spin-lattice relaxation time,has triggered a wealth of experimental and theoreticalresearch focused on the questions: How fast and by whatmeans can magnetic order be changed and manipulated?What are the channels of energy and angular momen-tum transfer between the photons and the spins and thevarious other degrees of freedom?

The goal of this review was to describe the progress inour understanding of femtosecond laser manipulation ofmagnetic order in the past 13 years after the originaldiscovery with a systematic review of the results ob-tained in a broad class of materials that includes bothferromagnetically and antiferromagnetically orderedmetallic, semiconducting, and dielectric materials. As amagnetic medium is characterized by a specific magneticordering, resulting from the interactions among elec-trons, spins, and the lattice, such a systematic approachmay help to reveal the roles of the electronic band struc-ture, the spin-orbit coupling, and the magnetic structurein this process. An excitation by a femtosecond laserpulse brings a medium into a strongly nonequilibriumstate, where a conventional description of magnetizationin terms of thermodynamics is no longer valid. The in-teractions between the relevant reservoirs are describedby the spin-orbit, spin-lattice, and electron-phonon in-teractions, which span a time scale from picosecondsto nanoseconds within these various materials, whileelectron-electron and exchange interactions set thelower boundary of the possible time scales.

The role of the electronic structure is demonstratedby a comparison of the demagnetization rates betweenmetals and dielectrics: the observed ultrafast subpico-second demagnetization in metals can be understood bythe heating of the itinerant electrons by the optical1–2 eV laser pulse. The hot electron gas plays the roleof a thermal bath for spins and thus facilitates both anintensification of spin-flip processes and the demagneti-zation. In contrast, in dielectrics, even the electrons thatare excited across the band gap cannot gain a lot ofkinetic energy and will thus not contribute in the processof demagnetization. There the latter occurs via laser-induced heating of the lattice, after the spins are heatedvia the spin-lattice interaction on a time scale of a100 ps.

Even without itinerant electrons, however, magnetiza-tion can be manipulated on a time scale faster than the



spin-lattice relaxation by optical manipulation of themagnetic anisotropy. It has been shown that this can bedone either via laser-induced heating, using the mostlymore i.e., than spin-lattice effective electron-phononinteraction 1 ps, or via direct photomagnetic opti-cal pumping of specific electronic transitions. The lattereffects have been mainly observed in materials with lowconductivity and long excited-state lifetimes, i.e., semi-conductors and dielectrics; the lifetime of this effect isexpected to be very short in metals Gamble et al., 2009.

In the past five years a totally new, so-called optomag-netic approach was introduced that depended on thepresence of strong spin-orbit interaction in the groundor excited electronic states. The corresponding magneto-optical susceptibility that is responsible for the magneto-optical Faraday effect, i.e., the effect of the magnetiza-tion on the polarization of light, was demonstrated to bealso responsible for the inverse Faraday effect: the ma-nipulation of the magnetization by circularly polarizedlight. Because of this optomagnetic effect, an intensefemtosecond circularly polarized optical laser pulse wasdemonstrated to act on spins similarly to an equivalentlyshort effective magnetic field pulse with a strength up to20 T. With such an optically induced magnetic field, onemay selectively excite different modes of magnetic reso-nance, realize quantum control of spin oscillations, andtrigger phase transitions nonthermally on a subpicosec-ond time scale. It was shown that a single 40 fs circularlypolarized laser pulse can controllably reverse magnetiza-tion, where the direction of this switching was solely de-termined by the helicity of the light pulse.

These results demonstrated that femtosecond laserpulses are indeed novel and effective stimuli for the ma-nipulation of magnetic order. Recent results that showthat even linearly polarized laser pulses can lead to simi-lar effects have indicated the importance of the detailsof the magnetic structure for the understanding of theseoptomagnetic results.

An additional comment should be made about the in-terpretation of the experimental results. Soon after theoriginal discovery of ultrafast demagnetization via theobservation of an ultrafast quenching of the magneto-optical Kerr signal, questions arose about the validity ofthe use of magneto-optical probes to follow changes inmagnetic order at such short time scales. Recent theo-retical work shows, however, that the correspondencebetween optical and magneto-optical response sensi-tively depends on wavelength and pulse length, with amaximum delay between the two on the order of tens offemtoseconds Zhang et al., 2009. There have been sev-eral attempts to describe ultrafast laser-induced spin dy-namics using ab initio theory and atomistic and micro-magnetic simulations. None of these approaches has sofar been successful in providing a complete theoreticalframework, suggesting that a solution might be found inan interdisciplinary multiscale approach.

The same can be said for the experimental ap-proaches: for a further and detailed understanding of“femtomagnetism,” a combination of experimental ap-proaches such as optical excitation and x-ray probing

will allow a nanometer-scale picture to be obtained ofwhat actually happens during and after the excitation ofa magnetic medium by a femtosecond laser pulseStamm et al., 2007; Lorenc et al., 2009. The develop-ment of novel femtosecond slicing approaches at syn-chrotron sources is therefore very promising. Moreover,this challenge is one of the motivations for the develop-ment of the next generation of x-ray sources–x-ray freeelectron lasers able to generate intense subpicosecondpulses of tunable x-ray radiation.

It may also be expected that the further progress inthe laser control of magnetism will be associated withpushing the duration of the optical stimulus down to theattosecond time scale. In this time domain, owing to thecoherence of optical excitations, the interaction of lightwith solids will drastically depend on the shape and du-ration of the laser pulse creating thus the new degrees offreedom.

Moreover, considering the progress in the develop-ment of compact ultrafast lasers Keller, 2003, opticalcontrol of magnetic order may potentially revolutionizedata storage and information processing technologies.Regarding these potential applications, for that it will beessential to extend the present state of optical manipu-lation and control of magnetic order toward smallernanoscale dimensions. Given the rapid developments innano-optics and plasmonics, such possibilities do notseem to be too far fetched. Either way, the present re-view shows that “femtomagnetism” is a vibrant new fieldwith a wealth of new phenomena waiting to be discov-ered and explored.

ACKNOWLEDGMENTS

This review has been possible only through the dedi-cated work of our co-workers A. M. Balbashov, F. Han-steen, A. Itoh, A. M. Kalashnikova, C. D. Stanciu, A.Tsukamoto, P. A. Usachev, and K. Vahaplar, supple-mented by daily technical support of A. J. Toonen andA. F. van Etteger. We are grateful to M. Aeschlimann, C.H. Back, J.-Y. Bigot, U. Bovensiepen, R. Chantrell, H.Dürr, V. N. Gridnev, B. A. Ivanov, M. I. Katsnelson, B.Koopmans, D. Mazurenko, A. Melnikov, M. Münzen-berg, U. Nowak, P. M. Oppeneer, R. V. Pisarev, I. Radu,P. van Loosdrecht, as well as to all Ph.D. students andpostdoctoral fellows of the Spectroscopy of Solids andInterfaces group for stimulating discussions. This re-search received funding from Nederlandse Organisatievoor Wetenschappelijk Onderzoek NWO, Stichtingvoor Fundamenteel Onderzoek der Materie FOM, theDutch Nanotechnology initiative NanoNed, and EC FP7contributions under Grants No. NMP3-SL-2008-214469UltraMagnetron and No. N 214810 FANTOMAS.

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