Low-Dimensional Magnetic Systems · Leonid Pryadko, USA Charles Rosenblatt, USA Alfonso San-Miguel,...

Low-Dimensional Magnetic SystemsGuest Editors: Roberto Zivieri, Giancarlo Consolo, Eduardo Martinez, and Johan Åkerman

Advances in Condensed Matter Physics

Low-Dimensional Magnetic Systems

Advances in Condensed Matter Physics


Guest Editors: Roberto Zivieri, Giancarlo Consolo,Eduardo Martinez, and Johan Åkerman

Copyright © 2012 Hindawi Publishing Corporation. All rights reserved.

This is a special issue published in “Advances in Condensed Matter Physics.” All articles are open access articles distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided theoriginal work is properly cited.

Editorial Board

A. Alexandrov, UKDario Alfe, UKBohdan Andraka, USADaniel Arovas, USAA. Bansil, USAWard Beyermann, USALuis L. Bonilla, SpainMark Bowick, USAGang Cao, USAAshok Chatterjee, IndiaR. N. P. Choudhary, IndiaKim Chow, CanadaOleg Derzhko, UkraineThomas Peter Devereaux, USAGayanath Fernando, USAJörg Fink, GermanyYuri Galperin, NorwayRussell Giannetta, USAGabriele F. Giuliani, USAJames L. Gole, USA

P. Guptasarma, USAM. Zahid Hasan, USAYurij Holovatch, UkraineChia-Ren Hu, USADavid Huber, USANigel E. Hussey, UKPhilippe Jacquod, USAJan Alexander Jung, CanadaDilip Kanhere, IndiaFeo V. Kusmartsev, UKRobert Leisure, USAHeiner Linke, USARosa Lukaszew, USADmitrii Maslov, USAYashowanta N. Mohapatra, IndiaAbhijit Mookerjee, IndiaVictor V. Moshchalkov, BelgiumCharles Myles, USADonald Naugle, USAVladimir A. Osipov, Russia

Rolfe Petschek, USAS. J. Poon, USARuslan Prozorov, USALeonid Pryadko, USACharles Rosenblatt, USAAlfonso San-Miguel, FranceMohindar S. Seehra, USASergei Sergeenkov, RussiaIvan Smalyukh, USADaniel L. Stein, USAMichael C. Tringides, USASergio E. Ulloa, USAAttila Virosztek, HungaryMarkus R. Wagner, GermanyNigel Wilding, UKGary Wysin, USAFajun Zhang, GermanyGergely Zimanyi, USA

Contents

Low-Dimensional Magnetic Systems, Roberto Zivieri, Giancarlo Consolo, Eduardo Martinez,and Johan ÅkermanVolume 2012, Article ID 671416, 1 page

Micromagnetic Study of Synchronization of Nonlinear Spin-Torque Oscillators to Microwave Currentand Field, M. Carpentieri, G. Finocchio, A. Giordano, B. Azzerboni, and F. LattaruloVolume 2012, Article ID 951976, 5 pages

Hamiltonian and Lagrangian Dynamical Matrix Approaches Applied to Magnetic Nanostructures,Roberto Zivieri and Giancarlo ConsoloVolume 2012, Article ID 765709, 16 pages

On the Formulation of the Exchange Field in the Landau-Lifshitz Equation for Spin-Wave Calculation inMagnonic Crystals, M. Krawczyk, M. L. Sokolovskyy, J. W. Klos, and S. MamicaVolume 2012, Article ID 764783, 14 pages

On the Travelling Wave Solution for the Current-Driven Steady Domain Wall Motion in MagneticNanostrips under the Influence of Rashba Field, Vito Puliafito and Giancarlo ConsoloVolume 2012, Article ID 105253, 8 pages

Micromagnetic Investigation of Periodic Cross-Tie/Vortex Wall Geometry, Michael J. DonahueVolume 2012, Article ID 908692, 8 pages

Static Properties and Current-Driven Dynamics of Domain Walls in Perpendicular MagnetocrystallineAnisotropy Nanostrips with Rectangular Cross-Section, Eduardo MartinezVolume 2012, Article ID 954196, 21 pages

Heterostructures for Realizing Magnon-Induced Spin Transfer Torque, P. B. Jayathilaka, M. C. Monti,J. T. Markert, and Casey W. MillerVolume 2012, Article ID 168313, 7 pages

Thermodynamics of the Classical Planar Ferromagnet Close to the Zero-Temperature Critical Point: AMany-Body Approach, L. S. Campana, A. Cavallo, L. De Cesare, U. Esposito, and A. NaddeoVolume 2012, Article ID 619513, 15 pages

Spin-Wave Band Structure in 2D Magnonic Crystals with Elliptically Shaped Scattering Centres,Sławomir Mamica, Maciej Krawczyk, and Jarosław Wojciech KłosVolume 2012, Article ID 161387, 6 pages

Gap Structure and Gapless Structure in Fractional Quantum Hall Effect, Shosuke SasakiVolume 2012, Article ID 281371, 13 pages

Semiclassical Description of Anisotropic Magnets for Spin S = 1, Khikmat Muminov and Yousef YousefiVolume 2012, Article ID 749764, 3 pages

Hindawi Publishing CorporationAdvances in Condensed Matter PhysicsVolume 2012, Article ID 671416, 1 pagedoi:10.1155/2012/671416

Editorial


Roberto Zivieri,1 Giancarlo Consolo,2 Eduardo Martinez,3 and Johan Åkerman4

1 Department of Physics, University of Ferrara, 44122 Ferrara, Italy2 Department of Sciences for Engineering and Architecture, University of Messina, 98166 Messina, Italy3 Department of Applied Physics, University of Salamanca, 37008 Salamanca, Spain4 Department of Physics, University of Gothenburg, 41296 Gothenburg, Sweden

Correspondence should be addressed to Roberto Zivieri, [email protected]

Received 2 August 2012; Accepted 2 August 2012

Copyright © 2012 Roberto Zivieri et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

The interest in the nanoscale properties of low-dimensionalmagnetic systems has grown exponentially during the lastdecades and has attracted the attention of both experimen-talists and theorists. The state of the art of these investiga-tions has indeed allowed to give valuable insights into theunderlying physics of complex magnetization dynamicsdriven by magnetic fields, electric currents and thermaleffects. At the same time, such studies have found, in rela-tively short times, several applications at industrial level inthe field of spintronics and magnonics as magnetic memo-ries, microwave oscillators, modulators, sensors, logic gates,diodes and transistors.

The goal of this special issue is to offer a variety of recentdevelopments on this topic by gathering contributions aris-ing from several specialists in the field of nanomagnetism.The strength of this issue lies indeed on its “variety”: theproperties of these systems are, in fact, investigated from theviewpoint of physicists, engineers and mathematicians. Also,the issue encloses studies carried out at both mesoscopic andatomic scales, as well as results of both theoretical approaches(analytical, numerical and, in some cases, even “hybrid”) andexperimental observations.

The covered topics range from the micromagnetic mod-eling of domain wall motion, dynamics of vortex structures,phase-locking phenomena in spintronic oscillators, exper-imental techniques for realizing heterostructures based onmagnon-induced spin transfer torque, band structure andexchange field in the Landau-Lifshitz equation for magnoniccrystals, gap and gapless structures in fractional quantumHall effect, semiclassical description of anisotropic magnetsand classical critical behaviour of Heisenberg ferromagnets.More specifically, within the subject dealing with domain

walls, for example, the structure of complex cross-tie/vortexwall structures in soft films has been studied in detail by usingmicromagnetic simulations whereas the influence of theRashba spin-orbit coupling on the current-induced dynam-ics has been investigated analytically. Regarding the exchangeinteraction governing the dynamics in magnonic crystals,a full analytical calculation of the exchange field acting onspin-wave dynamics from the microscopic Heisenberg modelhas been performed. Attention has been also devoted tothe study of thermodynamics in the case of classical planarferromagnets close to the zero-temperature critical point.

Two reviews are also included in this special issue. Thefirst one deals with two hybrid micromagnetic tools, basedon Hamiltonian and Lagrangian approaches, to model thespin-dynamics in laterally confined magnetic systems. Thesecond one is mostly devoted to the micromagnetic analysisof static and dynamic properties of magnetic domain walls inmaterials exhibiting perpendicular anisotropy.

Acknowledgments

This issue is the result of generous contributions to this newjournal from our colleagues. We are sincerely grateful to allof them for all their efforts. With papers by leading expertsin the field of micro- and nanomagnetism, we hope thatthis special issue will form a timely, open-access resource forexperts and novice alike.

Roberto ZivieriGiancarlo ConsoloEduardo Martinez

Johan Åkerman

Hindawi Publishing CorporationAdvances in Condensed Matter PhysicsVolume 2012, Article ID 951976, 5 pagesdoi:10.1155/2012/951976

Research Article

Micromagnetic Study of Synchronization of NonlinearSpin-Torque Oscillators to Microwave Current and Field

M. Carpentieri,1 G. Finocchio,2 A. Giordano,2 B. Azzerboni,2 and F. Lattarulo3

1 Dipartimento di Elettronica, Informatica e Sistemistica, University della Calabria, Via P. Bucci 42C, 87036 Rende, Italy2 Dipartimento di Fisica della Materia e Ingegneria Elettronica, University degli studi di Messina, C.da di Dio, 98100 Messina, Italy3 Dipartimento di Elettrotecnica ed Elettronica, Politecnico di Bari, Via Orabona 4, 70125 Bari, Italy

Correspondence should be addressed to M. Carpentieri, [email protected]

Received 30 March 2012; Revised 17 July 2012; Accepted 23 July 2012

Academic Editor: Giancarlo Consolo

Copyright © 2012 M. Carpentieri et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The nonautonomous dynamics of spin-torque oscillators in presence of both microwave current and field has been numericallystudied in nanostructured devices. When both microwave current and field are applied at the same frequency, integer phase lockingat different locking ratio is found. In the locking region, a study of the intrinsic phase shift between the locking force (current orfield) and the giant magnetoresistive signal as a function of the bias current is also exploited.

1. Introduction

In the last years, the effects due to a direct transfer of the spinangular momentum [1–4] in nanomagnets (magnetizationreversal [5, 6] or persistent oscillation of the magnetization[7, 8]) have opened new perspectives in the field ofnanotechnology. In particular, one of the most promisingapplications is the possibility to obtain a competitive gen-eration of nanoscale microwave oscillators, namely, spin-transfer torque oscillators (STOs) [9, 10]. Nowadays, STO ispromising from a technological point of view being one ofthe smallest auto-oscillators observed in nature. It exhibitsproperties such as frequency tunability on bias current andfield and narrow linewidth. To find practical application,STOs have to improve their output power. Some years ago,some researchers demonstrated that synchronized oscillatorsprovided increased output power [11, 12]. This effect, whichis receiving a great deal of attention due to its potentialapplications in telecommunications, has been studied bothexperimentally and theoretically for the purpose of under-standing and realizing mutual synchronization between twoor more STOs for microwave source applications. Later, otherstudies showed the synchronization of serially connected

STOs governed by phase locking to a microwave currentoffering a valid approach to fabricate STOs for output powerlevels closer to 1 μW [13, 14].

The synchronization phenomenon is based on the well-known property that when the external frequency fRF of a“weak” microwave current or field is close to the free runningoscillation frequency f0 of the STO, the self-oscillationmode moves and locks to the external frequency. Lockingphenomena are also present for the ratio r = fRF/ f0 closeto all integers (r = 1, 2, 3, etc.) and several rational values[15].

We recently studied the injection locking phenomenonbased on the application of a microwave field on per-pendicular materials [16]. These systems mainly operateunder “weak” microwave signal regime (the power ofthe microwave signal is negligible compared to the self-oscillation one) [9, 16–18].

In this work, by means of a micromagnetic study, thenonautonomous dynamical behavior of STO in presence ofmicrowave signal composed by the simultaneous applicationof microwave current density JAC and field hAC (both at thesame frequency) is studied. Also, the influence of a static fieldon the frequency behavior is investigated.

2 Advances in Condensed Matter Physics

120 nm 60 nm

3

4

5

6

7

8

Freq

uen

cy (

GH

z)

− 12 − 10 − 8 − 6 − 4

Cu

+M

+P

4 nm/Py

10 nm/Py

10 nm/Cu

8 nm/IrMn

x

yz

Current density (107 A/cm2)

Cu

J > 0

10

Happ

e− ,

(a)

− 12 − 10 − 8 − 6 − 4


0.8

1.2

1.6

2

2.4

2.8

3.2

Pow

er (

a.u

.)

(b)

Figure 1: (a) Frequency and (b) power amplitude versus dc current density when an out-of-plane field of 250 mT tilted 10◦ versus x-directionis applied. No microwave source is applied. Inset of (a): sketch of the structure.

2. Device and Numerical Details

We studied the dynamical behavior of exchange biasspin-valves composed by IrMn(8 nm)/Py(10 nm) (polar-izer)/Cu(10 nm)/Py(4 nm) (free layer) with elliptical cross-sectional area (120 nm × 60 nm) (see inset of Figure 1(a)).A Cartesian coordinate system has been introduced, wherethe x- and the y-axes are, respectively, related to the easy andhard in-plane axes of the ellipse. Our numerical experimentis based on the numerical solution of the Landau-Lifshitz-Gilbert-Slonczweski (LLGS) equation [1–3]. In addition tothe standard effective field (external, exchange, and self-magnetostatic), the Oersted field and the magnetostaticcoupling with the polarizer are taken into account. Theresulting equation (LLG + S) is expressed as

dmdτ

= − m× heff + αMs

(m× dm

dτ

)− χm× (m× p), (1)

where m = M/MS and heff = Heff /MS are the dimensionlessmagnetization vector and the effective field, respectively,τ = γ0Mst is the dimensionless time, γ0 is the gyromagneticratio, and Ms is the saturation magnetization of the freelayer. The first term on the right-hand side of (1) representsthe precessional torque around the effective field, whereasthe second one is the phenomenological dissipation term, αbeing the so-called Gilbert parameter. The third term in (1)is the dimensionless Slonczewski spin-transfer torque, wherep = P/Ms represents the dimensionless magnetization in thepinned layer, and the prefactor χ is given by

χ = μBγ0

JappM2s ed

g(

m, p), (2)

where e and μB are the electric charge and the Bohr magnetonrespectively, Japp is the current per unit area (density current),and d is the thickness of the free layer, and g(m,p) is thepolarization function depending on the relative orientationof the magnetizations [1–3].

For a complete model description of the numerical tech-niques see also [19–22]. Typical parameters for the Py havebeen used: saturation magnetization MS = 650 × 103 A/m,exchange constant A = 1.3× 10−11 J/m, damping parameterα = 0.02, and polarization factor η = 0.3 [1–3]. The bias fieldis applied out-of-plane (z-direction) with a tilted angle of 10◦

along the x-axis. The polarizer is considered fixed along thex-direction. To study the locking, we consider a microwavecurrent JRF = JM sin(2π fACt + π/2) (JM ≤ 2 × 107 A/cm2)and a microwave field linearly polarized at π/4 in the x-yplane hRF = hM sin(2π fACt + π/4)x̂ + hM sin(2π fACt + π/4) ŷ(hM ≤ 3 mT). This microwave field can be generated byusing the experimental technique developed in [25]. All thecomputations have been performed with no thermal effects.

3. Micromagnetic Results and Discussion

In order to characterize the device behavior, first of all weanalyzed the STO in the free running regime. We observedynamical regime in a wide range of current density for biasfield larger than 180 mT. Here we discuss in detail data for abias field of 250 mT, but qualitative similar results have beenalso observed for 200 and 300 mT.

In order to characterize the oscillator regime of thedevice, we swapped the dc current exceeding the criticalcurrent value (to obtain dynamics regime, threshold currentdensity was J = 3 × 107 A/cm2) up to current values,where oscillation regime is degraded by noise. Frequencyand power behavior of the nano-oscillator with respect tothe current density is shown in Figures 1(a) and 1(b) (nomicrowave signal). The frequency curve f0 as a function ofJ presents red shift from the critical current up to J1 =−3.5 × 107 A/cm2, where the dynamics is characterized byan in-plane oscillation axis. For |J| > |J1| the magnetizationprecesses around an out-of-plane axis and the blue shiftis achieved. The discontinuities observed in the oscillationfrequency are related to jumps of the oscillation axis that

Advances in Condensed Matter Physics 3

4 5 6 7 8 9 10 11 125.6

5.67

5.74

5.81

JRF

1 : 1 2 : 1

Pre

cess

ion

freq

uen

cy(G

Hz)

(a)

4.8

5.4

6

HRF

4 5 6 7 8 9 10 11 12

1 : 1 2 : 1

Pre

cess

ion

freq

uen

cy(G

Hz)

(b)

JRF +HRF

4 5 6 7 8 9 10 11 12

Microwave source frequency (GHz)

4.8

5.4

6P

rece

ssio

n fr

equ

ency

(GH

z)1 : 1

2 : 1

(c)

Figure 2: Response of the STO self-oscillation forHDC = 250 mT and JDC = −5×107 A/cm2 as a function of the microwave source frequencywhen (a) JRF = 2× 107 A/cm2, (b) HRF = 1 mT, and (c) RF current JRF = 1× 107 A/cm2 and RF field HRF = 1 mT are applied together at thesame frequency. The locking regions for the frequency ratio of 1 : 1 and 2 : 1 are also visible.

correspond to transitions between strongly nonlinear oscilla-tion modes. Similar discontinuities have been also observedin some experimental work [23]. In addition, we performedmicromagnetic simulations increasing the out-of-plane staticfield up to 600 mT, and we observed that for out-of-planebias field larger than 400 mT the red-shift zone disappears,jumps are softer, and the synchronization data achieved aresimilar to results already published in the literature (see e.g.,[10] for a review). At larger field (>600 mT), the polarizeris moved from the x-direction towards the out-of-plane z-direction that might generate additional noise showing notcoherent precessional states.

Figure 1(b) shows the power versus current behavior.Nonlinear power strongly increases at low current and forcurrent values greater than 4× 107 A/cm2 is about constant.

We systematically studied the locking to the first har-monic (the same of the self-oscillation) in the blue shiftregion as a function of the JM and hM . Figure 2 shows theprecession frequency of the GMR signal as a function ofthe microwave source frequency in two cases: RF currentonly and RF current and field together (for HDC = 250 mTand JDC = −5 × 107 A/cm2). We found different lockingregions at the locking ratio 1 : 1, 2 : 1, and 3 : 1 (in the lastcase only when the microwave field component is applied,not shown here). Typically, the locking region is much largerwhen the microwave force is a field (or a combinationof current and field). In fact, in the case of current wefound a locking region of about 150 MHz (1 : 1) and 50 MHz(2 : 1), whereas no locking on the third harmonic is found.Microwave field provides a locking region larger than 1 GHzand, since driving force breaks the oscillation symmetry, the1 : 1 synchronization region has a specific asymmetric shape.Then, whereas for small forcing signal the synchronizationregion can be described by an analytical theory (symmetrictongue where the locking region increases linearly with force

amplitude) [24], it cannot be described analytically whenincreasing the forcing signal and their precise determinationrequires necessarily specific numerical techniques.

Figure 3(a) summarizes the Arnold tongue (J = −5 ×107 A/cm2) computed up to JM = 1 × 107 A/cm2 (hM =0 mT) and then increasing hM up to 3 mT with JM =1 × 107 A/cm2 held unchanged. The border lines have beencomputed considering the lower (in the left part) andhigher (in the right part) microwave frequency where thephase locking is achieved. In the low regime of microwavesource, the Arnold tongue is related to the only applicationof the microwave current, which can be considered asa “weak” microwave signal. In fact, such a signal givesrise to symmetric synchronization region (no hysteresis isobserved) with a locking band linearly dependent on theforce locking.

When both microwave current and field are appliedsimultaneously at the same frequency, the nonautonomousresponse becomes more complicated. The presence of anadditional weak microwave field gives rise to increasing of thelocking region from 150 MHz at hM = 0 mT to 1.3 GHz forhM = 1 mT. As can be observed the locking region is stronglyasymmetric. This is caused by the strong nonlinearity ofthe dependence of the auto-oscillation frequency on theoscillation power [25].

Figure 3(b) shows the phase difference between themagnetization oscillation and the microwave source (neg-ative angle means magnetization in delay with respectto microwave source, Ψ being the phase of the naturalprecession of the magnetization and Ψe the force phase)when the microwave component is applied at the samefrequency of the free precession one. As shown, typically inboth cases (microwave field or current), the phase increaseswith current and decreases after a maximum value. Thephase shift depends on the initial detuning, and it goes to


4 4.5 5 5.5 6 6.50 0

01

1

1 1

1

2

3

Microwave frequency (GHz)

J M(1

07A

/cm

2)

hM

(mT

)

(a)

− 10 − 9 − 8 − 7 − 6 − 5 − 4


200

150

100

50

0

− 50

− 100

HRF

JRF

Ψ–Ψ

e(d

eg)

(b)

Figure 3: (a) Arnold tongue computed for HDC = 250 mT and J = −5×107 A/cm2 varying the magnitude of RF current and field. (b) Phasedifference between external force (current or field) and output magnetization as a function of dc current when an RF source at the samefrequency of the free precessional one is applied. Filled-red circle: RF current, white circle: RF field.

zero only if there is no detuning, that is, in the center ofthe synchronization region. It should be remembered thatthe phase difference depends on the initial phase of theforce and the way it affects the oscillator. We emphasizehere that the phase locking implies that phase difference willbe kept bounded inside a finite range of detuning, that is,within the synchronization region. In this case the microwavecurrent and field give the same qualitative behavior tothe oscillator, but the phase difference is translated byan angle of about 120◦. As expected, phase differencebetween microwave source and the magnetization precessionfollows the frequency behavior. It initially decreases with dccurrent amplitude (precession frequency decreases) and thenincreases with dc current (precession frequency increases)up to current values of the order of 7 × 107 A/cm2. Afterthat current value, phase shift jumps about 180◦, typicallythis is due to the different oscillation mode from in-planeto out-of-plane mode. Lastly, the phase difference graduallyincreases following the frequency slow rising with dc currentmagnitude [26].

In the locking region an intrinsic phase shift Ψi,computed as the difference between the phase of the self-oscillation Ψ and the phase of the microwave current Ψe, isfound.

Figure 4 summarizes Ψi as a function of the microwavefrequency, as can be observed a linear relationship betweenΨi and fAC is achieved with a range of Ψi which can cross0 or π/2 depending on the bias current density. As reportedby Slavin and Tiberkevich [10], analytical formulation of thephase difference in the locking region is given by

Φ0 = arcsin(fAC − f0Δ f

)− arctan(υ), (3)

where υ = (N/(G+ − G−)) is the nonlinear frequencyshift. We found N which is characterized from two differentvalues of nonlinear frequency shift N = 2π(df /dp), p isthe oscillation power, and G+ and G− are the non-linear

5.4 5.5 5.6 5.7 5.8

Microwave frequency (GHz)

100

50

0

− 50

− 100

− 150

JRF = 1 × 107 A/cm2

JRF = 2 × 107 A/cm2

HRF = 1 mT

HRF = 1 mT, JRF = 1 × 107 A/cm2

HRF = 1 mT, JRF = 1 × 107 A/cm2-analytical

Ψ–Ψ

e(d

eg)

Figure 4: Intrinsic phase shift Ψi between the phase of self-oscillation and microwave source (current or field) in the lockingregion for JDC = 5× 107 A/cm2.

damping coefficients as in [10]. Phase difference is stronglydependent on the frequency in the locking region [27]. Forall the cases, the phase difference decreases from left to rightin the Arnold tongue. In particular, the slope of the curve isstrong for RF current locking whereas it is more soft for RFfield, a result also in substantial agreement with the analyticaltheory. Considering (3) for RF applied field, the lockingbandwidth and nonlinear frequency shift are larger withrespect to ac current. A good agreement between analyticaland numerical data is found.


4. Conclusions

In summary, we have studied micromagnetically the non-linear behavior of spin-torque nano-oscillators in lockingregime driven by microwave current and field. We founda large locking region at different harmonics when RFfield is applied. The effect of the static applied field isalso studied. Finally, we showed and explained the intrinsicphase shift due to the difference between microwave sourceand magnetization precession inside the locking region,also comparing our numerical data with a recent analyticaltheory.

Acknowledgments

This paper was supported by Spanish Project under Contractno. MAT2011-28532-C03-01. The authors would like tothank Sergio Greco for his support with this paper.

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Hindawi Publishing CorporationAdvances in Condensed Matter PhysicsVolume 2012, Article ID 765709, 16 pagesdoi:10.1155/2012/765709

Review Article

Hamiltonian and Lagrangian Dynamical Matrix ApproachesApplied to Magnetic Nanostructures

Roberto Zivieri1 and Giancarlo Consolo2

1 CNISM Unit of Ferrara and Department of Physics, University of Ferrara, Via Saragat 1, 44122 Ferrara, Italy2 Department of Sciences for Engineering and Architecture, University of Messina, C.da di Dio, Villaggio S.Agata,98166 Messina, Italy

Correspondence should be addressed to Roberto Zivieri, [email protected]

Received 30 March 2012; Accepted 23 April 2012

Academic Editor: Eduardo Martinez Vecino

Copyright © 2012 R. Zivieri and G. Consolo. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

Two micromagnetic tools to study the spin dynamics are reviewed. Both approaches are based upon the so-called dynamicalmatrix method, a hybrid micromagnetic framework used to investigate the spin-wave normal modes of confined magnetic systems.The approach which was formulated first is the Hamiltonian-based dynamical matrix method. This method, used to investigatedynamic magnetic properties of conservative systems, was originally developed for studying spin excitations in isolated magneticnanoparticles and it has been recently generalized to study the dynamics of periodic magnetic nanoparticles. The other one, theLagrangian-based dynamical matrix method, was formulated as an extension of the previous one in order to include also dissipativeeffects. Such dissipative phenomena are associated not only to intrinsic but also to extrinsic damping caused by injection of a spincurrent in the form of spin-transfer torque. This method is very accurate in identifying spin modes that become unstable underthe action of a spin current. The analytical development of the system of the linearized equations of motion leads to a complexgeneralized Hermitian eigenvalue problem in the Hamiltonian dynamical matrix method and to a non-Hermitian one in theLagrangian approach. In both cases, such systems have to be solved numerically.

1. Introduction

In these last years, great attention has been given to the studyof magnetization dynamics in laterally confined magneticsystems. It is well know that spin excitations are quantizeddue to the lateral confinement. The oscillations are the so-called normal modes, which represent a pattern of motiongiven by all the parts of the system oscillating sinusoidallywith the same frequency and with the same phase relation.In this last decade, analytical models have given importantcontributions to understand the frequency spectrum ofnormal modes for different ground-state magnetizations [1–11]. However, some limitations due to the assumptions madefor the determination of the equilibrium state, the boundaryconditions, and the calculation of the energy contributionsto normal modes dynamics are still present.

On the other hand, a lot of efforts have been devoted todevelop micromagnetic codes having the aim of calculating

very precisely different ground states of nanometric particles[12]. Due to their accuracy, the developed micromagneticmethods have contributed to give additional informationabout the spin dynamics. The first micromagnetic calcula-tions were typically based upon codes developed to calculatein the first place the ground state of a given magnetic particle.Then the time evolution of the average magnetization of aparticle could be obtained and, from a subsequent postpro-cessing of these data (mainly using the Fourier transform ofthe magnetization), information could be extracted aboutmode frequencies and spatial profiles [13]. In the simplestapplication of the method, the limit of these calculations wasthe observation of modes with nonzero magnetization only.More recently, a micromagnetic method was extended tothe detailed calculation of eigenfrequencies and eigenvectorsunder the effect of an oscillatory in-plane small magneticfield [14]. Another recent micromagnetic method was alsodeveloped to study the quantized spin excitations in laterally


confined systems [15]. This is the so-called Hamiltonian-based dynamical matrix method (HDMM).

The HDMM was first formulated for isolated magneticnanoelements and then it has been generalized to the case ofinteracting nanoelements [16]. It is the prototype of finite-difference methods and represents an eigenvalue/eigenvectorproblem. The scope of this method is to find the frequenciesand profiles of the spin modes which are associated to theeigenvalues and the eigenvectors, respectively, of a dynamicalmatrix. It can be considered the analogous of the dynamicalmatrix formalism used to find atomic vibrations (phonons)in crystalline solids. The dynamical matrix contains thesecond derivatives of the density energy coming from asecond order expansion of the density energy around theequilibrium. This method was already used to study thespin excitations in magnetic multilayers with ferro- or anti-ferromagnetic coupling [17, 18]. Due to the translationalinvariance, the number of independent dynamic variableswas reduced to twice the number of the layers. The calcu-lated second derivatives are evaluated at equilibrium. Theeigenvalue/eigenvector problem can be set as a complexgeneralized Hermitian eigenvalue problem. The methodpresents several advantages: a single calculation yields thefrequencies and eigenvectors of all modes of any symmetry, itis applicable to a particle of any shape (within the nanometricrange), and the computation time is affordable. This meansthat by means of this micromagnetic approach, it is possibleto determine, after a single iteration, the frequencies and theprofiles of all spin-modes, independently of the ground-statemagnetization (e.g., vortex state, vortex in the presence of anexternal magnetic field, onion state, quasi-saturated state).The main restriction of the method is its applicability toconfined magnetic systems whose spin dynamics is assumedpurely precessional with no dissipative effects. Of course,this is true only in a first approximation, since in realmagnetic systems the intrinsic damping process plays animportant rule. In order to select the representative modesof the spectrum and to compare them with the onesobserved by means of the experimental techniques, thedifferential scattering cross-section has to be evaluated bothfor noninteracting and interacting magnetic particles.

The first applications of the HDMM were on chains ofdipolarly interacting rectangular dots representing a one-dimensional array [19] and of two-dimensional (2D) arraysformed by circular nanometric disks [20]. Very recently, thismethod was applied to study the collective mode dynamicsin arrays of holes embedded into a thin ferromagneticfilm. This calculation was done by including in the energydensity computation also the exchange interaction betweenmicromagnetic cells belonging to two adjacent primitive cells[21].

In order to overcome the above-mentioned restrictionsof the HDMM, Consolo et al. formulated very recentlythe so-called Lagrangian-based dynamical matrix method(LDMM) [22]. Such a method explicitly takes into accountthe intrinsic “positive” Gilbert damping and the current-induced spin-transfer-torque “negative” dissipation. Sincethe magnetic system so obtained is no more conservative,a Lagrangian formalism is necessary. Unlike the HDMM,

Ω

φ

ê1

ê2

ê3

ê3

θ

ψ

−→M

Figure 1: Reference frame used in the theory.

the LDMM cannot be cast as a classical (nongeneralized)eigenvalue/eigenvector problem, but it has to be formulatedas a complex generalized non-Hermitian eigenvalue prob-lem. The first application of the method was done on amagnetic nanopillar stack of circular cross-section subject toan external magnetic field directed a few degree away fromthe normal to the plane [22]. The analysis was then extendedto the case of external magnetic fields of variable intensityand orientation with respect to the plane of the nanopillar[23].

It is important to notice that both formalisms have beendeveloped up to now in the linear approximation, namely,considering small angular deviations of the magnetizationfrom equilibrium so that each spin excitation is a normalmode of the system.

The reference frame used in the micromagnetic calcu-lations performed both by means of HDMM and LDMMis illustrated in Figure 1. The z-axis is along the normal tothe particle and the x-y plane lies on the particle plane.According to this reference frame, the configuration of the

vector �M, representative of the magnetic dipole momentum,is identified through the polar angles, θ and φ, and theintrinsic rotation ψ. As it will appear clear in the sectiondevoted to the Lagrangian approach, this latter angle doesnot enter in the equation of motion, being the correspondingLagrange equation associated to a first integral of the motion(the conservation of the angular momentum). So that, as itis expected, because the modulus of the magnetization vectoris preserved in time, the dynamics can be described throughtwo degrees of freedom only (typically θ and φ).

If we assume that the magnetization vector �M is placedalong the generic direction given by the unitary vector ê′3, itcan be expressed in Cartesian coordinates by

�M(t) =Ms(sin θ(t) cosφ(t), sin θ(t) sinφ(t), cos θ(t)

),

(1)


being Ms the saturation magnetization value (the modulusof the magnetization vector) and the time dependence of theangles θ and φ expressed as

θ(t) = Θ + δθ(t),φ(t) = Φ + δφ(t).

(2)

In (2), we indicate by (Θ, Φ) the static (equilibrium)orientation of the magnetization obtained by solving the

stationary problem ( �̇M = 0) for whatever effective fieldand magnetization distributions. Instead, δθ(t) and δφ(t) arethe small polar and azimuthal deviations from equilibrium,respectively.

In the micromagnetic calculations, the magnetic systemis subdivided into rectangular cells. Each micromagnetic cellis identified by a single index k( j) that varies from 1 toN , where N denotes the number of cells. Hence, Mk is themagnetization in the kth cell and �ri j = �ri − �r j is the in-planedistance between the ith cell and the jth cell. The index hasbeen assigned so that the first line of the rectangular matrix(X × Y) corresponds to k = 1, . . . ,Y and the second lineto k = Y + 1, . . . , 2Y , and so on and so forth. X(Y) is thenumber of cells along x(y), while Z is the number of cellsalong z for a sample of thickness equal to d. We define for

each cell the reduced magnetization �mk = �Mk/Ms. Hence, ina polar reference frame for each cell

�mk(t) =(sin θk(t) cosφk(t), sin θk(t) sinφk(t), cos θk(t)

),(3)

where φk is the azimuthal angle and θk is the polar angle ofthe magnetization. The total energy density of the system,obtained by dividing the total energy by the volume of thecell, is a function of φk and θk :E = E(θk,φk) where k variesfrom 1 to N .

In the following sections we describe in detail thetwo micromagnetic methods. We do not illustrate theapplications of these methods to magnetic nanoparticles,because this aspect is not the purpose of this paper. We give asummary of the paper. In Section 2, we outline the formalismat the basis of the HDMM. First, we introduce the differentcontributions to the energy density and then we derive thesystem of linear and homogeneous equations of motion forthe isolated particle from the Hamilton equations. Finally,we present the generalization of the HDMM to the case ofinteracting elements. In Section 3, the formalism at the basisof the LDMM is presented. First, the Lagrangian equationsfor a macrospin system are derived. Then a generalization isgiven by considering N interacting momenta in an isolatedmagnetic element.

2. HDMM

This section deals with the review of the HDMM formalism.As stated above, the HDMM is a micromagnetic approachthat can be applied only to fully conservative systemswhich are supposed to have, in a first approximation, apurely (undamped) precessional motion of the magneti-zation about the effective field. It is a finite-difference

micromagnetic method developed in the linear regime ofspin dynamics by considering small deviations from theequilibrium magnetization. In the first subsection, the dif-ferent contributions of the energy density entering into thedynamical matrix are calculated and the equations of motionfor an isolated magnetic element are cast in the form ofa linear and homogeneous system that can be solved as acomplex generalized Hermitian eigenvalue problem. In thelast subsection, it is shown the generalization of the HDMMto interacting magnetic nanoparticles by including into thedynamical matrix the Bloch condition.

2.1. Energy Density. First, we give the explicit expressionsof the different interactions entering into the total micro-magnetic energy density E of a given confined magneticsystem simulated by using the HDMM: Zeeman energy, ex-change energy, demagnetizing energy, and anisotropy energy,respectively [24]. The energy density is defined as E = Ẽ/Vwhere Ẽ is the energy of the system and V its volume. In the

presence of an external magnetic field �H , the Zeeman energydensity can be written in the form

Eext = −μ0Ms �H ·N∑k=1�mi, (4)

being μ0 the vacuum permeability. In micromagnetic theory,the exchange energy can be expressed as a volume integral ofthe form

Ẽexch = A∫

part

3∑j=1

(�∇mj

)2dV , (5)

where the subscript “part” denotes the volume of a generalmagnetic particle, A is the exchange stiffness constant, and�∇ is the gradient applied to a given component of themagnetization. In this case the exchange contribution isindependent of z. Using the first-neighbours model, theexchange energy density can be written as follows:

Eexch = AN∑k=1

4∑n=1

1− �mk · �mna2kn

, (6)

where the variable akn is the distance between the centersof two adjacent cells of index k and n, respectively, k variesover all micromagnetic cells, and the sum over n rangesover the neighbours of the k-th cell. If the micromagneticcells k are situated at the edges, one must impose boundaryconditions. The cells on the edges interact with an externalrow of cells that have the same fixed magnetization, in thiscase the corresponding term in the sum must be weightedtwice.

In order to calculate the demagnetizing energy density,we have followed the method of the demagnetizing tensor.In the following, we use also the term dipolar in placeof the term demagnetizing, because the higher-order termsof the expansion vanish in the practical cases examined.Generally, within the framework of the demagnetizing tensor


method, the demagnetizing energy density can be written asfollows:

Edmg = 12μ0∑k j

�Mk ·N �Mj

= μ0M2s

2

∑k j

(mxk,myk,mzk

)×⎛⎜⎝Nxx Nxy NxzNyx Nyy NyzNzx Nzy Nzz

⎞⎟⎠⎛⎜⎝mx jmy jmz j

⎞⎟⎠.(7)

This equation includes the self-energy and Nαβ = Nαβ(�rk j)with α,β = x, y, z are the elements of demagnetizing tensor.Each component of the demagnetizing tensor is relatedto the interaction between two rectangular surfaces S andS′. Under the assumption made for the calculation of thedemagnetizing field for uniform magnetization, by using aversion of Gauss’s theorem, the demagnetizing tensor can bewritten as

N(�rk j)= 1V

∫Skd�S∫S′j

d�S′∣∣�r −�r′∣∣ , (8)where V = l2c d is the volume of the micromagnetic cell withlc the cell size and d is the cell height. Because of the four-fold C4 symmetry and since �rk j ≡ (x, y, z), all componentscan be expressed only as a function of Nxx(x, y, z) andNxy(x, y, z) components with suitable permutations of thevariables x, y, z.

The magnetocrystalline uniaxial anisotropy can belabeled with the symbol Eani. It is an energy density function,for a given micromagnetic cell, of the angle αk between themagnetization of the single cell �mk and the easy axis ofgeneric direction given by the unit vector û. We write

Eani =N∑k=1K (1)sin2αk =

N∑k=1K (1)

(1− cos2αk

)

=N∑k=1K (1)

[1− (�mk · û)2],

(9)

where K (1) is the first-order anisotropy uniaxial coefficient.As the dynamical matrix components are expressed in

terms of the second derivatives of the energy density, it isnecessary to calculate them from the above expressions. First,we calculate the second derivatives of the magnetization withrespect to the polar and azimuthal angles of the given micro-magnetic cell that represent the degrees of freedom of thesystem. Indeed, the second derivatives of the magnetizationappear in the final expressions of the second derivatives ofthe energy density. In particular

∂2�mk∂δφ2k

= (− sin θk cosφk,− sin θk sinφk, 0),∂2�mk

∂δφk∂δθk= (− cos θk sinφk, cos θk cosφk, 0),

∂2�mk∂δθ2k

= (− sin θk cosφk,− sin θk sinφk,− cos θk).(10)

The second derivatives of the energy density are calculatedat equilibrium. For the sake of simplicity, in the following,the derivatives are calculated with respect to δθk and δθlimplying that one or two of the two generic variables couldbe also δφ.

The second derivative of Zeeman energy density becomes

∂2Eext∂δθk∂δθl

=⎧⎪⎨⎪⎩−μ0Ms �H ·

∂2�mk∂δθk∂δθl

l = k0 l /= k.

(11)

As outlined previously, for the calculation of the ex-change contribution, the nearest-neighbour model is takeninto account. It is useful to give also the expression of thefirst derivative due to some important manipulations thathave to be performed. The first derivative with respect toδθk includes in the sum a term in which i = k and thusn /= k and also the other terms with n = k and with ione of the nearest neighbours. Thanks to a proper changeof indices in the second term, the following equation isobtained:

∂Eexch∂δθk

= −A4∑

n=1

1a2kn

∂�mk∂δθk

· �mn − A4∑

n=1

1a2kn�mn · ∂�mk

∂δθk

= −2A4∑

n=1

1a2kn

∂�mk∂δθk

· �mn,(12)

where the sum over n is made up over the nearest-neighbourmicromagnetic cells of the k-th cell. In the special case ofthe adopted first neighbours model, the second derivativesare

∂2Eexch∂δθk∂δθl

=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

−2A4∑

n=1

1a2kn

∂2�mk∂δθk∂δθl

· �mn l = k

−2A 1a2kl

∂�mk∂δθk

· ∂�ml∂δθl

l, k: nearest-neighbour

0k /= l and

not nearest-neighbour.(13)

We now pass to the calculation of the derivatives of thedemagnetizing energy density. Due to their rather compli-cated form, we give in the following the expression not onlyof the second derivatives of the demagnetizing energy densitybut also of the first derivatives with respect to the generic


variable δθi. The first derivative of the demagnetizing energydensity (7) is

∂Edmg∂δθi

= μ0M2s⎡⎣1

2

∑k /= i�mk ·N(k, i) ∂�mi

∂δθi+

12

∑j /= i

∂�mi∂δθi

·N(k, i) �mj + 12∂

∂δθi

(�mi · N̂(i, i) �mi

)⎤⎦

= μ0M2sN∑k=1k /= i

�mk ·N(k, i) ∂�mi∂δθi

+ �mi ·N(i, i) ∂�mi∂δθi

.

(14)

In the sum, the contribution of the terms with the same indexk = j = i has been separated; moreover, we have takeninto account that the tensor N fulfils N(k, i) = N(i, k) andis symmetric (Nαβ = Nβα).

Thanks to the previous consideration, it is possible towrite

∂�mk∂δθk

·N(k, i)�mi = �mi ·N(k, i) ∂�mk∂δθk

, (15)

and therefore,

∂

∂δθi

(�mi ·N(i, i)�mi

)= ∂�mi∂δθi

·N(i, i)�mi + �mi ·N(i, i) ∂�mi∂δθi

= 2�mi ·N(i, i) ∂�mi∂δθi

.

(16)

The second derivative must take into account the two cases:i /= l, i = l( j = i), namely,

∂2Edmg∂δθl∂δθi

=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

M2s

⎛⎝ N∑k=1�mk ·N(k, l) ∂

2�ml∂δθl∂δθi

+∂�ml∂δθl

·N(k, l) ∂�mi∂δθi

) l = iMs

2 ∂�ml∂δθl

·N(k, l) ∂�mi∂δθi

l /= i.(17)

The k = i term resulting from the derivative of the secondterm (in the expression of the first derivative given above)has been included in the sum over k.

The last step is the calculation of the term associated withthe anisotropy energy density. The second derivative of theanisotropy energy density can be written as

∂2Eani∂δφ j∂δθk

=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

−2K (1)[(

∂�mk∂δφ j

· û)·(∂�mk∂δθk

· û)

+(�mk · û

)

·(

∂2�mk∂δφ j∂δθk

· û)]

for δθ, δφ k = j

−2K (1)⎡⎣( ∂�mk

∂δφk· û)2

+(�mk · û

) · ( ∂2�mk∂δφk

2 · û)⎤⎦

for δφ, δφ or δθ, δθ k = j0 k /= j.

(18)

2.2. Equations of Motion for an Isolated Magnetic Particle.The derivatives calculated previously are included into thedynamic equations. Indeed, the equations of motion can becast into a linear and homogeneous system in which thesecond derivatives of the energy density calculated at equi-librium appear explicitly. It is well know that the equationof motion for a magnetic spin system which undergoes apurely precessional motion is the Landau-Lifshitz equation[25], expressed as a torque equation involving the effectivefield and the magnetization itself. Since our aim is to findthe energy density in a conservative system, we derive theequations of motion from the Hamilton equations.

2.2.1. HDMM for a Macrospin System. The equation ofmotion for the magnetic systems under investigation will bederived by following semiclassical approach. Also, the modelwill be first derived by considering the so-called macrospinapproximation, where the material is thought as uniformlymagnetized and represented by a single dipole momentum.

As known from classical mechanics, in the presenceof fixed constraints and conservative sources, the systemHamiltonian H coincides with the total mechanical energyẼ, namely, H ≡ Ẽ = T − U , where T is the kineticenergy and U is the potential expressed as the opposite ofthe potential energy V . By defining the Lagrangian variablesof the problem with qn, where n = 1, 2, . . . is the number ofdegrees of freedom corresponding to the dynamic variables,and the corresponding conjugate momenta with pn, theHamilton equations in the 2n canonical variables (qn, pn)take the form [15]

∂qn∂t

= ∂H∂pn

,

∂pn∂t

= −∂H∂qn

.

(19)

The direction of the magnetic dipole moment of thekth cellis given by (3). For the specific case, the dynamic variables


referred to the k-cell are the small deviations from equilib-rium of the azimuthal and polar angles given by

q1 = δφ, q2 = δθ, (20)where 1 (2) labels the first (second) variable.

To determine the conjugate momenta, the expression ofthe angular momentum is needed. We recall the relation

between the angular momentum �lΩ and the magnetic mo-mentum �μ by referring to a a rigid body with a fixed point Ω(see Figure 1), namely,

�lΩ = 1γ�μ = vcMs

γ�m, (21)

where γ is the gyromagnetic ratio and vc the volume of themagnetic moment to the case. Indeed, since q1 represents arotation about the z-axis of the magnetic dipole, its conjugatemomentum p1 corresponds to the z component of thevariation of the angular momentum, namely,

p1 = δlz = vcMsγ

δmz = −vcMsγ

sin θδθ, (22a)

where �m = (sin θ cosφ, sin θ sinφ, cos θ) is the unit magne-tization vector, normalized to the saturation magnetizationMs. Following an analogous argument, the other momentump2 can be determined. Indeed, q2 is a rotation of the dipolemoment about an axis of unitary vector φ̂′ = − sinφ ê1 +cosφê2 in the x-y plane at an angle φ̂′ = φ + π/2 fromthe x-axis. Therefore, p2 corresponds to the projection ofthe variation of the angular momentum along the φ̂′ vector,namely,

p2 = δ�lΩ · φ̂′ = vcMsγ

(δ�m · φ̂′

)

= −vcMsγ

(∂�m∂φ

δφ +∂�m∂θ

δθ

)· φ̂′

= vcMsγ

sin θδφ.

(22b)

By substituting (20), (22a), and (22b) into the firstHamilton equation (cf. (19)), the following system ofequations is obtained:

˙δφ = − γvcMs sin θ

Hδθ ,

δ̇θ = γvcMs sin θ

Hδφ,(23)

where the dot notation stands for the time derivative and, atthe right-hand side, the first derivatives of the energy withrespect to the mechanical variables appear. By introducingthe energy density E = Ẽ/V (keeping in mind that H = Ẽ)and expanding it in a power Taylor expansion around theequilibrium up to the second order, it yields

E = E0 + 12[Eφφ

(δφ)2 + 2Eφθδφδθ + Eθθ(δθ)2], (24)

where E0 is the constant zero-order term that is inessential,the first-order terms vanish at equilibrium, and Eαβ representthe second derivatives calculated at equilibrium (Eαβ =∂2E/∂δα∂δβ with α,β = φ, θ). By using (23) and (24), weobtain

˙δφ = − γMs sin θ

[Eθφδφ + Eθθδθ

],

δ̇θ = γMs sin θ

[Eφφδφ + Eφθδθ

].

(25)

By inserting the time dependence in the form eiωt, where ωis the angular frequency of the given collective mode, thesystem of equations of motion reads

− Eθφsin θ

δφ − Eθθsin θ

δθ − λ̃δφ = 0,

Eφφsin θ

δφ +Eφθ

sin θδθ − λ̃δθ = 0.

(26)

The linear and homogeneous system of equations expressedin (26) can be written as an eigenvalue problem

C�v = λ̃ �v, (27)

with λ̃ = i(Ms/γ)ω the complex eigenvalues of the problem,

C =

⎡⎢⎢⎢⎣− Eθφ

sin θ− Eθθ

sin θ

Eφφsin θ

Eφθsin θ

⎤⎥⎥⎥⎦, (28)being C a real, but not symmetric, matrix and �v = (δφ, δθ)T .

However, the system of motion equations (26) can bealso recast as a complex generalized Hermitian eigenvalueproblem

A�v = λ B�v, (29)where B is a Hessian matrix expressed by the second deriva-tives of the energy density at equilibrium. In particular

B =[Eφφ EφθEθφ Eθθ

],

A =[

0 i sin θ−i sin θ 0

].

(30)

It should be noticed that the matrix A is Hermitian, whereasB is real and symmetric, so that all the correspondingeigenvalues λ = γ/Msω are real quantities.

2.2.2. HDMM for an Isolated System Composed by N Inter-acting Magnetic Momenta. Equation (24) can be generalizedto the case of N interacting magnetic momenta (where eachmomentum is identified with a micromagnetic cell) takingthe form

E=E0 + 12N∑n=1

N∑l=1

[Eφnφl δφnδφl+2Eφnθl δφnδθl+Eθnθl δθnδθl

].

(31)


In (31), the total energy density is given by E = Eext + Eexch +Edmg +Eani where the different contributions are expressed in(4), (6), (7), and (9), respectively. By substituting (31) into(23), we get

˙δφk = − γMs sin θk

N∑l=1

[Eθkφl δφl + Eθkθl δθl

],

˙δθk = γMs sin θk

N∑l=1

[Eφkφl δφl + Eφkθl δθl

].

(32)

By introducing the time dependence in the form eiωt, thesystem of equations of motion is composed by the following2N linear and homogeneous equations for k = 1 · · ·N :

N∑l=1

(− Eθkφl

sin θk

)δφl +

N∑l=1

(− Eθkθl

sin θk

)δθl − λ̃δφk = 0,

N∑l=1

(Eφkφlsin θk

)δφl +

N∑l=1

(Eφkθlsin θk

)δθl − λ̃δθk = 0.

(33)

The unknown factors δφl, δθl represent the eigenvectors ofthe problem and are expressed by the small angular deviationfrom the equilibrium position of the azimuthal (φl) andpolar (θl) angles in the lth micromagnetic cell. The systemabove has a solution only if the determinant is zero. Bysuitable exchanges of rows (columns), the linear and homo-geneous system of equations expressed in (33) can be writtenas an eigenvalue problem in analogy with the case of amacrospin system

C�v = λ̃ �v. (34)

In (34), �v is the set of the unknown factors representing theeigenvectors of the problem that take the form

�v = (δφ1, δθ1, δφ2, δθ2, . . . , δφN , δθN)T . (35)C is the matrix whose elements are expressed as

C2k−1,2l−1 = −Eθkφlsin θk

C2k−1,2l = − Eθkθlsin θkC2k,2l−1 =

Eφkφlsin θk

C2k,2l =Eφkθlsin θk

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭k = 1 · · ·N , l = 1 · · ·N. (36)

Analogously to the case of a macrospin system, the matrixC is real but not symmetric, also in this case the eigenvalues(and not only the eigenvectors) are complex. This matrix canbe indeed seen as composed by two submatrices 2×2 for eachpair of values (k, l). In the diagonal submatrices (k = l) thefollowing relation is verified:

C2k−1,2k−1 = −C2k,2k. (37)

For the elements of two different submatrices (k, l) and (l, k),that are not diagonal (k /= l), the following symmetries hold:

sin θkC2k−1,2l−1 = − sin θlC2l,2k,sin θkC2k−1,2l = sin θlC2l−1,2k,sin θkC2k,2l−1 = sin θlC2l,2k−1,

sin θkC2k,2l = − sin θlC2l−1,2k−1.

(38)

As for the macrospin approximation, the equation of motioncan be recast as a complex generalized Hermitian eigenvalueproblem

A �v = λ B �v, (39)

where B is a Hessian matrix expressed by the second deriva-tives of the energy density at equilibrium. B is given by

B2k−1,2l−1 = EφkφlB2k−1,2l = EφkθlB2k,2l−1 = EθkφlB2k,2l = Eθkθl

⎫⎪⎪⎪⎬⎪⎪⎪⎭k = 1 · · ·N , l = 1 · · ·N , (40)where the matrix B is, again, real and symmetric. Moreover,since the static magnetization corresponds to a minimum ofthe energy and the matrix B is its Hessian, the matrix B isalso positive defined. Instead, the matrix A has the followingform:

A =

⎡⎢⎢⎢⎢⎢⎣0 i sin θ1 0 0 · · ·

−i sin θ1 0 0 0 · · ·0 0 0 i sin θ2 · · ·0 0 −i sin θ2 0 · · ·· · · · · ·

⎤⎥⎥⎥⎥⎥⎦. (41)

The matrix A is Hermitian. This allows us to solve the systemas a complex generalized Hermitian eigenvalue problemwhich admits only real eigenvalues. To further reduce thecomputational time, it is possible to evaluate only someeigenvalues and eigenvectors that are in a specific range.

Once the eigenvectors �v are obtained, the dynamicmagnetization δ�mk in the kth micromagnetic cell expressedin Cartesian coordinates and in unit of MS is given by

δ�mk=(− sin θk sinφkδφk + cos θk cosφkδθk,

sin θk cosφkδφk + cos θk sinφkδθk,− sin θkδθk).

(42)

For each solution of the eigenvalue problem, the collectionof all δ�mk defines the mode profile. It must be remarked thatδ�mk is a complex vector, because δθk, δφk are, in general,complex.

2.3. Equations of Motion for Interacting Magnetic Particles.Let us suppose to have a 2D periodic array of interactingnanodots characterized by the primitive vectors �a1 and �a2;


for example, for the specific case of a rectangular lattice theirvalues are

�a1 = λxx̂, �a2 = λy ŷ, (43)where λx and λy represent the periodicity along x-axis andy-axis, respectively. The primitive vectors of the reciprocal

lattice are �b1 and �b2

�b1 = 2π(�a2 × �a1

)× �a2(�a1 × �a2

)2 ,�b2 = 2π

(�a1 × �a2

)× �a2(�a1 × �a2

)2 .(44)

For the special case of a rectangular lattice the vectors become

�b1 = 2πλxx̂,

�b2 = 2πλy

ŷ.

(45)

Due to the analogy with the Bloch wave (analogy, not equal-ity, because the wave function has not a physical meaning,contrarily to the magnetization), it is possible to write thefollowing periodicity rule valid for the dynamic magnetiza-tion:

δ�m(�r + �R

)= ei�K·�Rδ�m(�r), (46)

where �R is a vector of the particle lattice given by

�R = i1�a1 + i2�a2 i1, i2 ∈ Z, i1, i2 = −Ni2 · · ·Ni2− 1,

(47)

and �r can be confined into the first primitive cell centered inthe first dot. The Bloch vector takes the following values:

�K = n1N1�b1 +

n2N2�b2, ni ∈ Z, ni = −Ni2 · · ·

Ni2− 1.

(48)

N1,N2 ∈ N indicate the number of primitive cells n indirection �a1 and �a2, respectively. In this scheme, both N1and N2 are taken as even numbers. In order to confirm thehypothesis on the dynamic magnetization N1,N2 must bevery large.

If the magnetizations of different primitive cells and thedifferent micromagnetic cells were independent, that is noperiodicity rule were present, then one would have a dynamicsystem with variables θk�R and φk�R, where the k index changes

inside the magnetic particle and �R can assume the valuesindicated in (47). In this case, the linear and homogeneoussystem of 2N equations of motion is

∑l,�R′

(−Eθk�Rφl�R′

sin θk�R

)δφl�R′ +

∑l,�R′

⎛⎝− Eθk�Rθl�R′sin θ jk�R

⎞⎠δθl�R′ − λδφk�R = 0,∑l,�R′

(Eφk�Rφl�R′sin θk�R

)δφl�R′ +

∑l,�R′

(Eφk�Rθl�R′sin θk�R

)δθl�R′ − λδθk�R = 0,

(49)

where k = 1 · · ·N and the sums over l and �R′ are on thesame values. Instead, thanks to the Bloch condition expressed

in (46), one can consider the equations only at �R = 0; more-over, taking into account the same condition, the variables

appearing for �R′ /= 0 can be replaced by using the samecondition. Now, when rewriting the system, the index �R isomitted when it has value equal to 0 or it is irrelevant.Owing to these considerations, the system given in (49) canbe rewritten in the form [16]

N∑l=1

⎛⎝−∑�R′ Eθkφl�R′ ei�K·�R′sin θk

⎞⎠δφl + N∑l=1

⎛⎝−∑�R′ Eθkθl�R′ ei�K·�R′sin θk

⎞⎠δθl− λ̃δφk = 0,

N∑l=1

⎛⎝∑�R′ Eφkφl�R′ ei�K·�R′sin θk

⎞⎠δφl + N∑l=1

⎛⎝∑�R′ Eφkθl�R′ ei�K·�R′sin θk

⎞⎠δθl− λ̃δθk = 0.

(50)

Equation (50) is similar to (33) by making the following re-placement:

Eαkβl −→∑�R′ei�K·�R′Eαk0βl�R′ , (51)

and recalling that now the energy is referred to the wholesystem of particles. Like for the case of the isolated magneticparticle, also for the case of interacting magnetic particles thesystem of linear and homogeneous equations given in (50)can be written as an eigenvalue problem which in turn can becast as a complex generalized Hermitian eigenvalue problem.

The symmetry for the matrix elements that was valid for

a single primitive cell now is not respected except for �K = 0or �K = �G/2 with �G a translational reciprocal vector:∑

�R′ei�K·�R′Eαk�0βl�R′ =

∑�R′ei�K·�R′Eβl�R′αk�0 =

∑�R′ei�K·�R′Eβl �0αk−�R′

=∑�R′e−i�K·�R

′′Eβl �0αk�R′′

/=∑�R′ei�K·�R′Eβl �0αk�R′ ,

�K /=�0, �K /=�G2

,

(52)

with �R′′ = −�R′. The primitive cell has at the centre a singledot that occupies only a part of it. The interdot exchangecoupling is zero. Thanks to the last consideration and to thefact that derivatives of Zeeman, exchange, anisotropy, energydensity are referred only to the cell of the first variable (αk�0)with αk�R = θk�R,φk�R or at most to the nearest neighbour,all terms of the sum in (51) with �R′ /= 0 are zero. Hence,for these energy density terms, the equations are the sameas those of the single particle case and the same occurs fortheir corresponding derivatives appearing in the equations ofmotion. The only energy density term that differs from the


one obtained for the isolated element is the demagnetizingenergy density. For a system of interacting nanoparticles, thedemagnetizing energy density can be written as

Edmg = 12μ0∑

�R,�R′,k,k′�mk(�R)N(�R, �R′, k, k′

)�mk′

(�R′). (53)

Due to its rather complex expression, it is useful to give thederivation also of the first derivative of Edmg like for the caseof the isolated nanoelement.

In order to calculate the first derivative, the properties ofthe demagnetizing tensor must be considered

〈v|N|ω〉 = 〈ω∣∣N+∣∣v〉∗ = 〈ω∣∣N+∣∣v〉= 〈ω∣∣Nt∣∣v〉 = 〈ω|N|v〉, (54a)

since N is real and symmetric and

N(�R, �R′, k, k′

)= N

(�R′ − �R +�rk′ −�rk

)= N

(−�R′ + �R−�rk′ +�rk

)= N

(�R′, �R, k′, k

),

(54b)

thanks to the inversion symmetry. Hence, the first derivativeof the energy is

∂Edmg

∂δαk(�0)

= μ0M2s

2

⎡⎢⎢⎢⎢⎣∑�R′k′

(�R′,k′) /= (�0,k)

�mk′(�R′)·N

(�R′,�0, k′, k

) ∂�mk(�0)∂δαk

(�0)

+∑�R′k′

(�R′,k′) /= (�0,k)

∂�mk(�0)

∂δαk(�0) ·N(�0, �R′, k′, k)�mk′(�R′)

+∂

∂δαk(�0)(�mk(�0) ·N(�0,�0, k, k)�mk(�0))

⎤⎥⎥⎥⎥⎥⎦

= μ0M2s

⎡⎢⎢⎢⎢⎣∑�R′k′

(�R′,k′) /= (�0,k)

�mk′(�R′)·N

(�R′,�0, k′, k

) ∂�mk(�0)∂δαk

(�0)

+ �mk(�0)·N

(�0,�0, k, k

) ∂�mk(�0)∂δαk

(�0)⎤⎥⎥⎥⎥⎥⎦.

(55)

The second derivative takes the form

∂2Edmg

∂δαk(�0)∂δβk

(�R)

=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

μ0M2S

⎡⎢⎣∑�R′k′�mk′

(�R′)·N

(�R′,�0, k′, k

)

×∂2�mk

(�0)


(�0)

+∂�mk

(�0)

∂δβk(�0)

·N(�0,�0, k, k

) ∂�mk(�0)∂δαk

(�0)⎤⎦

�R=�0, l=k

μ0M2s∂�ml

(�0)

∂δβl(�0) ·N(�R,�0, l, k) ∂�mk

(�0)

∂δαk(�0) (�R, l) /=(�0, k).

(56)

In the sum it is included the case in which (�R′, k′) = (�0, k)that is generated from the derivative of the second term inthe expression of the first derivative given above. Now it ispossible to calculate the terms that enter into the system of(50), starting with the one corresponding to l = k

∑�R

ei�K·�REαk�0βl�R =

∑�R

ei�K·�REαk�0βk�R

= μ0M2s

⎡⎢⎣∑�R′k′�mk′

(�0)·N

(�R′,�0, k′, k

)

×∂2�mk

(�0)


(�0) + ∂�mk

(�0)

∂δβk(�0)

·N(�0,�0, k, k

) ∂�mk(�0)∂δαk

(�0)⎤⎦

+ μ0M2s∑�R /=�0

ei�K·�R ∂�mk

(�0)

∂δβk(�0)

·N(�R,�0, k, k

) ∂�mk(�0)∂δαk

(�0)

= μ0M2s∑�R

⎡⎣∑k′�mk′

(�0)·N

(�R,�0, k′, k

)

×∂2�mk

(�0)


(�0)


+ ei�K·�R∂�mk

(�0)

∂δβk(�0)

·N(�R,�0, k, k

) ∂�mk(�0)∂δαk

(�0)⎤⎦.

(57a)

Equation (57a) is obtained by taking into account (46), by

including the term corresponding to �R = 0 into the sumperformed over �R /= 0 and by considering that both thestatic magnetization and the dynamic magnetization do not

depend on �R. Indeed, second derivatives are calculated atequilibrium and the exponential ei�K·�R appears on both thenumerator and the denominator of the derivative.

When l /= k, the term turns out to be∑�R

ei�K·�REαk�0βl�R

= μ0M2s

⎡⎢⎣∑�R

ei�K·�R ∂�ml

(�0)

∂δβl(�0) ·N(�R,�0, l, k) ∂�mk

(�0)

∂δαk(�0)⎤⎥⎦.(57b)

Due to the properties of the demagnetizing tensor, thesymmetry ∑

�R′ei�K·�R′Eαk�0βl�R′ =

∑�R′ei�K·�R′Eαl�0βk�R′ , (58)

is fulfilled when l = k, but it is not fulfilled when l /= k.The formalism previously developed for interacting

particles can be extended to a system of 2D antidots (ADs).In this case, it is necessary to add the exchange interactionbetween primitive cells [21]. In extended magnetic systemlike AD arrays, in addition to the usual nearest-neighboursexchange interaction between micromagnetic cells, theexchange contribution across the nearest-neighbours micro-magnetic cells belonging to adjacent surface primitive cellsmust be taken into account. Hence, we recall the exchangeenergy density of (6)

Eexch = A∑k

∑n

(1− �mk · �mn

)a2kn

. (59)

Here, the first sum runs over all the micromagnetic cellsof the primitive cell and the second sum runs over thenearest neighbours of the kth micromagnetic cell. Whenthe kth micromagnetic cell is on one of the edges (vertices)of the given primitive cell, the interaction with one (two)micromagnetic cell(s) belonging to the correct nearestprimitive cell must be added.

3. LDMM

This section is devoted to the review of the LDMM approachthrough which we derive the generalized Lagrange equation

−

+

I

Free layer

Spacer

Pinned layer

Metal

Metal

Figure 2: A schematic of a spin-valve nanopillar device.

in the presence of two dissipative effects arising from the“positive” intrinsic damping and the “negative” one inducedby the current-driven spin-transfer torque [22]. As for theHamiltonian approach, we limit our study to the dynamicstaking place in the linear and autonomous regime.

3.1. Description of the Magnetic System and Equation ofMotion. The magnetic systems in which such competingphenomena take place are generally referred to as spin-valvenanopillars. These are heterostructures composed by twoferromagnetic layers, having generally different thicknesses,separated by a nonmagnetic (metallic or insulating) spacer,which is used to decouple the exchange interactions betweenthem. The thinner magnetic layer is generally referred to as“Free Layer” (FL), whereas the thicker one is called “FixedLayer” or “Pinned Layer” (PL). By means of an externalvoltage source and metallic contacts applied at the top andbottom of the structure, a current flow traverses the structurealong the normal-to-plane direction (see Figure 2).

In this kind of device, conservative effects arise from thepreviously mentioned classical micromagnetic contributions(exchange, demagnetizing, magnetocrystalline anisotropy,and Zeeman fields) together with the Ampere (or Oersted)field due to the current flow. This latter contribution,however, will be neglected for simplicity. In fact, the maingoal of this section is to describe why and how a Lagrangianapproach needs to be take into account when noncon-servative forces act in the system. The inclusion of moresophisticated effects, such as the Ampere field, will becomerelevant when this approach will be further generalized todescribe the dynamics occurring in the nonlinear regime ofspin-wave generation.

Let us now briefly recall the governing equation ofmotion. When no dissipative contributions are taken intoaccount, a persistent precessional motion of the magnetiza-

tion vector �M takes place. It is described by classical Landau-Lifshitz equation �̇M = γ(�Heff × �M), where �Heff is theeffective field which accounts for all the above-mentionedcontributions.


On the other hand, nonconservative contributions arisefrom the material intrinsic dissipation and the spin-transfertorque induced by the current flow [26]. The former ac-counts for the phenomenologically introduced intrinsicGilbert dissipation [27], the phenomenon by which the pre-

cessional motion of �M, excited by a given stimulus, relaxestowards its equilibrium state. The relaxation rate is propor-tional to a scalar quantity, called Gilbert constant α. Thetorque exerted on the magnetization is generally represented

by �TID = (α/MS)( �M × �̇M).Concerning the dissipative effects induced by the current

flow, it has been extensively shown that this bias current canbecome spin-polarized in the direction of the magnetizationvector of the thicker magnetic layer and can then transferthis induced spin angular momentum to the magnetizationof the thinner magnetic layer. For a proper direction of thebias current I, this spin-transfer mechanism creates a torquewhich opposes to that induced by the Gilbert damping,creating an effective negative damping. The correspondingspin-transfer torque, derived by Slonczewski [28], can be

expressed as �TST = (σI/MS)[ �M × ( �M × �p)], where the unitvector �p defines the direction of the spin polarization (inturn defined by the magnetization vector of the PL) andthe constant σ modulates the strength of the spin-torqueeffect. It is equal to σ = εg0μB/2eMSSd, where ε is the spin-torque efficiency (defined in [29]), g0 is the Landè factor, μBis the Bohr magneton, e is the absolute value of the electroniccharge, d is the thickness of the magnetic layer, and S is thecurrent-carrying area [30].

It should be mentioned that, because of the larger valueof both saturation magnetization and thickness, the PL is notsubstantially affected by any current-driven magnetizationdynamics, so that it is generally treated as it were fixed (orpinned) along its equilibrium direction. On the contrary, theFL’s properties allow it to describe more easily several kindsof dynamics (e.g., switching [31], precession [32], domain-wall motion [33], gyrotropic motion of vortex state [34]), asit were “free” to move.

Under these circumstances, the magnetization dynamicsof the FL is governed by the Landau-Lifshitz-Gilbert-Slonc-zewski (LLGS) equation

�̇M = γ(�Heff × �M

)+

α

MS

(�M × �̇M

)+σI

MS

[�M ×

(�M × �p

)].

(60)

It has to be remarked that the equality (with opposite sign)

of the two torques (�TID = −�TST), achievable by means ofa proper intensity and sign of current (the positive one, I> 0, which corresponds to a current flow moving from thePL to the FL), which in turn implies the fully compensationof the two dissipation mechanisms, yields the system in anout-of-equilibrium zero-dissipation stationary state (a limitcycle, using the notation of dynamic systems) (see Figure 3).In such a regime, the excitation of microwave spin wavesbecomes physically conceivable.

To derive the mathematical formulation of the LDMM,for the sake of simplicity, we write, first, the generalizedLagrange equation for the case of an isolated magnetic

× × p

�H

�H

�H

�H

eff

effeff

×Meff × M ×˙M

Undampedprecessionprecession

Out-of-equilibrium

→M

→M

→M

→

→M→ → →

→M

Figure 3: Schematic representation of undamped (in the absenceof both Gilbert and current-induced damping) and out-of-equilibrium (in the presence of both Gilbert and current-induceddamping) precessions. In this latter case, a limit cycle is described aswell since the torques due to the intrinsic dissipation and the spin-transfer-induced one (for a proper intensity and direction of thecurrent) balance each other.

particle within the macrospin approximation. After that,we will generalize this approach for the case of an isolatedparticle composed by N interacting momenta.

3.1.1. LDMM for a Macrospin System. We will preliminarilyassume that the dynamics of the magnetization vector couldbe described through three degress of freedom (θ,φ,ψ), asshown in Figure 1. In such a framework the generalizedLagrange equations read

d

dt

(∂L

∂φ̇

)− ∂L∂φ

+∂�ST∂φ̇

+∂�ID∂φ̇

= 0,

d

dt

(∂L

∂θ̇

)− ∂L∂θ

+∂�ST∂θ̇

+∂�ID∂θ̇

= 0,

d

dt

(∂L

∂ψ̇

)− ∂L∂ψ

+∂�ST∂ψ̇

+∂�ID∂ψ̇

= 0,

(61)

where L = T + U represents the Lagrangian of the systemgiven by the sum of the kinetic energy T and the potential U,whereas �ST and �ID are the dissipation functions related tothe spin torque and the intrinsic damping, respectively.

Taking advantage of the explicit formulations of theenergy contributions given in Section 2.1, we can rewrite theprevious system by accounting for the relationship amongthe potential, the kinetic energy, and the conservative partẼ = T − U of the total mechanical energy (which in theconservative limit coincides with the Hamiltonian of thesystem H in the presence of fixed constraints). By sub-stituting L with 2T − Ẽ, we thus obtain

2d

dt

(∂T

∂φ̇

)− ddt

(∂Ẽ

∂φ̇

)− 2∂T

∂φ+∂Ẽ

∂φ+∂�ST∂φ̇

+∂�ID∂φ̇

= 0,

2d

dt

(∂T

∂θ̇

)− ddt

(∂Ẽ

∂θ̇

)− 2∂T

∂θ+∂Ẽ

∂θ+∂�ST∂θ̇

+∂�ID∂θ̇

= 0,

2d

dt

(∂T

∂ψ̇

)− ddt

(∂Ẽ

∂ψ̇

)− 2∂T

∂ψ+∂Ẽ

∂ψ+∂�ST∂ψ̇

+∂�ID∂ψ̇

= 0.(62)


To solve the proposed problem, we need thus to explicitlyfind the expressions of the variables T, Ẽ, �ID and �ST, shownin (62), in Lagrangian coordinates.

The kinetic energy T associated to the precessionalmotion of the magnetization vector can be computed byreferring to the case of a rigid body with a fixed point Ω (seeFigure 1). In this case, the rotational kinetic energy density isexpressed as

T = 12�lΩ · �ω, (63)

where �lΩ is the angular momentum and �ω is the angularvelocity vector. Notice that in the following the kineticenergy T, the dissipation functions �ID and �ST, the workdw, and the energy losses dw/dt are referred to theircorresponding quantities per unit volume. In addition, werecall the definition of the energy density E = Ẽ/V givenin Section 2.1. By virtue of the relationship between angular

momentum �lΩ and magnetic dipole momentum �μ (�lΩ =(1/γ)�μ), and considering that this latter is directed along ê′3(see Figure 1), the previous scalar product involves, in turn,

the only component of�lΩ along ê′3. It leads to

T = 12MSγ

(φ̇ cos θ + ψ̇

). (64)

Equation (64) therefore represents the rotational kineticenergy T = T(θ,φ,ψ) expressed in Lagrangian coordinateswithMS = μ/V . The conservative part E of the energy densityaccounts for all the standard micromagnetic contributionsdiscussed previously. As shown in Section 2.1, all the con-tributions appearing in E only depend on the Lagrangianvariables θ,φ, but not on their derivatives, namely, E =E(θ,φ).

To derive the Lagrangian formulation of the dissipativecontributions, let us start from the classical definition ofthe work dw carried out by a magnetic system subject to anonconservative force. As it is known, such a force has to bederived from the gradient of a dissipation function � and therate of energy losses associated to a dissipative torque can bethus expressed as

dw

dt= ± �̇M · ∂�

∂ �̇M, (65)

where the plus (minus) sign accounts for torques which actas “drain” (“source”) of energy and refers to the case � = �ID(� = �ST).

So, by multiplying the LLGS equation (60) by �Heff andassuming that the energy losses rates are small compared to

the conservative (precessional) part, namely, �̇M � −γμ0( �M×�Heff), we deduce the following expressions for the dissipativefunction densities

�ID = α2γM0�̇M

2

, (66)

�ST = σJγM0

�p ·(�M × �̇M

). (67)

It should be noticed that (66), which appears in the usualform of a Rayleigh-like dissipation function, is a positive-definite form, as expected for a power dissipated througha viscous friction mechanism, whatever the magnetizationconfiguration (α, γ, and Ms are positive constants). On theother hand, the spin-torque dissipation function of (67) isa non-Rayleigh one and strongly depends, apart from thedirection of the current flow J, on the relative magnetization

configuration of the ferromagnetic layers ( �M and �p).Finally, taking into account the expression of the time-

independent unit vector �p = (sinΘPL cosΦPL, sinΘPLsinΦPL, cosΘPL), the explicit expressions of the dissipativefunctions densities are

�ID = αMs2γ(θ̇2 + φ̇2sin2θ

)= �ID

(θ,φ

),

�ST = σJMsγ

[f1(φ)θ̇ + f2

(θ,φ

)φ̇]= �ST

(θ,φ

),

(68)

where

f1(φ) = sinΘPL sin(ΦPL − φ), (69)

f2(θ,φ

) = sin2θ cosΘPL − cos θ sin θ sinΘPL cos(ΦPL − φ).(70)

We proceed by evaluating first the third Lagrange equation

2d

dt

(∂T

∂ψ̇

)− ddt

(∂E

∂ψ̇

)− 2∂T

∂ψ+∂E

∂ψ+∂�ST∂ψ̇

+∂�ID∂ψ̇

= 0.(71)

In (71), we notice that the kinetic energy is the only termdependent on the intrinsic rotation, and, in particular, on its

velocity·ψ, so that the previous equation reduces to

2d

dt

(∂T

∂ψ̇

)= 0, (72)

which stands for a first integral of motion representingthe conservation of the (only component of the) angularmomentum

2∂T

∂ψ̇= MS

γ= constant ≡�lΩ ·�e′3. (73)

Such a first integral also points out that our system can bedescribed through only two degrees

Low-Dimensional Magnetic Systems · Leonid Pryadko, USA Charles Rosenblatt, USA Alfonso San-Miguel,...

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Transcript of Low-Dimensional Magnetic Systems · Leonid Pryadko, USA Charles Rosenblatt, USA Alfonso San-Miguel,...