silings nd silingsEinElwD my future stepftherD my motherEinElw nd my two prents s well s my lte nd living grndprentsF et lstD these knowledgments would e nothing if s filed to mention my long stnding friend nd life prtner gélineF s m deeply grteful for the lughterD the supportD the ful(llment nd so muh more tht her presene rought over the yersD nd s

pirstD s would like to express my sinere grtitude to my supervisor hilippe ernodD who mde this hh possile y (nding the fundingD nd who reted the reserh enviE ronment required for the development of the tivity on whih rests this workF he sme goes for my oEsupervisor xiols ierelinD who developed the expertise on rreEerthE sed mgnetoelsti thin (lms t the ore of this work nd mnged the overll reserh projetF s lso wnt to thnk my hh dvisor nnik hush who ws lwys ville from the strt to help me get sense of the siene nd the softwreD nd who set up the uerr mirosopeF pinllyD s m deeply indeted to tefno qiordno"without whom this thesis would not hve tken this ourse"for his unfiling support nd the ountless thoughtful disussions on vrious mttersD sienti( nd otherwiseF fesides my supervisors nd dvisorsD s m very muh oliged to enne fernndEwntelD hilippe veoeurD wihel rehn nd ldimir reorzhensky for their prtiiption s memers of my thesis ommitteeD nd to rsul iels nd endré hiville for kindly epting to review this mnusriptF s shouldD ut resonly nnotD mention every one of the permnent nd temporry memers of the eswexEpsvw group in whih s hve een fully integrtedF por those whom s hve worked withD s will spei(lly mention ssel for hving devised nd uilt onvenient sustrte mountD glire without whom s would never hve hieved the frition proess nd ylivier for llowing me to get tehing experiene t gentrle villeF yutside the groupD s m grteful to hominique for his time nd ptiene on the epwGwpw hrteriztionF s lso hve to knowledge the key ontriution of mny people in the len room personnel nd siwx s wholeF emong ollegues were lso friendsD eginning with the now legendry frrois tem with whih s hve hd so mny good moments t lunh time over those three yersF emong these tem mtesD s hve speil thought for tefnoD pioD slyesse nd grmeloD to whom s wish ll the est for wht lies hedF yutside workD mny thnks to the iene op gng nd the oreilles squdF xow on to fmily memersX thnks spei(lly to my une ouhe mgnétoélstique déposée dessusD qui régir mgnétiquementF gette opporE tunité notmment inspiré le onept derrière une mémoire mgnétoéletrique développée à l9siwx ptisée wivewF n élément mgnétoélstique à nisotropie unixile est soumis à un hmp de iis sttique perpendiulire à l diretion d9nisotropieD qui rise l symétrie qudrtiqueF gette on(gurtion rée deux étts stlesF ous l9e'et d9une ontrinte ppliquée selon une issetrie de es deux diretions @pr exemple pr un sustrt piézoéletriqueAD il est possile de fvoriser l9un d9eux et insi de déterminer l9étt dns lequel se trouver l9élément @voir pigF e pour une représenttion shémtiqueFA in onsidérnt des prmètres rélistesD l dissiption d9énergie ssoiée u sulement d9imnttion peut être réduite de plusieurs ordres de grndeurF pigure e ! rinipe de fontionnement de l mémoire mgnétoéletrique @wivewAD où une ontrinte uniforme permet de psser d9un étt stle à l9utre de mnière nonE équivoqueF ves sousE(gures A A et A présentent shémtiquement l9élément mgnétoéE lstique u repos et soumis à des ontrintes positives @respF négtivesAF ves sousE(gures dA eA nd fA sont des représenttions Qh de l9énergie en fontion des nglesF in se snt sur e oneptD e trvil de thèse propose une tehnique innovnte de déplement mgnétoéletrique de proi de domine à sse énergieF hns e systèmeD l ominison d9un hmp de iis sttique H 0 et d9une ontrinte unixile uniforme σ dns l même on(gurtion que pour l wivew permet le déplement unidiretionnel d9une proi u sein d9une struture mgnétoélstique à nisotropie unixile de onstnte K u @voir pigF fAF ve hmp de iis est un élément ruil responsle d9une risure de symétrieD r typiquementD une ontrinte uniforme seule ne peut ps engendrer le IH mouvement d9une proi à IVH • F v9pplition d9une ontrinte entrîne un ért d9énergie volumique entre les deux dominesD e qui pousse l9un d9entre eux à s9étendre ux dépends du seondD résultnt insi en un déplement de proi unidiretionnel prévisileF hes simultions numériques préliminires e'etuées à l9ide du pkge xmg ont permis d9pporter une preuve de prinipe de e phénomèneF pigure f ! hém du système onsidéré X une struture mgnétoélstique à deux doE mines omportnt une proi de domine trnsverseF our étudier le omportement sttique du systèmeD un modèle ad-hoc été onstruitF v géométrie générle hoisieD représentée sur l pigF gD est elle d9une nnostruture ferE romgnétique @sturtion M s A et mgnétoélstique @÷0eint λ s > 0A de file épisseur h ≤ 20 nmD de lrgeur onstnte ou vrile ≤ 100 nm et de longueur L > 400 nmF ve modèle est sé sur des hypothèses llégent onsidérlement le tritement numérique X on onsidère que l9imnttion M = M s w demeure dns le pln de l ouhe et qu9elle ne dépend que d9une vrile d9espe xD de sorte que l seule inonnue est Φ(x)F v première étpe onsiste à érire l forme de l9énergie en prennt en onsidértion toutes les ontriutions sous forme volumique X énergie d9nisotropie qudrtiqueD énergie eemnD énergie d9éhngeD énergie du hmp de désimnttion et énergie mgnétoélsE tiqueF einsiD l9expression de l9énergie totle U peut être exprimée omme suit X DW de l proi de domine en fontion de l ontrinte ppliquée pour trois pro(les de setion qudrtiques di'érentsF hes nlyses sur les di'érentes ontriutions énergétiques insi qu9un trvil sur un seond modèle enore plus simple ont permis de lri(er les ménismes à l9origine du mouvement de l proi et d9interpréter l reltion entre position d9équilire et ontrinteF he plusD une dpttion de l9éqution pour onsidérer les omposntes de ontrintes σ et τ orrespondnt à l9tionnement pr hmp életrique d9un sustrt piézoéletrique de wxE de oupe @HIIA est fournieF in(nD des luls sés sur des prmètres rélistes ont pu démontrer le gin énergétique ssoié à ette tehnique de déplement de proi de domine pr rpports ux tehniques usuellesF ve premier modèle présenté ne trite que de l9étt d9équilire du systèmeF our otenir des informtions sur l dynmiqueD et insi rtériser orretement le mouvement de l proiD il est néessire d9djoindre u modèle l9éqution de vnduEvifshitzEqilert @vvqA qui régit l dynmique des systèmes mgnétiquesF siD on doit onsidérer un veteur d9imnttion dérit pr deux ngles Φ et Θ lire de s9orienter dns l9espe @voir pigF iAF hns le s générlD l9éqution vvq donnnt le omportement de l9imnttion lolement soumise à un hmp mgnétique H peut s9érire de l sorte X ∂ w ∂t = -γ 0 1 + α 2 w × H eff + α w × w × H eff , @IA IQ pigure i ! hém du système étudiéF ve hmp életrique ppliqué selon z génére dns le sustrt piézoéletrique @type wxEA une distriution de ontrintes ve deux omposntes σ et τ de signes opposésF où α est le oe0ient d9mortissement de qilert et γ 0 est le rpport gyromgnétique de l9életronF he mnière similire à préédemmentD une méthode vritionnelle est suivie (n de déterminer l9expression du hmp mgnétique e'etif lol H eff prennt en ompte l9e'et de toutes les ontriutionsF v9intégrtion de e hmp e'etif dns l9éqution vvq donne le système d9éqution iEdessousF

F F F F F F F F F F F F F F F F F F F F F F F F F F F F F QP IFPFI yverview of tehnologil relevne F F F F F F F F F F F F F F F F F F QQ IFPFP he vnduEvifshitzEqilert eqution F F F F F F F F F F F F F F F F F QT IFPFQ pieldEsed motion F F F F F F F F F F F F F F F F F F F F F F F F F F F QV IFPFR gurrentEsed motion F F F F F F F F F F F F F F F F F F F F F F F F F RI IFQ wgnetoeletri mterils F F F F F F F F F F F F F F F F F F F F F F F F F F F RR IFQFI snstrinsi mgnetoeletri mterils F F F F F F F F F F F F F F F F F F RR IFQFP wgnetostrition F F F F F F F F F F F F F F F F F F F F F F F F F F F F RT IFQFQ iezoeletriity F F F F F F F F F F F F F F F F F F F F F F F F F F F F F RU IFQFR erti(il mgnetoeletri mterils F F F F F F F F F F F F F F F F F F
F F F F F F F F F F F F F F F F F F UP PFPFP i'etive nisotropy F F F F F F F F F F F F F F F F F F F F F F F F F F F US PFPFQ herivtion of the min eqution F F F F F F F F F F F F F F F F F F F F UT PFPFR foundry onditions nd stle sttes F F F F F F F F F F F F F F F F UW PFQ xumeril proedure F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F VQ PFQFI resenttion F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F VQ PFQFP gonvergene nd test ginst nlytil solution F F F F F F F F F F F VT PFR esults F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F VW PFRFI wotionD on(nement nd ejetion F F F F F F F F F F F F F F F F F F F WH PFRFP tressEposition reltionship F F F F F F F F F F F F F F F F F F F F F F F WQ PFRFQ inergeti interprettion F F F F F F F F F F F F F F F F F F F F F F F F WS PFRFR iezoeletri genertion of stress F F F F F F F F F F F F F F F F F F F F WW PFS gonlusion F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F
F F F F F F F F F F F F F F F F F F F IHT QFIFR herivtion of dynmil equtions F F F F F F F F F F F F F F F F F F F IHU QFP xumeril proedure F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IHW QFPFI resenttion F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IIH QFPFP gonvergene nd test ginst nlytil solution F F F F F F F F F F F IIQ QFQ hynmis in hourglss geometry F F F F F F F F F F F F F F F F F F F F F F F F IIS QFQFI hpe of domin wlls F F F F F F F F F F F F F F F F F F F F F F F F F IIT QFQFP rjetory of domin wlls F F F F F F F F F F F F F F F F F F F F F F F IIV QFQFQ eloity nd pplied eletri (eld F F F F F F F F F F F F F F F F F F F IIW QFR hynmis in n in(nite nnostripe F F F F F F F F F F F F F F F F F F F F F F IPI QFRFI imulting in(nite nnostripes F F F F F F F F F F F F F F F F F F F F F IPI QFRFP tedyEsttes in in(nite nnostripes F F F F F F F F F F F F F F F F F F IPR U QFRFQ iletri nd mgneti (eld dependene of veloity F F F F F F F F F F IPT QFRFR gode filure nd rekdown phenomenon F F F F F F F F F F F F F F F IPW QFS gonlusion F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IQI 4 Experimental investigations RFI wterils F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IQP RFIFI wgnetoelsti multilyers F F F F F F F F F F F F F F F F F F F F F F F IQP RFIFP wxE piezoeletri sustrte F F F F F F F F F F F F F F F F F F F F IQR RFIFQ wehnil oupling F F F F F F F F F F F F F F F F F F F F F F F F F F IQS RFP prition F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IQU RFPFI olishing of the wxE sustrte F F F F F F F F F F F F F F F F F F IQU RFPFP vithogrphy F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IQW RFPFQ heposition of multilyers F F F F F F F F F F F F F F F F F F F F F F F F IRI RFPFR viftEo' nd dryEething methods F F F F F F F F F F F F F F F F F F F F IRR RFQ ghrteriztion F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IRV RFQFI trin guge mesurements F F F F F F F F F F F F F F F F F F F F F F IRV RFQFP wgneti hrteriztion of sputtered lyers F F F F F F F F F F F F F ISI RFQFQ etomi nd wgneti pore wirosopy F F F F F F F F F F F F F F F F ISR RFQFR wgnetoEopti uerr mirosopy F F F F F F F F F F F F F F F F F F F F ITH RFR rospets F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F
U = L 2 -L 2 h (x) -K u cos 2 Φ + A dΦ dx 2 -µ 0 M s H 0 sin Φ + 3 2 λ s σ sin Φ cos Φ dx - V 2 1 2 µ 0 M 2 s w( r) • N ( r,
   Φ = 1 sin Θ γ 0 1 + α 2 (-s + αr), Θ =
γ 0 1 + α 2 (r + αs), @PA où l9on introduit les quntités r et s X r =sin Φ H dx + cos Φ H dy + cos Φ H 0 + 2 µ 0 M s (-K u cos Φ sin Φ sin Θ + 3 4 λ s (τσ) cos 2Φ sin Θ + A sin Θ Φ + 2 cos Θ Φ Θ + sin Θ Φ , @QA IR et s = cos Φ cos Θ H dx + sin Φ cos Θ H dysin Θ H dz + sin Φ cos Θ H 0 + 2 µ 0 M s K u cos 2 Φ sin Θ cos Θ + 3 4 λ s ((τ + σ) cos Θ sin Θ + 2(τσ) cos Φ sin Φ cos Θ sin Θ) + A Θ + Θcos Θ sin Θ Φ 2 . @RA he même que pour l9éqution sttiqueD le reours u tritement numérique est ii oligtoireF he pr les propriétés de l9éqution vvqD le hoix d9un lgorithme impliite est de rigueur (n de grntir stilité et préision du lulF insuiteD l méthode de xewton été intégrée pour l résolutionF our véri(er l vlidité des résultts de e nouveu progrmmeD églement développé sous wevefD on s9est d9ord tthé à omprer les positions d9équilire de l proi à l9issue de l simultion dynmique à elles otenues ve le progrmme de relxtion @cf. pigF hAF insuiteD on pu véri(er que l vitesse de l proi otenue dns le s d9un déplement induit pr un hmp mgnétique orrespond ien à l vleur prédite pr un modèle nlytique nlogue à elui de lkerF pigure p ! @A engles Φ et Θ de l9imnttion u ours d9une simultion dynmiueF @A eprésenttion Qh de l distriution d9imnttion en régime sttionnire le long d9un nnorun @ = 100 nm et h = 10 nmAD ve un hmp életrique de -1 wGmF ves études dynmiques se sont onentrées sur l9nlyse du omportement de l9imnE ttion u sein d9une nnostruture à lrgeur vrile identique à elle étudiée en sttiqueD insi qu9u sein d9un nnorun à lrgeur onstnteF illes ont pu mettre en évidene un mouvement unidiretionnel rtérisé pr une forme prtiulière de proi de domineF IS n omportement hors pln spéi(que est notmment remrquéD et eluiEi s9évnouit tndis que l proi s9pprohe de s position d9équilire dns le s d9une géométrie à lrgeur vrile on(nnt l proiF eu sein d9une struture à lrgeur onstnteD l9exisE tene d9un régime sttionnire été mis en évideneF geluiEi on(rme une forme de proi présentnt une omposnte hors pln signi(tive @voir pigF pAD dont l9mpleur vrie en pigure q ! itesse de l proi de domine en fontion du hmp életrique ppliqué et de l vleur du hmp mgnétique de iis pour un oe0ient d9mortissement de HDIPF ves di'érentes simultions ont montré que des vitesses très importntes @b VHH mGsA peuvent être otenues en hoisissnt ien les vleurs du hmp életrique ppliqué @et don des ontrintesA et du hmp mgnétique de iisD dont l ominison est à l9origine du mouvementF ges résultts sont présentés dns l pigF qF gomme évoqué plus hutD es performnes peuvent en prinipe être otenues pour de très files onsommtions d9énergieF sl fut églement noter que le ode ne fontionne plus uEdelà de ertines vleurs du ouple hmp életrique E hmp mgnétiqueF yn peut supposer que el est dû à un hngement qulittif de régime dynmiqueD à l9instr du lker rekdown oservé IT dns le dre du mouvement induit pr un hmp mgnétique longitudinlF in e'etD un omportement similire du ode été onstté dns e s lssiqueD où le ode éhouit à dérire l dynmique pour un hmp mgnétique supérieur u seuil du rekdownF ges trvux numériques ont permis de poser une se solide u onept de déplement de proi induit pr des ontrintes uniformesD en présene d9un hmp mgnétique de iisF hevnt l9intérêt que présente ette tehniqueD il serit intéressnt de l mettre en ÷uvre expérimentlementF our elD le hoix des mtériux s9est porté vers le wxE de oupe @HIIA et un multiouhe mgnétoélstique go 2 GpegoF v9ssoition des deux phses retenue pour e trvil de thèse est un dépôt d9une ouhe mine mgnétoélstique sur un sustrt piézoélétriqueF ges derniers sont des produits otenus dns le ommere spéiliséD néessitnt u prélle un polissge pour rendre possile l nnofrition sur leur surfeF v proédure développée permis d9otenir une rugosité de l9ordre du nnomètreF ne lithogrphie @résine positiveA don pu être mise u point sur es surfesD permettnt de développer les motifs du msque prépréD pour ensuite poursuivre ve un dépôt pr pulvéristion thodique du mtériu multiouhe @ PH nmAF eprès quelques justementsD il été possile d9e'etuer ve suès une proédure de liftEo'D de telle sorte que les motifs otenusD visiles sur l pigF rD re)ètent (dèlement eux du msqueF pigure r ! smge u mirosope életronique à lyge des nnostrutures mgnétoéE lstiques otenues pr liftEo'F v rtéristion des mtériux et strutures otenues donné lieu à plusieurs résulE ttsF out d9ordD les propriétés piézoéletriques du wxE ont été otenues à l9ide IU d9une juge de ontrinteF ves mesures révèlent un omportement nonElinéire ohérent ve e qui se trouve dns l littértureF ves propriétés mgnétiques et mgnétoélstiques des mtériux multiouhes utilisés ont été respetivement otenues pr des mesures w @nisotropieD imnttion à sturtionA et pr l méthode de l poutre enstrée )éhisE snteF qlolementD les vleurs mesurées sont ohérentes ve les prmètres hoisis lors des simultions numériquesF pigure s ! smges uerr d9un grnd motif mgnétoélstique soumis à un hmp mgnétique et des ontrintes méniques induites pr le sustrt piézoéletriqueF v on(gurtion multidomine est lirement visileF v topologie des nnostrutures été déterminée pr mirosopie à fore tomique @epwAD révélnt des pro(ls stisfisntsF r l suiteD l on(gurtion mgnétique u sein des nnostrutures mgnétiques pu être étudiée pr mirosopie à fore mgnétique @wpwAF ges oservtions ont mis en évidene le fit que les éléments étient qusiment systémtiquement de on(gurtion monodomineD en dépit des e'orts pour induire l rétion de prois de domineF ur des éléments plus grndsD il été possile de fire des

Introduction

wgnetism is known to everyoneD if only through the widespred use of simple fridge mgnetsF roweverD reltively few people relize how hevily ll the tehnology essentil to our modern lifestyle relies on mgnetismF prom eletri genertors to hrdEdrive disks nd funtionl mgneti resonne imgingD the use of mgneti mterilsD tools nd devies is uiquitousF yf ourseD ll these fruitful pplitions of mgnetism were mde possile y the work of genertions of reserhers dediting their reers to these mttersD from yerstedD empère nd prdy onwrdF ynly frtion of them ould foresee tht their work would led to suh spetulr developmentsF odyD oherent nd omprehensive ody of knowledge helps us understnd the origin of mgnetism nd suessfully ounts for the ehvior of mgneti mterilsF sn prtiE ulrD domin wlls"whih seprte regions of uniform mgnetiztion in ferromgnets" hve long een sujet of interest euse of their importne from the fundmentl point of view s well s their relevne in mgneti deviesF heir mnipultion is linked to dynmi nd ompetitive re of reserhF wgneti (eldEdriven domin wll motion hs een thoroughly investigtedD nd with the dvent of spintronisD the vrious wys of induing domin wll motion with n eletril urrent hve een widely studied nd reported in the literture in reent yersF wotivted y potentil dvnes on the fronts of energy onsumptionD filitted implementtion or opertionD innovtive methods s well s re(nements of wellEknown tehniques llowing ontrol of mgnetiztion nd domE in wlls re onstntly put forwrd y reserhersF emong the proposed ndidtesD mgnetoeletri mterils hve een reently gining tremendous ttentionF he min reson for this is tht the use of n eletri (eld insted of n eletri urrent is n elegnt wy to drmtilly redue the energy required to ontrol mgnetiztionF hile mgE netoeletri e'ets n exist in singleEphse mterilsD the superior opportunities o'ered y heterostrutures tend to diret reserh e'orts in this diretionF sn this thesisD we present n innovtive pproh to domin wll mnipultion sed on stressEmedited mgnetoeletri e'etF e uild on onepts developed t siwxD IW more spei(lly within the viwegGvsg snterntionl essoited vortoryF hese previous works ulminted with the development of mgnetoeletri memories implemenE ted through the mehnil oupling of piezoeletri sustrte nd mgnetoelsti lyerF e ontend tht the omintion of uniform stress nd is mgneti (eld reking the symmetry n result in the ontrolled motion of trnsverse domin wllF e show tht suh motion is ssoited with unusul physil feturesD ompetitive perE formnes nd low energy onsumptionD while its experimentl implementtion ppers relistiF he (rst hpter provides the neessry kgroundF he sis of mgnetism required re rie)y presentedD followed y n ount of the literture on domin wll motionF he proposed onept is then introdued in the ontext of urrent reserh trendsD with fous on mgnetoeletri mterilsF his hpter lso ontins preliminry numeril vlidtion of the oneptF he seond hpter rings new evidene through n ad hoc one dimensionl model le to lulte the equilirium mgnetiztion distriution in systems with nnostripe geometryF fsed on vritionl methodsD it tkes into ount the relevnt physil ontriutionsF efter short presenttion of the odeD the results otined on nnostrutures of vrile width re desried in greter detilF he third hpter goes on to ddress the dynmis of the motionF he outome of vritionl methods is omined with the vnduEvifshitzEqilert eqution nd is solved using nother ad hoc model developed over the ourse of the hhF he pproh hosen is rie)y explined nd results re disussed in more depthD s the dynmis hppen to e hrterized y interesting feturesF he fourth hpter dels with experimentl investigtions undertken during the hhF he mterils hosen re desriedD nmely the ommeril piezeoeletri sustrtes nd the rreEerthEsed multilyers routinely used t siwxF hetils re given on the frition proessF he susequent e'orts to hrterize the mterils nd devies re reportedF pinllyD generl onlusion loses this mnusriptF st reviews the generl pproh dopted s well s its most prominent (ndingsF e disussion on the possile lines of reserh for future investigtions is lso inludedF PH Chapter 1

Fundamentals and context he work tht is to e presented in this thesis involves vrious onepts of miromgneE tismD nmely si properties of ferromgneti mterils t the mirosopi sleF hey re prt of omprehensive nd onsistent theoretil frmework whih suessfully desE ries wide rnge of physil phenomen enountered experimentllyF sn this hpterD we will (rst im t providing some physil kground nd desription of the uilding loks on whih our developments dependD inluding mgneti domin wlls whih re of ritil importne in ferromgnetismF peil ttention will e pid to mgnetoelstiity @or mgnetostritionAD sine this property is essentil to the phenomen studied in this thesisF prom these fundmentlsD n overview of the very dynmi reserh re of domin wll motion nd its pplitions will e givenF he di'erent tehniques used to mnipuE lte domin wlls will e disussed in terms of the physil mehnisms t ply s well s their potentil nd weknessesF imphsis will e put on mgnetoeletri oupling pE rdigmsD inluding the emergent stressEsed pprohesF sn this ontextD we will present the ide of omining uniform stress nd symmetryEreking mgneti (eld to ontrol mgnetiztionF his omintion hs een proposed nd suessfully implemented in the ontext of the mgnetiztion swithing of monodomin mgnetoelsti prtilesF henD we will present the sujet mtter of this thesisD tht is the extension of this prdigm to twoEdomin mgnetoelsti elements in order to drive domin wll motionF he dvntE ges of this method will e detiled nd preliminry numeril proofs of the onept will e shownF

1.1 Brief introduction to (micro)magnetism his (rst setion onsists in short introdution to si mtters in mgnetismF hese deE velopments re essentil to the susequent desription of the ontext of the work presented PI in this thesisF efter presenting the min onepts nd physil quntities of mgnetism in mterilsD the link etween mrosopi ehvior in mgnetism nd mirosopi phenoE men is explored nd mgneti domin wlls re therey introduedF henD some physil detils on domin wlls re givenF 1.1.1 Magnetization and energy contributions he sis for the understnding of miromgnetism omes from pioneering work y vnE du @most remrkly in his lndmrk IWQS pper with vifshitzAD fekerD eissD flohD xéel nd frown mong othersF he ltter provided omprehensive ount of the suE jet in his IWTQ ook entitled Micromagnetics ID s ghikzumi lter did PF ell the onepts nd developments elow n e tred to the minstrem theory of miromgE netism desried in these referene ooksF sn this workD we will exlusively del with ferromgneti mterilsF yther forms of mgneti ehvior suh s prmgnetism or dimgnetism will therefore not e ddressedF sn ferromgneti mterilD individul toms hve spontneous mgneti moment m @unit em 2 AF st is mostly due to eletronsD through their intrinsi property lled spin @riE sing from the omintion of reltivisti nd quntum e'ets QA nd to their net oritl ngulr momentF his n onfer net mgneti moment to the tom depending of the prtiulr rrngement of eletronsF elsoD the spin of neutrons nd protons mens there is lso smll ontriution of the nuleus to the overll mgneti momentF perromgneti mterils re hrterized y the ft tht these moments intert with their immediE te neighorsF his intertion of quntum nture is positive in ferromgneti mterils @respF negtive in ferrimgneti nd ntiferromgneti mterilsAD so tht neighoring toms will tend to hve the sme spin orienttion @respF opposite spin orienttionAF e mrosopi olletion of mgneti moments oriented in the sme diretion n give rise to the mrosopi phenomen everyone hs witnessed in mgneti mterilsF qiven the length sles involved nd ording to the lssil pproh of miromgE netisD we n de(ne vetor (eld of quntity ommensurle with volumi mgneti moment @eGmAD i.e. density of mgneti momentF st is lled mgnetiztion"referred to s M "nd is thus de(ned within the mgneti ody t ll points of its volume V ⊂ R 3 F e si ssumption of miromgnetism is tht the mgnitude of M in ferromgneti mteril depends only on temperture nd is onstnt in time nd uniform in speD i.e. Anisotropy energy he rystlline rrngement of toms in mgneti mteril n give rise to nisotropi ehvior in ferromgneti mterilsD for instne through spinE orit intertionF pei( mgnetiztion orienttions re therefore energetilly fvored depending on the ssoition of the hemil elements onstituting the mterilD nd its ristllineGmorphous strutureF he nisotropy found in mterils is t the origin of some of the hystereti ehviors oserved nd of the distintion etween soft @low nisotropyA nd hrd @high nisotropyA mgneti mterilsF Exchange energy golletive ehvior in ferromgneti mterils rises from the inteE rtion etween djent mgneti momentsF his intertion is positive in ferromgneti pigure IFS ! @A hemti of two ntiprllel domins in nnostripe geometry with inE plne nisotropyF op view of mgnetiztion distriution relting to @A trnsverse nd @A vortex domin wllsF pigure from IUF PW here re two sutypes of trnsverse domin wllsD whih n lso e hyridized IVF he defult domin wll type nd (rst to e desriedD is the Bloch wallF vet us for instne onsider two ntiprllel domins seprted y domin wll in mrosopi smpleF he need to redue demgnetiztion energy puts onstrint on the sptil vrition of the mgnetiztion from one domin to the otherF st is est interpreted s the tendeny to void the retion of mgneti hrges ρ m within the domin wllF ine ρ m = -div(M )D this ondition is equivlent to dwx dx = 0 @if we onsider the xes of pigF IFTAF hereforeD there is no vrition of the mgnetiztion omponent orresponding to the xis perpendiulr to the domin wllD s n e visulized in pigF IFTF sn this on(gurtionD the mgnetiztion rottes round the xExisF sn thin (lmsD exept in the se of perpendiulr mgneti nisotropy @weAD the mgnetiztion tends to remin in the plne of the (lm to minimize demgnetiztion energy @cf. soElled shape anisotropyAF his onstrint 'ets the energeti trdeEo' of the domin wllF sndeedD there is strong inentive for the mgnetiztion in the domin wll to lso sty inEplneD whih mens inEplne rottion ross the domin wll @pigF IFTA to prevent the pperne of lrge mgnetostti energyD s seen in pigF IFTF prom hereD the ft tht the ondition dwx dx = 0 nnot e respeted in thin (lms is quite strightforwrdF e(ned onsidertions s well s sutegories within xéel nd floh wlls n e found in IWF pigure IFT ! hemti of @A floh nd @A xéel wll in ferromgneti lyerF pigure from TF qiven trnsverse domin wll on(gurtion @e it of floh or xéel typeAD the omiE ned in)uene of ompeting physil onstrints governs the distriution within the domin QH wllD most notly the rte of sptil hnge from one mgnetiztion diretion to the otE herF yn the one hndD the exhnge energy tends to disourge ny rupt grdient of mgnetiztion nd thus fvors wide domin wllsF yne the other hndD the mgnetiztion hs to devite from the xis fvored y the nisotropyD long whih is the mgnetiztion within the two dominsF hepending on the reltive weights of these two ontriutions to the domin wll energyD the width of trnsverse domin wll n vry s shown in pigF IFUF he exhnge energy is proportionl to n exhnge onstnt AD @unitsX tGmA while the unixil nisotropy is proportionl to onstnt noted K u @unitsX tm -3 AD or equivlentlyD n nisotropy (eld H a @so tht K u a 1 2 µ 0 M s H a D where µ0 is the vuum perE meilityAF he A/K u rtio therefore plys n importnt role in determining the sptil extension of domin wllF pigure IFU ! hemti of omprtively wide nd thin domin wllsF sn ferromgneti odyD this is ditted y the prtiulr lol set of physil onstrints to whih the domin wll is sujetedF pigure from TF elthough domin wlls usully llow for sustntil redution in overll energyD they re nonetheless ssoited with lol inrese of energyF sndeedD in the domin wll viinityD ontriutions of exhngeD eemnD demgnetiztion nd nisotropy my e inresedF hether or not the width of the domin wll is negligile with respet to its other dimensionsD surfe energy n e de(ned nd e expressed s funtion of A nd K u F pirst proposed in IWQP y flohD the floh wll is treted nlytilly y vndu nd vifshitz in their IWQS pper PHF heir wll lultion hs een rried out multiple times therefter with smll hnges nd dpttionsF hey onsidered n in(nite ferromgnet whih mgnetiztion only vries with one dimension @let it e lled xAD with only unixil nisotropy nd exhnge involvedF he inEplne mgnetiztion unit vetor w n e desried using one ngle Φ ∈ [0, 2π]F et the oundries @x = -∞ nd x = +∞AD vlues of Φ re ntiprllel long the xExisX Φ(-∞) = 0 nd Φ(+∞) = πF sn this seD the QI lultion is reltively strightforwrd nd n e rried out nlytillyF he solution Φ funtion is s follows @see ppendix e for derivtionAX

Φ = π -arccos K u A x , @IFQA
whih n lso e written

Φ = 2 arctan exp K u A x .
@IFRA egin one n see the importne of the reltive mgnitude of nisotropy nd exhnge through the rtio of the orresponding onstntsF he lrger the rtioD the thinner the domin wllD s is visile in pigF IFVF homin wll thikness in this se is of the order of π A Ku TF sn generlD the vlue onsidered for the exhnge onstnt A is round 9 × 10 -12 tGm for ll mterils PI while the strength of the nisotropy n vry from negligile to very lrge vlues in hrd ferromgnets @nisotropy (eld in the order of mgE nitude of 10 6 eGmAF hereforeD domin wll width typilly rnges from less thn nnometer to tens of nnometersF e hve seen tht the mgneti distriution in very soft mteril is often smoothly vryingD so tht the notion of domin wll eomes meE ningless in prtieF here is therefore lurry line for some mterilsD where it n e di0ult to distinguish etween lerly de(ned domin wll nd lrge res of slowly vrying mgnetiztionF es lredy mentionedD domin wll is not rigorously de(ned ojet ut is used to help the understndingF sf the system hs (nite lengthD the oundry onditions t its extremities led to nother solution involving ellipti funtionsD s demonstrted in ppendix eF hespite the utter simpliity of this modelD the lultion ove n tully e deent pproximE tion depending on the on(gurtion studiedF his nlytil distriution hs therefore een used extensively to this dy in reserh deling with domin wlls in ferromgnetsD inluding the dynmis of domin wll motionD sujet of tremendous interest whih will e ddressed elowF

∂ M ∂t = -γ 0 M × H + α M s M × ( M × H) .
@IFSA rereD γ 0 = µ 0 γ nd γ is the gyromgneti rtio of the eletronD α is the dmping oe0ient nd M hs lredy een de(ned s the mgnetiztion vetorF he form of the eqution QT rought y qilert is slightly di'erentX

∂ M ∂t = -γ G 0 M × H + α M s M × ∂ M ∂t . @IFTA
st n e shown tht the two equtions re mthemtilly equivlentD y tking the ross produt of iqF @IFSA with M F henD one n determine the following reltion etween the oe0ients y identi(tion TRD TSX

γ G 0 = (1 + α 2 )γ 0 . @IFUA
reneD the vnduEvifshitzEqilert eqution n e written s followsX

∂ M ∂t = - γ G 0 1 + α 2 M × H + α M s M × ( M × H) . @IFVA pigure IFII !
v(H) = - γ 0 H α A K u + µ 0 M 2 s cos 2 φ 0 , @IFWA sin 2φ 0 = H 2αM s = H H c .
@IFIHA e n see tht the ngle φ is uniform nd ssumes the vlue φ 0 given y iqF @IFIHAF H c is ritil (eld eyond whih this eqution hs no solutionF gontrry to wht is often sttedD the reltion etween mgneti (eld nd veloity is not liner nd tht is euse of the in)uene of φ 0 whih vries with HF he nonlinerity is rely visile if there is strong nisotropy VPD ut otherwiseD it n e signi(nt @see pigF IFIQAF feuse of the resulting demgnetizing (eldD the geometry onsidered lso hs n e'et kin to n nisotropyD whih will in)uene the dynmis VQF enother point tht is not often stressed is tht the lker solution is not lwys stle VRD depending on nisotropy prmetersF yther nlytil solutions exist"inluding the uolmogorovEetrovskiiEiskunov @uA solution"whih hve shown to mth the results of numeril simultions VSD VTF hen the pplied (eld is inresed eyond the soElled Walker breakdown @i.e. H > H c AD the system enters n osilltory regimeF his omplex ehvior requires further pproximtions or the use of numeril tretmentF sn ny seD strong redution of veloity @negtive moility dv dH A is oservedF yverllD oth numeril nd experimentl investigtions @see pigF IFIRA hve supported the existene of n rupt trnsition round ritil mgneti (eld VU!VWF he motion eyond the rekdown (eld is hrterized y n osillting veloity"to the point of sometimes using retrogrde motion"s well s internl dynmis of the domin wll UTDVVDWHDWIF hey re lso reports tht osilltions my even our efore rekdown WP nd in the se of spin wve emissions WQF eprt from the two regimes desried ove @stedyEstte )ow regime nd osilltory regimeAD experimentl investigtions hve shown the existene of nother regime t low (eldsF veled creep regimeD it is hrterized y very smll veloities nd stohsti proess of step y step domin wll motion WRD see pigF IFISF et the mirosopi sleD this is interpreted y the pinning of the domin wll on lol defets or inhomogeneities of vrious origins in mterilsF prom hereD models hve een developed WS desriing this phenomenonD yielding veloity proportionl to exp(-Cf -µ /T ) where f is the externl fore nd µ is the ritil exponentF he motion is thus thermlly tivted proessX veloity is deresed with lower temperture in thin (lms WTF he mehnism ehind depinning hs lso een studied WU nd there ppers to e liner reltionship etween QW pigure IFIQ ! et of urves representing the veloity s funtion of the pplied mgneti (eld for severl vlues of the rtio M 2 s Ku F ith reltively wek nisotropyD the nonlinerity yields onve shpe whih n exhiit mximum elow H c F pigure from USF depinning (eld nd edge roughness WVF hile it is often modeled y elsti interfes WWD this is vlid only for domin wlls in Ph on(gurtionsF sn on(ned geometries suh s nrrow ferromgneti stripesD the reep regime turns linerX onsidering the domin wll s rigid or pointElike entity propgting in one dimensionl disordered medium shows good greement with experimentl dt while the regulr sling lw reks down IHHF elthough most of the reserh fouses on domin wll motion in thin (lms or nnowire geometries with onstnt setionsD delierte engineering of ptterns to in)uene domin wlls is lso present in the litertureF pirstD it is interesting to note tht widespred tehnique used to generte domin wlls in nnowires is to use speil shpes in the design of the extremities IHIF yne retedD domin wlls n e pinned t lol onstritions or nothes long the nnowireD s hs een shown experimentlly IHP!IHSD whih is n exmple of shpeEindued potentil lndspe engineeringF sn the se of rel pinning @i.e. with trpped domin wllAD the depinning (eld inreses with the size of the RH pigure IFIR ! ixperimentl results showing the omplex dynmis of (eldEdriven domin wll motionD s funtion of mgneti (eldF yne n lerly identify the rekdown (eld round 10 yeF pigure from VWF nothD nd depends on the hirlity of the domin wll IHTD lthough the prtiulr shpe of the noth does not seem to mtter systemtilly IHUD IHVF e olletion of nothes n e used to rrnge stility t multiple lotions IHWF es will e explined in susetion IFRFID engineering of the ross setion n lso e used in tEeyeEshped elements in order to rek the symmetry IIHF idge roughness nd pinning e'ets in generl do 'et the dynmis eyong the reep regime IIID IIPD ut does not seem to hnge the rekdown (eld WV @lthough it hs een reported tht this n e hieved y the delierte engineering of ross setion IIQDIIRAF roweverD there is evidene tht trnsverse (eld hs n in)uene on the vlue of the lker rekdown (eld IISD llowing signi(nt inreses of the domin wll veloity IITF he sme n e sid of domin wll motion in mterils with strong hzyloshinskiiEworiy intertion IIUF 1.2.4 Current-based motion ith disoveries suh s gint mgnetoresistne IIV nd spinEpolrized urrentsD ouE pled to new opportunities provided y nnotehnologyD the (eld of spintronis hs een growing very rpidlyF e mjor reserh re of spintronis dels with urrentEindued domin wll motionD (rst proposed in IWUV IIWF sn this susetionD we will give very RI pigure IFIS ! ypil dependne of veloity on the pplied (eldD showing the reep regime t low (elds @rekdown (eld not shownAF pigure from WUF rief ount on this sujetF por more detiled informtion on theoretil mttersD see efsF VID IPHD nd for n experimentl review efF IPIF he reserh for full understnding of the physil mehnisms ehind urrentE indued motion is still ongoingD ut lot hs een explored nonetheless through miromgE neti studies IPPD IPQD theoretil IPR!IPT nd experimentl investigtions IPUD IPVD in prtiulr in thin (lms nd nnostripesF he motion is indued y torques pplied to the domin wll y the urrent )owing through it @e inEplne or outEofEplneAF sndeedD urrent )owing in ferromgnet tends to e polrized long the lol mgnetiztionD nd will exert torque on lol mgnetiztion in se it is not prllel to its spin polriztionF wo types of torques hve een well identi(edX n diti torque nd nonditi torqueD whih reltive importne depends on the vlues of prmeters suh s domin wll width nd typil lengths desriing spin trnsferF he diti torque domintes in the se of wide domin wllsD while nonditi torques re notly involved with thin domin wlls nd strong pinning IPSF eltively high veloities @hundreds of mGsA hve een hieved experimentlly in urrentEdriven motion IPVD IPWD espeilly in ultrthin nnostripes IQHF he veloity will lso depend on domin wll on(gurtion @i.e. the type of domin wllA nd urrent injetion sheme IQIF here re indeed di'erent wys to injet urrent to move domin wllX in nnostripeD the urrent n )ow long the nnostripe ut lso perpenE diulrly to itD e it inEplne or outEofEplne @vertillyA RTF yn this mtterD it is lso worth mentioning tht while the motion n e indued y lol eletri urrent )owing RP through the domin wllD it n lso e done remotely with spinEurrent injetion IQP @lso knwon s nonElol spin injetionAF sndeedD onversion etween eletri nd spin urrent through the spinErll e'et is possileD nd the use of pure spin urrent hs the dvntge of drmtilly reduing lol hetingF es n sideD therml grdients"for inE stne generted y toule heting"hve lso een used to mnipulte domin wlls IQQF pigure IFIT ! ypil veloityEurrent reltionshipD from one dimensionl model y hiville et lF pigure from IPQF sn ny seD the dynmis of urrentEdriven domin wll motion hve mny fetures in ommon with (eldEdriven motion @see pigF IFITAF he urrent dependne of veloity exhiits lkerElike delinetion @urrent density limitA eyond whih the veloity is signi(ntly redued IPQD IQRD IQSF he stedyEstte regime is repled y n osilltory ehvior IQTF felow threshold urrentD one n lso notie reep regime similr to wht n e oserved with (eldEdriven motion IQUD where the nonditi torque seems to ply mjor role IQVF here hs een sustntil dete over the universlity of reep regime nd therml tivtion IQW!IRQF he question is out whether or not the stohsti proess governing reep motion in urrentEdriven motion is the sme s with (eldEdriven motionF RQ 1.3 Magnetoelectric materials homin wll motion nd ontrol of mgnetiztion t smll sles in generl is thus very tive re of reserhF sn this thesisD the min gol is to explore the possiilities regrding domin wll motion nd mgnetoeletri e'et medited y mehnil stress in rti(il multiferroi struturesF sndeedD this pproh o'ers signi(nt dvntges over stndrd methodsD suh s enhned funtionlity nd lower power onsumptionF feuse it relies on the mgnetoelsti ouplingD susetion is dedited to short desription of mgnetostritionF 1.3.1 Instrinsic magnetoelectric materials he mgnetoeletri e'et ws (rst proposed y F gurie in IVWR IRR nd lter oserved experimentlly y estrov IRSDIRT in hromium oxide gr 2 y 3 D lso following the predition of vndu nd vifshitz IRU tht this mteril should exhiit mgnetoeletri e'et in its ntiferromgneti phse IRVF his phenomenon dels with the oupling etween eletri nd mgneti quntitiesF he djetive diret or inverse is sometimes used to identify if the stimulus is mgneti or eletriF he pplition of mgneti (eld @respF eletri (eldA will result in n eletri @respF mgnetiA e'etF his oupling n e found in some mterils where it rises spontneouslyF hile the term mgnetoeletri is often onfused with multiferroiD they refer to distint propertiesF sndeedD mgnetoeletri mterils re thus sensitive to oth mgneti nd eletri (eldD ut need not show ny mgneti or eletri order @see pigF IFIUAF pigure IFIU ! hemti showing the reltionship etween lsses of mterilsF pigure from IRWF RR here re two resons whyD despite the interesting potentil of intrinsi mgnetoeletri mterilsD these hve not yet enountered muh suessF yne of them is the ft tht in ny se the intrinsi mgnetoeletri e'et is reltively wekD with fundmentl upper ound for the omponents of the mgnetoeletri suseptiility tensor ISHD s is predited for spei( mterils ISIF purthermoreD most mterils only exhiit interesting properties t low temperturesD whih is for exmple the se of gr 2 y 3 ISP or fipey 3 ISQF yne wy of getting strong mgnetoeletri response is to work in the viinity of phse trnsitionsF he pplition of mgneti or eletri (eld n then indue lrge mgnetoeletri e'etD inluding t room temperture ISRD ISSF hile it does not involve the mgnetoeletri e'et per seD nother quite reent tivity dels with ultrthin (lms IST nd mterils with perpendiulr mgneti nisotropyD where it is possile to in)uene nisotropy nd mgnetiztion ISU!ITR through sutle interfe e'etsF iletri (elds pve the wy for ontrol of mgneti properties"inluding the phse"without resorting to nneling or other kinds of tempertureEsed proedures ITSF sn the se of domin wll motionD nuletion nd pinning hve een otined or in)uened y eletri (elds ITRDITTDITUD s well s modultions of domin wll dynmis in the reep ITV nd )ow ITRD ITWD IUH regimesF hese e'ets re sometimes lled charge-mediated magnetoelectric e'etsF pinllyD interesting developments onern the voltgeEontrol mgneti nisotropy in mgneti tunnel juntionsF unnel juntions re ruil element of spintronis nd the ft tht eletri (elds n to some extent 'et their ehvior hs not gone unnotied y the reserh ommunity IUI!IUQF snterestinglyD the possiility to indue domin wll motion in intrinsi mgnetoeletri ntiferromgnets @suh s gr 2 y 3 A using n eletri (eld hs een theoretilly investigted IURF es in mgneti (eldEdriven motionD there seems to e lkerElike rekdown nd mximum veloity hieved t or efore this pointF he uthors lso desrie how the sitution n e improved y dding n inEplne stress orD for tht mtterD ny kind of inEplne nisotropy in order to hinder preessionF everl reviews hve overed the sujet of mgnetoeletri mterilsF hey knowE ledge the weknesses of the intrinsi e'etD ut lso present the opportunities nd potentil pplitions of more reent reserhD espeilly in thin (lms IRWDIUSF ome even mention revival of the mgnetoeletri e'et IUTD in light of ll the pplitions onsideredF sn prtiulrD vrious opportunities ome from the use of ntiferromgnets nd the rpid development of rti(il mgnetoeletri mterilsF RS 1.3.2 Magnetostriction fefore disussing rti(il mgnetoeletri mterilsD it is importnt tht we hve short desription of mgnetostrition nd its modelingF sndeedD rti(il mgnetoeletri mE terils often involve mgnetostrition euse they rely on the mehnil oupling of ferroeletri nd ferromgneti phsesF his is lso the se in this thesisF wgnetostrition hs numerous pplitionsD most notly in sensing nd tutionD nd is lso responsile for the typil 4humming4 noise herd ner eletril power trnsformersF fsillyD mgnetostrition is the proess y whih mgneti mteril will undergo deformtion following the modi(tion of its mgnetiztion @for exmple y the pplition of mgneti (eldAF he onverse e'etD sometimes lled inverse mgnetostrition or illri e'etD dels with the hnge of the mgneti suseptiility of mteril on whih given mehnil stress is ppliedF st is importnt to point out tht the trnsformtion used y mgnetostrition is ssoited with nisotropi deformtions ut is isovolumiF hile mgnetostrition ws (rst desried y toule in the IWth entury IUUD its physil origin is still sujet of theoretil investigtionsF he phenomenon is etter understood for rreEerth elements @from ge to vuAD ltE hough most of them hve gurie tempertures elow room tempertureF heir mgneE tostritive properties stem from the ngulr distriution of oritls in their un(lled Rf sushellD whih is prtly responsile for the overll tomi mgneti momentF hese orE itls re signi(ntly spheril @see pigF IFIVAF herefore when the mgneti moment is rottedD the strong spinEorit oupling will tend to lso reorient the oritlsF his results in hnge of hrge distriutions ndD euse of the doumented spheriity of the orE itlsD this will hve diret in)uene on eletrostti fores nd strong mgnetostritive strins IUVF qiven the olte @s in e.g. teriumA or prolte @s in e.g. smriumA spet of the oritlsD positive nd negtive mgnetostrition will e respetively mesuredF yf ll elementsD terium hs the lrgest mgnetostrition known todyD whih leds to n elongtion of nerly I7 t H u IUWF hile there re still some unknowns on the physil workings of mgnetostritionD espeilly in lloys with trnsition metlsD one ould rgue tht e'orts to formlize IRUD IVID IVP this phenomenon hve een quite suessfulF e (rst level of desription dels with the intertion energy of neighoring mgneti momentsD for instne s proposed y xéel IVQ sed on the osines of the ngle etween mgnetiztion nd the diretion etween two tomsF e ylor expnsion of this expression will yield fmilir forms of the mgnetoelsti intertionD depending on the symmetry onsidered TD IVRF ytherD more RT pigure IFIV ! 4f eletron hrge density in rreEerth elementsF pigure from IVHF phenomenologilD pprohes fous on the onstitutive lws onsidered in the ehvior of mgnetoelsti mterilsF his leds to the following expression of the energy densityX

u me = -T ik µ ik , @IFIIA
where T refers to the lol guhy stress tensorD nd ˆ µ is the free strin of the mteril @i.e. the strin mesured if the ody is perfetly free to deform itselfAF prom thereD it follows tht in n isotropi mteril nd in the presene of unixil stressD the energy density ssoited with the mgnetoelsti intertion is often written @see hpter P for detiled derivtion of this expression from iqF @IFIIAAX

u me = - 3 2 λ s σ cos 2 Φ, @IFIPA
where Φ is the ngle etween the mgnetiztion nd the diretion of the stress σF his expression will e used throughout the work presented in this thesisF rereD one n see tht mthemtillyD it is tntmount to the retion of n indued qudrti nisotropyF he end physil result will depend on the sign of λ s X if positive @respF negtiveAD the mgnetiztion will tend to lign long @respF perpendiulr toA the diretion of the stressF • inergy onsumptionX the min dvntge of wivew nd indeed stressEmedited memories in generl lies in the lowEpower opertion tht they requireF wost of the energy dissipted orresponds to the hrge nd dishrge of the piezoeletri phseD whih is dieletriF he gp is sustntilX proponents of mgnetoeletri memories lim tht the power requirements n e redued y severl orders of mgnitudeF rious estimtes hve put the totl energy onsumption for writing one it t less thn 10 3 k B T while )sh memory sed on xexh logi gtes needs 10 ntD or 10 13 k B T PTHF por the system to workD there re importnt onstrints on the vlue of the pplied mgneti (eld whih must sty within ertin rngeF sf it is too lrgeD the eemn e'et eome strong enough to sideline nisotropy @i.e. H 0 > H a AD nd only one stle stte remin for the mgnetiztionF sf it is too smllD the opertion does not work properly s the mgnetiztion will not swith to the desired sttes PSIF he threshold vlue for the (eld here is Ha √ 2 D whih orresponds to stle sttes whih re WH • prt from eh otherD symmetrilly with respet to the mgneti (eldF herefore we hve the following √ 2 ( yx) is then trnsmitted to the mgnetoelsti lyer y piezoeletri sustrte sujeted to n pproprite eletri (eldF he rtionle ehind the proposition of suh system to hieve domin wll motion"lthough less menle to qulittive desription"is lose to the mehnism implemented in wivew nd involves mking one stte more energetilly fvorle thn the otherF he result of the interply of ompeting energeti ontriution on the stle sttes of mgnetiztion re illustrted in pigF IFPUF he overll di'erene in volumi energy is t the origin of motionX the domin with lower volumi energy will expnd t the expense of the otherF SU resumlyD the mplitude of the energy gp should hve n in)uene on the motionD this will e explored in lter developmentsF

onstrintX Ha √ 2 < H 0 < H a F ST 1.4.3
H 0 M 1 M 2 M ′ 1 M ′ 2 M ′′ 1 M ′′ 2 π/4 3π/4 σ > 0 H 0 M 1 M 2 M ′ 1 M ′ 2 M ′′ 1 M ′′ 2 π/4 3π/4 σ < 0
pigure IFPU ! tle mgnetiztion sttes within the nnostripeF he unixil nisotropy without H 0 nd the stress σ indues two opposite sttes M 1 nd M 2 F hen H 0 is ppliedD we hve M 1 nd M 2 D still symmetriF purtherD the mehnil stress leds to the tilted sttes M 1 nd M 2 D whih re symmetri for oth positive nd negtive stressF yne wy to expose the mehnism ehind motion in more intuitive fshion is to detil sequentilly the e'et of the stress nd only then tht of the is mgneti (eldD insted of the reverseF pigF IFPV shows how the twoEdomin mgnet retsD nd helps grsp the ruil role of the is mgnet (eld s symmetryErekerF he omined e'et of unixil nisotropy nd mgnetoelsti intertion is n equivlent nisotropy tilted with respet to the xExisF his tilt"etter explored in hpter P"llows for the disrimintion etween the two sttes sed on their eemn energyD see susetion PFPFPF es suhD the phenomenon is stress-triggeredD euse the tilt is indued y the stressD ut eld-driven euse the eemn energy gp is responsile for the motionF eginD these mtters re detiled in the following hpterF he reson for onsidering uniform stress stems from the ft tht setup with multiple eletrodes somewht limits the downsling of the whole devieF purthermoreD if lrge piezoeletri sustrte is used @ommeril or grownAD the hypothesis of uniE form stress is resonleF tress grdients n indeed exist in ptterned nnostrutures depending on the properties of the mterils used nd the dimensions of the lyersF rue enoughD moderte stress grdients would ertinly 'et the ehvior of the domin wll nd its dynmisD ut would not neessrily prelude opertion nd therefore do not diE minish the relevne of the mehnism proposedF edmittedlyD onsidering the e'et of stress grdients would e useful in terms of representing rel devie in the ontext of n engineering endevorF roweverD it would dd unneessry omplexity in the pursuit of physil insight euse there would e risk of distint phenomen eing onfoundedF es for the is mgneti (eldD it n e generted y neighoring urrowed permnent SV pigure IFPV ! hemti of twoEdomin ferromgnet @A t restD @A with the pplition of stressD @A with the pplition of stress nd mgneti (eld @A where the symmetry is rokenD induing domin wll motion s result @dEeAF mgnets PTI @nd thus does not ontriute to the overll energy onsumptionAF roweverD it is importnt to keep in mind tht resorting to in situ strutures is not essentil to the prtil implementtionD nd mrosopi strutures n e usedD s the mgneti (eld is simply stti is reking the symmetryF he sustrte n indeed e pled in the ir gp of surrounding mgneti iruit whih inludes n rrngement of permnent mgnetsF qiven the lengthsles involved @mirometers t mostA one n expets tht the homogeneity of the mgneti (eld generted y suh devie would e seured over the sptil extent of the systemF SW feuse this system is essentilly sed on the sme oneptD there re severl fetures in ommon with wivewD inluding the dvntge in terms of energy onsumption ompred to other domin wll motion tehniques @see hpters P nd Q for detilsAD s well s the unequivol opertion @the diretion of motion n e fully predited from the stress pplied on the mgnetoelsti stripeAF roweverD there re some importnt detils on whih the two systems di'erF sn prtiulrD the onstrints on the vlue of the mgneti (eld mentioned for the se of wivew re somewht liftedF sndeedD onsequene of the shpe nisotropy rising from the stripe geometry is to strengthen the unixil nisotropyD so tht the existene of two stle sttes is ensured well eyond wht would e predited if only nisotropy nd eemn energy re tken into ount @see susetion PFPFR of hpter PAF fesidesD the lower mgneti (eld limit for the wivew system to work does not exist hereD s the movement of the domin wll does not rely on the trajectory of mgnetiztion ut ultimtely depends only on the dierence in volumic energy of the two dominsF 1.4.4 Engineering of cross section es mentioned in susetion IFPFQD the ptterning or engineering of the edges in nE nostruture hs n in)uene of domin wll motionF his work inludes the study of systems with vrile ross setion @through vrile widthA on the ehvior of domE in wll moved y uniform stressD s depited in pigF IFPWF o the est of our knowledgeD disrete pinning sites hve een investigted theoretilly nd experimentllyD ut no syE stemti study of the generi e'et of ltering the potentil lndspe of the domin wll through engineering of the ross setion tully exists @inluding in the se of (eldE nd urrentE driven motionD for tht mtterAF imply putD mgneti domin wll will hve the tendeny to shift towrd regions of lower width @onstritionsA euse of the energy pyEo' relted to redued exhnge nd nisotropy energyF sndeedD s explined in susetion IFIFRD domin wll is ssoited with lol inrese of volumi energyF hereforeD the lrger the domin wllD the lrger the inrese of energyF st follows tht ny lol minimum of width will represent potentil well"nd hene stle lotion"for domin wllF gonverselyD ny lol mximum will represent potentil rrierF es resultD omplex potentil lndspe n in priniple e designedF his leds to mny interesting lines of investigtion in reltion to domin wll motion in nnostripes of vrile widthF ith regrd to pplitionsD one n think of devies with multiple lol minim in whih domin wlls will e trppedD onept of possile interest for logi tsksF fesidesD on more theoretil onsidertionsD mny TH pigure IFPW ! sn this workD we onsider nnostripe with width free to vry with xF unnswered questions lie in the in)uene of vrile width in the dynmis of domin wll motionF sndeedD it is estlished tht the dimensions of the medium in whih the domin wll trvels do 'et the dynmis @stedyEstte veloityD lker rekdown (eldFFFAF ht hppens in the di'erent dynmi regimes with vrile widthD for instne onsidering low nd high sptil periodiityD remins n open questionF sn this workD we fous on two sesX onstnt setion nd proli setion @symmetriE l with respet to the middle of the stripeAF nder some onditionsD the ltter n ensure on(nementD with stle position in the middle nd potentil rriers to the ejetion of the domin wllD i.e. its dispperne from the stripeF es n sideD we onsidered pro(les whih were perfetly smoothD nd we did not investigte the in)uene of rough edgesF resumlyD the (ndings of suh n nlysis would e similr to wht is oserved for (eldE nd urrentEdriven motion @see susetions IFPFQ nd IFPFRAF 1.4.5 Proof of concept through micromagnetic simulations he xmg free nd openEsoure pkge PTP sed on (nite elements ws used to provide proof of the onept proposed oveF he softwre llows the user to perform miromgE neti simultions with wide rnge of mteril prmeters nd informtion desriing the prolemD ll spei(ed in ython sript loted in dedited (leF snformtion out the geometry nd the mesh is provided y mesh(le generted for instne using the gmsh softwre PTQF TI he minimum length of the mesh elements in ll our simultions ws S nmF ine the gol ws not to perform detiled nlyses ut to gin insight in the vlidity of the oneptD no omputtions with etter resolution thn this were rried outF enywyD the result of simultions showed tht the typil width of domin wlls in this ontext ws severl times the size of elementsD so tht domin wlls ould e desried orretlyF everl geometries hve een onsidered @minly the two ses mentioned in susetion IFRFRAD ll with their lrgest dimension long xD nd their smllest long z @vetors ontined in the xyEplne re lled in-planeAF sn the ython sriptD one n speify n ritrry nisotropyF sn the present seD we study the ehvior of mterils with unixil nisotropy long x @qudrti nisotropy energy proportionl to w 2

x AF en externl mgneti (eld of given mplitude nd long given diretion n e esily implemented in xmgD in our se it ws introdued long the yExisF es for the mgnetoelsti energyD there is no dedited sn stndrd miromgneti softwre suh s nmgD Qh ojet is divided into (nite numer of smll elements with hosen geometry @oftentimes tetrhedrAF hen when prompted to (nd n equilirium distriutionD n lgorithm usully sed on vritionl methods is used to determine n pproximted solution to oundry vlue prolemF he equtions solved involve quntity tht n e lled the energy of the systemD whih will e minimum"t lest lolly"when the system hs rehed its equilirium stteF he lulus of vritions provides set of tehniques imed t (nding the extrem of funtionl PTVF sn our se we wnt to (nd the minimum of quntity whih n e equted with the totl energy of the ferromgneti region @whih inludes the relevnt ontriutions suh s nisotropyD eemn etAF he (rst step is therefore to de(ne rigorously wht is the energyD whih orresponds to our funtionlF his is not s strightforwrd s it seemsD nd sometimes the logi of the sutle hoies mde in the expression of the ontriutions is not fully explinedF rere in prtiulr we give detiled desription of how the prolem onerning the demgnetiztion vetor (eld is ddressedF yne the energy of the system is determinedD pproprite onstrints hve to e tken into ountF he outome of the vritionl proedure @minimiztion proessA yields n eqution of whih the equilirium mgnetiztion distriution is solutionF

V ⊂ ℜ 3 ∂V n M ( r) = M s w( r) ferromagnetic material vacuum µ 0
pigure PFI ! qeometry of the ferromgneti mteril oupying region V with externl surfe ∂V nd norml unit vetor nF he system tken into onsidertion is region V ⊂ R 3 with externl surfe ∂V nd norml unit vetor n @see pigF PFIAF e suppose tht the region V is (lled with TU ferromgneti mteril with mgnetoelsti @or mgnetostritiveA propertiesD while the externl region R 3 \ V is omposed of liner mgneti mteril hrterized y the vuum permeility µ 0 F es desried in the previous hpterD the ferromgneti mteril is desried y mgnetiztion vetor M D de(ned within the region V y its spe vrying diretion w( r)F st mens tht M ( r) = M s w( r)D where M s represents the mgnetiztion t sturtionF 2.1.2 Denition of the energy functional he funtionl to e minimized here is the totl energy of the systemF sn the followingD the di'erent energeti ontriutions relevnt to our prolem re de(nedF he energy density is the sum of these terms written in their most generl formD s followsX

u = u an + u ex + u me + u Ze .
@PFIA vet us now de(ne these ontriutions nd give generl expressionsF

• u an represents the nisotropy energy density nd tkes into onsidertion spei( diretions where the mgnetiztion is preferly orientedF his term is due to the rystlline struture of the ferromgneti mteril PHF e onsider here n ritrry form

u an = f an ( w). @PFPA
• u ex tkes into onsidertion the exhnge intertion mong mgneti dipoles of the mgnetiztion distriutionF sing the iinstein nottionD its generl form follows IRU

u ex = 1 2 β ij ∂w l ∂x i ∂w l ∂x j , @PFQA
where w is the mgnetiztion diretion nd β ij is symmetri tensorF

• u me is the energy ssoited to the mgnetoelsti e'etF sts generl form n e written s ID IRUD IVP

u me = -a iklm T ik w l w m = -T ik µ ik . @PFRA
rereD T ik is the guhy stress tensorD whih is onsidered known nd imposed to the strutureF he prolem of determining the tul stress distriution in V D tking into ount the oupling etween mgnetostrition nd elsti quntities when the ody is emedded in di'erent elsti environment is treted elsewhere TV nd not onsidered in the present development @see e.g.D efF PSSAF a iklm is tensor with the symmetry properties a iklm = a kilm nd a iklm = a ikml F he quntity µ ik ( w) = a iklm w l w m is the soElled eigenstrin hrterizing the mgnetoelsti e'etF st mens tht the onstitutive eqution from the elsti point of view is given y T = L -µ D where T nd re the tul stress nd strin tensorsD nd L is the elsti sti'ness tensor @stisfying the sme symmetries s a plus the dditionl property L iklm = L lmik AF his onstitutive eqution must e interpreted s followsF sf the stress is zero @elstilly free odyAD then we (nd = µ D i.e. the rel strin orresponds to the eigenstrinD whih ssumes the hrter of strin imposed y the mgnetiztionF st mens thtD when w is (xed in given regionD then will tend to e equl to µ ( w)F sn the sitution where T is not zero @region V onstrined or emedded in given elsti mtrixAD the tul strin nnot ssume the vlue µ D nd the system (nds ompromise etween the e'ets of mgnetiztion nd elsti intertions with the mtrixF

• u Ze is the energy orresponding to the lol intertion etween mgnetiztion nd mgneti (eldF st is lled the eemn term nd its generl expression is PTW

u Ze = -µ 0 M S H l • w. @PFSA
st is importnt to underline tht H l is the local mgneti (eld tht one n mesure t ny given point rF ypillyD it is omposed of two ontriutionsX n externlly pplied (eld H 0 nd mgneti (eld H d generted y the mgnetiztion distriution M ( r) itselfF he ltter is referred to s the demgnetizing (eldF hereforeD we hve

H l = H 0 + H d D
V X U = V u d r U = V f an ( w) + 1 2 β ij ∂w l ∂x i ∂w l ∂x j -T ik µ ik -µ 0 M s H li w i d r. @PFTA

Derivation of classical equations

ritten s suhD U depends on the funtion w = w ( r)D while T nd H l re imposed quntitiesF he ehvior of the ferromgneti mteril in the region V n e summrized TW through the following sttementX t equiliriumD the mgnetiztion distriution within V is found y minimizing U with respet to w ( r)D with T ( r) nd H l ( r) (xed eforehndF prom the mthemtil point of viewD this is prolem of the lulus of vritionsF roweverD the minimiztion s formulted ove is hrdly pplile to rel situtionsD nd therefore it is not the prolem we hve to solveF hile it is possile to (x the externl (eld H 0 D it is typilly not the se for the totl (eld H l F he reson for this is tht the demgnetizing (eld H d itself depends on w = w ( r)D eing the sum of ll ontriutions generted y the elementry dipoles of the overll distriution PTWX

H d ( r) = M s V N ( r, r 0 ) w ( r, r 0 ) d r 0 , @PFUA with N ( r, r 0 ) = 1 4π 3 ( r -r 0 ) ⊗ ( r -r 0 ) r -r 0 5 - I r -r 0 3 . @PFVA
sn iqF @PFVAD a ⊗ b represents the tensor produt etween two vetors a nd bD i.e. we hve a ⊗ b ij = a i b j D nd I is the identity opertorF he nottion • is used for the eulidin normF prom this expressionD one n prove tht N exhiits the following symmetry propertiesX

N ( r, r 0 ) = N T ( r, r 0 ) , @PFWA N ( r, r 0 ) = N ( r 0 , r) . @PFWA
he priniple desried ove suggests to minimize U with T nd H l (xedF ine this is not suitle for our physil prolemD we hve to (nd n equivlent result sed on di'erent funtionl Ũ minimized with stress tensor T nd n externl (eld H 0 @insted of lol (eld H d A (xedF fesidesD there is onstrint on the norm of the unit vetor wF sn other wordsD we hve to (nd nother funtionl Ũ stisfying the following equivlene

min w : w =1 U H l fixed ⇔ min w : w =1 Ũ H 0 fixed . @PFIHA
e prove in ppendix f tht the ext mthemtil form of Ũ is the following

Ũ = V f an ( w) + 1 2 β ij ∂w l ∂x i ∂w l ∂x j -T ik µ ik -µ 0 M s H 0i w i d r - V 2 1 2 µ 0 M 2 s w( r) • N ( r, r 0 ) w( r 0 )d r 0 d r, @PFIIA UH
where the lst term represents the demgnetiztion energyD with the noteworthy presene of ftor 1 2 F he minimiztion of Ũ with respet to the diretion wD with T nd H 0 imposedD leds to n eqution giving tul equilirium mgnetiztion distriution within the ferromgneti odyF snidentllyD it is interesting to point out tht this mthemtiE l prolem ompletely desries the emergene of the domins struture typil of the ferromgneti mterilsF he result of this minimiztionD proved in ppendix fD follows

w × H eff = 0, @PFIPA
where

H eff = H 0 + H d - 1 µ 0 M s ∂f an ∂ w + β ij µ 0 M s ∂ 2 w ∂x i ∂x j + 2 µ 0 M s a : T w. @PFIQA
ht this eqution mens is tht t equiliriumD the torque generted y ll lol e'etive (elds is zeroF hese results re in perfet greement with lssil developments ID IRUD IVPD PUHD PUIF yne n lso onsider the lst three terms s e'etive (elds"whih ll depend on w"nd lel them with the energy from whih they derive @ H an D H ex nd H me AF o omplete the disussionD oundry onditions must e imposed on the externl surfe of the region V @see gin ppendix f for detilsAX

β ij lst w t ∂w l ∂x j n i = 0 ∀s on ∂V. @PFIRA
sn prtiulrD with isotropi exhnge @β ij ∝ δ ij D the uroneker deltA this n e further simpli(edF es w is lwys perpendiulr to ∂ w ∂ n @onservtion of mgnetiztion mplitudeAD the oundry onditions simplify to

∂ w ∂ n = 0 on ∂V. @PFISA
o onludeD the ehvior of the ferromgneti region V n e summrized y oserving tht iqF @PFIPA must e stis(ed within V with oundry onditions stted in iqF @PFIRA for the generl seD or in iqF @PFISA for isotropi exhngeF

Equilibrium distribution in nanostructures

e now use n dpted version of the ove generl proedure to demonstrte the posE siility to indue the motion of domin wll through uniform mehnil stress in UI unixil ferromgnetsF he sme vritionl method is usedD ut with the spei(s of the geometry nd hypotheses integrted from the strtF he eqution otined is tht on whih the numeril work presented in the rest of this hpter is sedF 2.2.1 Denition of the energy functional he geometry of the ferromgneti region V onsidered is s shown on pigF PFPF st is onstituted y nnostruture of thikness hD length L nd vrile width (x)F o simplifyD the geometry is onsidered symmetril with respet to the yzEplne so tht the region V (lled y the mgnetoelsti mteril is de(ned y

V = - L 2 ≤ x ≤ + L 2 , - (x) 2 ≤ y ≤ + (x) 2 , - h 2 ≤ z ≤ + h 2 . @PFITA
wo mjor hypotheses will e ssumed for our nlysisX @iA the mgnetiztion M = M s w lies in the xyEplneD nd @iiA the mgnetiztion depends only on xF reneD we hve

w = w(x) = (cos Φ(x), sin Φ(x), 0) , @PFIUA
where Φ(x) is the ngle etween w nd the xExisF hereforeD our hypothesis implies tht M is onsidered onstnt within yz ross setions t x onstntF his is resonle s long s the thikness nd width of the nnomgnet re reltively smll ompred to its length @nnostripe geometryAF roweverD it is worth keeping this strong ssumption in mindX this model nnot desrie vortex wllsD or for tht mtter the tilt of trnsverse wllsD s n e seen for instne in the miromgneti results otined with nmg in hpter IF es for hypothesis @iAD it is justi(ed y the ft tht the geometry tht we pln to study is suh tht the dimensions long the zExis is muh smller thn the othersF hereforeD t equilirium it is unlikely tht the mgnetiztion will exhiit (nite omponent w z F prom the point of view of the physil response of the ferromgneti mterilD we ssume unixil ehvior with n esy xis long x desried y the energy densityX

u an = -K u w 2 x = -K u cos 2 Φ. @PFIVA
feuse of the unixil symmetry of the ferromgnetD the exhnge energy desried in iqF @PFQA simpli(es y onsidering two di'erent onstntsX

β 11 = A Y β 22 = β 33 = B nd β ij = 0, ∀ i = jF wore expliitelyD we hve u ex = A ∂ w ∂x 2 + B ∂ w ∂y 2 + ∂ w ∂z 2 . @PFIWA UP σ σ y(HA) x(EA) H 0 M Φ(x) z ℓ(x) h L
pigure PFP ! perromgneti nnostruture hrterized y vrile width (x) @hereD prol is shown s n exmpleAF he ferromgneti esyExis @ieA is ligned with the xExisD while the hrdExis @reA orresponds to the yExisF he mgnetiztion M is desried y the ngle Φ(x) @mesured ntilokwiseA nd the mgneti (eld H 0 nd mehnil tion σ re pplied to the systemF ith the ssumption in iqF @PFIUAD we esily get

u ex = A ∂ cos Φ ∂x 2 + ∂ sin Φ ∂x 2 = A -sin Φ ∂Φ ∂x 2 + cos Φ ∂Φ ∂x 2 u ex = A ∂Φ ∂x 2 .
@PFPHA he pplied mgneti (eld onsidered is long the yExisD leding to the ontriutionX

u Ze = -µ 0 M s H 0 • w = -µ 0 M s H 0 sin Φ. @PFPIA
ettention must e pid to the ft tht this energy only overs the intertion with the pplied (eldF es desried in hpter ID this (eld is le to rek the symmetry nd to modify the two stle positions of the mgnetiztion indued y the nisotropyF sndeedD with the sole in)uene of the nisotropyD we hve the ouple of positions (0, π)F edding the mgneti (eldD we otin the tilt of the stle sttes towrd the mgneti (eld @hereD long + yAF UQ e now introdue the mgnetoelsti intertionF pirst of ll we de(ne the mthemE til form of the mgnetostritionF µ ( w) orresponds to the strin tensor ssoited with the mgnetiztion diretion w for free ody @no stressAF e point out tht the displeE ment (eld desriing uniform deformtion long w n e written s u l ( x) = l ( x • w) wD where l is the soElled longitudinl deformtionF yn the other hndD displement (eld orresponding to uniform trnsversl deformtion t is given y u

t ( x) = t [ x -w( x • w)]F
he sum of the two ontriutions leds to strin tensor ij = l w i w j + t (δ ijw i w j )F st is well estlished ft tht mgnetostrition is isovolumiD nd thus we hve

Tr µ = 0D i.e. µ
ii = l + 2 t = 0D from whih we n derive tht t =l 2 F fy de(ning λ s = l D typilly referred to s the mgnetostrition oe0ientD we hve

µ ij = λ s 2 (3w i w j -δ ij ), @PFPPA
ndD introduing positive mgnetostrition oe0ient λ s the generl from of the mgE netoelsti energy is

u me = -T : µ = -T ij µ ij = - λ s 2 (3w i w j -δ ij )T ij u me = - λ s 2 (3w i w j T ij -T ii ) . @PFPQA
sn our seD we pply unidiretionl stress long the diretion identi(ed y Φ σ = 3π 4 F st is thus onvenient to work with xes (x , y ) tilted with respet to the (x, y) frme y n ngle of π 4 @see (gureA in whih we write w = (w x , w y )F henD if T y y = σ nd ll of its other omponents re zeroD we otin

u me = - λ s 2 (3σw 2 y -σ) = - 3 2 λ s σw 2 y + const. @PFPRA
sn order to otin the orresponding energy densityD we hve to ome k to the originl (x, y) system of xesF e hve w = cos Φ e x + sin Φ e y = cos(Φ -π 4 ) e x + sin(Φ -π 4 ) e y . @PFPSA xowD in the (x , y ) frme we hve unidiretionl stress long the seond xis (y ) ndD thereforeD the energy density is given y iqF @PFPRAD with

w 2 y = sin 2 (Φ - π 4 ) = (sin Φ cos π 4 -sin π 4 cos Φ) 2 = 1 2 sin 2 Φ + 1 2 cos 2 Φ -sin Φ cos Φ w 2 y = -sin Φ cos Φ + const.
@PFPTA UR pinllyD we hve proved tht

u me = 3 2 λ s σ sin Φ cos Φ. @PFPUA
st is strightforwrd to hek tht positive stress will tend to fvor mgnetiztion long diretion de(ned y n ngle 3π 4 + kπD nd negtive stress will fvor ny mgnetiztion perpendiulr to this xisF sn summryD the energy density of the system isX

u = -K u cos 2 Φ + A dΦ dx 2 -µ 0 M s H 0 sin Φ + 3 2 λ s σ sin Φ cos Φ - 1 2 µ 0 M s H d • w, @PFPVA
where the lst term represents the demgnetiztion energy densityD s shown in iqF @PFIIAF e determine the totl energy U = V ud r y integrting iqF @PFPVA in the region V F por the (rst four termsD one n note tht they depend only on xF

U = L 2 -L 2 h (x) -K u cos 2 Φ + A dΦ dx 2 -µ 0 M s H 0 sin Φ + 3 2 λ s σ sin Φ cos Φ dx - V 2 1 2 µ 0 M 2 s w( r) • N ( r, r 0 ) w( r 0 ) d r 0 d r. @PFPWA

Eective anisotropy

fefore getting to the derivtion of the equtionD it is pproprite to re)et on the physis ehind some of the energy termsF st is key to understnd the reson for domin wll shift in simple wyF sf we onsider only the mgneti nisotropy nd the mgnetoelsti energy terms oveD it is ler tht it is equivlent to n e'etive nisotropyD sine oth re qudrti quntities @see pigF IFPVAF herefore we n go further y tully writing this e'etive nisotropy u eff omprising the omined e'et of mgneti nisotropy nd mgnetoelsti energyX u eff ≡ u an + u me F vet us de(ne n ngle Φ eff hrterizing the diretion of this new nisotropyD nd K eff its onstnt @unitX tm -3 AF heir form n e found y writing the followingX

-K eff cos 2 (Φ -Φ eff ) = -K u cos 2 Φ + 3 2 λ s σ cos Φ sin Φ.
@PFQHA e few lines of si trigonometry lultion yields to the expression of

K eff nd Φ eff y identi(tionX US K eff = K 2 u + 3 2 λ s σ 2 , @PFQIA Φ eff = - 1 2 arctan 3 2 λ s σ K u . @
d dβ U (Φ(x) + βk(x)) β=0 = L 2 -L 2 h (x) + 2K u cos Φ sin Φk(x) -µ 0 M s H 0 cos Φk(x) + 3 2 λ s σ cos(2Φ)k(x) + 2A dΦ dx dk dx dx - 1 2 µ 0 M 2 s   V 2 ∂ w ∂Φ (x) • N ( r, r 0 ) w(x 0 )k(x)d r 0 d r + V 2 w(x)N ( r, r 0 ) ∂ w ∂Φ (x 0 )k(x 0 )d r 0 d r   . @PFQQA UT o ontinueD the exhnge term is integrted y prtD giving L 2 -L 2 h (x)2A dΦ dx dk dx dx = 2h (x)A dΦ dx k(x) L 2 -L 2 - L 2 -L 2 2hA d dx (x) dΦ dx k(x)dx. @PFQRA
he demgnetiztion term is nonElol in tht its vlue for eh point depends on the mgnetiztion distriution in the whole regionF ymmetries of the tensor N D s shown previously in iqF @PFWA nd iqF @PFWAD llow to equte the lst two integrls of iqF @PFQQA

involving N F   V 2 ∂ w ∂Φ (x) • N ( r, r 0 ) w(x 0 )k(x)d r 0 d r + V 2 w(x)N ( r, r 0 ) ∂ w ∂Φ (x 0 )k(x 0 )d r 0 d r   = 2 V 2 ∂w ∂Φ (x) • N ( r, r 0 ) w(x 0 )k(x)d r 0 d r. @PFQSA
his n e rewritten with expliit integrtion dominsD for the mgneti (eld integrlsF ine

H d = M s V N ( r, r 0 ) w( r 0 )d rD we eventully otin V 2 M s ∂ w ∂Φ (x) • N ( r, r 0 ) w(x 0 )k(x)d r 0 d r = + L 2 -L 2 + (x) 2 - (x) 2 + h 2 -h 2 ∂ w ∂Φ (x) • H d k(x) dxdydz = + L 2 -L 2 h (x) ∂ w ∂Φ (x) • H d y,z k(x) dx, @PFQTA
where

H d yz = 1 h + (x) 2 - (x) 2 + h 2 -h 2 H d ( r)
dzdy @ funtion of xAD is the verge vlue of the (eld H d over setions t x onstntF fy sustituting iqsF @PFQRA nd @PFQTA in iqF @PFQQA we UU eventully get

d dβ U (Φ(x) + βk(x)) β=0 = + L 2 -L 2 h (x)k(x) 2K u cos Φ sin Φ -µ 0 M s H 0 cos Φ + 3 2 λ s σ cos 2Φ dx - + L 2 -L 2 2hA d dx dΦ dx + (x) d 2 Φ d 2 x k(x) dx + 2hA (x) dΦ dx k(x) + L 2 -L 2 -µ 0 M s + L 2 -L 2 h (x) ∂ w ∂Φ (x) • H d y,z k(x) dx. @PFQUA
ine we wish to study the motion of domin wll in the intervl -L 2 , + L 2 D we (x the vlue of Φ -L 2 nd Φ + L 2 in order to hve domin wll t x = 0 when σ = 0F row these onditions re hosen will e explined in susetion PFPFRF he perturtion funtion k must vnish t the intervl endEpoints x = ± L 2 F reneD iqF @PFQUA ppers with single integrl nd it is zero for ny rel funtion k when

h (x) 2K u cos Φ sin Φ -µ 0 M s H 0 cos Φ + 3 2 λ s σ cos 2Φ -2hA d dx dΦ dx + (x) d 2 Φ d 2 x -µ 0 M s h (x) ∂ w ∂Φ (x) • H d y,z = 0, @PFQVA orD dividing y -2Ah (x) Φ (x) + Φ (x) (x) (x) - 1 2A 2K u cos Φ sin Φ -µ 0 M s H 0 cos Φ + 3 2 λ s σ cos 2Φ + 1 2A µ 0 M s ∂ w ∂Φ (x) • H d y,z = 0. @PFQWA
xowD w eing equl to (cos Φ, sin Φ, 0)D we n (nlly write

Φ (x) + Φ (x) (x) (x) - 1 2A 2K u cos Φ sin Φ -µ 0 M s H 0 cos Φ + 3 2 λ s σ cos 2Φ +µ 0 M s sin Φ H dx y,z -µ 0 M s cos Φ H dy y,z = 0. @PFRHA
(x)Φ (x) + (x)Φ (x) = 0 ⇔ d dx ( (x)Φ ) = 0 ∃ C 0 Φ (x) = C 0 (x) @PFRIA
ith this reltionship etween Φ nd D one n see tht if the funtion is not onE stntD the regions of greter sptil hnge of mgnetiztion @for instne domin wllA will tend to e onentrted in onstritionsF e region of lower width is thus tntmount to lol stle position for domin wllF o omplete the desription of the modelD we need to speify the oundry onditionsD nd to desrie the numeril pproh used to solve the prolemF 2.2.4 Boundary conditions and stable states gonsistently with the vritionl proedure followedD the oundry onditionsD i.e. the vlues of the mgnetiztion ssigned to oth ends of the regionD re (xed quntitiesF hereforeD they will not hnge throughout the simultionsD nd s resultD the hoie of their vlues will e ritrry to some degreeF sn order to put relevnt vlues with regrd to the expeted ehvior of the systemD we hose to hve the oundry vlues equl to UW the ngles Φ s @s ∈ 1, 2A of the two stle sttesF sndeedD in the sitution simultedD we n expet distriution with domin wll etween two 4plteu4 regions with ngles lose to those of the stle sttesF his is orroorted y the result of our preliminry numeril investigtions presented in hpter IF pigure PFQ ! i'et of the mgneti (eld on the stle sttes of the system @with σ = 0AF he two ngles get loser s H 0 inreses until only one stle position remins @Φ = π 2 A for H 0 ≥ H a F sn dshed lines re represented the stle sttes when simpli(ed demgnetizing (eld is tken into ount @N y = 0.225AF

H 0 (10 3 Am -1 ) Φ s Φ 1 (no demag) Φ 2 (no demag) Φ 1 = Φ 2 Φ 1 (with demag) Φ 2 (
vet us now lulte those two vluesX we hve to (nd the solutions to the minimiztion of the energy density in the simple se of uniformly mgnetized systemD tht is with Φ (x) = 0 nd Φ (x) = 0F woreoverD we (rst neglet the demgnetizing (eld in order to simplify the lultionF he eqution for Φ eomes

2K u cos Φ sin Φ -µ 0 M s H 0 cos Φ + 3 2 λ s σ cos 2Φ = 0. @PFRPA
he solutions of this eqution re the ngles of two mgnetiztion diretions generted y the omintion of nisotropyD externlly pplied (eld nd mehnil stressF o solve for Φ we use the representtions cos Φ = 1-t 2 1+t 2 nd sin Φ = 2t 1+t 2 D where t = tan Φ 2 F he (nl result is the following fourthEdegree lgeri equtionX

-µ 0 M s H 0 + 3 2 λ s σ + 4K u t -9λ s σt 2 -4K u t 3 + µ 0 M s H 0 + 3 2 λ s σ t 4 = 0. @PFRQA VH
feuse it is fourthEdegree equtionD nd given the hoie of prmetersD there re potentilly four distint solutionsD only two of whih re relevnt to our physil prolemF roweverD it is not di0ult to identify the orret solution from the numeril point of viewF

Φ 1 (no demag) Φ 1 (with demag) Φ 2 (no demag) Φ 2 (with demag) π 5π 4 3π 4 π 2 π 4 0 -π 4 -200 -100 0 σ (MPa) 100 200
Φ s pigure PFR ! engles Φ 1 nd Φ 2 of stle mgnetiztion orienttions s funtion of the mehnil stress σD with nd without demgnetiztionF o eginD let us look t the prtiulr se where σ = 0X

4K u t -µ 0 M s H 0 1 + t 2 = 0. @PFRRA
his eqution gives two symmetri solutions with respet to Φ = π 2 tht re distint provided tht the mgneti (eld is not too strong @see hpter IAF sndeedD eyond H 0 = H a D the only stle equilirium stte is de(ned y Φ = π 2 F olutions Φ s re plotted in pigF PFQD with the trnsition t H 0 = 93 × 10 3 eGmF elso mentioned in the previous hpter is the ft tht the demgnetizing (eld in nnostripe geometry will dd shpe nisotropy tht will inrese the unixil nisotropyF his e'et is highly signi(ntD s it ppers from our lultions tht it n inrese the pprent nisotropy (eld more thn twofold from the lulted vlue H a F his n e seen in pigF PFQ where simpli(ed demgnetizing (eld hs een introdued in the lultion of the stle sttes @hypothesis VI of uniformly mgnetized ellipsoid of in(nite lengthX H d = N w with the digonl tensor N AF pigure PFS ! inergy lndspe s funtion of the mgnetiztion ngle Φ nd pplied mgneti (eld H 0 F he pplition of stress modi(es the stle mgnetiztion sttes nd their energy densitiesD fvoring given stte over the other oneF sn the (rst pnel we onsidered σ = 0D in the seond one σ > 0 nd in the third one σ < 0F es for the se where σ = 0 with H 0 elow H a D there re lso two stle sttesF pigF PFR shows these ngles for |σ| < 200 w nd H 0 = 20 × 10 3 eGmD with nd without demgE netiztion @sme pproximtion s in pigF PFQAF es the stress inresesD there is shift of the stle stte ngles nd sturtion tht is quiker in the sene of demgnetiztionF sn this hpterD in order to simplify the simultions we hose to ignore demgnetiztion in the lultion of the oundry onditionsD s it is geometryEdependentF sn ny seD smll mismth etween the plteu ngle nd the oundry onditions do not seem to lter signi(ntly the ehvior of the system @see susetion PFRFIAF o understnd the interply etween (eld nd stressD we n look t the energy lndsE pe represented in pigF PFSF he three su(gures orrespond to di'erent vlues of stress VP @one is zeroD the other two re ±20 wAF e n oserve the intrinsi istility of the system under the threshold H a of mgneti (eld nd the e'et of the mehnil stressD whih genertes n energy gp etween the stle sttes @only when H 0 = 0AF 2.3 Numerical procedure his setion desries the numeril pproh for solving the eqution iqF @PFRHA sine it nnot e treted nlytillyF st inludes quik desription of the tehnique s well s the wy it ws heked for onvergene nd uryF 2.3.1 Presentation he im of this setion is to desrie the ad hoc numeril method proposed to solve iqF @PFRHAF his is seond order nonliner integroEdi'erentil eqution with oundry onditionsF hereforeD the stndrd iuler method or the more dvned ungeEuutt tehniquesD whih re relevnt for prolems with initil onditionsD re not suited to this prolemF therD the knowledge of the oundry onditions"see iqF @PFRQA"suggests the tht nonliner relxtionD or itertiveD method would e pproprite PUPF he implementtion ws done with the wevef softwre @PHIR versionAF o introdue this tehniqueD we onsider disretiztion of the intervl -L 2 , + L 2 in N points nd N -1 intervlsF ell quntities will therefore e disretized nd de(ned over uniform regions

[x i , x i+1 ] × [-l(x i ) 2 , + l(x i ) 2 ] × [-h 2 , + h 2 ]
for i ∈ {1, . . . , N -1}F he quntity ∆x will e tken equl to L N -1 F iqF @PFRHA n e written in disretized wyX

Φ m (x i+1 ) -2Φ m+1 (x i ) + Φ m (x i-1 ) (∆x) 2 + Φ m (x i+1 ) -Φ m (x i-1 ) 2∆x × (x i ) (x i ) - 1 2A K u sin 2Φ m+1 (x i ) -µ 0 M s H 0 cos Φ m+1 (x i ) + 3 2 λ s σ cos 2Φ m+1 (x i ) +µ 0 M s sin Φ m+1 (x i ) H dx y,z -µ 0 M s cos Φ m+1 (x i ) H dy y,z = 0. @PFRSA
he unknown is Φ m+1 D wheres quntities with the m index re given y the previous itertionF yne n see tht in order to ompute the vlue of Φ m+1 t lotion iD knowledge of the previous vlues t neighoring points is requiredF o solve this eqution we n use n itertive proedure sed on the isetion methodD whih llows for oth fst onvergene nd ritrrily high preision PUQF VQ pigure PFT ! woEpss sheme used in the proedure with N = 8F e (rst pss performs the isetion for even indies @in lueA using vlues t odd indies @in orngeA from the previous stepD nd the seond pss performs the isetion for odd indies using vlues t even indies tht hve just een omputedF lues t the extremities do not hnge sine they re (xed oundry onditions @see dshed rrowsAF olid line rrows indite the inputs for the lultionF foth psses use demgnetiztion dt from the previous stepF e n strt the proedure with given guess funtion Φ 0 (x i ) @e.g.D the liner funtion imposed y Φ(x 1 ) nd Φ(x N ))D nd then proeed y itertionsF e possile sheme to go from m to m + 1 is represented in pigF PFT nd ws used in the proedureF es Φ(x 1 ) = Φ(-L 2 ) nd Φ(x N ) = Φ(+ L 2 ) re (xedD vertil dshed rrows orresponding to Φ 1 nd Φ N represent the (xed oundry onditions whih re the sme for ny step of the proedureF es for other vluesD odd nd even indies re omputed seprtely in two suessive sustepsF por eh even @respF oddA pointD Φ m+1 (x i ) is omputed using the vlues of its odd @respF evenA neighorsF felow is n exerpt of the wevef routine showing the lssil isetion methodD where r is the unkwown nd err is the threshold vlue for onvergene @sed on the soluteEvlue normAF VR % d e f i n i t i o n of the function in ppendix gF es mtter of ftD the demgnetizing (eld t eh point depends on the mgnetiztion of the whole ferromgneti odyD whihD in dditionD hnges t eh itertionF hereforeD nive implementtion is likely to e omputtionlly intensiveF rereD we onsider the ontriutions of the N -1 prllelepipedl regionsD whih n e lulted through losed form expressionsD s demonstrted in ppendix gF henD the totl demgnetizing (eldD mesured t ny given pointD is simply the sum of ll ontriutiE ons generted y ll prllepepipedl regionsF yf ourseD we introdue smll systemti error due to the ft tht the system with vrile width n e only pproximtely reE presented y the juxtposition of ll the prllelepipedsF roweverD the pproximtion is very good for low derivtive (x) reltively to the disretiztionF he terms H dx y,z nd H dy y,z n then e omputed with the mgnetiztion distriution t the mEth stepD whih y nd lrge represent the most omputtionEintensive elements from numeril point of viewF prom omputtionl point of viewD the key thing to notie is tht in VS iqsF @gFIWA nd @gFPHA geometry @derivtives of F funtionA nd physis @mgnetiztionA n e deoupledF he former n e omputed one nd for ll o1ine efore the iterE tion proedure tully eginsD nd then omponents of the demgnetizing (eld n e omputed from the urrent mgnetiztion distriutionF he result is simple produt etween mtrix with @(xedA geometryEdependent ftors nd the mgnetiztionD whih represent n enormous ene(t in terms of omputtion timeF o ompute the men vlue of the demgnetizing (eld on the ross setion plne of eh pointD in this hpter we onsidered the demgnetizing (eld lulted on the xExisF his pproximtion requires to lulte the (eld t only one point per lotion iD whih n e interesting depenE ding on the time onstrints of the o1ine lultionF elterntivelyD one ould lso try to mp more preisely the prllelepipedl regionD for instne using dequte qussin qudrturesF 2.3.2 Convergence and test against analytical solution xumeril lultions hve een rried out with given set of physil properties orresponding to the mgnetoelsti mteril erfenolF eordinglyD we used mgE netiztion sturtion M s = 64 × 10 4 eGm PSSD n exhnge intertion oe0ient A = 9 × 10 -12 tGm PIF e lso dopt n nisotropy onstnt K u = 37.5 × 10 3 tGm 3 orresponding to n nisotropy (eld H a = 92 × 10 3 eGm @de(ned through the expression K u = 1 2 µ 0 M s H a AD whih n e esily otined in rel ferromgneti lyers PSQF e emphsize tht H a represents the ifurtion threshold shown in pigF PFQF he strong mgnetostrition in erfenol is hrterized y oe0ient λ s = 1 × 10 -3 PUTF woreoE verD n pplied (eld H 0 = 20 × 10 3 eGm is used throughout ll simultionsF ithin the odeD onvergene n e ontrolled using two stopping riterisF yne will ontrol the residul error in the isetion methodD tht is n upper ound for the distne etween the solution found nd the tul solutionF he other is for the itertive proedure itselfD it is the eulidin norm etween vlues t steps m nd m + 1D written δF pirstD it seemed ler tht for resonle input prmetersD the ode did not rsh or give surd results for the ngle ΦF sn prtiulrD the pplition of stress results in the motion of region of high sptil vrition @domin wllA in preditle diretionD while in the sene of stressD the distriution remins symmetriF sn order to hek tht the result otined is orretD we n ompre it to solution tht n e hndled nlytilly in simple seF por tht purposeD we onsider region with nisotropy nd exhnge VT onlyD other terms eing negleted nd the width supposed onstntF he totl energy n e derived from iqF @PFPWA pigure PFU ! wgnetiztion ngle distriution Φ(x) onverging towrd the nlytil soE lution @red urveA for the simple system with only exhnge nd nisotropy ontriutionsF he itertion proess strts from liner guessF he inset shows the reltive error versus the numer of itertionsF sn this se the solution is given y the following losed form expressions

U = + L 2 -L 2 h (x) -K u cos 2 Φ + A ∂Φ ∂x
K u A x + L 2 = 1 ξ F arcsin ξ sin Φ ξ 2 -cos 2 Φ , 1 ξ , @PFRUA 1 ξ K 1 ξ = L 2 K u A , @PFRVA
whih re proved in ppendix e @where we lso de(ne the ellipti funtions F nd KAF he seond eqution must e solved with respet to the prmeter ξ nd then the (rst one gives the reltion etween x nd Φ hrterizing the spei( domin wll under investigtion @for detils see ppendix eAF e underline tht this result is onsistent with pigure PFV ! esolute error etween numeril nd theoretil solutions versus the numer N of points used in the disretiztion @ll simultions hve een performed with the sme stopping riterion sed on δ min = 10 -20 A he onvergene of the proedure n e more losely monitored y quntifying the solute error de(ned s the di'erene etween the numeril nd the theoretil soluE tionF e rn simultions with di'erent levels of disretiztion ut with the sme stopping riterion δ min = 10 -20 F sn pigF PFV we show s funtion of the numer of points N used in the disretiztionF he reltionship etween the two is lerly represented y power lwD the exponent eing out -4F st ppers tht this vlue of the exponent remins onstnt y vrying the prmeters A nd K u of the prolemD therey proving universl VV nd fst onvergeneF sn pigF PFWD we show the ompromise etween ury nd omE puttionl lodD whih should ome s no surpriseF o do soD we look t the reltive error δ versus the numer of itertions for di'erent levels of disretiztionF fy inresing the numer of points we improve the ury ut we slow down the onvergeneD s is visile from the slopes in the (gureF he ost of ury here is twofoldX (rstD with more points eh itertion does tke more time to ompute nd seondD the onvergene is otined pigure PFW ! eltive error δ s funtion of the numer of itertions for di'erent vlues of N F sn ll sesD the onvergene is de(nitively exponentilF woreoverD for lrger numer of points the onvergene is slowerF prom these resultsD the proposed method ppers to e relileD e0ientD nd suitE le for our purposesF sn the simultions presented elowD the numer of points N ws systemtilly tken equl to RHHD whih mens tht ∆x ≈ 1 nmF 2.4 Results yne some on(dene ould e pled in the output of this numeril proedureD mE pign of tests ws rried outF he min results from the opertion of the ode will herey VW e presentedF pirstD the possiility to indue motion is demonstrtedF hepending on the geometry studiedD the ontinued pplition of stress n led to the ejetion of domin wll from the simulted ferromgnet @leving single dominAD or to n equilirium poE sitionF sn the ltter seD n explortion of the reltionship etween equilirium position nd pplied stress will e presentedF he setion ends with n energyEsed interprettion of this reltionshipD nd disussion on the piezeoeletri option for the implementtion of stressEmedited domin wll ontrolF 2.4.1 Motion, connement and ejection sn the previous setionD it ws shown how the proedure onverged towrd the orret domin wll shpe in the sene of (eld or stressF hen is mgneti (eld is pplied long yD the on(gurtion is slightly hnged @the gp etween the plteu regions is more nrrowA nd the domin wll remins stle t x = 0F his solution n e used s n initil guessF es the simultion is rried out in mteril of positive mgnetostrition nd with positive @respF negtiveA pplied stressD the domin wll shifts to the left @respF rightAF enother importnt spet is the quik dpttion of the plteu vlues when the system is sujeted to stressF e n verify here tht the hoie of oundry onditions is ppropriteF sn prllelepipedi geometry @onstnt width nnostripeAD the sustined pplition of onstnt stress leds to singleEdomin on(gurtionF sndeedD s long s the stress is ppliedD there is n inentive for domin wll shiftD only eventully stopped y the x 2 where (0) = 40 nm @entrl widthAD (±L/2) = 70 nm @width t extremitiesA nd L = 400 nmF trting with the sme initil guess used in pigF PFIHD we pply σ = -100 w nd we otin (nl position t out 60 nmF es eforeD we note two sling regimes for the reltive errorD nmely @iA modi(tion of the shpe nd @iiA displement of the wllF he following setion explores the in)uene of the stress vlue on the equilirium positionF st is importnt to point out tht the distriution otined t equilirium is di'erent from lssil domin wll shpesD due to the in)uene of oth the mgnetoelsti energy nd the vrile width of the nnostrutureF es n sideD our simultions showed tht this ehvior @on(nementA n lso e oserved with mgneti (eld H 1 long the x diretion in the sene of trnsverse WP mgneti (eld nd mehnil stressX elow threshold (eld the domin wll n move while remining on(nedF he results llow to tlk out n equivlent mgneti (eldD de(ned s the mgneti (eld long x neessry to result in the sme domin wll displE ementF his shows the universlity of the potentil rrierEgenerting phenomenon of vrile widthF 2.4.2 Stress-position relationship he knowledge of the mgnetiztion distriution for severl vlues of σ llows us to determine the reltionship etween position of the domin wll x ∞ DW nd pplied stressF edmittedlyD the determintion of the ext lotion of the domin is somewht ritrryD espeilly here where the width of the wll is signi(nt with respet to the extent of the nnostrutureF rereD we hose to onsider the lotion where the slope is mximum to e the position of the wllF sn other wordsD its position x ∞ DW is determined with max dΦ dx F σ (MPa)

x ∞ DW (nm)

ℓ 1 (x) ℓ 2 (x) ℓ 3 (x) 80
D 2 nd 3 @ 1 (0) = 2 (0) = 3 (0) = 40 nmD 1 (±L/2) = 2 (±L/2) = 70 nmD 3 (±L/2) = 80 nmD h 1 = 10 nmD h 2 = h 3 =
20 nmAF efter quite liner region for low vlues of σD the response sturtesD exhiiting displement rnge depending on the geometryF elso visile is the @ntiAsymmetry of the equilirium positionD suh tht

x ∞ DW (-E 0 ) = -x ∞ DW (E 0 )F
e performed this nlysis for three slightly di'erent geometries s shown in pigF PFIPF WQ es reminderD the importnt quntity whih will in)uene the ehvior of the domin wll is the rtio F sn ll ses we used proli pro(le (x) de(ned in the ptionF fetween 1 nd 2 D the thikness is hngedD while etween 2 nd 3 D the slope of the vrile width is hngedF he response exhiits quite liner ehvior for smll vlues of σ nd sturtion of the displement for high vlues of σF e underline tht lrger setion t the extremities of the nnostruture redues the slope of the x ∞ DWσ urve nd mximum displement of the domin wll @it n e dedued y ompring the results for 2 nd 3 AF his is due to the totl exhnge energy ssoited with the domin wllF imilrlyD lrger thikness of the nnostruture redues the slope of the urve nd mximum displement of the domin wll @it n e dedued y ompring the results for 1 nd 2 AF his is due to the demgnetiztion energyD whih is lower with thinner nnostrutures @for dditionl informtion see susetion PFRFQAF pigure PFIQ ! @A wgnetiztion distriutions Φ(x) for severl vlues of the pplied stress σ @X σ = 60 wY X σ = 20 wY X σ = 0 wY dX σ = -20 wY nd eX σ = -60 wAF olid lines orrespond to solutions with oundry onditions lulted s in iqF @PFRPAD wheres dshed lines represent the solutions with Φ(-L/2) = 0 nd Φ(+L/2) = π s oundry onditionsF yne n notie tht the in)uene of the di'erene etween the two sets of oundry onditions is rely visile hereF @A gorresponding energy density pro(les u(x)X the energy gp etween the two plteu regions is lerly visileF elsoD the region of inresed energy mrks the presene of the domin wllD whih is thinner when stress is ppliedF -L/2 u(x)h (x)dxF st is not di0ult to identify the regions orresponding to the two domins @with di'erent energy densitiesA nd the region relted to the wll etween them @peksAF st is interesting to note tht n pplied stress σ hs the e'et of mking the domin wll thinnerF 2.4.3 Energetic interpretation st is ler tht the motion is driven y the gp in energy density etween the two dominsF roweverD it is not strightforwrd to identify the role of eh energy ontriutionF sn order to etter understnd the interply etween ll ontriutionsD it is useful to look t their dependne on σF rereD we used one gin the oundry onditions s proxy for the stle sttes of the systemF es shown in susetion PFPFRD the di'erene etween the two sets of ngles stems from the introdution of the geometryEdependnt e'et of the demgnetizing (eld ndD while signi(ntD does not hnge the overll ehvior of the systemF he quntities u re de(ned s u Ze + u an + u me F sn the (rst four su(gures in pigF PFIRD individul ontriutions re plotted ginst σD s well s the energy densityF heres oth nistropy nd mgnetoelsti energy densities exhiit similr trends for Φ 1 nd Φ 2 D this is solutely not the se for the eemn energyF sndeedD it is inresed for Φ 1 nd deresed for Φ 2 F his is highlighted in the (fth su(gureD where the di'erene in energy density for those sme ontriutions is plotted ginst σF e striking oservtion is tht the nisotropy nd mgnetoelsti gps lmost lne eh other for ny vlue of σ @they oth reh their extremum when 3 2 λ s σ = K u D whih orresponds to σ = 25 wAF es resultD the sum of these four terms is lmost equl to the eemn energy gpF hereforeD s this gp is the use of the motionD one ould here tlk out stress-triggered eld-driven domin wll motionF elsoD s the stle sttes get lose to {-π 4 , 3π 4 }D the sturtion is ssoited with vnishing nisotropy nd mgnetoelsti energy gps s those two sttes hve the sme energy density vlues for those two ontriutionsF WS pigure PFIR ! @A wgnetoelsti energyD @A nisotropy energyD @A eemn energy nd @dA the sum of those three ontriutions ssoited with ngles Φ 1 nd Φ 2 D plotted ginst σ for H 0 = 20 × 10 3 eGmF @eA hi'erene in energy density u 2u 1 = u (Φ 2 )u (Φ 1 )D nd individul ontriutionsD s funtions of σF sn order to etter ppreite the overll ehviorD σ is tken etween H nd QHH w @there is omplete ntisymmetry for negtive vluesAF WT sn order to try to understnd the link etween the energy gp to the equilirium position x ∞ DW D we n develop n even simpler modelF vet us onsider ferromgnet with vrile width nd twoEdomins hrterized y the ngles Φ 1 nd Φ 2 D negleting the demgnetiztion nd with mgnetiztion distriution suh tht

θ(x) = Φ 1 ∀x ∈ -L 2 , x 0 Φ 2 ∀x ∈ x 0 , + L 2
his mens tht there is n in(nitely thin domin wll t the lotion x = x 0 F heE refore the totl energy of the system n e written s the sum of three termsX

U = L 2 -L 2 u(x)S(x) dx @PFRWA U = x 0 -L 2 u 1 S(x) dx + L 2 x 0 u 2 S(x) dx + + L 2 -L 2 u d dx + U DW (x 0 ). @PFSHA
yviouslyD the nive hypothesis of wll with no width would yield n in(nite exhnge energyD whih is not physilF he quntity U DW introdued here is (nite nd represents the energy ost of the wll region @exhngeD nisotropy nd so forthAF woreoverD we introdued the demgnetiztion energy density u d = -1 2 µ 0 M s H d • wF o ontinueD let us rell tht u 1 nd u 2 do not depend on x euse the mgnetiztion is onsidered uniformF he equilirium position for the domin wll is found when U is minimumF his n e written s followsX

d dx 0 u 1 x 0 -L 2 S(x) dx + u 2 L 2 x 0 S(x) dx + + L 2 -L 2 u d dx + U DW x 0 =x ∞ DW = 0 ⇔ (u 1 -u 2 )S(x ∞ DW ) + dU d dx 0 (x ∞ DW ) + dU DW dx 0 (x ∞ DW ) = 0.
sntroduing the surfe energy of the domin wll e DW D then U DW = S(x 0 )e DW D we n write the eqution giving the vlue of

x ∞ DW X S (x ∞ DW ) S(x ∞ DW ) + ∆u e DW + 1 S(x ∞ DW )e DW dU d dx 0 x 0 =x ∞ DW = 0. @PFSIA
eginD one n see tht the relevnt quntity regrding the vrile ross setion is the rtio S S = F he only free prmeter here is e DW whih n e estimted experimentllyD theoretillyD or even numerilly for instne y using the relxtionEsed numeril WU proedure presented in this hpter @y (nding the vlue of e DW for whih the solution of iqF @PFSIA mthes the x ∞ DWσ reltionship otined using the relxtion methodAF sn prinipleD this tremendously simple model ould help design the ross setion for desired stti response t miniml omputtionl ostF por tht purposeD it is neessry to hve good estimte of the demgnetizing term dU d dx 0 F sndeedD without tking into ount the demgnetizing (eldD it ppers tht proli pro(le @hourglssEshpedA suh s those we hose in susetion PFRFP is not le to on(ne the domin wll for resonle vlues of e DW F sn lssil domin wll with only exhnge nd nisotropyD the surfe energy n e written 4 √ AK u TD whih in our se gives is 2 × 10 -3 tm -2 F ith this order of mgnitudeD the pro(le required to prevent singleEdomin outome exhnge lone is so sti' tht overll the geometry nnot resonly e onsidered kin to nnostripeF his ft is importnt from the physil point of view hereD s it mens tht the demgnetiztion is ritil for the on(nementF ithout demgnetiztionD there n e no on(nement for the geometries presented in this hpterD whih mens tht there is no solution to iqF @PFSIA for |x 0 | < L 2 @ ft on(rmed y independent testing with the relxtion proedure stripped of the in)uene of demgnetiztionAF pigure PFIS ! homin wll displement s funtion of the pplied stress otined y solving iqF @PFSIAF hysil prmeters re the sme of those used previouslyD the geometry orresponds to the 1 width pro(le nd e DW = 5 × 10 -3 tm -2 F WV olving iqF @PFSIA for x 0 yields the equilirium position of the domin wll for given vlue of σF he result orretly predits rightwrd motion for σ < 0 nd leftwrd motion for σ > 0D s well s the sturtion s σ inreses if e DW is lrger thn threshold vlueF he order of mgnitude required hppens to e very muh onsistent with si theoretil preditions @≈ 10 -3 tm -2 AF roweverD the rude ssumption regrding the mgnetiztion distriution @two uniform regionsA yields rther poor quntittive greement etween these results @see pigF PFISA nd those previously otined @see for instne pigF PFIPAF sn prtiulrD the sturtion is not estlished s quiklyF xeverthelessD this smll model shows the relevne of onsidering the energy density gp etween the two domins s the min drive for domin wll displementF 2.4.4 Piezoelectric generation of stress sn these simultionsD we supposed tht uniform stress is pplied to the systemF por tehnologil onsidertionsD it is interesting to tully onsider the possile wys to pE ply uniform stress on nnostrutureF hepositing the mgnetoelsti lyers on piezoeleE tri sustrtes is n ovious option @see pigF PFITAF por tht purposeD ermi mterils suh s led mgnesium nioteEled titnte solid solution @wg f rac13 x 2 3 Ay 3 (1-x) E iy 3 x @0 < x < 1A"often revited wxE"hs een drwing muh ttentionF he reson for this is tht its dieletri nd piezoeletri properties re very interestingF

σ (MPa) x ∞ DW (nm) ℓ 1 (x) ℓ 2 (x) ℓ 3 (x) 100
Φ (x) + Φ (x) (x) (x) - 1 2A [2K u cos Φ sin Φ -µ 0 M s H 0 cos Φ + 3λ s µ(d 32 -d 31 )E 0 cos 2Φ +µ 0 M s sin Φ H dx y,z -µ 0 M s cos
∂ w ∂t = + γ 0 1 + α 2 w × H eff -α w × w × H eff , @QFIA
where α is the qilert dmping oe0ientD nd γ 0 = µ 0 γD γ eing the gyromgneti rtio of the test prtile @oth quntities should e negtive to desrie the eletron preessionAF he e'etive (eld H eff lredy ppered in the previous developments nd gthers within single vetor ll the physil ontriutions involvedF he sme hypothesis of oneE dimensionl system is ssumed hereD i.e.

u me = - 3 2 λ S τ w 2 x + σw 2 y , u me = - 3 4 λ S (τ + σ) w 2 x + w 2 y + 2(τ -σ)w x w y . @QFPA
with the sme nottions s used previously @w x = w• e x = ( e x + e y )/ √ 2 nd w y = w• e y = ( e ye x )/ √ 2AF hile w 2 x + w 2 y ws onstnt in hpter PD sine we hd Θ = π 2 D here this term will ontriute to the qâteux derivtive nd thus to the dynmisF he demgnetiztion ontriution eing unhngedD U is the funtionl of w nd reds

U = + L 2 -L 2 h (x) -K u w 2 x + A ∂ w ∂x 2 -µ 0 M s H 0 w y - 3 4 λ S (τ + σ) w 2 x + w 2 y + 2(τ -σ)w x w y dx - 1 2 µ 0 M 2 s V 2 w( r 0 ) • N ( r, r 0 ) w( r 0 )d rd r 0 , @QFQA
where the ftor 1 2 in the demgnetiztion term is still present for the resons desried in susetion PFIFPF IHS 3.1.3 Determination of eective eld es previouslyD this quntity hs to e minimized with the onstrint w = 1 @see ppenE dix fAX

min w : w =1 U H 0 fixed ⇔ min w U + V λ( w • w -1)d r H 0 fixed , @QFRA
where we introdued the vgrnge multiplier λ(x) lredy mentioned in ppendix fF vet us onsider n ritrry rel vetor k = (k x , k y , k z ) nd ontinue with the qâteux derivtiveD using the sme properties nd simpli(tions s previouslyX

d dβ U w(x) + β k(x) β=0 = + L 2 -L 2 h (x) -2K u k x w x + 2A ∂ w ∂x • d k dx -µ 0 M s H 0 k y - 3 2 λ s ((τ + σ)(k x w x + k y w y ) + (τ -σ)(k x w y + k y w x )) + 2λ(x) w • kdx -µ 0 M 2 s V 2 k • N ( r, r 0 ) w(r 0 )d rd r 0 . @QFSA
sing integrtion y prt gin on the exhnge termD nd introduing the quntity H d yz s in susetion PFPFQD one is left with single integrl

d dβ U w(x) + β k(x) β=0 = + L 2 -L 2 h (x) k • -2K u w x x -2A ∂ 2 w ∂x 2 + (x) (x) ∂ w ∂x - 3 2 λ s ((τ + σ)(w x x + w y y) + (τ -σ)(w y x + w x y)) -µ 0 M s H 0 -µ 0 M s H d yz + 2λ w dx. @QFTA
his hs to e true for ny vetor kF herefore we n write the following equtionX

-2K u w x x -2A ∂ 2 w ∂x 2 + (x) (x) ∂ w ∂x -µ 0 M s H 0 + H d yz - 3 2 λ s ((τ + σ)(w x x + w y y) + (τ -σ)(w y x + w x y)) + 2λ( r) w = 0. @QFUA IHT
fy tking the rossEprodut of iqF @QFUA with wD the vgrnge multiplier term n e deleted nd one otins w × H eff = 0, @QFVA with

H eff = H 0 + H d yz + 2 µ 0 M s K u w x x + A (x) ∂ ∂x (x) ∂ w ∂x + 3 4 λ s ((τ + σ)(w x x + w y y) + (τ -σ)(w y x + w x y)) . @QFWA
his is the e'etive (eld whih hs to e used in the vnduEvifshitzEqilert equtionF 3.1.4 Derivation of dynamical equations he expression of the e'etive (eld in iqF @QFWA hs to e sustituted in iqF @QFIAF he

ltter n e deomposed into the following three equtionsX

∂w x ∂t = - γ 0 1 + α 2 ((w y H z -w z H y ) + α(w y (w x H y -w y H x ) -w z (w z H x -w x H z ))) , @QFIHFIA ∂w y ∂t = - γ 0 1 + α 2 ((w z H x -w x H z ) + α(w z (w y H z -w z H y ) -w x (w x H y -w y H x ))) , @QFIHFPA ∂w z ∂t = - γ 0 1 + α 2 ((w x H y -w y H x ) + α(w x (w z H x -w x H z ) -w y (w y H z -w z H y ))) , @QFIHFQA
where we used H insted of H eff to simplify expressionsF o proeed from hereD it is useful to develop the derivtives of the left side into expliit formsD using the expressions of w omponents given in susetion QFIFIF e dopt the ommon nottion

˙ ≡ ∂ ∂t F ∂w x ∂t = ∂ (cos Φ sin Θ) ∂t = -sin Θ sin Φ Φ + cos Φ cos Θ Θ, @QFIIFIA ∂w y ∂t = ∂ (sin Φ sin Θ) ∂t = sin Θ cos Φ Φ + sin Φ cos Θ Θ, @QFIIFPA ∂w z ∂t = ∂ (cos Θ) ∂t = -sin Θ Θ. @QFIIFQA
ustituting iqF @QFIIFQA in iqF @QFIHFQAD s well s iqF @QFIIFIA nd iqF @QFIIFPA in iqF @QFIHFIA nd iqF @QFIHFPAD respetivelyD we n write

IHU sin Θ Φ = γ 0 1 + α 2 ( -cos Φ cos Θ H x -sin Φ cos Θ H y + sin Θ H z + α(-sin Φ H x + cos Φ H y )), @QFIPA Θ = γ 0 1 + α 2 ( -sin Φ H x + cos Φ H y + α(cos Φ cos Θ H x + sin Φ cos Θ H y -sin Θ H z )), @QFIQA
where iqF @QFIPA is otined y lulting cos Φ ∂wy ∂tsin Φ ∂wx ∂t F es n sideD onsiE dering tht

1 sin Θ ∂ w ∂Φ = (-sin Φ, cos Φ, 0) nd ∂ w ∂Θ = (cos Φ cos Θ, sin Φ cos Θ, -sin Θ)D the equtions n e rewritten s        sin Θ Φ = γ 0 1 + α 2 - ∂ w ∂Θ • H + α 1 sin Θ ∂ w ∂Φ • H Θ = γ 0 1 + α 2 1 sin Θ ∂ w ∂Φ • H + α ∂ w ∂Θ • H . @QFIRA
e more expliit form of this eqution requires the expressions of the three omponents of HF fsi lultions not worth showing here give the followingX

H x = H dx + 2 µ 0 M s K u cos Φ sin Θ + 3 4 λ s ((τ + σ) cos Φ sin Θ + (τ -σ) sin Φ sin Θ) +A -sin Φ sin Θ ∂Φ ∂x + cos Φ cos Θ ∂Θ ∂x -cos Φ sin Θ ∂Φ ∂x 2 + ∂Θ ∂x 2 -2 sin Φ cos Θ ∂Φ ∂x ∂Θ ∂x -sin Φ sin Θ ∂ 2 Φ ∂x 2 + cos Φ cos Θ ∂ 2 Θ ∂x 2 , @QFISFIA H y = H 0 + H dy + 2 µ 0 M s 3 4 λ s ((τ + σ) sin Φ sin Θ + (τ -σ) cos Φ sin Θ) +A cos Φ sin Θ ∂Φ ∂x + sin Φ cos Θ ∂Θ ∂x -sin Φ sin Θ ∂Φ ∂x 2 + ∂Θ ∂x 2 + 2 cos Φ cos Θ ∂Φ ∂x ∂Θ ∂x + cos Φ sin Θ ∂ 2 Φ ∂x 2 + sin Φ cos Θ ∂ 2 Θ ∂x 2 , @ QFISFPA
H z = H dz + 2A µ 0 M s sin Θ ∂Θ ∂x -cos Θ ∂Θ ∂x 2 -sin Θ ∂ 2 Θ ∂x 2 . @QFISFQA IHV efter simpli(tionsD iqF @QFIRA n e rewritten    Φ = 1 sin Θ γ 0 1 + α 2 (-s + αr), Θ = γ 0 1 + α 2 (r + αs), @QFITA
where we introdued the two quntities r nd s suh tht

r = -sin Φ H dx + cos Φ H dy + cos Φ H 0 + 2 µ 0 M s (-K u cos Φ sin Φ sin Θ + 3 4 λ s (τ -σ) cos 2Φ sin Θ + A sin Θ Φ + 2 cos Θ Φ Θ + sin Θ Φ , @QFIUA nd s = cos Φ cos Θ H dx + sin Φ cos Θ H dy -sin Θ H dz + sin Φ cos Θ H 0 + 2 µ 0 M s K u cos 2 Φ sin Θ cos Θ + 3 4 λ s ((τ + σ) cos Θ sin Θ + 2(τ -σ) cos Φ sin Φ cos Θ sin Θ) + A Θ + Θ -cos Θ sin Θ Φ 2 . @QFIVA
sn iqF @QFIUA nd @QFIVAD we used the nottion ≡ ∂ ∂x for Φ nd Θ s ws hitherto the se for F snidentllyD it is interesting to point out tht the equtions tht ditte the equiliE rium stte re given y r = 0 nd s = 0 @tking Φ = Θ = 0AF sn the prtiulr se where Θ = π 2 D we re left with r = 0 whih is equivlent to iqF @PFRHAF sndeedD those sE sumptions re impliit in the hypotheses hosen in the eginning of setion PFPF imilrly to iqF @PFRHAD iqF @QFITA is suh tht theoretil insights on stedyEstte solutions kin to lkerElike lultions pper out of rehF hereforeD the numeril pproh remins the only wy to investigte the physis of stressEindued domin wll motionF sn ftD we n identify the reson why the prolem does not lend itself to nlytil tretmentD onE trry to (eldEdriven motion in the se explored y hryer nd lker USF he prolem lies with the tilted nisotropy introdued y the stress whih prevents from otining the sme kind of simple result of (eldEdriven motion s in ppendix hF wenwhileD this feE ture hs frErehing onsequenes on the physil di'erenes in the dynmis tht will e explined in setions QFQ nd QFRF 3.2 Numerical procedure his setion desries the numeril pproh for solving eqution iqF @QFITA sine it nnot e treted nlytillyF st inludes quik desription of the tehnique s well s IHW the wy it ws heked for onvergene nd uryF 3.2.1 Presentation he most ovious sheme tht omes to mind in the ontext of the numeril tretment of di'erentil equtions is the expliit iuler methodF roweverD impliit shemes re fr superior to expliit shemesD whih often pper insu0ient in solving nonliner prolemsF he reson for this is tht the propgtion of errors leds to strong onstrints in terms of temporl disretiztion in order to prevent physilly nd numerilly errnt evolutiE onsF ypillyD in the se of the het eqution for instne"whih shres some fetures with the vnduEvifshitzEqilert equtionD nmely the order of the derivtives with reE spet to spe nd time"the time step dt should not e greter thn threshold vlue proportionl to dx 2 @sptil disretiztionA PUQF he omputtionl ost of these onE strints mke onvergene hrd to hieve in prtieF he hoie ws to use n impliit (nite di'erene shemeD or kwrd iuler methodF es for the solver hosenD despite our erly ttempts it ppers tht this system of two nonEliner integroEdi'erentil equtions nnot e solved using the uiltEin tools in wtlF his led us to develop dedited lgorithmD nd we hose to implement xewton9s method whih proved to work quite wellF e generl introdution to xewton9s method is followed y some detils out our ad hoc implementtionF xewton9s method is routinely used in numeril nlysis to (nd the root of di'erenE tile relEvlued funtionF vet us de(ne F : R k → R k vetor funtion of rel vriles x = [x 1 , . . . , x k ] for whih we hve to (nd the rootX

F (x) = 0.
@QFIWA e (rst estimte of the solution is found from guess x 0 D y lulting the vlue x 1 where the (rstEorder pproximtion to F t point x 0 is zeroF his involves the lultion of the toin J of the F funtionX

J(x 0 )(x 1 -x 0 ) = -F (x 0 ), @QFPHA
where J ij = ∂F i ∂x j is kEyEk squre mtrix of whih ll elements re lulted t the guess vlues x 0 F he proess is itertiveX the solution x 1 otined is etter pproximtion of the rootD nd is used to (nd new solution x 2 D etF por n ritrry step m in this loopX

J(x m )(x m+1 -x m ) = -F (x m ).
@QFPIA IIH he proedure is stopped fter numer of steps when su0ient preision is otinedD i.e. when the norm of F (x m ) is less thn predetermined vlueD nd the solution x new is thus ssigned the vlue x m F sn our se k = 2N D sine we hve two vriles Φ nd Θ of size N F vet us ontente them nd de(ne the following vrileX

X =        Φ 1 Θ 1 F F F Φ N Θ N        . @QFPPA
et ny time stepD the previous stte is used s n input guess for (nding the nextD nd there re 2N equtions to e solvedD

F (X new ) = 0, @QFPQA with F (X new ) = X new -X old -∆tG(X new ). @QFPRA
rereD X new is the unknownD nd this eqution shows tht the proedure hosen is impliitF he vetor funtion G orresponds to the rightEhnd side of iqF @QFITAF por eh index i in {1, . . . , 2N }D G i depends on the vrile X i D ut lso on neighoring vriles @suh s X i-1 nd X i+1 A euse of the presene of sptil derivtives X nd X in iqF @QFITA @expressions of disretized derivtives re identil to wht is shown in iqF @PFRSAAF he proedure followed is itertive nd inludes onvergene hek sed on the omprison of the eulidin norm of F with onvergene prmeter ritrrily tken equl to 2 × 10 -30 F e introdued here n exponent m s in iqF @QFPIA to refer to the steps in the xewton itertive loopF por ny step mX

J(X m )X m+1 = J(X m )X m -F (X m ), @QFPSA
where J is the toin mtrixF he right hnd side n e expndedX

J(X m )X m -F (X m ) = X old + ∆t G(X m ) - dG dX X m @QFPTA
he derivtive is de(ned s the sum i ∂G ∂X i F sn prtieD it is typilly omposed of six terms @sine it depends on

Θ i D Θ i-1 D Θ i+1 D Φ i D Φ i-1 nd Φ i+1 AF
ettention should lso e rought on the ft tht the (rst term of the rightEhnd side in iqF @QFPTA @X old i A is mere III prmeter in the equtionF sn ny seD the solution n then e found y lulting

X m+1 = J(X m ) -1 B(X m )D with the vetor B is s followsX B(X m ) = X old + ∆t G(X m ) - dG dX (X m )X m . @QFPUA
felow is n exerpt of the wevef odeD whih is the ontent of one step @from m to m + 1A in the xewton proedure for given time stepF he (rst loop is the de(nition of the toin mtrixF st should e noted tht the (rst two nd lst two elements of the digonl re treted seprtely s Φ 1 Θ 1 D Φ N nd Θ N re unhnged through the simultion @oundry onditionsAF xmes of vriles re the sme s oveD exept for dsrphiD drstheta nd the like whih refer to spei( prtil derivtives of GF % the Jacobian matrix J(1 ,1)=1; J(2 ,2)=1; for i = 3:2:2 * Npoints-3 % odd indices : dot phi equations he end result is then used to ompute the vlue of F nd hek it ginst the onvergene prmeterF e should lso mention tht the demgnetizing (eld is held onstnt within the xewton9s method proedure of time stepF st is relulted efore entering the proedureD hene it is not fully impliitF he ode n e stopped either fter given mount of timeD when the domin wll hs rehed given positionD or when the system hs rehed equilirium @from riteri similr to tht used in the relxtion odeAF 3.2.2 Convergence and test against analytical solution sing this omputtionl shemeD onvergene is hieved t eh time step for resonle inputs of prmetersF st is intriguing to note tht ny ttempt to remove demgnetiztion or introdue simpli(ed demgnetiztion in the presene of stress entils the eventul filure of the odeF rowever when onvergene is hievedD it is very quikD demonstrting the e0ieny of xewton9s methodF his llowed us to use very smll internl onvergene prmeter @2 × 10 -30 AF he physil prmeters onsidered here re the sme thn in hpter PD exept for λ s whih is tken equl to 2 × 10 -4 hereF es mtter of ftD it is this wy loser to the vlue of mgneti multilyers suh s those used in our lortory nd desried in hpter RF sn ny seD this hnge mounts to simple resling of stress vluesF purthermoreD we hose to use the eletri (eld s the vrile ditting the mehnil stress pplied to the systemF husD the vlues of σ nd τ in iqF @QFITA re given y expressions in susetion PFRFRF yne dditionl prmeter tht ws not present in the previous study is the qilert dmping onstntF sts vlue is known to e typilly higher in thin (lms thn in ulk mterils PVHF sn ll our simultionsD it ssumes four di'erent vluesD from 0.12 to 0.06F e higher dmping onstnt mens less pronouned preession nd smller domin wll veloities for given stimulus @see iqF @IFWAAF st should lso e noted tht throughout these simultionsD the gyromgneti rtio γ ws tken equl to the opposite of its rel vlue @≈ -1.76 × 10 11 s -1 -1 AD nd is thus positive insted of negtiveF st results from this tht the preession is rried out in the other diretionD nd tht we re IIQ looking t mgnetism for positive hrgeF hile this mistke 'ets ll the resultsD the dynmis will e un'etedF he onsequenes of reversed preession will e mentioned s we go long in the disussionD every spet of whih will remin vlidF sn hpter P we ompred the result of the ode to known nlytil solution for the mgnetiztion distriutionF e n hek tht this numeril tool deling with mgneE tiztion dynmis lso psses this testD nd tht it onverges to the orret distriution with the pplition of stress @s ompred to the result of the previous odeAF hile this turns out to e the seD it is not su0ient in itselfF es mtter of ftD even if the ode is le to (nd n equilirium distriutionD it only shows tht the proedure is le to relx to distriution given y r = 0 nd s = 0 ut it does not gurntee tht the pth followed is orret from the dynmil point of viewF hereforeD there is ruil need to hek the proedure ginst se where n nlytil solution is knownF es shown in susetion IFPFQD the se of (eldEdriven motion n e hndled nlytillyF hereforeD we hose to ompre the output of the ode with the veloity of the (eldEdriven motion of hedEtoEhed domin wllF his se is di'erent from wht hd een treted y hryer nd lker who onsidered n esy xis perpendiulr to the diretion of motionF rereD the nisotropy is long the xExisD s is the diretion of motionF gontrry to the ount given in iqsF @IFWA nd @IFIHAD this sitution requires the ontriution of demgnetiztionD introdued here in the simpli(ed form H d = -N M D s if it were generted y n in(nitely long ellipsoidF N is digonl @with tr(N ) = 1A nd the elements of the digonl re noted N x D N y nd N z F his point is ddressed further in ppendix hF he result is the followingX

J( i , i)=1-dt * gamma0/(1+alpha^2) * dsrphi ( f l o o r ( i /2)); J( i , i+1)=-dt * gamma0/(1+alpha ^2) * ( dsrtheta ( f l o o r ( i /2))... -((-s ( f l o o r ( i /2))+alpha * r ( f l o o r ( i / 2 )) ) . . . . * cos (ph( f l o o r ( i /2))))./ sin (ph( f l o o r ( i /2))).^2); J( i , i+2)=-dt * gamma0/(1+alpha^2) * dsrphip ( f l o o r ( i /2)); J( i , i+3)=-dt * gamma0/(1+alpha^2) * dsrthetap ( f l o o r ( i /2)); J( i , i-1)=-dt * gamma0/(1+alpha^2) * dsrthetam ( f l o o r ( i /2)); J( i , i-2)=-
v(H 1 ) = - γ 0 H 1 α A K u -µ 0 M 2 s /2(N z -cos 2 φ 0 N x -sin 2 φ 0 N y ) , @QFPVA sin 2φ 0 = 2H 1 αM s (N y -N x ) = H 1 H c . @QFPWA
sn pigF QFP we show two sets of omprisons tht hve een mdeF he ode is lso suited to the omputtion of (eldEdriven dynmisD when E 0 = 0D H 0 = 0D nd mgneti (eld H 1 = H 1 y is introduedF por di'erent vlues of H 1 we ompred the output of the ode with the nlytil expressions in iqsF @QFPVA nd @QFPWAF e dmping oe0ient α of HFI is ssumed nd M s is tken equl to 5 × 10 5 eGmF he two vlues of nisotropy onsidered re K u = 10 3 tm -3 nd K u = 10 5 tm -3 F es for the demgnetiztionD we hose N y = 0.995 nd N z = 1 -N y = 0.005 @the hypothesis eing tht of n in(nitely long @iA in the model presented in susetion PFPFIX s expetedD there is no outEofEplne omE ponent in the equilirium distriution @i.e. Θ = π 2 AF his is not true of the mgnetiztion during the whole dynmisD howeverF here is indeed notiele outEofEplne ehviorD nd this phenomenon is trnsientX from ompletely inEplne initil distriution t the eginnig of the simultionD w z inreses stedily in the viinity of the domin wll @i.e. Θ devites from π 2 A nd vnishes s the domin wll rehes its (nl position x ∞ DW F his outEofEplne exursion is upwrd or downwrd depending on the sign of E 0 F pei(llyD when the motion is rightwrdD the outEofEplne ehvior is downwrdD nd vieEversF et this stgeD it is importnt to note tht the opposite outome is otined when the sign of γ is hnged to desrie eletronEsed mgnetism insted of positive hrgeE sed mgnetismX rightwrd motion is ssoited with upwrd outEofEplne ehviorD nd vieEversF eprt from this hngeD there is no di'erene to mentionF hese outEofEplne exursions look like mnifesttion of distintive spet of the di'erentil equtions solvedD whih di'ers from stndrd domin wll motion equtions in fundmentl wyF sndeedD in the se of (eldEdriven motionD ll the mgnetiztion distriution is ontined in single plne @see ppendix h or efF USAF his mens thtD while the mgnetiztion n e out of the plne of the simulted ferromgnetD there is lwys tilted plne tht ontins ll the mgnetiztion vetors long the xExis IIT pigure QFR ! he two ngles desriing the mgnetiztion vetor @A nd the outEofEplne mgnetiztion omponent @A re plotted every tenth of nnoseond while n eletri (eld E 0 of ±0.8 wGm nd mgneti (eld H 0 of 20 × 10 3 eGm re ppliedF here is notiele devition of Θ from π 2 D whih trnsltes into nonEzero outEofEplne omponentF his ehvior hs no equivlent in the stedyEstte regime desried y hryer nd lkerF @de(ned y φ 0 in iqF @IFIHA or iqF @QFPWAAF roweverD t this point it is not possileD stritly spekingD to ssign this peulir ehvior to n intrinsi e'et of the prtiulr omintion of (eld nd stress studied hereF st my e used y the vrile widthD whih is nother di'erene with lssil developments suh s those of hryer nd lkerF purther nlyses on onstntEsetion nnostripes will help settle this question in setion QFRF here is nonetheless nother key (nding tht onstitutes strong evidene tht these outEofEplne exursions re somehow linked to the pplition of mehnil stressF sn pigF QFRD the symmetry with respet to the eletri (eld is quite onspiuousF yne n oserve tht the outEofEplne exursion is more pronouned with positive eletri (elds thn negtive eletri (eldsD lthough it is present in oth sesF he di'erene etween those two ses is the lne etween ompressive nd tensile omponents of the stress tensorF his intriguing ehvior explored in susetion QFQFQ stems from the disprity d 32 = -d 31 nd hene the reltion |τ | > |σ| @nd they re lwys of opposite signAF st mens tht if E 0 > 0D the domin wll moves to the left @x < 0A with ompression |τ | lrger thn the tension σY onverselyD if E 0 < 0D the domin wll moves to the right IIU @x > 0A with tension τ lrger thn the ompression |σ|F ine ompression indues planar anisotropy from the mgneti point of view @perpendiulrly to its diretionA nd trtion indues n axial anisotropy for the mgnetiztion @long its diretionAD the motions to the left nd to the right re not dynmilly equivlentF e stronger plnr nisotropy filittes the exursion of mgnetiztion out of the xyEplneF

Trajectory of domain walls

α 1 α 2 α 3 α 4 100 -100 -50 0 50 E 0 = -0.4 MV/m E 0 = -0.9 MV/m E 0 = 0.4 MV/m E 0 = 0.9 MV/m x DW (nm) 0 1 2 3 4 t (ns)
pigure QFS ! he trjetory of the domin wll for severl vlues of pplied eletri (elds nd qilert dmping is plottedF he equilirium position depends on E 0 ut not on αF roweverD the dynmis is in)uened y othF sn these simultions nd throughout this workD the vlues of dmping oe0ient re s followsX α 1 = 0.06D α 2 = 0.08D α 3 = 0.10 nd α 4 = 0.12F es stted in susetion QFQFID the equilirium distriution oinides with tht of the relxtion ode for the sme set of inputsF ht is new here though is the desription of the rel timeEresolved evolution of the mgnetiztion distriution towrd this equiliriumF he generl proedure hosen to study the dynmis ws s followsF pirstD the initil distriution for Φ is tken from the relxtion ode with E 0 = 0F Θ is tken uniform nd equl to π 2 F henD the dynmis re omputed from this stte following the instntneous pplition of n eletri (eld of (nite vlueD nd the position of the domin wll is onsidered to oinide with the lotion where Φ is mximumF IIV es shown in pigF QFSD wht is oserved fter short period of time due to preession is the motion of the domin wll towrd its equilirium positionF he equilirium position of the domin wll depends on the mplitude of the pplied eletri (eld nd the reltionship is ntisymmetril s lredy shown in hpter P @mening x ∞ DW (-E 0 ) = -x ∞ DW (E 0 )AF es n sideD it does not depend on αD nd this ws expeted sine this oe0ient is only involved in the dynmis @for instneD it does not pper in iqF @PFRHAAF roweverD the dynmis re 'eted y oth the eletri (eld nd the strength of the dmpingF he in)uene of α is indeed quite lerF por instne t low dmping oe0ientsD there is tendeny to 4overshoot4 efore the domin wll rehes its (nl position with inresing eletri (eldsF e lrge oe0ient is ssoited with lrger dmpingD nd therefore less preessionF oughly spekingD it mens tht the mgneti system is more 4rigid4 nd will exhiit only moderte osilltionsF yn the ontrryD wek dmping will result in more drmti vritions of mgnetiztion in time ndD potentillyD longer time to reh equilirium when the eletri (eld is suddenly ppliedF his is lerly the se here with lrger overshoots oserved when the dmping is wekF elsoD the di'erene etween the dynmis indued y positive nd negtive eletri (eld is lerly visileF he question of the origin of this di'erene hs lredy een ddressed in susetion QFQFI nd will e explored further in susetion QFQFQF 3.3.3 Velocity and applied electric eld hen deling with the dynmis of domin wll motionD it is only fir to mention domin wll veloityF ine the veloity is not onstnt in this system @see trjetory in pigF QFSAD we hose to de(ne the quntity v DW s the verge veloity over the pth from the origin to the position 2 3 x ∞ DW F st is plotted ginst E 0 in pigF QFT for severl dmping oe0ientsF rereD lower dmping is ssoited with lrger veloitiesF sn the four urvesD when the eletri (eld is inresed from 0D the vlue of | v DW | is inresedF his inrese is quite liner t the eginningD until the veloity rehes mximumF henD while for E 0 < 0 slight rte of derese is oservedD for E 0 > 0 the minimum of v DW is immeditely followed y strong veloity redutionF elsoD the mximum veloity is rehed t lower eletri (elds for E 0 > 0 ompred to E 0 < 0F hese remrks re dditionl signs of the symmetry of the system regrding the eletri (eldF en explntion of the oserved symmetry sed on the reltive in)uene of ompresE sion nd tension ws put forwrd in susetion QFQFIF hen the ompression is lrger thn the tension @E 0 > 0AD the previling plnr nisotropy indues outEofEplne exursiE ons with onsiderle devition of Θ from π/2D nd domin wll propgtion is sensily

IIW E 0 (MV/m) α 1 α 2 α 3 α 4 100 0 -100 0 -1 -2 1 < v DW > (m/s)
pigure QFT ! eloity is plotted ginst E 0 for the sme four vlues of α s in pigF QFSF hinderedF yn the other hndD when the tension is lrger thn the ompression @E 0 < 0AD pigure QFW ! wgnetiztion distriution exhiiting the di'erene etween the oundry onditions with nd without the simpli(ed ontriution of demgnetiztionF he width of the nnostripe is (±L/2) = 40 nm nd the orresponding demgnetiztion tensor omponent is N y = 0.225F

-µ 0 M s H 0 + 3 2 λ s σ + (4K u + 2µ 0 M 2 s N y )t -9λ s σt 2 -(4K u + 2µ 0 M 2 s N y )t 3 + µ 0 M s H 0 + 3 2 λ s σ t 4 = 0. @QFQHA
por n in(nite ellipsoid of semiEmjor xis length 2 nd semiEminor xis h/2D the demgE netiztion ftor N y is equl to e/(e + 1) where e = /hF sn prtieD the vlue of N y ws determined y tril nd error from the guess e/(e + 1)D so tht there is no visile di'erene etween the oundry onditions nd the vlue of the plteu regionsF he di'erene etween this djusted oundry ondition nd the regulr oundry ondition exluding demgnetiztion is shown in pigF QFWD with the distriution of mgnetiztion t rest @E 0 = 0A s n exmpleF wore informtion on the in)uene of demgnetiztion on the oundry onditions n e found in susetion PFPFRD in prtiulr through pigF PFQF smportntlyD sine the N y ftor is purely geometrilD its vlue will e the sme whtever the eletri (eld or mgneti (eld pplied to the systemF e should mention here tht the numer of points on the yzEplne used to ompute the men vlue of demgnetiztion t eh point ws the sme for ll geometriesF es the width is hnged etween the three geometriesD the preision is 'eted euse the IPQ resulting disretiztion is di'erentF sn prinipleD it my e use for onern s it is now ler tht the demgnetiztion hs signi(nt in)uene on the systemF rowever we veri(ed tht the hnge in outome ws very limitedX n overestimtion of only few perents on the vlue of the veloity is to e expetedF 3.4.2 Steady-states in innite nanostripes he study of in(nite nnostripes with onstnt width ws rried out y (rst tking the domin wll shpe ssoited with moving domin wll from the relxtion ode s the initil distriutionF o do thisD the relxtion ode is run for very short period of time so tht the domin wll ssumes its shpe without trvelling fr from the enter of the simulted ferromgnetF his distriution is used s the input of the dynmil ode whih will ompute the dynmis from tht initil stteF yf ourseD the relxtion ode only overs the distriution of ΦD therefore the ngle Θ is initilly set to the vlue π 2 over the whole nnostripeF pigure QFIH ! nel @A fetures the distriution of ngle Φ s funtion of timeF yne n lerly oserve the onset of stedyEstteF nel @A shows the trjetories of the domin wll for severl eletri (elds E 0 nd qilert dmping αF he position of the domin wll tend to e liner s the stedyEstte is estlishedF he pplied (eld is still H 0 = 20 × 10 3 eGmF pigF QFIH shows exmples of the time evolution of mgnetiztion from the initil stte desried oveF yne n see tht Θ hs to ssume prtiulr distriution with IPR devition from π 2 D whih is onsistent with previous (ndingsF henD one mgnetiztion hs rehed given distriutionD it ppers tht stedyEstte is estlishedF he motion is kin to the shift of rigid domin wll strutureF he presene of outEofEplne exursions is on(rmed with these simultionsD proving tht they re fundmentl feture of the equtions solvedF pigure QFII ! isuliztion of the stedyEstte shpe of the trvelling domin wll using the rview visuliztion softwre for two vlues of eletri (eld E 0 F he olor orreE sponds to the vlue of w x F he simulted ferromgnet hs IH nm thikness nd IHH nm widthF egrding the eletri (eldD we n lso on(rm here tht its in)uene on the shpe of the domin wll is twofoldX strength nd sign of E 0 ply mjor role in determining the stedyEstte shpe dopted y the domin wllF hen the mplitude of E 0 is inresedD the width of the domin wll dereses nd the vlues of w z is inresedF his is illustrted on pigF QFIID whih shows the mgnetiztion distriution in the domin wll region using Qh vetorsF yne n see the peulir shpe of the domin wll with more lrityD s well s the e'et of n inresed mplitude of eletri (eld on the outEofEplne exursionF he seond importnt spet of the eletri (eld dependene of the domin wll is tht positive eletri (eld is ssoited with more pronouned outEofEplne exursionsF his is shown on pigF QFIP for n eletri (eld of ±0.8 wGmF rere gin we n on(rm the oservtion mde in susetion QFQFIF IPS pigure QFIP ! rnsverse view of the mgnetiztion distriution for positive nd negtive eletri (eldF he di'erene in outEofEplne ehvior is lerly visileF yther prmeters in)uening domin wll shpe inlude the qilert dmping oe0E ient α @the lrger the dmpingD the smller the devition of Θ from π 2 A nd the externl mgneti (eld @the lrger the mgneti (eldD the lrger the devitionAF yf ourseD mteril prmeters suh s nisotropy must ply role tooD ut this ws not investigted euse the in)uene ws either preditle or deemed of lesser interestF 3.4.3 Electric and magnetic eld dependence of velocity he existene of stedyEstte in the dynmis mens tht for eh ouple (E 0 , H 0 )D there will e orresponding domin wll shpe ut lso veloity ssoited with the motionF o numerilly 4mesure4 the veloityD we tke dvntge of the ft tht the shpe ssumed y the domin wll is (xed when the stedyEstte is rehedF sndeedD if we mke the simple hypothesis tht during the motion we hve Φ(x, t) = Φ(ξ)D with ξ = xv DW t @introduing v DW the veloity of the domin wllAD we n then writeX

α 1 α 2 α 3 α 4 x (nm) Θ (rad) Φ (
IPT ∂ ∂t Φ(x, t) x = 1 L ∂ ∂t + L 2 -L 2 Φ(x, t)dx = 1 L + L 2 -L 2 ∂ Φ ∂t (ξ)dx = 1 L + L 2 -L 2 ∂ξ ∂t d Φ dξ (ξ)dx = - v DW L + L 2 -vt -L 2 -vt d Φ dξ (ξ)dξ ∂ ∂t Φ(x, t) x = v DW L Φ - L 2 -v DW t -Φ + L 2 -v DW t ≈ Φ 1 -Φ 2 @QFQIA
es resultD preise estimte of the veloity n e determined numerilly y simply lulting the verge vlue of Φ s the simultion is running nd tking its vrition in timeX

v DW = L (Φ 1 -Φ 2 ) ∂ ∂t Φ(x, t) x .
@QFQPA sn order to let the stedyEstte developD the simultion is stopped only when the domin wll hs rehed point lose to the edge of the simulted nnostripeF hen the veloity t the end of the trjetory is omputed using iqF @QFQPAF he dependene of the veloity with the eletri (eld hppens to e very similr to wht is otined with the vrile width in pigF QFTD lthough ll vlues re higher sine the stedyEstte regime hs time to fully developF ht n e explored further here is the in)uene of the mgneti (eldF st hs een shown in pigF PFIR tht the energy gp etween the two stle sttes of mgnetiztion is dominted y the eemn energyD hene it is resonle to think tht modifying the mgneti is will neessrily in)uene the dynmisF pigF QFIQ shows the veloity s funtion of the mgneti (eldD for severl vlues of eletri (eldsF he reltionship is not simpleD s we n seeF hile there is generl upwrd trendD the onvity is highly dependnt on the dmping oe0ient nd the eletri (eldF elsoD depending on the mgneti (eldD it ppers tht lower dmping is not systemtilly ssoited with higher veloitiesD espeilly t high mgneti (eldsF strins @see iqF @PFSPAAF sn other wordsD the inEplne piezoeletri response should e nisotropiF pigure RFP ! @A exes nd smple on(gurtion showing the diretion of the eletri (eld @etween iGt eletrodesA nd the plne in whih the relevnt strins pperF @A epresenttion of rhomohedrl unit ell whih ppers to e present in wxED long with the di'erent diretions @the zExis orresponds to the @HIIA diretion of growth tht is of interest hereAF ossile polriztion diretions re long the red nd lue dotEdshed rrowsF pigure from PVRF ery good ndidtes from this perspetive re relxor single rystls of @HIIAEut @wg f rac13 x 2 3 Ay 3 (1-x) Eiy 3 x @orienttion shown in pigF RFPAF heir properties re espeilly impressive ner the worphotropi hse foundry @wfAD tht is with the vlue of x in given rnge @0.3 < x < 0.4AF wxE single rystls of vrying ompositiE onsD growing methods @)uxEgrown or meltEgrownA nd orienttions hve een investigted he vyoutiditor softwre ws used to design eletron em msk (lesF everl versiE ons were mde nd we will go on to rie)y desrie the shpes onsideredF pirstD ellipses were inludedD the gol eing to look t the ehvior of singleEdomin elementsF yf ourseD the msk lso ontined elongted strutures @retngulr or hourglssEshpedA orresponE ding to the geometries studied theoretillyF ings of onstnt nd vrile width @re etween onentri or nonEonentri irlesAD s well s long 4pier4Elike nnostripe nd other shpes were drwnF sndividul elements re pled su0iently fr from eh other @severl mirons or tens of mironsA to prevent mgnetostti intertionF he dimensions @length nd widthA of the nnostrutures were vrileD ut in n e'ort to inrese the likelihood of oserving on(gurtion with mgneti dominsD ll were signi(ntly greter thn wht ws simultedD from PHH nm to P µm in widthD nd ten times these vlues in lengthF fy the end of the testingD the writing prt of the frition ws well mstered nd yielded ompletely stisfying resultsF he di'erent steps re listed elowX IRH

IPU α 1 α 2 α 3 α 4 (b) (a) H 0 H 0 v DW v DW v DW E 0 = -0.2MV/m E 0 = -0.4MV
• hehydrtion of the smple

• pinEoting with losed lid @single lyerA E esistX wwe WSHu R7 nd preEke

• iletronEem lithogrphy

• ostEexposure ke

• hevelopment E heveloper I wsfu nd P ethnol pigure RFW ! esist ptterns oserved t the optil mirosope feturing @A ellipsesD hourglssEshped nd retngulr nnostripesD s well s @A urves nnostrutures nd 9pierElike9 strutures @see insetAF overed with mgnetoelsti mterilF sn the ltter seD the mgnetoelsti mteril is sputtered diretly on the sustrteF henD the lithogrphy is performed on the multilyer nd is followed y the deposition of solid msk @usully metlliAF his msk is removed y liftEo' nd the smple is sujeted to dryEething of the mgnetoelsti mterilF he msk is susequently removed nd the expeted ptterns re pprentF pigF RFIR illustrtes these two pprohes step y stepF ine the liftEo' solution is more strightforwrd nd required less testingD it ws tried (rstF he proedure desried in the previous susetions ws followedD lthough IRR pigure RFIR ! teps orresponding to the methods of liftEo' nd plsm ethingF tep @A is spinEotingD @A is eletronEem exposureD @A is developmentD @dA is sputter deposition nd @eA is the liftEo' of the multilyers per seF por ethingD note tht step @A9 orresponds to steps @AD @A nd @A ominedF initilly the spinEoting speed ws lowerD resulting in ISH nmEthik otingF viftEo' ws performed y pling the smple in gEIR solvent t UH • g for severl hoursF mooth ultrsound tretment ws lso helpfulD while stronger exittions proved hrmful to the mgnetoelsti strutures @smller elements re destroyedAF e iw imge of retngulr mgnetoelsti element is shown on pigF RFISF hile the shpe is reognizleD the ontour is fr from smoothF sn ftD the lighter ontour hints t strong 9wllElike9 umps on the edges of every mgnetoelsti elementF he likely explntion for these irregulrities is quite strightforwrdF gontrry to evportionD the motion of the inoming prtiles in sputtering is isotropiX the diretion of motion of ejeted mteril is not direted t the sustrteF fesidesD the deposited thikness of resist is ISH nmD while only PH nm IRS pigure RFIS ! @A iw imge of mgnetoelsti element otined y liftEo'F @A snterE prettion of the presene of strong irregulrities on the edgesF of mgnetoelsti mteril is depositedF es resultD there is signi(nt deposition of mteril on the vertil side of the resistD nd these irregulrities remin one the resist is removed @see pigF RFISAF he piezeoeletri hrteriztion of our wxE sustrte ws performed using strin gugesF fsillyD strin guge onsists in long nd thin metlli stripe whih resistne hnges with its elongtion in liner fshion through the guge ftor kX ∆l l k = ∆R R F hereforeD the hnge in resistne provides preise informtion on the tul strin tht uses the elongtion of the metlli wireF e hetstone ridge is often used to llow the detetion of minute vritions of resistne nd turns them into voltge signlsF he strin guges used were ought from rfw nd n mesure the strin in two perpendiulr diretions simultneously @ setupAF ih diretion is hrterized y given guge ftorF e used ynorylte to tth the strin guges to wxE suE strtes on whih fullEsurfe eletrodes were deposited on oth sides @see pigF RFIVAF o filitte the estlishment of ontt points on the surfe s well s the onnetion with the strin guge eletril outputD mount ws speilly frited to (x the sustrte s prt of n internshipF st is shown on pigF RFIVF he experiment set up (rst involves the lirtion of the hetstone ridge nd ll the required eletril pprtusD in order to know how to quntittively trnslte vritions of voltge into vritions of resistneF sn prinipleD the strin is then esily lulted using the guge ftors tht were pproximtely equl to IFUF rere howeverD we only know the vlue of the strin within n dditive onstntF sndeedD when the strin guge is tthed to the sustrte the tul strin t tht moment @orresponding to IRV pigure RFIV ! @A hotogrph of strin guge tthed to smpleF @A mple with strin guge tthed pled on the mount enling esy (xing nd onvenient ontt pointsF zero elongtion of the gugeA is unknownF sn ny seD sine the mgnetoelsti lyer is lso deposited on preEstrined sustrteD it is the vrition mesured tht mttersF huring preliminry tests it ws notied tht t some points tht ppered like shrp trnsitionsD dynmil ehvior of the system ould e oserved nd it took time for the mesured vlues to settle to n equiliriumF he generl protool for the experiment ws tht for eh mesurementD we wited for n equilirium to e rehedF sn this senseD the pproh hosen ould e onsidered qusiEsttiF he pplied voltge went from EPHH to CPHH D nd then me k to EPHH @sine the sustrte re QHH nmEthikD the orresponding rnge for the eletri (eld is ±0.67 wGmAF hepending on the lol steepness of the urveD points were tken every IH or S F mples from oth vendors were hrterizedF he tul reltionship etween estiE mted strin @within n dditive onstntA nd eletri (eld is shown in pigF RFIWF he otined response is quite omplex nd requires some desriptionF hile liner portions do existD the overll ehvior is highly nonlinerF prom one poled stte nd deresing the mplitude of the pplied eletriD liner region is desriedF st is worth mentioning tht in ordne with prior disussion the vrition of the strin is of opposite signs for the orthogonl diretions nd @the piezoeletri oe0ients re indeed of opposite signs lollyAF qoing to negtive @respF positiveA eletri (eld on n initilly positively @respF negtivelyA poled sustrteD there is (rst ritil (eld where the y strin vries ruptly to reh n extremumF sn the ontext of rhomohedrl phse s shown in pigF RFPD this n e interpreted s the (eld t whih the polriztion swithes from diE retions with positive or negtive z omponent @red respF violet dotEdshed rrowsAD IRW pigure RFIW ! eltionship etween pplied eletri (eld nd mesured strin in oth diretions for smples from @A gq edvned wterils nd @A ehnologiesF he ft tht the loop does not lose in @A my e ttriutle to historyEdependent ehvior or hnges in experimentl onditions over the ourse of the experimentF to polriztion diretions in the xyEplne @lue dotEdshed rrowsAF hen the mplitude of the (eld is further inresedD the y strin suddenly jumps k to vlues similr to norml poled responseF his new trnsition onstitutes the poling of the sustrte in the other diretion @polriztion diretion long the violet respF red dotEdshed rrowsAF enother liner region is then desriedF he x strin lso shows unusul ut more omE plex ehvior round these ritil (eldsF st should e noted tht within these trnsitionsD the dynmis of the mesured strins re suh tht severl minutes re t times neessry for the system to reh equiliriumF edmittedlyD the modeling presented in previous hptersD in whih liner ehvior ws ssumedD is insu0ient to relly desrie the ehvior of wxE sustrteF he piezoeletri oe0ients used in the simultions orrespond to the response t high freE quenies PUVD nd do not orrespond to wht is mesured in the liner regions t low frequenies PVRF sn ny seD this spet only 'ets tehnologil mtters nd from the physil point of viewD the more interesting ontent is the ehvior of the mgneti system onsidering given strin or stressF imilr mesurements re reported in the litertureF sn efF PVRD the uthors pplied n eletri (eld with tringulr wveform t frequeny of HFHI rz nd the dt ws olE leted y digitl quisition systemF his point mkes their pproh slightly di'erent from oursD where eh mesurement ws mde fter the system rehed equiliriumF st is di0ult to omment eyond mere speultion on the impt of suh di'ereneF sn ny ISH pigure RFPH ! iletri (eldEstrin reltionship reported in the litertureF flk nd red urves orrespond to mesurement loop similr to wht we didF qreen nd lue urves orrespond to unipolr mesurementsF pigure from PVRF seD the results @lk nd red urves inluded in pigF RFPHA re nonetheless qulittively s well s"to some extent"quntittively similr nd on(rm the strongly nonliner eE hvior of @HIIAEut wxEF he di'erenes with our mesurements my e ttriutle to the protool followed nd norml vritions etween smplesF heir investigtions lso inluded yles with eletri (elds in ertin rnge so tht the sustrte is never fully poled in the opposite diretionF sn this wyD they showed tht it is possile to swith eE tween two sttes of the sustrte tht re ssoited with very di'erent strin vluesD s illustrted y the green nd lue urves on pigF RFPHF fesidesD dditionl nlyses in this pper suggest tht the sustrtes do not seem prone to ftigue nd tht the permnent strins sttes re very stle in timeF ell these properties re interesting in the prospet of memory pplitionsD for instneF 4.3.2 Magnetic characterization of sputtered layers he lyers deposited with the reipe desried ove n e hrterized mgnetilly using the irting mple wgnetometer @wA set up t siwx @iW model from ehi shown on pigF RFPIAF sn wD the smple is tthed to n osillting rod in the ir gp of mgneti iruitF he response of the smple to mgneti (eld generted is then deteted y dditionl oils using synhronous detetionD given the smll mount of mgneti mteril involvedF sn this mnnerD the di'erent omponents of the mgneti ISI pigure RFPI ! hotogrph of the irting mple wgnetometerF moment n e estimtedF sn our seD hysteresis loop etween P nd EP is performed on the squre silion smple on whih the deposition ws mde @see pigF RFIQAF he frequeny of the vertil osilltions to whih the smple is sujeted is US rzF he mplitude of mgnetiztion is inferred y estimting the volume of mteril from the known surfe of the squre nd the thikness depositedF pigF RFPP shows n exmple of the hysteresis loops long the esy nd hrd xis for the typil multilyers depositedF prom the dt t high (elds in these plotsD one n dedue tht the mgnetiztion is out 930 × 10 3 eGmF hese two grphs lerly show the presene of lerEut unixil nisotropy through typil esy nd hrd xis urvesF he points t whih the mgnetiztion rehes the mgnetiztion t sturtion on oth plots de(nes the nisotropy (eldD whih hppens to e etween RH nd SH ×IH 3 eGmF e n lso use de)etometry tools ville t siwx to mesure mgnetostrition in these smples PVQD PWUF huring sputter depositionD long nd nrrow frgment @emA of silion wfer is put in the hmer for tht purpose @see pigF RFIQAF o mesure mgnetostritionD the em is (xed t one of its extremities in dedited setup enling the pplition of mgneti (elds in order to mesure the orresponding displement using de)eted lser emF por tht purposeD the shpe of the em sujeted to the strin indued y the mgnetoelsti mteril is supposed to e proliF unowing the mount of mgneti mteril s well s the dimensions nd elsti prmeters of the silion F he vlues ssumed re respetively ISQ E = 80 q nd ν = 0.25 of the oung9s modulus nd oisson9s rtioF hespite the wekness of the signl ssoited with suh thin multilyersD it ws possile to perform preise mesurement of mgnetostritionF he result of (eld sn long the hrd xis is shown on pigF RFPQF sn these multilyersD the lrger proportion of pego redues mgnetostritionD with b γ,2 ≈ 6.6 w @λ s ≈ 0.7×10 -4 AF ypillyD mesurements on multilyers with lyers of pego nd go 2 of equl thikness yield mgnetostrition oe0ient λ s ≈ 1 × 10 -4 D i.e. round IHH ppmAF

Atomic and Magnetic Force Microscopy

he tehnique referred to s wgneti pore wirosopy @wpwA is vrint of etomi pore wirosopy @epwAD whih hs to e suintly desried efore introduing wpwF en tomi fore mirosope is essentilly type of snning proeD in whih piezoeletri element is used to ontrol the position or tute the motion of ntilever pled ner the surfe of the smpleF he proximity etween the tip of the ntilever @proeA nd the smple indues de)etion of the ntilever due to intertion foresD in whih vrious phenomen re involved @eletrostti foreD vn der ls foreFFFAF his de)etion n e mesured with detetor using re)eted lser em @see pigF RFPRA or through the feedk signl of the piezoeletri elementF here re three min modes of imgingD with the tip either in onstnt ontt with the surfeD in intermittent ontt @tpping modeA or without ny onttF he (rst of these three modes is now rrely used euse of the superior performnes of the other twoF sn these sesD the ntilever is mde to osillte round its resonne frequenyF ith wpwD the tip used hs mgneti oting @oftentimes mde of oltAD nd is hene sujet to mgneti intertions tht n e deteted nd yield informtion out the lol stte of mgnetiztionF vet us onsider uniform mgnetiztion of the tip nd the orresponding mgneti moment noted mF nder some ssumptions @reltively hrd mterilsD uniformity of mgnetiztion nd of mgneti (eld ross the mgneti tipAD the fore exerted on the tip is the followingX

F i = µ 0 m j ∂H j ∂x i .
@RFIA sn most sesD s in oursD the tip is initilly mgnetized long z @m = m z AD tht is perpendiulrly to the surfe of the smpleF he expression ove n simply e written ISR pigure RFPR ! orking priniple of epwF e ntilever interts with the smple surfe with the tipF he position of the ntilever is mesured using re)eted lser emF pigure from freesiFhF sX

F = F z z = µ 0 m ∂H z ∂z z. @RFPA
pigure RFPS ! sllustrtion of the vift wodeF he seond pss tres the topogrphyD dding n o'set heightF pigure from frukerF he fore is proportionl to the vertil (eld grdientF ht we re looking t is ISS therefore relted to the outEofEplne omponents of the stry (eldD nd simple domin on(gurtions n redily e inferred from this informtionF st is worth noting tht it is the phse dt tht ontins mgneti informtionF sn the se where wpw imges is miguous or n e tied to severl di'erent solutions of miromgneti equtionsD re(ned tehniques tht hve een desried to help reonstrut mgneti distriutions from wpw dt PWVF wore detils out wpw nd its pplitions in generl n e found in dedited ook hpters PWWD QHHF pigure RFPT ! hotogrph of the epw usedF huring ny opertionD the imge is formed y snning the smpleF sn the se of wpwD there re two psses for eh line @tht isD two traces nd two retracesAF he (rst pss extrts topogrphi informtion in tpping modeF sn the seond pss the tip is rised to given height nd is mintined to predetermined distne from the surfeF his is lled the Lift Mode nd ensures tht there is only miniml in)uene of topogrphy on mgneti imging @see pigF RFPSAF sn the se of the viner wodeD the proe is kept t onstnt ltitudeF yne of its dvntge is its reltive insensitivity to rtefts relted to strongly uneven surfesD whih will mtter to our sitution s explined lterF pigure RFPW ! wpw dt of hourglssEshped mgnetoelsti nnostrutures in @A vift wode in whih strong topogrphyErelted rtefts re oserved nd @A nd @A viner wodeD whih prevents suh rteftsF sn @AD the ontrst is onentrted on the extremiE tiesF golorr legends hve een removed for lrity on this qulittive disussionF wpw oservtions of the multilyers of elongted shpe @hourglss nd onstnt setion nnostripesA led to more omplex resultsF pirstD it ws found tht the topogrphy hd tremendous e'ets on the phse signl oserved s visile in pigF RFPWD despite the use of vift wodeF his turned out to e n rteft linked to the topogrphyX while the tip lwys stys t the sme distne from the point of the surfe loted diretly eneth itD it my get loser or frther from other nery points of the surfeF rolems of this kind re to e expeted espeilly when the relief is steepD whih is lerly the se ISV in our (rst smples @see susetion RFPFRAF st ppered tht swithing to viner wode signi(ntly redued these unwnted e'ets @see pigF RFPWAF enother striking feture of preliminry mesurements ws the presene of four lerly de(ned high nd low pthes in signl ontrstD visile on the (rst two sns of pigF RFPWF his ws initilly onsidered evidene of twoEdomin mgneti on(gurtionF roweverD upon re)etion the oserved ptterns ws not omptile with this hypothesisD nd the resulting skeptiism motivted further srutinyF edditionl mesurements showed instnes where the ontrst in the middle of the mgnetoelsti element ws strongly diminished @see pigF RFPWAD so tht these phenomen re now suspeted of lso eing rtefts of uiltEin (lters nd vergingF sn the endD the most prole hypothesis is tht mgnetiztion in the nnostrutures is in ft uniform like in the ellipsesD or tht t lest there re no domin wllsF pigure RFQH ! @A opogrphil nd @A wpw dt in portion of I µmEwide ringF he ptterns oserved my e due to the presene of domin wllF he only instne where we relly think tht wpw dt hinted t the possile presene of domin wll ws in ring of I µm widthF pigF RFQH shows the topogrphil nd phse dtF sn this prtiulr instneD the pro(le on the edges ws quite poorF roweverD the pttern within the surfe of the mgneti mteril is reminisent of wht n e oserved in the viinity of domin wlls in similr strutures IUF e did not hve the opportunity to look t the smples while mgneti (eld nd mehnil stress re ppliedF ossile options for improvement on this spet re ddressed in susetion RFRFPF roweverD further mesurements were rried out using uerr mirosopeD s explined in the following susetionF ISW 4.3.4 Magneto-optic Kerr microscopy pigure RFQI ! hemti of the setup needed to oserve mgnetoEopti uerr e'etF xote tht the presene of photoelsti modultor @iwA nd ompenstor is optionlF pigure from QHI snformtion out lol mgnetism n lso e otined using mgnetoEoptil tehniE quesD inluding the mgnetoEopti uerr e'et @wyuiAF hen light is re)eted on mgE netized surfeD it undergoes some hnge in intensity nd polriztion due to intertion with lol mgnetiztionF his hnge depends on the orienttion of the mgnetiztionD so tht in the presene of multiple mgneti domins or inhomogeneous mgnetiztion distriutionsD there is ontrst in the re)eted signlF his llows for the diret oservE tion of the mgnetiztion distriution of smplesF he generl setup required for uerr mirosope is shown on pigF RFQIF he light is polrized @usully linerlyD whih is equivE lent to the sum of two irulrly polrized signlsA efore getting to the mgneti surfeF sn the generl seD the re)etion will rotte the xis of polriztion @due to di'erent phse veloitiesA nd indue smll elliptiity @due to di'erene of sorptionAF he vrition of the polriztion of the re)eted light is then mesured using n nlyzer pled efore the detetorF por mximum sensitivityD the nlyzer diretion should e roughly orthogoE nl to tht of the polrizerD whih neessrily implies lowEintensity signlsF hepending on the reltive orienttion of the inident light with respet to lol mgnetiztionD one dels with polr @outEofEplne mgnetiztionAD longitudinl @mgnetiztion long the inidene diretionA or trnsverse @mgnetiztion orthogonl to the inidene diretionA wyuiF yf ourseD the strength of the mgnetoEoptil response depends on the mterilsF e uerr mirosope ws set up t siwx over the ourse of the hhF st uses wRSSvQ horls mounted vih of short wvelength @RSS nmAD whih is interesting from the resoE lution point of viewF roweverD it ppers tht green hppens to e the est when it omes to strength of mgnetoEoptil response for pe nd go @whih onstitutes the uppermost ITH pigure RFQP ! hotogrph of the uerr mirosope set up t siwxF lyer in our smplesA QHPF es for the detetorD ygeEplshRFH Q rmmtsu E mer ws usedF es uerr imging involves light of low intensityD the properties relevnt for uerr imging inlude lowEnoise opertionD high sensitivity nd lrge dynmi rngeF he performne of the softwre ssoited with the mer is lso ruil s it is often neessry to perform signi(nt imge proessing in order to get ler resultsF st ppered tht some ontrst linked to mgnetiztion ould e hieved using mE terils similr to the multilyers we usedD lthough it ws di0ult to stilize n imge with nonEuniform mgnetiztion distriution using n pplied mgneti (eldF fesidesD mehnil virtions from the tle dded signi(nt noise to the imgesD lthough there

were suessful e'orts to redue itF ine these mterils did exhiit mgnetoEoptil responseD we tried to look t our smples with this new mirosopeD using the mount shown in pigF RFIVF feuse it is essentilly n optil mirosopeD only elements tht were mirons ross length nd width put on the msk were visileF hile smll nnostrutures were reogniE ITI pigure RFQQ ! uerr imges of multilyer sujeted to mgneti (eld nd mehnil stress indued y the piezoeletri sustrteF he di'erent mgneti domins re lerly visileF zleD their size ws too smll for the mirosope to give vlule insight on their internl mgneti strutureD espeilly onsidering the noise indued y virtions on the setupF roweverD lrge mrks of isoseles tringles shpe @hypotenuseX VH µm nd therefore ltiE tude of RH µmA were lrge enough to look t their mgneti domin on(gurtionF iven thenD during yles of pplied mgneti (eld etween sturting vluesD they mere most of the time monodominD lthough the mgnetiztion swithing ws lerly hieved y domin wll propgtionF he proedure followed ws the followingF pirstD sturting @for instne negtiveA mgneti (eld is pplied long the esy xisD so tht the mgneE tiztion of the tringle is uniformF he mplitude of this mgneti (eld is then redued until it rehed positive vlue lose to the oerivityF sn this regionD the mgneti (eld is inresed very slowly until nuletion of smll dominF et this point the mgneti (eld is kept t its vlueD nd n eletri (eld is pplied on the wxE sustrte to geneE rte stress omponentsF e showed tht it resulted in domin wll propgtion euse proly due to the size of the sustrteF ynly limited progress is to e expeted in this diretionD sine the spinEoting rottion speed is lredy lose to the mximum villeF woreoverD the thinner the lyer of resistD the more di0ult the lifto' will eF sing nother resist is n optionD lthough the lithogrphy proess would hve to e overhuledF sn this ontextD mny would lso onsider usul ilyer eEem strutures @for instne spinning wwe on top of opolymerAD with signi(nt underut QHQF en even simpler ide is to deposit thiker multilyers in order to redue the gp etween the thikness of the multilyer nd tht of the resistF por wide nnostripesD this would proly not e prolem from the mgneti point of viewF ytherD more omplited options inlude the use of smooth nisotropi ething with tilted ngle to rek the umps nd spikes on the edges fter liftEo'D for instne using etive son fem ithing @sfiAF enother prolem ws the dmge used y the neessry ultrsoni lening used during liftEo'F iven t minimum powerD there re signi(nt risks tht smller elements get dethed while long nnowires tend to rekF st is possile tht (neEtuning the lening prmeters @mode of opertionD powerD timeD tempertureA my yield etter resultsF elsoD the hoie of nother metl for the eletrode on whih the multilyers re deposited my help inrese their dhereneF vet us not forget tht these issues dispper if we onsider ething insted of liftEo' @see pigF RFIRAF roweverD other prolems need to e solved to otin smooth pro(les s well s len smplesF rere gin lot of prmeters ome into ply @gsD )ow rteD powerD metl used for msk nd so onAF elsoD the sfi tool t siwx is speilly suited ITQ for ething very thin (lmsF his solution ws tried oneD with mixed results so tht more time would e neessry to optimize the proessF roweverD it remins n interesting option s this tool is le to remove in relEtime the impurities generted y ethingF pinllyD the omintion of mterils hosen in this thesis @go 2 Gpego sputtered multilyers on @HIIA wxE ulk sustrteA should not e seen s the only pir worth onsidering to hieve stressEindued domin wll motion s presented hereF pirstD using

U = h + L 2 -L 2 -K u cos 2 Φ + A dΦ dx 2 dx.
@eFIA he vgrngin funtion of the vritionl prolem is therefore

L = -K u cos 2 Φ + AΦ 2 , @eFPA
where Φ ≡ dΦ dx F he ssoited rmiltonin funtion is

H = ∂L ∂Φ Φ -L = AΦ 2 + K u cos 2 Φ. @eFQA
ine ∂L ∂x = 0D we hve the onservtion of HD leding to the simpli(ed di'erentil eqution

Φ = - K u A cos 2 Φ + C 1 , @eFRA
where C 1 is n integrtion onstntF he oundry onditions

Φ(- L 2 ) = 0 , Φ(+ L 2 ) = π, @eFSA
IUQ dopted to nlyze the prolemD gurntee the existene of the ferromgneti domin wllF eprting the vriles in iqF @eFRAD we otin

Φ 0 dΦ C 2 + sin 2 Φ = x + L 2 K u A , @eFTA
where we used the (rst oundry ondition nd we introdued C 2 = AC 1 /K u -1F he oe0ient C 2 n e lulted y onsidering the seond oundry onditionF e get π 0 dΦ

C 2 + sin 2 Φ = 2 π 2 0 dΦ C 2 + sin 2 Φ = L K u A .
@eFUA e now introdue the integrl QHT

dx 1 + p 2 sin 2 x = 1 1 + p 2 F α, p 1 + p 2 , @eFVA where α = arcsin √ 1+p 2 sin x √ 1+p 2 sin x
nd F (ν, q) is the inomplete ellipti integrl of the (rst kindD de(ned s QHUD QHV

F (ν, q) = ν 0 du 1 -q 2 sin 2 u = sin ν 0 dx (1 -x 2 ) (1 -q 2 x 2 ) .
@eFWA e therefore otin the eqution for C 2 in the form

1 √ 1 + C 2 K 1 √ 1 + C 2 = L 2 K u A , @eFIHA
where we lso used the omplete ellipti integrl of the (rst kind K(q) QHUD QHV

K(q) = F ( π 2 , q) = π 2 0 du 1 -q 2 sin 2 u @eFIIA
imilrlyD we n rewrite iqF @eFTA in terms of ellipti integrls

K u A x + L 2 = 1 √ 1 + C 2 F α Φ , 1 √ 1 + C 2 @eFIPA
where

α Φ = arcsin √ 1 + C 2 sin Φ C 2 + sin 2 Φ . @eFIQA

IUR

iqutions @eFIHA nd @eFIPAD lthough in impliit formD solve the prolem of the (niteE length stripe with exhnge nd nisotropy energiesF e (nlly oserve tht y de(ning ξ = √ 1 + C 2 D these equtions n e further simpli(ed s follows

K u A x + L 2 = 1 ξ F arcsin ξ sin Φ ξ 2 -cos 2 Φ , 1 ξ , @eFIRA 1 ξ K 1 ξ = L 2 K u A . @eFISA
st is interesting to prove thtD for L → ∞D we otin the lssil wll lultion s found in PHF o eginD we note tht ξ → 1 + nd therefore η ≡ 1/ξ → 1 -when L → ∞F iqution @eFIRAD in terms of ηD eomes

K u A x + ηK(η) = ηF arcsin sin Φ 1 -η 2 cos 2 Φ , η .
@eFITA reneD the limiting se for L → ∞ is not trivil sine oth rguments of the ellipti funtion F in iqF @eFITA depends on η → 1 -F o ope with with this prolemD we use the following property of the funtion F QHUD QHV

F (ϕ, sin α) + F (ψ, sin α) = K(sin α) if cos α tan ϕ tan ψ = 1, @eFIUA with sin α = η nd ϕ = arcsin sin Φ √ 1-η 2 cos 2 Φ
F hen iqF @eFITA n e eventully rewritten s

K u A x + ηF (ψ, η) = 0, @eFIVA
where ψ n e found through the reltion

tan ψ = 1 cos α tan ϕ = 1 -η 2 cos 2 Φ -sin 2 Φ sin Φ 1 -η 2 , @eFIWA
leding to sin ψ = cos ΦF gonsequentlyD sustituting the expression of ψ into iqF @eFIVAD we hve Ku A x + ηF (arcsin cos Φ, η) = 0F hereforeD for L → ∞ @iFeF η → 1AD we hve tht

Φ(x) = 2 arctan exp K u A x = π -arccos tanh K u A x , @eFPIA
whih is the wellEknown solution for the in(nitely long stripe with exhnge nd nisotropy energies PHF IUT

1 2 µ 0 M 2 s w( r) + β k( r) • N ( r, r 0 ) w( r 0 ) + β k( r 0 ) d r 0 d r β=0   = 0. @fFUA
where we used the symmetries desried in iqsF@PFWA nd @PFWAF e n then de(ne n

uxiliry funtion Ũ Ũ = U ( w) - V µ 0 M s H 0 • w d r - V 2 1 2 µ 0 M 2 s w( r) • N ( r, r 0 ) w( r 0 ) d r 0 d r, @fFVA
whih n e minimized with H 0 (xedF e hve therefore proved this series of equivlenes

min w : w =1 U H L fixed ⇔ min w U H L fixed ⇔ min w Ũ H 0 fixed ⇔ min w : w =1 Ũ H 0 fixed , @fFWA
where Ũ is de(ned in iqF @PFIIAF pinllyD the minimiztion of Ũ with respet to the diretion wD with T nd H 0 imposedD leds to the tul mgnetiztion of the ferromgE neti odyF e n therefore pply the methods of the lulus of vritions to otin the eqution for wF es eforeD we work with the qâteux derivtive

d dβ Ũ w + β k β=0 = 0, @fFIHA where Ũ ( w) = Ũ + V λ( r)( w • w -1) d r. @fFIIA IUV e otin V   ∂f an ∂ w • k - 1 2 β ij ∂k l ∂x i ∂w l ∂x j + ∂w l ∂x i ∂k l ∂x j + 2λk l w l -µ 0 M s H 0i k i -a iklm T ik (k l w m + w l k m ) -µ 0 M 2 s V N ij ( r, r 0 )k i ( r)w j ( r 0 )d r 0   d r = 0, @fFIPA
where we used gin oth symmetries of tensor N F o proeed from hereD we use the divergene theorem in the form

V ∂h ∂x i d r = ∂V h n i dS. @fFIQA xowD if h = f gD we hve V f ∂g ∂x i d r = ∂V f g n i dS - V g ∂f ∂x i d r. @fFIRA
his property n e used on the exhnge terms to further elorte iqF @fFIPAD s follows

V   ∂f an ∂ w • k -β ij k l ∂ 2 w l ∂x i ∂x j + 2λk l w l -µ 0 M s H 0l k l -2a iklm T ik k l w m -µ 0 M 2 s V N ij ( r, r 0 )k i ( r)w j ( r 0 )d r 0   d r + ∂V β ij k l ∂w l ∂x j n i dS = 0, @fFISA
where we exploited the symmetry of tensor β ij F he (rst integrl of iqF @fFISA eing true for ny smooth funtion k ∈ R 3 D we n write

∂f an ∂ w -β ij ∂ 2 w ∂x i ∂x j -µ 0 M s H 0 + 2λ w -µ 0 M s H d -2a : T w = 0, @fFITA
where a : T w l = a iklm T ik w m F he vgrnge multiplier λ n e (nlly eliminted y pplying rossEprodut with wD whih yields iqsF @PFIPA nd @PFIQAF prom iqF @fFISAD we n lso dedue the oundry onditions whih n e pplied to minimize the energy funtionlX β ij k l ∂w l ∂x j n i = 0 on ∂V D where the perturtion vetor k is not free sine it must verify the ondition w + β k • w + β k = 1 when w • w = w 2 = 1F o the (rst order in β this trnsltes to w • k = 0 ndD thereforeD we n sy IUW tht k = w× w for n ritrry vetor wF henD k is not ritrry ut w is ompletely freeF fy onsidering k l = lst w s w t D we hve β ij lst w s w t ∂w l ∂x j n i = 0 ∀w s D or β ij lst w t ∂w l ∂x j n i = 0 ∀s on ∂V @ lst eing the veviEgivit permuttion symolAF his ondition n e strongly simpli(ed when β ij ∝ δ ij @isotropi exhngeAF sn this se we hve lst w t ∂w l ∂x i n i = 0 ∀sD where ∂w l ∂x i n i is the diretionl derivtive of w l long nF oD it orresponds to w × ∂ w ∂ n = 0F e lso oserve tht w is lwys perpendiulr to where

φ( r) = V 1 4π M ( r 0 ) • ( r -r 0 ) r -r 0 3 d r 0 .
@gFPA e useful development n e performed when the mgnetiztion M is uniform in given regionF sn this se we n pply the divergene theorem

V div w d r = ∂V w • n dS, @gFQA
where we onsider w = M r-r 0 D with onstnt (eld M F sndeedD we otin

div w = M • ∇ 1 r-r 0 = M • r-r 0 r-r 0 3 , @gFRA nd therefore iqF @gFQA reds V M • r -r 0 r -r 0 3 d r = ∂V M • n r -r 0 dS. @gFSA
es onlusionD for uniform mgnetiztion M D the slr potentil is given y

φ( r) = 1 4π M • ∂V n dS r -r 0 , @gFTA

IVI

where n is the unit vetor norml to the externl surfe ∂V with re element dSF o lulte the totl demgnetiztion (eld in given pointD we n prtition the whole region in given numer of prllelepipedl lyersF es disussed elowD the prllelepipedl geometry llows for quik nd simple omputtion of the demgnetiztion (eld through the slr potentil φF henD we n dd ll the ontriutions to otin the (nl resultF reneD we suppose M uniform within the ritrry prllelepiped

[x 1 , x 2 ]×[y 1 , y 2 ]×[z 1 , z 2 ]D
nd we lulte φ through iqF @gFTAF e de(ne the integrl I = ∂V n dS r-r 0 nd we write the x omponent s

I x = z 2 z 1 y 2 y 1 dy 0 dz 0 (x -x 2 ) 2 + (y -y 0 ) 2 + (z -z 0 ) 2 - z 2 z 1 y 2 y 1 dy 0 dz 0 (x -x 1 ) 2 + (y -y 0 ) 2 + (z -z 0 ) 2 .
@gFUA e hnge of vriles leds to

I x = z-z 2 z-z 1 y-y 2 y-y 1 1 (x -x 2 ) 2 + ξ 2 + η 2 - 1 (x -x 1 ) 2 + ξ 2 + η 2 dξdη.
@gFVA xowD let us de(ne the funtion F (A, B, C, D, E, F ) s

F = B A D C 1 E 2 + ξ 2 + η 2 - 1 F 2 + ξ 2 + η 2 dξdη, @gFWA
nd we otin the demgnetiztion potentil in the form

φ( r) = 1 4π [M x I x + M y I y + M z I z ] , @gFIHA
where

I x = F(z -z 1 , z -z 2 , y -y 1 , y -y 2 , x -x 1 , x -x 2 ), @ gFIIA
I y = F(x -x 1 , x -x 2 , z -z 1 , z -z 2 , y -y 1 , y -y 2 ), @ gFIPA
I z = F(y -y 1 , y -y 2 , x -x 1 , x -x 2 , z -z 1 , z -z 2 ). @gFIQA
o lighten the nottionD we hose to write it in more onise form

φ( r) = 1 4π M x F 321 + M y F 132 + M z F 213 . @gFIRA

IVP

where the symol F ijk mens tht the funtion F is lulted with vriles spei(ed in iqsF @gFIIAD @gFIPA nd @gFIQAF pinllyD the omponents of the demgnetiztion (eld n e derived s @gFPHA o omplete this disussionD one n oserve tht the integrl in iqF @gFWA n e lulted in losed formF his is very useful for the numeril implementtion of the proposed proedure

H dx = - ∂φ ∂x = - 1 4π M x ∂F
F =A ln √ E 2 + A 2 + D 2 + D √ F 2 + A 2 + D 2 + D √ F 2 + A 2 + C 2 + C √ E 2 + A 2 + C 2 + C + B ln √ E 2 + B 2 + C 2 + C √ F 2 + B 2 + C 2 + C √ F 2 + B 2 + D 2 + D √ E 2 + B 2 + D 2 + D + C ln √ E 2 + A 2 + C 2 -A √ F 2 + A 2 + C 2 -A √ F 2 + B 2 + C 2 -B √ E 2 + B 2 + C 2 -B + D ln √ E 2 + B 2 + D 2 -B √ F 2 + B 2 + D 2 -B √ F 2 + A 2 + D 2 -A √ E 2 + A 2 + D 2 -A + E arctan E 2 + A 2 -A √ E 2 + A 2 + D 2 D E -E arctan E 2 + A 2 -A √ E 2 + A 2 + C 2 C E -E arctan E 2 + B 2 -B √ E 2 + B 2 + D 2 D E + E arctan E 2 + B 2 -B √ E 2 + B 2 + C 2 C E -F arctan F 2 + A 2 -A √ F 2 + A 2 + D 2 D F + F arctan F 2 + A 2 -A √ F 2 + A 2 + C 2 C F + F arctan F 2 + B 2 -B √ F 2 + B 2 + D 2 D F -F arctan F 2 + B 2 -B √ F 2 + B 2 + C 2 C F . @gFPIA
woreoverD ll its derivtives n e omputed in order to implement iqsF@gFISAD @gFITA nd @gFIUAF his n e esily done in symoli environmentF IVR Appendix D Analytical treatment of domain wall motion his ppendix ontins short derivtion of the lssil lker solution for the motion of IVH • domin wll in slightly dpted versionF sndeedD sine in this thesis we work with hedEtoEhed domin wlls in thin (lmsD this is the on(gurtion hosen here insted of the originl domin wll studied y hryer nd lkerD in whih the esy xis is perpendiulr to the diretion of motion USF sn order to gurntee simpliity in the derivtionD we dopt the onvention of spheril oordintes used y hryer nd lker for the ngles desriing the diretion of mgnetiztionF hereforeD there will e some key di'erenes etween the onvention used here @ngles noted φ nd θA E whih mirrors tht of hryer nd lker E nd wht is used in hpters P nd Q @ngles noted Φ nd ΘAF he diretion of motion is long z so tht the mgnetiztion w n e written (cos φ cos θ, cos φ sin θ, cos θ)F e onsider ferromgneti mteril with unixil nisotropy long the zExis @onE stnt K u AD forming two domins with ntiprllel mgnetiztion long the diretion of motionF e uniform mgneti (eld H 1 = H 1 z is ppliedF he exhnge intertion is lso tken into ountF es for the demgnetiztionD simpli(ed form is introduedD orreE sponding to the demgnetiztion of uniformly mgnetized ellipsoidF et ny pointD its diretion is opposed to lol mgnetiztion so tht H d = -M s N wD with N digonl tensor with unit treF fy nlogy with iqF @QFWAD we n esily write the orresponding e'etive (eldX

H eff = H 1 -M s N w + 2 µ 0 M s -K u w z z + A ∂ w ∂z . @hFIA

IVS

pollowing the sme pth s in susetion QFIFRD we otin the dynmil equtionsX φ = -1 sin θ γ 0 1+α 2 (-s w + αr w ), θ = -γ 0 1+α 2 (r w + αs w ), @hFPA he quntities r w nd s w re s followsX

r w = 2A µ 0 M s (sin θφ + 2 cos θφ θ ) + M s cos φ sin φ sin θ(N x -N y ), @hFQA s w = -sin θH 1 + 2 µ 0 M s -K u sin θ cos θ + A θ -sin θ cos θφ 2
+ M s sin θ cos θ(N zcos 2 φN xsin 2 φN y ). @hFRA sn the se of n in(nite ellipsoid long the zExisD the element N z of tensor N is zeroF xowD two ssumptions will e mde on the shpe of the domin wllD whih llow us to (nd solution of the dynmil system oveF he (rst ssumption is tht φ = φ 0 is onstnt nd uniformD this rings some simpli(tion to the equtionD sine s result φ D φ nd φ re ll equl to HF he seond ssumption is tht θ ssumes the lssil domin wll shpe derived in ppendix e nd trvels t onstnt veloity vX θ(x, t) = θ ∞ (ξ(xvt)) = 2 arctan(exp ξ), @hFSA where we introdued Ku = K u + µ 0 M 2 s 2 (cos φ 0 N x + sin 2 φ 0 N y ) nd ξ = Ku A (xvt)F hese ssumptions led to signi(nt simpli(tions of the dynmil equtionsD due to the reltionship etween θ nd its derivtivesX

dθ ∞ dξ = 2 exp ξ 1 + (exp ξ) 2 = sin θ ∞ , @hFTA sine exp ξ = tan θ ∞ 2 F herefore it follows tht ∂θ ∂t = ∂ξ ∂t dθ ∞ dξ = - Ku A v sin θ ∞ . @hFUA vikewiseD ∂θ ∂x = Ku A sin θ ∞ , @hFVA and ∂ 2 θ ∂x 2 = Ku A sin θ ∞ cos θ ∞ . @hFWA
IVT he (rst prt of iqF @hFPA reds

αM s cos φ 0 sin φ 0 sin θ ∞ (N x -N y ) + sin θ ∞ H 1 - 2 µ 0 M s Ku sin θ ∞ cos θ ∞ -Aθ = 0,
@hFIHA nd then using iqF @hFWA αM s cos φ 0 sin φ 0 sin θ ∞ (N x -N y ) =sin θ ∞ H 1 . @hFIIA his eqution yields the desired reltionship giving the ngle φ 0 X

sin 2φ 0 = 2H 1 αM s (N y -N x ) = H 1 H c , @hFIPA
where H c = αMs(Ny-Nx)

2
is the soElled lker rekdown (eldF es the vlue of sin 2φ 0 nnot exeed ID there is solution to this eqution only when H 1 < H c F es for the seond prt of iqF @hFPAD

θ = - Ku A v sin θ ∞ = - γ 0 1 + α 2 (α(-sin θ ∞ ) + M s cos φ 0 sin φ 0 sin θ ∞ (N x -N y )) .
@hFIQA sing iqF @hFIPA @sin φ 0 cos

φ 0 = 1 2 sin 2φ 0 A nd dividing y -sin θ ∞ X v = - Ku A γ 0 1 + α 2 αH 1 + H 1 α . @hFIRA
sn the endD the lker solution is given y the following systemX The manipulation of magnetic domain wallsthat separate regions of uniform magnetizationis associated with both fundamental and technological research interests. A large part of the literature on domain wall motion deals with the use of magnetic elds and electric currents. However, several concernsmost notably energy dissipationmotivates the search for alternatives. Among potential candidates, the mechanical stress-mediated magnetoelectric coupling in magnetoelastic/piezoelectric heterostructures seems promising. In this thesis, it is shown that the combination of a bias magnetic eld and uniform mechanical stress can induce unidirectional domain wall motion in nanostructures with uniaxial anisotropy. Static and dynamic aspects of this phenomenon are studied by means of ad hoc numerical procedures simulating the mechanical coupling of 011-cut PMN-PT generating the stress, and TbCo 2 /FeCo multilayers magnetoelastic nanostructures. The design of the cross section prole in nanostructures allows to tailor the response of the system, enabling for instance the control of domain wall position in conned geometries. The associated dynamics stands apart from known regimes because of the shape of the domain wall. The existence of steady-state regimes in nanostripes of constant width shows that velocities comparable to those of other techniques can be obtained, for a fraction of the energy required. Experimental investigations resulted in the development of a successful fabrication process on PMN-PT and the exploration of the magnetoelectric eect. Keywords domain wall, magnetoelastic, magnetoelectric, mechanical stress, piezeoelectric, spintronic, nanostructures, nanotechnology Manipulation magnétoélectrique de parois de domaine transverses dans des nanostructures magnétoélastiques La manipulation de parois de domaine magnétique, qui séparent des régions d'aimantation uniforme dans les matériaux, est associée à des enjeux à la fois fondamentaux et technologiques. De nombreux travaux portent sur le déplacement de parois par champs magnétiques et courants électriques. Cependant certaines préoccupations, notamment la dissipation d'énergie, motivent la recherche d'alternatives. Parmi les solutions potentielles, le couplage magnétoélectrique par l'intermédiaire de contraintes mécaniques dans des hétérostructures magnétoélastique/piézoélectrique paraît prometteur. Dans cette thèse, il est montré que l'association d'un champ magnétique de biais et de contraintes mécaniques uniformes peut engendrer le déplacement unidirectionnel d'une paroi de domaine transverse dans des nanostructures à anisotropie uniaxiale. Les considérations statiques et dynamiques de ce phénomène sont étudiées par le biais de procédures numériques ad hoc simulant le couplage mécanique entre substrat de PMN-PT de coupe 011 générant des contraintes, et nanostructures multicouches magnétoélastiques TbCo 2 /FeCo. Le design du prol de section des nanostructures permet de moduler la réponse du système, par exemple pour contrôler la position de parois connées. La dynamique du système se distingue des régimes habituels de par la forme de la paroi de domaine. L'atteinte de régimes permanents dans des nanorubans montre que des vitesses comparables aux autres techniques sont obtenues, pour une dissipation d'énergie beaucoup plus faible. Des travaux expérimentaux ont permis de mettre au point une procédure de fabrication sur PMN-PT et d'explorer l'eet magnétoélectrique. Mots-clés paroi de domaine, magnétoélastique, magnétoélectrique, contrainte mécanique, piézoélectrique, spintronique, nanostructures, nanotechnologie PIW

v = - γ 0 H 1 α A K u + µ 0 M 2 s /2(cos 2 φ 0 N x + sin 2 φ 0 N y ) , @hFISA sin 2φ 0 = 2H 1 αM s (N y -N x ) = H 1 H c . @

  to @miroAmgnetism F F F F F F F F F F F F F F F F F F F F PI IFIFI wgnetiztion nd energy ontriutions F F F F F F F F F F F F F F F PP IFIFP wrosopi ehvior of ferromgneti odies F F F F F F F F F F F F PR IFIFQ wgneti on(gurtions nd domin wlls F F F F F F F F F F F F F F PT IFIFR ypology nd physil fetures of domin wlls F F F F F F F F F F F PV IFP homin wll motion F

  RVIFR wotion indued y uniform stress in mgnetoelsti mterils F F F F F F F SI IFRFI he need for symmetry rekingF F F F F F F F F F F F F F F F F F F SI IFRFP ymmetry reking sed on is mgneti (eld F F F F F F F F F F F SS IFRFQ hesription of the proposed system F F F F F F F F F F F F F F F F F F SU IFRFR ingineering of ross setion F F F F F F F F F F F F F F F F F F F F F F TH IFRFS roof of onept through miromgneti simultions F F F F F F F F TI IFS gonlusion F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F TR T 2 Static behavior in magnetoelastic nanostructures 66 PFI ritionl pproh to miromgnetism F F F F F F F F F F F F F F F F F F F TT PFIFI yverview F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F TU PFIFP he(nition of the energy funtionl F F F F F F F F F F F F F F F F F F F TV PFIFQ herivtion of lssil equtions F F F F F F F F F F F F F F F F F F F F TW PFP iquilirium distriution in nnostrutures F F F F F F F F F F F F F F F F F F UIPFPFI he(nition of the energy funtionl F

  IHP 3 Dynamics of stress-induced domain wall motion 103 QFI iqutions governing domin wll dynmis FF F F F F F F F F F F F F F F F F IHQ QFIFI yverview F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IHQ QFIFP he(nition of the energy funtionl F F F F F F F F F F F F F F F F F F F IHSQFIFQ hetermintion of e'etive (eld F F

  ITQ RFRFI yptimiztion of frition proess F F F F F F F F F F F F F F F F F F ITQ RFRFP gontrol of domin wlls in nnostrutures F F F F F F F F F F F F F F ITR RFRFQ wxE studs FF F F F F F F F F F F F F F F F F F F F F F F F F F F F ITS RFS gonlusion F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F Fpossédnt un ordre mgnétiqueD l distriution sptile d9imntE tion n9est ps toujours uniformeF ypiquementD des régions d9imnttion uniformes de diretions données sont séprées pr des parois de domaineD u sein desquelles l vriE tion sptile d9imnttion est importnteF ves outils issus de l théorie du miromgnéE tisme permettent d9interpréter et prédire es on(gurtions qui orrespondent à des miE mimums d9énergie mgnétiqueD en onsidérnt l9ensemle des ontriutions énergétiques pertinentes @eemnD nisotropieD éhnge etFA v mnipultion de strutures mgnétiques à l9éhelle mirométrique ou nnoméE trique revêt un grnd intérêt à l fois fondmentl et tehnologiqueF in prtiulierD le déplement de prois de domine est le sujet de nomreuses reherhes réentesF ve déE veloppement rpide de l spintronique et les vnées tehnologiques ssoiées lissent envisger de nomreuses pplitionsD notmment en miroéletroniqueF v9pplition de hmps mgnétiques et de ournts életriques o're des possiilités de ontrôle ujourE d9hui mîtrisées et de mieux en mieux omprises sur le pln théoriqueF r exempleD des modèles simples permettent de prédire le omportement d9une proi de domine soumise pr exemple à un hmp mgnétiqueF tusqu9à un hmp seuilD le mouvement est rtéE risé pr un régime sttionnire et une vitesse ugmentnt ve le hmp ppliquéF euEdelà d9une limite ommunément ppelée Walker breakdownD l moilité de l proi hute de fçon rupte et l dynmique devient plus omplexeF h9utres méthodes moins lssiques de mnipultion lole de l9imnttion ont fit l9ojet de reherhes explortoiresF gellesEi sont entre utres motivées pr des onsidérE tions de onsommtion d9énergieD puisque l9utilistion de ournts életriques @que e soit diretement ou pour générer des hmps mgnétiquesA est reltivement peu performnte de e point de vueF in prtiulierD l9pprohe mgnétoéletriqueD qui onsiste à ontrôler l9imnttion pr des hmps életriques plutôt que des ourntsD représente un ndidt prometteurF in utilisnt des hétérostrutures mgnétoélstiqueGpiézoéletriqueD il est posE sile d9otenir de ons ouplges mgnétoéletriquesF einsiD un sustrt piézoéletrique W pourr générer une déformtion sous l9e'et d9un hmp életrique et l trnsmettre à

Φ

  r 0 ) w( r 0 ) d r 0 d r. gon(gurtion et géométrie de l struture ferromgnétique onsidéréeF ve dernier terme de ette éqution est reltif u hmp de désimnttionF our trouE ver l on(gurtion mgnétique orrespondnt à l9étt d9équilireD une méthode vriE tionnelle été doptéeF gelleEi onsiste à luler l9expression de l dérivée de qâteux de l9énergieD qui doit être nulle pour l on(gurtion d9équilireF Φ sin Φµ 0 M s H 0 cos Φ + 3 2 λ s σ cos 2Φ +µ 0 M s sin Φ H dx y,zµ 0 M s cos Φ H dy y,z = 0. sl est lir qu9une telle éqution intégroEdi'érentielle nonElinéire ux dérivées prE tielles ne peut être résolue nlytiquementF n tritement numérique est don néessireF v9pprohe hoisie est sée sur une proédure itértiveD ou de relxtionD où l9on herhe le zéro d9une fontion pr l méthode de l dihotomieD implémentée sous wevefF ne méthode originle et très e0e de lul du hmp de désimnttion pr illeurs été développéeF v proédure numérique insi onstruite été vlidée pr une omprison ve des solutions nlytiques onnuesF ves résultts otenus ont on(rmé l possiilité de dépler une proi de ette mnièreF hns un nnorun @lrgeur onstnteA l proi se déple jusqu9à être éjetée et lisser une on(gurtion monodomineF hns une géométrie à lrgeur vrileD il est possile de on(ner l proi qui trouve une position d9équilireF gelleEi dépend de l ontrinte ppliquéeD omme illustré sur l pigF hF IP σ

  fontion de l9mplitude du hmp életrique ppliqué et de son signeF ge phénomène est imputle à l9importne reltive des nisotropie unixile et plnire ssoiées ux ontrintes positive et négtiveD respetivementF E 0 (MV/m)

  mterils nd therefore tends to lign djent mgneti momentsF hespite its quntum originD it n e written in terms suitle for ontinuous modelingF fy putting ost to inhomogeneous distriutionsD this ontriution tends to lign ll mgneti moments in the sme diretionF Magnetoelastic energy wgnetoelsti mterils re suset of ferromgneti mE terils with signi(nt oupling etween mehnil nd mgneti quntitiesF ghnges in mgnetiztions in mgnetoelsti ody re ssoited with hnges in strin nd PQ vieEversF his physil phenomenon is of prmount importne in this work nd thus will e disussed in further detil in susetion IFQFPF edditionl ontriutions my pper nd hve in some ses ritil in)uene on the energy lndspe of mgneti struturesF por instneD interfe intertions etween two mterils s well s other forms of exhnge energy re sujeted to inresing reserh interestF roweverD the phenomenon studied in this work only involves the ontriutions listed oveF hynmi ehviors n e studied y solving the vnduEvifshitzEqilert equtionD introdued in susetion IFPFPF 1.1.2 Macroscopic behavior of ferromagnetic bodies perromgneti ehvior n e oserved in some mterils elow ritil temperture T c D lled the gurie pointF feyond this tempertureD the mteril eomes paramagneticD whih mens tht euse of therml motion the mgneti moment is liner with respet to ny pplied mgneti (eld nd the mgnetiztion drops to zero in the sene of (eldF sn other wordsD moments will lign with strong mgneti (eldD nd will hve rndom distriution t zero (eldF st is importnt to note tht in generlD the term 4ferromgneti mterils4 ritrrily refers to mterils whih gurie temperture hppens to e ove room tempertureF veving lloys sideD the most ommon hemil elements exhiiting ferromgneti ehvior re ironD olt nd nikelF yther forms of ordered mgnetism existD suh s in ferrimgneti nd ntiferromgneti mterilsD whih re hrterized y distriutions with opposing mgneti momentsF nlike ntiferromgnetsD ferrimgneti mterils retin spontneous mgnetiztion euse the ntiprllel mgneti moments re unequlF he gurie tempertures of severl mterils re given in pigF IFIF prom n empiril point of viewD the ehvior of mrosopi ulk ferromgneti mteril n e studied through the pplition of n externl (eld long ny given diretionF e typil smple mgnetiztion s funtion of the pplied mgneti (eld is represented in pigIFPF prom demgnetized initil stteD the progressive pplition of @for instneA positive mgneti (eld yields the urve of (rst mgnetiztionD until the smple rehes sturtionF he mplitude of mgnetiztion t this point is wht hs een referred to s mgnetiztion t sturtion in susetion IFIFI nd is noted M s F hen the pplied (eld is redued to zeroD one n mesure the remnent mgnetiztion of the smpleF he pplition of negtive mgneti (eld will redue the mgnetiztion until it eomes zeroD whih de(nes the oeritivity of the smpleF his quntity is importnt in tht it is the sis for the distintion etween hrd nd soft mgnetsF rrd PR pigure IFI ! le showing the gurie point of severl mterilsF he xéel point is the ferrimgneti equivlent of the gurie point @note tht some elements n e ferromgnetiD ferrimgneti or prmgnetiD depending on tempertureAF pigure from RF @respF softA mgnets hve strong @respF wekA oerivityD mening tht reltively strong @respF wekA mgneti (eld is required to swith most of the mgneti moments in its diretionF yf ourseD this ll depends on the xis long whih the mgneti (eld is ppliedD nd therefore the onept of oeritivity is relted to the notion of nisotropyF hen sujeted to the reverse opertion @strting from strong negtive (eldA the smple will follow symmetril pth towrd sturtionF prom sturted stteD smple n e sujeted to demgnetiztion loopD y performing hysteresis loops with diminishing (eld mplitudes until the vnishing of the overll mgnetiztionF hepending on the nture nd size of the smple sujeted to the hysteresis loopD the hnges in mgnetiztion re rought out y omintion of di'erent phenomen t the mirosopi sleF he most intuitive of them is the oherent rottion of ll mgneti momentsF yther phenomen involve the lol vrition of mgnetiztionD with regions of nonuniform distriutionF his leds to the notion of domin wllsD introdued historilly in order to ount for the oveEdesried mrosopi ehvior of ferromgnetsF PS pigure IFP ! ypil hysteresis loop plot feturing urve of (rst mgnetiztionD sturtion nd oreitivityD s desried in the min textF rereD the )ux density is shown insted of the mgnetiztionD ut the disussion is unhngedF pigure from TDK-LambdaF 1.1.3 Magnetic congurations and domain walls et restD the ontriutions ited in susetion IFIFI @s well s others like tempertureD distriution of defets etcFA in)uene the equilirium distriutionD whih is ssoited with minimum vlue of the free energyF ken togetherD the joint ontriutions sum up nd re then ssoited with n eective eldD noted H eff F henD the equilirium distriution within the volume V of the ferromgneti ody stis(es the equtionX w × H eff = 0. @IFIA his eqution sttes tht the lol torque t ny point vnishes when the equilirium is rehedD so tht the diretion of mgnetiztion within the mteril is tht of the lol e'etive (eld H eff F yn the surfe ∂V of the mteril @if we note n the surfe norml vetorA the oundry ondition isX the mterils whih will pply in our seF qreter detils on these equtions will e given in hpter PFPF pinding the equilirium distriution of mgnetiztion involves the resolution of these equtionsD oftentimes using vritionl tehniquesF sn ferromgneti odyD ompromise hs to e found etween sometimes on)iting ontriutionsD even in the sene of ny externl stimulusF es stted in susetion IFIFID the exhnge energy will hve the PT tendeny to lign ll mgneti momentsF hepending on the geometry nd lengthsles involvedD howeverD the demgnetiztion energy ssoited with uniform mgnetiztion n e very lrgeF ptil vrition of mgnetiztion is thus ound to rise in some onditions to void suh mgnetostti issuesF sn soft ferromgnetsD the redution of demgnetiztion energy n e hieved y )ux losuresD i.e. forming loops to lose mgneti (eld lines reted y the distriution or more generlly with smooth vritions throughout the ferromgneti ody @see pigF IFQAF his on(gurtion n e prtiulrly fvored in ring geometry SF pigure IFQ ! hen the nisotropy is lowD the mgnetiztion is free to ssume ny diretionF st n therefore rete vortexElike ptterns like on this shemtiF pigure from TF sn the presene of some form of nisotropyD the mgnetiztion n vry more ruptly within the ferromgnetF he overll energy of the system is thus drmtilly ut y the retion of multiple distint domins inside whih the mgnetiztion is uniformF hey re usully lled magnetic domains or Weiss domains nd re seprted y domain wallsD where the mgnetiztion rpidly hnges from one diretion to nother one nd forms peulir ptterns suh s in pigF IFRF heir existene hs een postulted y eiss in order to ount for the mrosopi ehvior of ferromgneti mterilsF sndeedD the distriution in mrosopi ferromgnet n onsist of multitude of smll domins with unorrelted orienttionsD so tht the overll mgnetiztion n e zero even though M s is notF es is lso the se of soft mgnetsD the sptil vrition of mgnetiztion rings stisfying explntion to the sene of net mgnetiztion generlly oserved in mE rosopi smplesF wore diret experimentl vlidtions egn with visuliztions using mgneti powders nd lter mgnetoEoptil tehniques @uerr e'etAD mgnetoresistneD mgneti fore mirosopy @wpwA nd Ery mgneti irulr dihroism mirosopy @wghAF PU pigure IFR ! wgnetoEoptil visuliztion of typil rinElike ptterns of mgneti domE insD here with mgneti grnetF ih olor orrespond to one mgnetiztion diretionF pigure from TF wgneti domin wlls hve then een sujeted to extensive theoretil nd experiE mentl investigtions nd the physil theory of mgneti domins is long estlished U pillr on whih modern miromgnetis is sed VF 1.1.4 Typology and physical features of domain walls qenerlly spekingD domin wll n e de(ned s the mgneti struture etween two djent domins of di'erent mgnetiztion diretions w 1 nd w 2 D whih need not e oppositeF he sptil delimittion of domin wll is neessrily ritrry due to the ontinuous nture of the mgnetiztion distriution in the miromgneti formlismD nd domin wll is not well de(ned entity stritly spekingF his does not prelude the use of the onept of domin wllD referring to the region of greter mgnetiztion grdients within the mteril s mgneti ojetF sn this thesisD we will only over ferromgneti domin wllsD nd it is worth mentioning tht the ehvior of domin wlls in ntiferromgnets is qulittively di'erent W!IIF elthough it is esier to desrie nd depit domin wlls etween two domins of opposite mgnetiztionD this needs not e the seF he orienttion of mgnetiztion in domins depends hevily on lol nisotropy nd other prmetersF homins with ntiprllel mgnetiztions re ound to rise in mterils with strong unixil nisotropyD ut other on(gurtions suh s orthogonl domins n exist in mterils with ui nisotropyF PV hepending on prmetersD tempertureD externl stimuli nd the geometry of the onE sidered ferromgneti odyD whole vriety of domin wlls nd mgneti ojets n e oservedF sn prtiulrD tremendous interest hs reently een drwn on skyrmionsD whih re very smll topologil oddities IPF sn thin nd nrrow lyers of ferromgneti mterilsD there re two generl types of domin wlls"lthough others hve een desriE ed IQ"whih n e lled vortex nd transverse wlls @see pigF IFSA @the ltter n e symmetri or symmetri IRAF sn the sene of ny externl in)ueneD exhnge energyD mgnetoristlline nisotropyD nd demgnetiztion energy re the sole plyers nd will determine whih one of the vortex or trnsverse wll will e the fvoredF es n sideD in mny wys the in)uene of the ltter is kin to form of nisotropyF por instneD in very elongted nnostruture"with one of its dimensions very lrge with respet to the other two"the demgnetizing (eld will rete n pprent unixil nisotropyF ome uthors refer to this s the shape anisotropyF st n thus e onsidered muh in the sme wy s is the mgnetoristlline nisotropyD s long s we keep in mind tht in doing so we fous on the form it ssumes rther thn on its physil originF wny investigtions on domin wll on(gurtions hve thus foused on permlloy @xi 81 pe 19 A euse of its negligile mgnetoristlline nisotropyF hereforeD the role of geometry n e studied independently of other ftors for mny purposesD leding to the onstrution of ever more re(ned phse digrms to determine the most stle type of wll IR!ITF e good rule of thum is tht the wider nd thiker the lyerD the more energetilly fvored vortex wll will eF

1. 2

 2 Domain wall motionhomin wlls re fundmentl to the modern understnding of mgnetism in mterilsF his setion egins with n ount of their importne in modern tehnologil ppliE tionsD nd the resulting interest in their mnipultionF fefore moving on to the min QP wgnetiztion distriutions long the xExis s originlly lulted y vnE duD where eh urve is ssoited with given K u /A rtioF lot @A shows the ngle ΦD while the mgnetiztion vetor omponent w x = cos Φ nd w y = sin Φ n e seen respetively in plots @A nd @AF tehniques to indue domin wll motionD rief presenttion of the equtions governing mgnetiztion dynmis is providedF 1.2.1 Overview of technological relevance wgneti domin wlls nd their mnipultion hve drwn signi(nt interest nd hve een widely studied oth for fundmentl nd pplitionEoriented purposesF yver the lst dedesD domin wll motion hs een very dynmi re of reserhD from theE oretil explortions to experimentl investigtionsF gompred to ferroeletri domin wllsD mgneti domin wlls re quite omplex ojetsF hey possess internl degrees of freedom nd ome in vriety of types s we hve shown ove PPF his omplexity is one of the resons for the widespred interest in domin wlls from fundmentl physis point of view s well s for pplitionsF he strting point to understnd where lrge prt of the tehnologil importne of domin wlls nd miromgnetism in generl omes from is the ft tht informtion n e stored in mgneti mediumF sn other wordsD its of dt n e written @nd QQ redA in the diretion of lol mgnetiztionF he lssil nd most fmous exmple is the mgneti hrd drive tehnology widely used in omputers over the lst dedesF hile this tehnology hs seen drsti hnges in terms of performnes sine eing (rst introdued y sfw in the IWSHsD the si working priniple remins the smeF he writing prt is hieved y pplying mgneti (eld lolly on ferromgneti mteril through smll oil mounted on ferromgneti hedD see pigF IFWF pigure IFW ! orking priniple of hrd disk driveF e moving hed writes nd reds using urrentEindued mgneti (elds pplied lolly on ferromgneti stripF pigure from Public domainF hile this tehnology is now quite mtureD there is still reserh foused on optimiE ztionD for instne illustrted y reent developments of hetEssisted reording PQ!PS nd erlierD itEptterned medi PTF elsoD it hs reently enountered ompetitionD most notly from olidEtte hrives @hsA whih rely on nonEvoltile xexhEsed )sh meE mory @ tehnology tht n lso e found in f drivesAF elthough still signi(ntly more expensive thn rhhs per stored itD hs re likely to ontinue gining trtion s pries drop euse of severl dvntges tht will not e disussed hereF sn ny seD the study of domin wlls is nowdys dynmi reserh re with reE levne in vrious (eldsF eent tehnologil developments mde the designD proessingD QR nd hrteriztion of mgneti thin (lms nd nnosle strutures inresingly onveE nientF his turned out to e gret soure of knowledge nd insight into the physis of mgneti domins nd domin wllsF sndeedD efore the dvent of nnotehnology only the surfe of ulk mterils ould e hrterizedD while preise mgneti struture ptE terns n now e studied with more ontrol on geometryD phse nd omposition PUF sn reent yers for instneD smples with perpendiulr mgneti nisotropy @whih enles greter densityA hve een seen s very promising PVF feuse of the mny dvntges suh s reliilityD fst opertionD low power onE sumption nd nonEvoltilityD devies sed on domin wlls @nd spin systems in generlA re widely seen s promising tools for vrious pplitionsD inluding memoryD sensing nd logi PSD PW!QIF e notorious exmple of suh innovtive proposls is the retrk memory QPD QQF he ide is to use mgneti domin wlls in nnowire to enode dtD with n eletri urrent )owing through the nnowire induing ontrolled motion of domin wlls @see pigF IFIHAF yther memories sed on domin wlls QRD QS hve een proposed in reent yersF pigure IFIH ! hemti of the retrk memoryF pigure from IBMF enother (eld of interest onerns llEmgneti logiF sndeedD logi gtes nd whole rhitetures for logi opertions nd nnoomputing sed on mgneti domins mniE pulted with mgneti (elds hve een proposed QT!RPF he mnipultion of domin wlls hs lso een put forwrd in the ontext of unonventionl prdigms of informtion proessingD inluding neuromophi omputing systems sed on memristive deviesF he QS ltter re seen s good ndidtes for rti(il synpsesD nd n e sed on domin wlls RQ!RTF e wide vriety of other relted pplitions re onsidered for spinEsed devies RUD RVF pinllyD the mgneti grdient reted y stry (elds in the viinity of domin wll hs the potentil to ttrt mgneti eds down to the nnosleD mking their ptureD mnipultion RW!SQ or detetion SR possileF 1.2.2 The Landau-Lifshitz-Gilbert equation he dynmis of h motion is ultimtely governed y single strongly nonliner difE ferentil prtil eqution desriing the time evolution of single mgneti moments in the presene of given e'etive mgneti (eldF pirst desried in IWQS y vndu nd vifshitz PHD this eqution hs een re(ned over time s more insight ws gined into the underlying physisF edditionl terms to e inluded to the e'etive (eld were proposed to ount for spei( phenomen @e.g. spinEorit torquesD hzyloshinskiiEworiy inteE rtionFFFAF sn its simplest formD two terms in)uene the evolution of mgneti moment in the presene of mgneti (eldF he (rst one is the torque used y tht exterE nl mgneti (eldD nd the other is phenomenologil term introduing dmpingD on the sis of the oservtion tht mgneti moment oviously does not preess forever round the diretion of e'etive (eld ut eventully ends up ligned with itF pigF IFII is shemti of typil dmped preessionF qilert9s work on this sujet led him to propose form of the dissiptive term SS!SUF reneD one either tlks out the vnduE vifshitz @vvA eqution or vnduEvifshitzEqilert @vvqA equtionF yther vrints exist for the tretment of dissiptionD inluding the soElled vvfr eqution @vnduEvifshitz frykhtr equtionA SVD SW nd generliztion of the qilert model to desrie more diverse dissiption phenomenD e.g. dry frition TH!TPF sn ny seD this eqution exhiE its interesting properties nd its tretment remins hllenging even in the simplest systems TQF he originl version of the eqution y vndu nd vifshitz in the presene of mgneti (eld H n e written s followsX

  ypil dmped preession of mgneti moment in the presene of mgneti (eld H eff F he trjetory of mgnetiztion is the dshed lue lineD with the terms of preession in red nd the dmping in lueF pigure from Public domainF here hs een sustntil dete on the reltive relevne of these two equtionsD in prtiulr whih dmping term should e used TT!UIF hey re equivlent when deling with onservtive torquesD nd while they my e eqully vlid in desriing rel phenomenD the 4est4 version would e tht whih orretly seprtes dmping nd preession termsF his ould in priniple e determined experimentlly with p mgneti (elds TRF st is not in the sope of this thesis to propose de(nitive nswer to this questionF roweverD primrily euse it is more widespredD we hose the vvq eqution @i.e. with qilert dmpingA in this work @see hpter QAF QU 1.2.3 Field-based motion he most studied nd simplest wy to indue domin wll motion is to use @usully sttiA mgneti (eld tht will energetilly fvor one of two domins of opposite mgnetiE ztionsF he di'erene of eemn energy resulting from the presene of the (eld will mke one domin expnd t the expense of the otherF he dynmis of this phenomenon hve een extensively studied theoretilly nd experimentlly over severl dedes in di'erent mterils nd geometriesF snitil work pulished in the IWTHs nd IWUHs pved the wy for future investigtions of (eldEindued domin wll motionD most notly the lndmrk ontriution of hryer nd lker UP!UTF sndeedD under simple pproximtionsD it ws shown tht n nlytil tretment of the vvq eqution in one dimension ould e rE ried outF he ehvior of moving domin wll in the presene of n externl mgneti (eld ould then e desried nd is quite intriguingF nder threshold (eldD the motion rehes stedyEstte regimeD with veloity whih n e determined nlytillyD s well s the mgnetiztion distriutionF sndeedD the vndu solution for the resting wll @shown in susetion IFIFRA is lso solution of the dynmil systemF en pproximtion of the trnsient regime hs lso een proposed UUF golletive oordintes pprohes introdued y hiele in the IWUHs UV lso desrie the sme stedyEstte dynmis nd hve een generlized nd re(ned over time UW!VIF rere is the solution of the dynmil system s provided in US @see pigF IFIPD for the geometry onsideredAX pigure IFIP ! gon(gurtion onsidered in lker9s originl pperD with ngles φ nd θF pigure from USF QV

  Description of the proposed system he funtionlity of wivew rests on the possiility to fvor energetilly one mgneE tiztion diretion over notherD whih is possile euse of the is mgneti (eldF sn oneEdomin nnomgnetD this n led to the ontrol of mgnetiztion nd mgnetiztion reversl s previously shownF he question we rise in this work is out the opportuniE ties o'ered y this mehnism in the ontext of twoEdomin mgnetoelsti elementD i.e. the e'et of the omintion of is mgneti (eld nd uniform stress on domin wllF pigure IFPT ! hemti of the generl system onsideredX twoEdomin mgnetoelsti ferromgnet with trnsverse domin wllF e therefore onsider unixil ferromgnet with mgnetoelsti properties elongted in the diretion of its esy xis @let us ll it xAD within whih the mgnetiztion is distriuted in two distint domins seprted y trnsverse domin wllD s shown in pigF IFPTF e is mgneti (eld is pplied perpendiulr to the esy xisD i.e. long the yExisF e uniform stress σ orresponding to the tension or ompression of the diretion de(ned y the vetor 1

(

  tool in xmg to tke it into ountF roweverD it n esily e integrted into the ritrry nisotropy de(ned y the userX s we hve shown in susetion IFQFPD the mgnetoelsti energy n e written s simple qudrti nisotropyF felow is n exerpt of the ython input (le to xmgD showing the de(nition of the ritrry nisotropyF (xx , m) ay = scalar_product (yy , m) az = scalar_product (zz , m) return -K_u * ax * * 2-3.0/2.0 * lambda_S * sigma * (ax * * 2+ay * * 2-ax * ay) he vrious prmeters hosen @nisotropy onstntD mgnetiztion t sturtion etA orrespond to the wellEknown mgnetoelsti mteril erfenolEhF hese will e detiled in the following hpterF e qilert dmping of HFI ws usedF rereD one needs to work with n initil sitution where domin wll is presentF his is firly strightforwrd with xmg sine the user n deide wht the initil distriution of mgnetiztion will eF ypillyD speifying two regions of opposite mgnetiztion @long + x nd -xA seprted y smll re of intermedite mgnetiztion @long ± yA nd llowing the system to relx will seure the desired outomeF enother option ked y experimentl dt PTR is to pply strong mgneti (eld long ± y nd then reduing it to smll vlueF yne the initil mgnetiztion is otinedD the dynmil study n egin with the pplition of stressF hedited routines help sve relevnt dt t (xed time intervlsF TP pigure IFQH ! he omintion of stti mgneti (eld in the y diretion nd in plne uniform stress is le to indue the motion of domin wll in preditle diretionF he (rst imge shows the distriution of w x t rest with stti mgneti (eldD the seond imge shows the distriution of w x I ns fter pplying onstnt negtive stressF rereD the nnostripe is P µmElongD IH nmEthik nd IHH nmEwideF isuliztion of the dt ws done using riew whih is n open soure softwre supporting intertive visuliztion of vtk (lesF he (elds tht re of primry interest here re the three omponents of mgnetiztionF he simultions showed tht the pplition of stress in the presene of trnsverse mgneti (eld does indeed result in domin wll motionF sn the se of onstnt setionD the domin wll moves towrd the end of the nnostripe until there is only one domin left @see pigF IFQHAF sn the se of proli pro(le of ross setionD the domin wll moves until it rehes n equilirium position @see pigF IFQIAF sf relxedD the domin will go k to its initil position in the middle of the simulted mgneti elementF roweverD nother ehvior is oserved if the mgneti (eld ndGor the pplied stress is too lrgeX the domin wll shift leftwrd or rightwrd until it rehes the extremity of the simulted mgnetD muh like in the se of onstnt setionF he intuitive interprettive of these two di'erent situtions is tht if the 4driving fore4 ehind the motion of the domin wll is too strongD it will go pst the exhngeErelted energy rrier formed y the geometryF his is ll tht n e sid here on this sujetD s these time onsuming preliminry studies were not performed to ddress ny other mtter thn the vlidity of the oneptF sndeedD sine xmg simultions work like lk ox to lrge extentD it is often di0ult to get n understnding of the physil phenomen involvedF sn order to etter grsp the underlying physis nd to ensure tht ll prmeters re within ontrolD simpli(ed models were uilt nd will e presented in the following hptersF TQ pigure IFQI ! ith proli pro(leD the domin wll n e on(ned in regionF ith the pplition of stressD the domin wll will reh n equilirium positionF 1.5 Conclusion sn this hpterD we sw tht domin wlls re prominent elements of modern reserh in res suh s miromgnetismD miroeletronis nd nnotehnologyF heir mnipulE tion is sujet of gret interestD most notly in reltion to their potentil in vrious pplitionsF hile domin wll motion nd ontrol of mgnetiztion is most often meE dited y mgneti (elds or eletri urrentsD there is wide interest in the serh for lterntivesF emong wht hs een put forwrdD stressEmedited rti(il mgnetoeleE tri mterils pper promisingF sndeedD the onveniene of n eletril ontrol"tht isD using voltge"over mgnetiztion is often highlightedF wore importntlyD the ultrElowE power requirements of suh systems mke them good ndidtes for future tehnologil nd ommeril developmentsF sn this ontextD the present thesis proposes new prdigmti system involving the pplition of uniform stress on twoEdomin mgnetoelsti nnostrutureF roweverD uniform stress lone nnot indue motion of IVH • domin wllsD hene the need to rek the symmetryF e wy of doing tht ws introdued y the eswexEpsvw group t siwx severl yers go in the ontext of mgnetiztion swithing of unixil mgnetoelsti mgnetsF st involves the pplition of stti mgneti (eld perpendiulr to the esy xisD whih enles susequent uniform stress to rete n energy gp etween the two TR stle sttesF sn twoEdomin ferromgnetD this would mount to domin wll motionD sine one domin will expnd t the expense of the otherF his is the ide proposed hereD nd it hs een shown how preliminry Qh simultions showed tht this ide seemed vlidF he following hpter will detil how we studied this phenomen with ad hoc theoretil we mde n innovtive proposl for domin wll motionF reliminry results sed on stndrd numeril simultions suggest tht the ide presented in the previous hpter might led to new wys of moving domin wllsF roweverD while miromgneti simultions re interesting in tht they re n esy wy to study omplex systems in Qh geometryD there re two signi(nt downsides to their systemti useF pirstD the omputtionl e'ort neessry to rry out those simultions is enormousD espeilly for timeEresolved omputtions PTSF eondD lthough these softwres re esy to hndleD there re neessrily used s lk oxes to some extentF he user usully spei(es inputs nd ollets outputs with only limited ontrol on the omputtion itselfF por these resonsD nother pproh sed on new numeril tool hs een devised nd will herey e presentedF ilored to mth the requirements of nnostripe geometryD this Ih model yields stisfying results with moderte omputtionl e'ortF woreoverD the totl ontrol over the lultion helps understnd the underlying physil mehnisms t plyF he work presented in this hpter ws the sujet of two pulitions PTTD PTUF 2.1 Variational approach to micromagnetism sn this setion we present rief introdution of the lssil equtions derived from vritionl pproh ID IRUD IVPF hile the following development does not introdue nything tht hs not een previously reportedD it is n ttempt to present proper syntheti derivtion of frown9s eqution from mthemtil point of viewF sn prtiulrD n emphsis is put on the requirements of the vritionl proedure regrding the mgneti

  where H d diretly depends on w ( r)F es (rst stepD the hoie is mde of joining demgnetiztion nd externl (eld in the sme eemn energy term to etter introdue neessry disussion proposed in the next susetionF e n (nlly introdue the totl energy stored within the region

  2.2.3 Derivation of the main equatione follow the sme pproh s in susetion PFIFQ to derive the eqution tht the mgE netiztion distriution should stisfy t equiliriumF rereD we will expliitly give the developments tht led to the desired equtionF vet us onsider U s funtionl of Φ(x)F e n pply the qâteux derivtiveD y introduing perturtion funtion k nd onstnt β s follows

  function f = f1 ( r ) f=(Phi( i+1)-2 * r+Phi( i -1))/ dx^2-1/(2 * A) * (-mu0 * H0 * Ms * cos( r ) +2 * K_u * cos( r ) * sin ( r)+3/2 * lambda_s * sigma * cos(2 * r )) +ld ( i )/ lx ( i ) * ( Phi( i+1)-Phi( i -1))/(2 * dx) +1/(2 * A) * mu0 * Ms * (-sin ( r ) * Hdx+cos( r ) * Hdy); end % bisection procedure i f f1 (a) * f1 (b)>0 disp ( 'Problem : f (a) * f (b)>0 ' ) e l s e c = (a + b)/2; err = 1; while err > 1e-15 i f f1 (a) * f1 (c)<0 b = c ; e l s e a = c ; end err = abs (a-b ); c = (a + b)/2; end end he only nonEtrivil term to ompute is the demgnetiztion ontriutionF sndeedD ltE hough tools hve een proposed PURD PUS for the purpose of simplifying the omputtion of the demgnetiztionD it often remins the lest strightforwrd prtF he numeril determintion of the ext demgnetizing (eld is hieved through sheme explined

  VU lssil expressions otined onsidering L → +∞ PH lredy disussed in hpter 1Fhis solution is now used to hek the numeril tehnique y diret omprisonF e onsider uniform nnostripe with onstnt thikness h = 20 nmD onstnt width = 40 nm nd length L = 400 nmF es one n see in pigF PFUD the numeril proedure orretly onverges towrd the expeted solutionD from n initil liner guess etween two oundry onditions @hereD 0 nd πAF sn the inset of pigF PFU we hve lso shown the reltive error δ @the di'erene in norm etween steps m nd m + 1A whih pprohes zeroD proving the onvergene of the sequene PUUF he stopping riterion of the itertive proess is sed on threshold vlue of δD referred to s δ min @here we used δ min = 10 -20 AF

(

  xed oundry onditionF en exmple of suh n 4ejetion4 is shown in pigF PFIH where we onsidered uniform nnostripe with h = 20 nmD = 40 nm nd L = 400 nmF e dopted the oundry onditions desried y iqF @PFRPAF he symmetril urve etween -L/2 nd +L/2 represents the solution with σ = 0D used s initil guess for the itertive proedure when σ = -100 wF ell the dshed lines represent itertions of the relxtion methodF e plotted urve every IHHH itertions to etter show the evolution of the proess nd we used δ min = 10 -10 for the stopping riterionF he domin wll rrives t the right extremityD eliminting the seond mgneti dominF sn the inset we lso show the reltive error δ versus the itertion numerF e n identify two relxtion regimesX in the (rst one the reltive error dereses nd the mgnetiztion distriution ssumes the orret shpe @e.g.D in terms of the vlues in the plteu regions s well s the slope of the domin wllAY in the seond one the reltive error is quite onstnt nd the wll moves until ejetionD orresponding to the (nl error dropF edditionllyD we n WH pigure PFIH ! stertive proess showing the ejetion of the domin wll in ferromgnet with onstnt retngulr setion @h = 20 nm nd = 40 nmAF sn the inset the reltive error δ is plotted versus the itertions numerF e used δ min = 10 -10 nd σ = -100 wF notie tht the gp etween the plteu vlues nd oundry onditions @due to the e'et of demgnetiztionA is onsistent with wht ws found in pigF PFR in tht regrdF elthough there my e oinidentl reltionship etween the speed t whih the equilirium distriution is rehed nd the intensity of the stress ppliedD the ode is not le to provide ny informtion out motion dynmisD whih will e overed in hpter QF feuse this relxtion proedure is only mde to (nd the mgnetiztion distriution of the equilirium stteD the intermedite mgnetiztion distriutions s the simultions proeeds hve no physil relityF es resultD simultions suh s those with onstnt setion"whih inevitly led to the ejetion of domin wll"re of limited interest prt from showing tht the numeril proedure devised to solve the eqution pper to work nd tht its results support the onlusions of preliminry investigtions of hpter IF he end result will lwys e singleEdomin on(gurtion in this seF roweverD the possiility to introdue vrile width in the model pves the wy for interesting nlysesF es desried in hpter ID n inresed width t the extremities retes rrier for the domin wllF ith n pprioprite shpeD it is therefore possile WI to mnipulte on(ned domin wllF sn the rest of this hpterD we fous on the study of on(ned domin wllsF x (nm) stertive proess showing the pth towrd the equilirium position of the domin wll for hourglssEshped nnostruture with entrl width (0) = 40 nm nd width t extremities (±L/2) = 70 nm @h = 20 nmAF sn the inset the reltive error δ is plotted versus the itertions numer @log sleAF e used δ min = 10 -10 nd σ = -100 wF sn pigF PFII we oserve n equilirium position otined in nnostruture with vrile width @prolEshpedAF e onsidered nnostruture with thikness h = 20 nm nd width (x) = (0) + 4 (±L/2)-(0) L 2

  homin wll displement t equilirium x ∞ DW s funtion of the pplied stress for three di'erent proli pro(les 1

  he mgnetiztion distriutions orresponding to di'erent vlues of the pplied stress re shown in pigF PFIQ together with the relted energy density distriutions within the nnostrutureF hese results hve een otined with the pro(le 1 (x)F he (rst plot dels with the mgnetiztion distriutionsD with solid lines orresponding to solutions otined WR with the oundry onditions in iqF @PFRPA nd dshed lines to the oundry onditions Φ(-L/2) = 0 nd Φ(+L/2) = πF e n oserve tht the oundry onditions do not in)uene the overll ehvior of the domin wll in drmti fshionD lthough the e'et is notieleD espeilly for lrge stress vluesF he sene of signi(nt e'et will e espeilly true for long stripes with two lrge nd lerly de(ned dominsF he di'erenes n indeed e only oserved in the regions lose to the extremitiesD nd those fr from them re hrdly 'etedF he seond plot shows the orresponding energy density pro(les u(x) de(ned in susetion PFPFID suh tht U = +L/2

  hemti of mgnetoelsti nnostruture @wiA mehnilly oupled to piezoeletri sustrte @iAF feuse of these piezoeletri propertiesD the stress tensor pplied to the mgnetoeE lsti nnostruture will not e s simple s wht ws onsidered eforeF vet us reE WW write the mgnetoelsti term so tht this is orretly tken into ountF es in suE setion PFPFQD the generl form of the energy density desriing the mgnetoelsti inteE rtion is u me = -T ij µ ij F sts mthemtil form is µ ij = λs 2 (3w i w jδ ij )F sf we onsider @HIIAEut wxE sustrteD we hve T x x = τ nd T y y = σF hereforeD we otin u me = -3 2 λ s (σ -τ )w 2 y +const. xowD within the piezoeletri mteril we suppose tht the reltionship etween the strins nd the eletri (eld is linerF edmittedlyD while most piezoeletri mterils do exhiit region where this ssumption is vlidD nonliner ehE vior should e expeted in experimentl works nd is reported in hpter RF sn ny eD s result of this hypothesisD one n write 11 = d 31 E 0 nd 22 = d 32 E 0 F sn wxE ermi sustrteD d 32 = 600 pgGx nd d 31 = -1900 pgGx PUVF yn the other hndD in the mgnetoelsti mteril we hve T ij = 2µ ij + λ kk D where λ nd µ re the vmé oe0ients @the strin within the sustrte is supposed to e ompletely trnsmitted to the mgnetoelsti lyer whih is very thinAF fy onsidering T 33 = 0 nd 11 nd 22 imposed y the piezoeletri sustrteD we esily otin T 11 = τ = 2µ 2(λ+µ)d 31 +λd 32 2µ+λ E 0 nd T 22 = σ = 2µ 2(λ+µ)d 32 +λd 31 2µ+λ E 0 F he orresponding energy density is therefore given y u me = -3λ s µE 0 (d 32d 31 )w 2 y + const. he rotted frme de(ned y the diretions noted I nd P is identil to wht ws used in susetion PFPFIF hereforeD we must sustitute w 2 y = sin 2 (Φ -π 4 ) =sin Φ cos Φ + const.D to eventully otin u me = 3λ s µE 0 (d 32d 31 ) sin Φ cos Φ. @PFSPA snsted of deling diretly with the pplied stressD we hve the eletri (eld whih genertes two orthogonl stress omponents of opposites signsD σ nd τ F his is linked to the ft tht d 31 nd d 32 re negtive nd positiveD respetivelyF e n now see the potentil of suh sustrte for tehnologil pplitionsF rereD while tensile stress fvors one dimensionD n orthogonl ompressive stress fvors plne ontining this dimensionF ith this modi(tionD iqF @PFRHA eomesX

  Φ H dy y,z = 0. @PFSQA st is ler from this eqution tht stress σ = στ would then hve the ext sme e'et on the stti ehvior of the systemF por this reson nd euse of the liner hypothesisD it should ome s no surprise tht the reltionship etween eletri (eld nd IHH pigure PFIU ! wgnetiztion ngle distriutions θ(x) within the nnostruture for (xed mgneti (eld @H 0 = 20×10 3 eGmA nd severl vlues of eletri (eld @IU vlues uniformly distriuted in -3.5 wGm ≤ E 0 ≤ +1.1 wGmAF e oserve tht the domin wll moves to the left for E 0 > 0 nd to the right for E 0 < 0F displement is the sme s tht etween pplied stress nd displement lredy otined nd shown in pigF PFIPF por instneD pigF PFIU shows the ngle Φ of the mgnetiztion distriution t equilirium for severl vlues of pplied the eletri (eld pplied to the systemF his pprohD sed on the eletri (eld nd wxE sustrte is dopted in the work presented in hpter QF sntroduing piezoeletri quntities llows the estimte of the energy onsumption ssoited with the proposed tehniqueF e n then drw omprison with other methods sed on the pplition of n externl eletri urrent nd show tht stressEsed systems re promising from n energeti perspetiveD s stted in hpter IF he totl energy for moving the domin wll etween two positions is the sum of the eletrostti energy stored within the piezoeletri sustrte @CV 2 A nd the mgneti energy dissipted within the mgnetoelsti nnostrutureF xeverthelessD the ltter is typilly negligile nd the totl energy is pproximtely equl to the eletrostti ontriution PSPD PSUF sf we onsider piezoeletri wxE sustrte @length L = 400 nmD width b = 80 nmD nd thikness d = 1 µmA with reltive dieletri onstnt ε r = 5500 for the struture desried y 1 (x)D we otin n energy ∆E = ε r ε 0 LbdE 2 0 = 1.5 ft = 3.6 × 10 5 k B T for moving the IHI wll etween the mximum vlues of ±x ∞ DW @we used E 0 = ±10 6 Gm generted y n eletri potentil di'erene of ±1 AF por omprisonD we n ite the energy dissiption ∆E = 10 4 ft for one logi opertion in urrentEdriven gte sed on the domin wll motion RHF sn this se we numerilly proved tht the mgneti dissiption @ 10 -17 tA is two order of mgnitude smller thn the eletrostti ontriution @ 10 -15 tAF 2.5 Conclusion sn this hpterD we presented omplete model desriing the stti ehvior of mgneE tiztion in mgnetoelsti nnostrutures with vrile width sujeted to n externl mgneti (eld nd mehnil stressD using n ad hoc numeril proedureF sn prtiulrD we wrote the eqution governing the mgnetiztion distriution y tE king into ount nisotropyD exhngeD demgnetiztionD mgnetostrition nd eemn e'etsF he pproh hosen onsisted in writing the totl energy of the system in terms of the diretion of the mgnetiztion nd then in pplying the tehniques of the lulus of vritions in order to minimize the totl energy itselfF he end result is nonliner integroEdi'erentil prtil eqution whih solution orresponds to the equilirium stte of the mgneti systemF o solve this equtionD we proposed n e0ient numeril tehniE que in order to solve the min nonliner integroEdi'erentil eqution with relxtionD or itertiveD methodF e studied the onvergene properties of this lgorithm y pplying it to prtiulr seD whih n e hndled nlytillyF ith this onvenient toolD we demonstrted the possiility to indue domin wll motionX in onstnt setion nnosE tripeD pplying uniform stress in time leds to the ejetion of the domin wll so tht the mgnetiztion distriution eomes uniformF ith n pproprite design of the ross setion pro(leD we lso demonstrted tht it is possile to ontrol mgneti domin wll position in ferromgneti nnostruture with externl mehnil tionsF e showed tht the origin of motion lies in the gp in eemn energy of the two domins indued y n e'etive nisotropy (eld modulted y the stressF his stress n e generted in multiferroi heterostrutures @omposed of piezoeletri nd mgnetoelsti susystemsAF yne of the key dvntges of suh tehnique of domin wll motion is its relevne from the energeti point of viewF IHP Chapter 3 Dynamics of stress-induced domain wall motion sn the previous hpter we presented model whih gives the equilirium mgneti onE (gurtion of Ih nnomgnet sujeted to mgneti (eld nd mehnil tionF roweverD ll intermedite mgnetiztion distriutions tken from these simultions do not possess ny physil relity nor do they represent tul mgnetiztion sttes of the nnomgnetF his is euse the pth followed y the system s the simultion is rE ried out nnot e equted with temporl evolutionF por this resonD it is interesting to onsider the possiility to devise model whih inludes the vnduEvifshitzEqilert eqution tht governs mgnetiztion dynmisF his is the sujet mtter of this hpterD whih inludes the method followed to (nd the dequte equtionD the numeril pproE h hosen to solve itD s well s the min resultsF smportnt physil insights pertining to stressEindued mgnetiztion dynmis re lso presentedF he work of this hpter is the sujet of one pulition PUWF 3.1 Equations governing domain wall dynamics his setion ontins the pth towrd (nding the right dynmil eqution for the prolem we hose to studyF efter the hoie of nottion re explined nd energy ontriutions de(nedD the expression of the e'etive (eld is soughtF prom thereD the orresponding vnduEvifshitzEqilert eqution is writtenF 3.1.1 Overview he logi ehind the pproh is similr to wht ws desried in hpter PD in tht vritionl proedure is followed to otin the e'etive (eldF roweverD this is not the IHQ pigure QFI ! hemti of the system studiedF he eletri (eld pplied long z genertes in the piezoeletri sustrte stress distriution with two omponents σ nd τ of opposite signF end result s the eqution w × H eff = 0 only desries the equilirium stteF rereD the expression of the e'etive (eld hs to e tken s n input to the vnduEvifshitzEqilert @vvqA eqution tht governs mgnetiztion dynmisF he version of the eqution used is s follows

  quntities depend only on xF yne importnt hnge with respet to the previous study is tht it is now ompulsory to onsider mgnetiztion (eld whih diretion is not ound to the xyEplneD de(ning new ngle Θ ∈ [0, π] llowing the representtion of outEofEplne mgnetiztion omponents @see pigF QFIAF rowever the pproximtion stting tht M depends only on x is preservedF es resultD one n write w = (cos Φ sin Θ, sin Φ sin Θ, cos Θ)F yviouslyD these two ngulr vriles Φ nd Θ now depend on x nd tF e minimiztion proedure similr to the previous exmple hs to e rried outF roweverD the vrile of interest in the funtion will e w = (w x , w y , w z ) insted of ΦF sn IHR this ontext it is thus useful to rewrite ll ontriutions in terms of this vrileD with more spei( forms for eh term in reltion to the on(gurtion studiedF elsoD we onsidered the mehnil oupling of mgnetoelsti lyer with wxED lredy mentioned in hpter PF 3.1.2 Denition of the energy functional everl of the energy terms re quite strightforwrd to write from prior expressionsF he unixil nisotropy is simply u an = -K u w 2x D while the eemn ontriution eE omes u Ze = -µ 0 M s H 0 w y F he exhnge energy n e written u ex = A ∂ minor dpttion hs to e mde for the developE ment of the mgnetoelsti energyF sndeedD s w is not restrited to the xyEplneD the expression is slightly di'erentX

  dt * gamma0/(1+alpha^2) * dsrphim( f l o o r ( i /2)); % even indices : dot theta equations J( i +1, i+1)=1-dt * gamma0/(1+alpha^2) * drstheta ( f l o o r ( i /2)); J( i +1, i+2)=-dt * gamma0/(1+alpha^2) * drsphip ( f l o o r ( i /2)); J( i +1, i+3)=-dt * gamma0/(1+alpha^2) * drsthetap ( f l o o r ( i /2)); J( i +1, i)=-dt * gamma0/(1+alpha^2) * drsphi ( f l o o r ( i /2)); J( i +1,i-1)=-dt * gamma0/(1+alpha^2) * drsthetam ( f l o o r ( i /2)); J( i +1,i-2)=-dt * gamma0/(1+alpha^2) * drsphim( f l o o r ( i /2)); end J(2 * Npoints -1 ,2 * Npoints -1)=1; J(2 * Npoints ,2 * Npoints)=1; Npoints-3)=phi_old (2: Npoints-1)+dt * gamma0/(1+alpha ^2)...

  ellipsoid we hve N x = 0AF he resulting rekdown (eld is 24.75 × 10 3 eGmF IIR H 1 (10 3 A/m) hiret omprison of the output of the ode with lker9s nlytil soluE tion in the se of (eldEdriven motionF he greement is very good for oth vlues of nisotropyF M s is hnged here ompred to other simultions nd is equl to 5×10 5 eGmF es visile in pigF QFPD there is n exellent greement for oth vlues of K u etween the numeril nd nlytil veloitiesF his gives on(dene in the pity of this numeril proedure to orretly represent the dynmil trjetories of domin wllsF roweverD it should e noted tht the ode systemtilly rshes for H 1 eomes superior to H c = 24.75 eGmD even only slightly superiorF his indites tht there is fundmentl hnge in the dynmis eyond the lker rekdown whih nnot e ounted for y our numeril implementtionF3.3 Dynamics in hourglass geometryhis setion dels with the results otined onerning the dynmis of the system with vrile width studied in hpter PF he trjetory nd shpe of domin wlls in suh strutures re exploredF henD the dependene on the eletri (eld of domin wll veloity is studied nd disussedF elsoD the symmetry tht ppers in the dynmis with respet to the eletri (eld is explined in terms of the lne etween ompressive nd tensile omponents of the orresponding pplied stressF IIS 3.3.1 Shape of domain walls pigure QFQ ! hemti of the proli geometry onsidered in the dynmil studyF fefore getting to the nlysis of the dynmisD it is ressuring to note tht the equiliE rium mgnetiztion distriution given y this dynmil ode is the sme s the relxE tion ode for given vlue of eletri (eldF his ft lso strongly sustntites hypothesis

  the outEofEplne exursions re omprtively reduedD nd domin wll motion is fiE littedF sn order to test this hypothesis furtherD n investigtion ws rried out with di'erent piezoeletri prmetersF he originl oe0ients were d 31 = -1900 pgGx nd d 32 = 600 pgGxD orresponding to @HIIAEut wxE ermi sustrte PUVF pour dE ditionl sets of inputs were hosenD keeping the sum |d 31 | + |d 32 | onstntX one ompletely symmetril se with d 31 = -2000 pgGx nd d 32 = 500 @suh tht σ = 0AD one omE pletely symmetril se with d 31 = -d 32 = -1250 pgGx @suh tht σ = -τ AD s well s two intermedite on(gurtionsF esults re shown in pigF QFUF yne importnt ft is tht ll urves oinide in the liner portion @for low vlues of E 0 AF henD it is ler tht the lrger the gp etween the mplitude of the stress omponentsD the lrger the symE metry in the dynmis for lrge vlues of E 0 F sn prtiulrD totl symmetry is otined when |d 32 | = |d 31 |F st is lso pprent tht the wxE sustrte is lose to the fully symmetril piezoeletri mterilD so tht in ft |τ | is lwys muh lrger thn |σ|F sn the endD these results lend more support to the interprettion sed on the reltive importne of mehnil ompression nd tension in the dynmisF IPH E 0 (MV/m) pC/N d 31 = -1250 pC/N pigure QFU ! eloity s funtion of the eletri (eld for di'erent vlues of the piezeoeleE tri oe0ients d 31 nd d 32 F his enles to explore the vrious degrees of symmetry in the dynmis indued y the reltive in)uene of ompressive nd tensile stress omponentsF 3.4 Dynamics in an innite nanostripe sn the previous setion we explored the dynmis in geometry with vrile widthF enother interesting se from fundmentl point of view is the (lm with uniform widthF sndeedD if the e'ets of vrile setion re removedD one n fous on the spei( fetures of stressEindued domin wll motionF his is espeilly the se in nnostripe geometryD whih is hrterized y high spetErtios nd thus vnishing edge e'etsF his setion presents the wy we simulted the evolution of domin wll in in(nite nnostripesF sn these onditionsD the study of stedyEsttes is oneivle nd is of gret interestF sn prtiulrD the reltionship etween domin wll veloity nd pplied eletri nd mgneti (eld is mtter tht demnds investigtionF e disussion on the instnes where the ode rshes loses the setionF 3.4.1 Simulating innite nanostripes he study of n in(nitely long nnostripe with onstnt width"s in pigF QFV"is inteE resting from the physil point of viewF gonsidering suh geometry in iqF @QFITA n e done y simplifying demgnetiztion @to model n in(nitely long ellipsoid of ompE IPI rle dimensionsA nd removing terms relted to the vrile widthF his ws lredy done in susetion QFPFPF nfortuntelyD even then the eqution otined is still suh tht no nlytil investigtion seems possileF eginD one must rely only on numeril tretmentF pigure QFV ! hemti of the onstnt setion geometry onsidered in the dynmil studyF yviouslyD the numeril pproh entils the use of (nite geometriesF roweverD we n devise wy to simulte n in(nite nnostripe with the existing odeF sf we onsider rel system with in(nite lengthD it is sfe to ssume thtD fr from the domin wllD mgnetiztion will e (xed nd equl to the stle mgnetiztion sttes @inluding the in)uene of demgnetiztionAF herefore we n fous on omputing the evolution of mgnetiztion in region round the domin wllD nd onsider the ontriution to the demgnetizing (eld of two lrge prllelepipeds eyond ± L 2 F sn prtieD this ws done y simulting the demgnetizing (eld generted y two regions of I mm length on eh side in the lultion of the demgnetiztion ftorsF rving done thisD it is suitle to dpt the oundry onditions Φ 1 nd Φ 2 whihD until nowD were lulted only y tking into ount nisotropyD externl mgneti (eld nd pplied stressF he gol is to (nd oundry onditions tht losely mth the vlue of the lerly identi(ed plteu regionsF o ompute the stle sttesD we del with uniform mgnetiztion nd thus the simpli(ed demgnetiztion H d = -N M @s in susetion QFPFPA is suitleF he pproprite terms re introdued in the lultion of the oundry onditionsD giving modi(ed version of iqF @PFRPAX IPP

  pigure QFIQ ! hependene of the veloity of the domin wll on the mgneti (eldD for severl eletri (elds nd dmping oe0ientsF snsets represent the resling of the xExisD showing universl response t low (eldsF he width of the nnostripe is RH nmF e n onlude tht the mehnilly indued stedyEstte regime yields propgtion veloities lrger thn SHH mGsD whih re omprle to those otined y urrentEdriven domin wll motion IPVD IPWF sn the lssil (eldEindued lker propgtion the dependene of the domin wll veloity on H 0 nd α is medited y the single vrile H 0 /α @see iqF @IFWAAF reneD we plot v DW versus the rtio H 0 /α in the insets of pigF QFIQF e oserve tht the urves orresponding to di'erent α ollpse to single universl response in the liner regionF e lso note tht ∂v DW /∂ρ @for low vlues of ρ = H 0 /αA is n inresing funtion of E 0 D whih is not surprisingF roweverD for higher vlues of the mgneti (eldD v DW depends on oth H 0 /α nd α nd the urves do not oinideD proving one gin the essentil di'erene etween the mehnillyEindued nd the (eldEindued domin wll dynmisF o get more systemti view of the omined in)uene of the mgneti nd eletri (elds on the veloity of the domin wllD Ph sn ws performed on @H 0 DE 0 A gridD for ll four dmping oe0ients α i F sn order to remove the neessity to ompute the dynmis for positive nd negtive vlues of E 0 D symmetri piezoeletri sustrte ws simulted @d 31 = -d 32 so tht σ = -τ AF esults for lrge dmping re shown in pigF QFIRD where the veloity is plotted s surfe over the @H 0 DE 0 A gridF esults for width of RH nm IPV E 0 (MV/m) Qh representtion of the veloity of the domin wlls funtion of the mgneti nd eletri (elds for α = α 4 = 0.12F he width of the nnostripe is IHH nmF re very similr qulittivelyD lthough the vlues of veloity otined re signi(ntly smllerF et low mgneti (eldsD the vrition of the eletri (eld hs miniml in)uene on the veloityF et higher (eldsD the grdients re muh higherF e n on(rm here tht the vlue of the mgneti (eld hs huge impt on the veloity rehed y the domin wllF his oservtion"omined with the remrk tht the energy density gp etween the two domins is equl to the eemn gp"relly vindites the view tht the motion of the domin wll is stress-triggered nd eld-drivenF he ft tht the surfe does not over the whole grid is due to the filure of the ode in some instnes @depending on the vlues of the (elds used s inputsAF st is lso the reson why some of the urves in previous (gures stopped t some pointF his phenomenon is disussed in the following susetionF 3.4.4 Code failure and breakdown phenomenon sn some instnesD the dynmil ode fils to desrie the stedyEstte orresponding to the inputs givenF hese rshes lso hppen with the vrile widthD s the ode fils IPW to desrie the motion of the domin wll towrd its equilirium positionF xo solution ould e found to prevent this from hppening @fourfold inrese of sptil disretiztionD s well s the redution of time step ∆t hve een triedAF pigF QFIS shows the re where the ode runs normlly @olored regionsA nd the limits fter whih there is rshF he di'erent olors refer to the di'erent dmping oe0ientsF pigure QFIS ! golored res orrespond to instnes where the ode suessfully desries the stedyEstte ssoited with the orresponding inputsF he surfes for the four vlues of α re superimposedF e lredy enountered limittions of the numeril proedure in lssil (eldEdriven motion @see susetion QFPFPAD where the ode ws unle to rry out the simultion for longitudinl mgneti (eld H 1 lrger thn the lker rekdown (eldF he similrity in the symptoms re intriguingF elthough it would still e speultion to sy tht this numeril phenomenon is n tul sign of rel physil rekdown in the system stuE diedD we hve to knowledge the hypothesis is quite proleF por strtD it ers mny of its pprent feturesF he errors tht umulte nd eventully led to the filure re reminisent of wht is oserved eyond the lker (eld in the ontext of regulr (eldEdriven motionF elsoD there re smll signs of osilltory ehvior whih ould e ssoited with the onset of new regime of motionF he ft tht the filure hppens t lower (elds when the dmping is wek is lso quite tellingF es n sideD the numeril IQH filure hppens irrespetively of the tul vlue of the veloityD nd so it is not due to lrge rte of mgnetiztion vritionF fesidesD we n rgue on physil grounds tht the existene of phenomenon similr to the lssil lker rekdown should e expeted hereF sn hpter ID we sw tht the rekdown ws quite universlD sine it is present in oth (eldE nd urrentEdriven motionF rereD we del with motion tht is essentiE lly (eldEdrivenD this implies tht the existene of sudden trnsition in the dynmil ehvior is plusileF sn ny seD in the sene of nlytil resultsD more inEdepth nlysis of the oserved phenomenon is needed in order to rule out the hypothesis of numeril rteftF3.5 Conclusionsn this hpterD we numerilly studied the motion of domin wlls sujeted to the omE intion of is mgneti (eld nd n pplied stress in mgnetoelsti nnostruturesF e devised Ih model omining the e'etive (eld derived from vritionl methods with the vnduEvifshitzEqilert equtionF he outome ws implemented numerilly using impliit sheme sed on xewton9s methodF hrough the exmple of the proE li geometryD we explored how the tiloring of the vrile width llows one to preisely design stti nd dynmi feturesF he study of stedyEstte regimes in nnostripes of onstnt width on(rmed tht even though it is ultimtely driven y mgneti (eldD the resulting moving mgneti struture is fundmentlly di'erent from usul domin wlls in nnostripesF sn prtiulrD the motion is hrterized y spei( outEofEplne pheE nomen whih do not exist in (eldEdriven domin wll motionF he in)uene of severl prmeters on the extent of these outEofEplne exursions hs een reviewedF fesidesD this new type of domin wll motion is ssoited with ompetitive veloities @hundreds of mGsAF he numeril studyD s well s physil onsidertions lso rought severl hints tht phenomenon kin to lker rekdown my e involved in the dynmisD lthough more reserh is needed to settle this questionF wny lines of investigtion remin open in the study of suh systemF por instneD it is possile to look t dynmil input (elds @vrile E 0 (t) or H 0 (t) signlsAF elsoD new types of geometries n e onsideredD suh s elements with two minim or moreD exhiiting hystereti ehviorF homin wll pinning s well s edge roughness or ptterE ning my lso in)uene the dynmis of this system s it is the se for stndrd (strong theoretil nd numeril sis to stressEtriggered domE in wll motionD it is worth investigting this phenomenon experimentllyF his hpter inludes desription of the di'erent mterils onsidered for the mgnetoelsti nd piezoeletri phsesF st lso gives n ount of the preprtion in the len room of thin (lms mde of mgnetoelsti mterils with pproprite size nd geometryD s well s the preprtion of suitle piezoeletri sustrtesF pinllyD the tehniques used to hrteE rize oth phsesD the orresponding results nd some of the prospets re disussedF 4.1 Materials his setion overs the hrteristis of the mterils hosen in the implementtion of the mgnetoeletri oupling involved in the physil phenomenon investigtedF yn one hndD teriumEsed lloys re rrnged in multilyers to onstitute the mgnetoelsti phseF yn the other hndD ommeril wxE sustrtes with spei( growth orienttion onstitutes the piezoeletri phseF 4.1.1 Magnetoelastic multilayers he eswexEpsvw group t siwx hs extensive experiene with the use of rreE erthEsed mgnetoelsti mterilsF sn prtiulrD pe x @erfenolA nd go x lloys @x ≈ 2A re routinely used nd exhiit strong mgnetoelsti properties @positive mgE netostritionAF hey re usully mde y sputteringF roweverD euse of the presene of teriumD the nisotropy in the resulting lyer is often very lrge PVID hindering the mnipultion of mgnetiztionF fesidesD in the se of go 2 D the vlue of the stuE rtion is quite wek euse of the ferrimgneti orderF o redue the nisotropy nd gurntee sustntil mgnetiztion t sturtionD these lloys re ompnied y IQP lyers of ferromgneti lloys sed on metls suh s pe nd go in multilyered strutuE res @see pigF RFIAF sndividul deposited lyers re very thin @typilly less thn IH nmAF he mgneti oupling of suh strutures n e studiedD with key insights thtD for inE stneD led to the design of exhngeEspring mgnets PVPF he mgneti ehvior n e omplited @with di'erent lyers swithing t di'erent pplied (eldsAF sn previous hptersD we ssumed tht they ehved like single equivlent homogeneous mterilD whih is likely to e the se with very thin nd thus strongly oupled lyersF uh mixing llows stisfying ompromise etween nisotropyD mgnetiztion t sturtion nd mgnetostritionD to ensure filitted mnipultion nd interesting performnesF pigure RFI ! rnsmission iletron wirosope imge showing IIH nmEthik deposition of multilyers otined t siwxF yne of the interesting fetures of this multilyered mteril is the possiility to ontrol the diretion nd"to some extent"the strength of the nisotropy PVQF sn ny seD the lyers otined y sputtering re typilly morphousF his is ssoited with redued nisotropyD lredy smller in thin (lms thn in ulk mterilsD whih is helpful in our seF hespite the morphous struture nd the thin (lm geometryD signi(nt mgnetostrition is hievedF IQQ 4.1.2 PMN-PT piezoelectric substrate iezoeletri mterils hve found multiple pplitionsD for instne in sensing nd tuE tionF his led to signi(nt dvnes in the preprtion of highly e0ient piezoeletri mterilsF sn prtiulrD perovskite ermis suh s led zironte titnte @A hve drwn tremendous interestF sn our seD we need to (nd piezoeletri sustrte geneE rting strong inEplne strinsF por them to indue mgnetoelsti nisotropy in the ferromgneti mterilD the two inEplne diretions should not e hrterized y equl

  for wide rnge of exittion frequenies PVS!PVWF elsoD the properties of single PWH nd multidomin PWI wxE hve een studied nd ompredF he very strong pieE zoeletri properties exhiited y wxE re est understood s onsequene of the morphotropi phse oundry @wfA whih hppens for given ompositions of the solid IQR solutionF he origin of this phenomenon is the sujet of more generl investigtions PWPF sn prtiulrD it seems tht round the wfD di'erent phses n simultneously e preE sentD nmely rhomohedrlD monoliniD orthorhomiD tetrgonl phses PVRF e firly omprehensive set of mesurements ws rried out PUV nd shows the promising poE tentil of @HIIAEut wxE in the ontext of the mgnetoeletri oupling sought hereF sndeedD the piezoeletri oe0ients d 32 nd d 31 re of opposite signs nd therefore sustrte n generte simultneous ompressive nd tensile stress omponents long ortE hogonl diretionsF sn dditionD its hystereti ehvior is suh tht it is possile to ene(t from two distint permnent strin sttesF his point is ddressed in greter detil in susetion RFQFIF yne should mention tht while ommeril ulk wxE is redily ville in the form of singleErystl sustrtesD reserh hs lso shown the possiility to mnufture wxE thin (lms PWQ!PWTF 4.1.3 Mechanical coupling pigure RFQ ! wo di'erent strtegies"@A longitudinl nd @A trnsversl"for the loE tion of eletrodes reting the eletri (eld within the piezoeletri sustrteF es shown in previous hptersD the mgnetoelsti nd piezoeletri phses hve to e oupled mehnillyF hile mgnetoelsti prtile emedded in piezoeletri mtrix likely represents the most e0ient mehnil ouplingD its tehnologil implementtion is not strightforwrdF e more fesile lterntive is to deposit the mgnetoelsti lyer on the piezoeletri mterilF es for the positions of the eletrodes tuting the piezoeletri sustrteD there re two ovious possiilitiesF he (rst solution is to ple the eletrodes next to the mgnetoelsti elementD so tht the eletri (eld is inEplne @see imge @A of IQS pigF RFQAF sn the ontext of ulk sustrtesD the min dvntge of suh n rrngement is the proximity of the two eletrodesD mening weker voltges re needed to indue the desired eletri (eld within the piezoeletri mterilF elthough this option is sometimes onsideredD there is mjor issue with regrd to the pth of the eletri (eldF ine the multilyer in our se is metlli while the piezeoeletri sustrte is dieletriD there is n inentive for the eletri (eld to go through the ferromgneti metl insted of the sustrte eneth itF here is s result signi(nt risk of hving wek strinF he other option is therefore to hve eletrodes on oth sides of the piezoeletri sustrteD reting n eletri (eld long the vertil diretion @see imge @A of pigF RFQA long the whole thikness of the sustrteF pigure RFR ! hird option for the rrngement of the eletrodes involving some struturing @ethingA of the piezoeletri sustrteF enother issue reltes to the possiility of the piezoeletri sustrte to e free of tully generting the strinD espeilly in the ontext of strong piezeoeletri responseF sf eletri (eld lines re on(ned to smll portion of piezoeletri mteril surrounded y the gist of the sustrte where the eletri (eld is zeroD then there is risk tht it will prevent the retion of the expeted strinF orseD where strins re negtive it n led to frtures nd mehnil dmge of the mterilF e solution to this prolem is to ese the portion of wxE y engineering the surfe of the sustrte to rete prllelepipedi studs @for instne with squre ross setionAD s illustrted on pigF RFRF xote tht this solution is lso good ide in the prospet of integrtion nd lowEpower opertionD s it onsiderly redues the eletrostti energy CV 2 tht is proportionl to the volume of dieletri involvedF his will e explored in susetion RFRFQF sn the followingD we dopted solution @A of pigF RFQD s onerns of integrtion nd energy were seondryF IQT 4.2 Fabrication his setion overs the proess of fritionD from the preprtion of ommerillyE otined piezoeletri sustrtes to the design nd lithogrphyD deposition nd liftEo' of mgnetoelsti elementsD inluding the relevnt experimentl prmeters involvedF ell steps of the frition were performed within siwxD nd in prtiulr using the len room filitiesF 4.2.1 Polishing of the PMN-PT substrate yver the ourse of the hhD we worked with two di'erent sets of wxE sustrtesF pigure RFS ! sndividul wxE sustrtes otined from gq edvned mterilsD delivered with euGgr eletrodesF he (rst set of PH wxE sustrtes ws otined from gq edvned wterilsD formerly known s rFgF wterilsF pigF RFS shows the sustrtes in their individul pkgingF ih smple hd smll red point intended s n indition of the diretionof initil poling @through the thiknessAF he smll dimensions @IHxIHxHFQ in millimetersA mde their mnipultion quite di0ult nd mde them very frgileF foth fes of the smples were lredy overed with euGgr eletrodesF roweverD the sustrtes were not polished nd the surfe roughness ws very poor @R a ≈ 50 nmAF he imge in pigF RFT gives n ide of the initil surfe qulityF qiven the lyer thiknesses involved in future stepsD it ws ovious tht some polishing proess ws neessry efore proeeding ny furtherF IQU pigure RFT ! etomi pore wirosopy @epwA imge showing the poor surfe qulity of the smplesF he liner pttern of irregulr relief ws visile t the nked eyeF et ething of the euGgr eletrodes is quite strightforwrd nd fortuntely does not seem to eth or 'et the ermiF elsoD stndrd lening of IS minutes t UH • g in iug ws done efore ny polishingF his step ws routinely used t vrious points of the frition proessF eginD the ermi ws seemingly un'eted y this strongly si environmentF yne oth sides of the smples were free from metlD n e'etive polishing method ws developed using the vogiteh wS system t siwxF he (rst step is to ond the smple to Q inh or R inh glss plte using speil pste nd het them in dedited vogiteh mhine during pproximtely one hourF henD the glss plte is pled t the ottom of ylindril struture tht n modulte the weight put on the smpleF st is then put on lrge rotting felt plteD with n dditionl sweeping hieved y roller rmF sn our seD the rottion speed is inresed until the mximum vlue @UH rpmA for totl polishing time of S minutesF hroughout the proessD smll quntities of pI 4slurry4 re poured on the felt plteF ixperiments with the ghemlox slurry were not onlusiveD nd the poor results otined with wter showed the importnt polishing role of the pI slurryF he results otined fter polishing re quite impressiveD over IHxIH µm windowD roughness of pproximtely P nm is ensuredD with sunnometer roughness rehed on smller res @see pigF RFVAF IQV pigure RFU ! hotogrphs of the equipment used to polish the wxE sustrtesF he seond set of IH sustrtes ws otined from nother ompny lled ehnologiesD with ISxIHxHFQ dimensions @mmAF he slightly di'erent olor nd trnspE reny for the sme thiknessD s ompred to gq wterils smples hints t possile smll di'erenes in ompositionF hese sustrtes re lredy polished y the mnufE turerD nd hene signi(ntly more expensiveF roweverD the surfe qulity ws not s good s our homeEpolished smples to the point tht we reEpolished some of themF ith len nd polished sustrtesD new eletrodes ould e deposited on oth sides of the smplesF feuse the gol is eventully to trnsmit the strin generted y the sustrte to the mgnetoelsti lyer loted ove the eletrodeD it is importnt to hve thin eletrode @≤ IHH nmAF elsoD hoosing sti' metls is importnt to ensure strong mehnil oupling @oung modulus etween those of the ermi nd the multilyersAD so we minly onsidered tGi or uGi eletrodes nd voided euF 4.2.2 Lithography he need to design mgnetoelsti strutures of vrile size nd geometry pointed t the use of lithogrphy insted of foused ionEem @psfA tehniques whih re sometimes used to rete nnostripe geometriesF enother reson for not using psf is tht it is oftentimes di0ult to ontrol the depth of ething nd the extent to whih the integrity of lol mteril is mintinedF his is prolem sine we del with thin (lms nd tht eletrode eneth the mgnetoelsti mteril should not e dmgedF he two lterntives were liftEo' nd dryEething @or plsm ethingAD whih were oth exploredF sn ny se it is neessry to use n eletronEem mskD given the lengthsles involvedD s well s IQW pigure RFV ! epw imge of the polished surfeF ettention should e rought to the rnge of the olorr ompred to pigF RFTF positive resistF he lithogrphy ws initilly tried on silion wfers in order to (nd the pproprite prmetersF

  pollowing these stepsD the end result is of very high qulity nd reprodues urtely wht is on the mskF feuse of the ovious impt of eletrons on the resist efore developmentD we do not hve iw imges showing the developed ptternsF roweverD pigF RFW shows sreenshots tken from optil mirosope oservtionsF 4.2.3 Deposition of multilayers he tehnique hosen to rete multilyers of mgnetoelsti mterils is sputteringF st is widely used for the preprtion of thin (lms s it provides nnometer preision on the otined thiknessF puttering refers to the proess of ejetion of toms from trget using energeti gs ions @plsmAD see pigF RFIHF he plsm my or my not e hemiE lly retiveF he plsm is reted y pplying strong stti or p eletri (eld tht will ionize some of the gs prtilesF sing n osillting (eld hs the dvntge of inE resing the numer of ollisions etween hrged nd neutrl prtiles in the gs so tht IRI pigure RFIH ! hemti view of sputter depositionF uli dominF it inreses the proportion of hrged prtiles nd therefore speeds up the depositionF elsoD it redues the umultion of hrges on the trgets mde of insulting mterilsF sn the se of mgnetron sputteringD mgneti (elds re introdued to rete helioidl motion of hrged prtiles round mgneti (eld lines t the vrmor frequenyD therey inresing the numer of ollisionsF roweverD relying on mgnetron sputtering my sigE ni(ntly derese the lifespn of the trget euse of nonEuniform wer on its surfe due to the distriution of mgneti (eld linesF pigure RFII ! hotogrph of the veyold SSHF et siwxD di'erent equipments exist for sputter depositionF he equipment used for the multilyers in our se is veyold SSHD with p ergon plsm deposition @sputter IRP pigure RFIP ! R inh trgets of @A terium nd olt nd @A iron nd olt usedF downA shown on pigF RFIIF everl trgets re inluded in the systemD nd eh one tht we used is omposed of prts mde of di'erent metls @for instne piees of pe glued on go diskA in order to otin the desired lloys @see pigF RFIPAF he ompounds of di'erent trgets n e susequently deposited on sustrte during runD in pttern of deposition progrmmed eforehndF he plsm is reted ner the trgets whih re (xedD wheres the plte on whih the sustrte is pled is moileF 9eipes9 re prepred nd onsist of series of osilltions of the plte eneth the trgetsF yviouslyD prior lirtions re required to know with stisfying preision how muh is deposited in given numer of osilltions through the plsmF por ny depositionD frgments of silion wfer re pled ner the sustrte for suE sequent nlyses @mesurement of thikness tully deposited using pro(lometerD hE rteriztion in irting mple wgnetometer nd mesurements of mgnetostritionAF hese re visile in pigF RFIQF huring depositionD the estlishment of n esyExis is enE sured y the presene of stti inEplne mgneti (eld reted y mrosopi mgnets pled round the sustrteD lso present on pigF RFIQF he physil explntion of this phenomenon is still elusive to some extentF hue to the sene of irlokD long period of time hs to elpse efore su0ient vuum is hieved in the hmerF sn order to improve the seondry vuumD titnium is sputtered nd liquid nitrogen is used in ryogeni trpD s they respetively help redue the mount of oxygen nd wter in the hmerF elsoD n idle run on teriumEontining trgets is neessry to remove the super(il oxidized mterilD s terium is very muh prone to oxidiztionF huring the hhD we worked with PH to PT nmEthik multilyers of go 2 GpegoF he lyer put on top is lwys pego to limit oxidiztionD so tht the strutures were lwys IRQ pigure RFIQ ! mple put in the hmer of the veyold SSH for sputter depositionF onstituted of Q lyers of go 2 nd R lyers of pegoF he deposition proess itself tkes pproximtely ten to (fteen minutes overllF 4.2.4 Lift-o and dry-etching methods rving independently developed suessful lithogrphy nd sputter depositionD it is worth pusing to re)et on the most onvenient nd e0ient wy to get the mgnetoelsti ptterns on the wxE sustrteF he two lterntives mentioned in RFPFP re liftEo' nd dryEethingF sn the former seD the lithogrphy is performed (rst on the sustrte nd the mgnetoelsti lloy is sputtered on the resistF he liftEo' onsists in removing the resist nd the multilyers on its surfe so tht only res open during development remin

  pigure RFIT ! iw imge of the mgnetoelsti strutures otined y liftEo' with updted prmetersF everl solutions to this prolem my e put forwrdD the most ovious of whih is to spin thinner resistF his n e hieved y setting higher rottion veloityF eduing the thikness from ISH to SH nm s well s other minor (neEtuning of the liftEo' proess helped signi(ntly improve the resultsD s pigF RFIT showsF his ws repeted severl timesD so tht there is some on(dene tht the proess developed is ppropriteF IRT roweverD results proved less stisfying on wxED proly euse of the smple size ompred to lrger frgments of wfersF pigure RFIU ! iw imge showing the result of ething using gl 2 GerF sn prllelD we ttempted to hieve the formtion of mgnetoelsti strutures using dryEethingF edmittedlyD these e'orts did not yield similr suessF he dryEething ws performed in sgEsi @sndutively goupled lsm E etiveEson ithingA enteh systemF mples with fullEsurfe deposition of IPH nmEthik multilyers were prepred to test plsm ethingF wo di'erent gses were onsideredX grp 3 nd gl 2 GerF urprisinglyD the former seemingly filed to eth nythingD s no e'et ws notied fter IH minutes of ethingF ith gl 2 GerD whih hs retive omponent due to the presene of hlorineD the multilyer ws suessfully ethedF fesidesD the optil signl used for end point detetion exhiited osilltion ttriutle to the suessive ething of individul lyersD whih is very interesting in itself nd for preise ontrol over the ething proessF e test ws then rried out to eth the multilyer deposited under pttern metlli mskF xikel is most often used for the protetive msk in dryEethingF rowever in our se this ws not n option s the metlli multilyers re lso ethed y the nitri idEsed wetEething solution used to remove nikelF he sme goes for hromeF es resultD luminum ws hosen instedF efter ething nd liftEo'D we otined mixed resultsF pirstD the ething seemed to hve dversely 'eted the multilyers nd the msk in the sense tht it ws impossile to remove oth y wetEethingF his my e due to impurities from retive ething of the msk nd the multilyerF elsoD these impurities were sttered on the IRU surrounding surfe nd did not dispper fter stndrd leningF es rther stisfying results were otined y liftEo'D there ws no inentive to spend muh more time trying to (gure out why this hppened nd how to prevent itF 4.3 Characterization his setion overs the vrious mesurements rried out on the mterilsF he piezoeleE tri sustrte response to n eletri (eld ws otined using strinEguge mesurementsD nd ompred to dt ville in the litertureF es for the mgnetoelsti lyersD their mgneti properties re extrted from experiments in irting mple wgnetomeE terF hetiled informtion out the topogrphy nd lol mgnetism re otined y etomi nd wgneti pore wirosopyD respetivelyF uerr mirosopy ws used to gin diret insight on the ehvior of the mgnetoelsti mteril sujeted to mgneti (eld nd mehnil stressF 4.3.1 Strain gauge measurements

  sustrteD it is possile to estimte the mgnetostrition oe0ient b @in wA nd thus ISP pigure RFPP ! w hysteresis loops performed on the test smple ontining thin (lm of the mgnetoelsti multilyers depositedF pigure RFPQ ! rrd xis mgnetostrition dt for the test smple ontining thin (lm of the mgnetoelsti multilyers depositedF b γ,2 = b satb sat ⊥ F qiven the morphous struture of the mterilD this physil quntity n e relted to the mximum reltive displement λ s in simple wyD involving elsti prmeters of the thin (lm PX λ s = -

  he epw we used t siwx is himension son mde y frukerF st is shown on pigF RFPTF he (rst epw tests yielded informtion out the lol topogrphy on the multilyers ptternsF sn prtiulrD muh of the disussion of susetion RFPFR revolved round the pro(les on the edges of the mgnetoelsti strutures otined fter liftEo'F epw dt ws olleted on the smples otined fter lithogrphy of di'erent resist IST pigure RFPU ! Qh imges llowing the omprison of the liftEo' otined efore nd fE ter updting the prmeters to spin thinner resist lyerF he pro(le is drmtilly improvedF thiknesses in order to test the explntion for the presene of these wllElike umpsF pigF RFPU ontins Qh imges tht turn out to e typil of the two smplesF st is quite ovious tht deresing the thikness of the resist hd drmti e'et on the pro(leF his supports the interprettion developed in susetion RFPFRF pigure RFPV ! @A opogrphil nd @A wpw dt over n elliptil mgnetoelsti elementF he wellEde(ned dipole on(gurtion is synonymous with monodomin distriE utionF xote tht the ontrst otined orrespond to out I • in phse shiftF por mgneti imgingD severl tips with di'erent mgnetiztion nd oerivity re villeF e used the wiEP tht hs medium oerivity nd mgneti momentD with whih very ler mgneti signls re esily deteted @for instne on the test mgE neti tpe provided y frukerAF fefore ny epw mesurementD the smples re sturted roughly long the hrd xis WH • in the w in the hope of induing nonEuniform mgE netiztion distriutions nd hene domin wlls in the nnostruturesF vooking t the ISU elliptil struturesD good mgneti signl is otinedF pigF RFPV shows wpw dt for typil elementF st seems ler tht the mgnetiztion is uniformD with two ler nd distint mgneti polesF st is interesting to note tht the ellipses tht we looked t ll hd the sme leftEright pttern of light nd drk pthesD suggesting prllel mgnetiztion mong themF

  of the oservle inrese in re of the dominD with pinned domin wllsF hile this experiment does not involve the sme geometry nd mgneti domin struture s wht ws intendedD it does show mgnetoeletri e'et induing domin wll propgtion through modi(tion of nisotropy indued y mehnil stressF ITP 4.4 Prospects his setion ontins disussion deling with the wys in whih the experimentl stte of 'irs n e improvedF rious onsidertions on how to (nd the optiml prmeters pertining to the qulity of the lithogrphyD liftEo' nd ething re put forwrdF ome thoughts on the filure to preisely mnipulte domin wlls in the proposed on(gurtion followF pinllyD prllel work tht we hve een involved in on wxE ething in the prospet of integrtion nd e0ient opertion is presentedF 4.4.1 Optimization of fabrication process elthough signi(nt progress hs een mde on the liftEo' proess sine the (rst testsD the pro(les otined still exhiit steep relief on the edges of the mgnetoelsti struturesF he pproh hosen here ws to redue the thikness of the resistF his turned out to work quite wellD lthough results were less stisfying on wxE thn on silion wfersD

  wxE thin (lms ould e n opportunity with regrd to integrtion nd slingD under the ssumption tht pproprite rystlline phse nd orienttion s well s good surfe qulity n e otined on silion wfersF fesidesD other mterils my exhiit superior intrinsi propertiesD etter ouplingD or llow more onvenient proessing or opertionF e should lso er in mind tht the reserh is not done in the vuum ut within regultory nd soil frmework tht tend to re)et eonomiD strtegi nd eologil onernsF uli helth risks ssoited with the use of led in ommeril produts resulted in stringent worlwide regultion of ledEsed produtsF es resultD there is strong inentive for reserhers to (nd ledEfree lterntivesD espeilly in the industry QHRD QHSF nfortuntelyD ledEfree ndidte with ompetitive properties hs yet to e foundF e n mke similr point on the use of rreEerth sed mteril for the mgnetoelsti phseF st remins to e seen how more undnt mgnetostritive elements"suh s nikel"ould reple themF he key issue here lies with the possiility to e0iently tilor the mgneti nisotropy s it is urrently done in nnostrutured teriumEsed multilyers sputtered in the presene of mgneti (eldF 4.4.2 Control of domain walls in nanostructures he filure to otin n experimentl vlidtion of the phenomenon studied numerilly should not shed dout on the possiility of suessfully oserving domin wll motion indued y the omintion of uniform stress nd is mgneti (eldF he si phyE sil mehnism involved"stressEtriggered vrition of e'etive nisotropy generting eemn energy gp etween djent domins"is quite simple nd ompletely onsistent with existing knowledgeF gomining this strong a priori plusiility with the numeril evidene presented in this thesisD it would e surprising tht suh phenomenon ould not e oserved experimentllyF st is possile tht the phenomenon would not e esily oserved euse of the reltive wekness of the energy gp tht ould fil to overome domin wll pinning strength t room tempertureD ut this seems rther unlikelyF pirstD with modern tehniques nd ITR knowEhowD it is possile to hieve very limited roughness in nnostruturesF eondD the energy gp depends hevily on the strength of the mgneti (eldD whih n e inresed s high s the e'etive nisotropy long x @unixil nisotropy C shpe nisotropyA to fore the depinning of the domin wllF yne importnt issue we fed reltes to the retion of domin wllsD whih is mtter seldom overed in ppers deling with domin wll motionF pirstD it is prole tht the ptterns hosen for the nnostrutures do not lend themselves to domin wll genertion in suh mterilsF eondD the proess y whih we thought domin wlls ould pper @sturtion with orthogonl mgneti (eld nd relxtionA my not e suited to suh on(gurtionsD nd in ny se ws implemented with poor preision over the sturtion ngleF elizing thisD we frited n eletromgnet (xed on horls mount tht n rotte with preision of less thn I • F roweverD to e of ny useD suh devie would hve to e set up in the wpw so tht fter eh sturtion it would e possile to hek for the presene of domin wll in the exmined nnostripeF nfortuntely this ould not e hievedD nd there were douts tht the (eld generted y the mgneti iruit would e strong enough to overome the e'etive nisotropy of individul elementsF fesidesD the onsequenes of the proximity of the mgneti tip nd vrile mgneti (eld in the xyEplne remins n unsettled mtterF wgneti (elds up to Q n e pplied in ultrE high vuum equipments t siwxD ut only perpendiulrly to the smple nd for the time eingD there re no dequte mgneti tips villeF 4.4.3 PMN-PT studs sn prllel with the work on mgnetoelsti struturesD there were e'orts direted t more pplitionEoriented mttersF wore preiselyD investigtions hve looked t the onditions in whih the ontrol of individul mgnetoelsti strutures ould e implemented in n rry of suh elements from n integrtion perspetiveF es disussed in susetion RFIFQD the risk of inhiiting piezoeletrillyEindued displement s well s lolly dmging the sustrte suggests tht some proessing of the wxE sustrte my e usefulF his is the resoning ehind the proposed ptterning shown in pigF RFRD in whih stud of sustrte is retedF e should lso note tht this solution drmtilly redues the dissipted energy s the volume of dieletri involved is reduedF sn ny seD the deveE lopment of n ething proess of wxE is neessryF sn prtieD given the smll size of the stud @only mirons rossAD it is neessry to rrnge 4ridge4 to ensure eletriE l ontt of the surfe of the studF rior multiphysis studies using the gywyv R softwre showed tht n spet rtio of I @width to thiknessA ws good ompromise to ITS llow the piezoeletri mteril to e'etively deform itself on its upper prtF hereforeD quite deep ething is neessryF pigure RFQR ! @A psf n @A plsm ething of the wxE sustrtes to rete SxS µm studsF e (rst step onsisted in depositing pproprite eletrodes on wxE sustrteF henD preliminry tests involved poused son fem @psfA to eth the sustrte @see pigF RFQRAF roweverD sine ething in this fshion n only e done one site t timeD it is not suited to the eventul purpose of hving rrys of studsF lsm ething ppers like etter optionD nd proved possile lthough the ething pro(le on wxE sustrtes turned out to e muh less stright thn on silion wfersF grystlline orienttions my e to lme on this issueF en exmple of I µm deep ething is shown on pigF RFQRD mde possile y nikel mskF igni(nt wer of the ridge ws oservedD nd onern over its eventul dispperne euse of the longer periods of ething neessry eventully led to modi(tion of the designF e simple prllelepipedi 9pierElike9 struture of onstnt width is now onsideredF qiven the depth of ething sought @severl mironsAD thik nikel protetive msk is requiredF he most onvenient tehnique to use in suh ses is eletrolysisF rious senrii hve een put forwrd for the whole proessD nd one urrently reE mins under onsidertionF hey re summed up in RFQSF yther ndidtes were eliminE ted euse they proved too omplited from n experimentl point of view or tht they yielded poor resultsF sn the greedEupon viewD eletrodes would hve to e deposited (rstD efore the liftEo' of mgnetoelsti elements is doneF por the eletrolysisD it is neessry to hve fullEsurfe deposition of metl @so tht the smple surfe is onduting nd ts s the thodeAF ine it will e on top of the multilyersD gold ws hosen s it n e wetEethed without dmge to the multilyersF e negtive resist is then spun to over ITT pigure RFQS ! intire proess roken down into simple stepsD from re sustrte to ethed surfe with eletrodes nd mgnetoelsti strutures on studsF the res where the sustrte is supposed to e ethedF P µm of nikel re deposited during the eletrolysisF efter liftEo'D res of the sustrte to e dryEethed re openF pinllyD fter the deep ething of wxED the nikel nd gold n e wetEethed one fter the otherF 4.5 Conclusion sn this hpterD we presented the vrious investigtions pertining to the ttempt to experimentlly oserve the phenomenon studied numerilly in previous hptersF he mterils hosen for the implementtion of the mgnetoeletri oupling re @HIIAEut wxE for its prtiulr piezoeletri propertiesD nd rreEerthEsed mgnetoelsti multilyers routinely used t siwxF yn the frition proessD few ostles hve een enounteredD nd t this pointD stisfying proess hs een developedF he suessful ITU polishing of wxE sustrtes llowed gret surfe qulity on whih lithogrphy ould e performed esilyF go 2 Gpego multilyers were deposited y sputteringF he hoie of liftEo' of the multilyers yielded highEqulity ptterns with only limited normlitiesF o egin on the hrteriztion frontD the piezoeletri properties of wxE suE strtes were mesuredF he suessful mesurements hinted t omplexD nonliner ehvior tht turned out to e onsistent with dt reported in the litertureF he hrteriztion of the multilyers deposited ws done in the wD nd on(rmed the presene of lerly de(ned unixil nisptropyF es for the mgnetoelsti struturesD epw nd wpw oservtions were rried out nd we onluded tht for the most prt mgnetoelsti elements hd uniform mgnetiztionF he use of newly ssemled uerr mirosope provided informtion out the mgnetiztion distriution of lrger elementsF st ws possile to oserve mgnetoeletri e'et on smple sujeted to mgneti (eld nd n eletri (eldF pinllyD severl ides on the possile opportunities for improE vement on ll these mtters re disussedF he pth towrd integrtionD inluding work on deepEething of wxE sustrtes ws lso ddressedF ITV Conclusion sn this thesis we hve proposed nd studied in detil n innovtive tehnique to indue the motion of trnsverse domin wlls in nnostripes of onstnt nd vrile widthF es shown in hpter ID this endevor (ts very well into urrent reserh trends in mgnetism pertining to the mnipultion of mgnetiztion nd domin wlls t the nnosleF wgneti (elds nd eletri urrents re routinely used to indue domin wll motion experimentllyD nd the study of the dynmis unveiled the omplex ehvior of mgneti domin wllsD inluding the wellEknown lker rekdownF he need to redue energy onsumption ssoited with devie opertion hs led to e'orts to (nd lowE energy lterntivesF wgnetoeletri mterils re promising exmpleD sine the use of n eletri (eld o'ers dvntges over eletri urrentsF sn prtiulrD the mehnil oupling of piezoeletri sustrte nd mgnetoelsti lyer provides n interesting implementtion of the mgnetoeletri e'etF qenerlly spekingD the pplition of stress on mgneti system is tntmount to the retion of n dditionl nisotropyFemong other worksD previous investigtions t siwx showed the possiility to reversily swith the mgnetiztion of mgnetoelsti element with unixil nisotropy using uniform stress nd stti is mgneti (eld rrnged so tht it reks the qudrti symmetryF wore preiselyD if the esy xis is long xD the mgneti (eld is generted long y while the diretions of unixil inEplne stress re given y the ngle isetorF everl fetures onferred to this system relevnt properties in the ontext of memory pplitionsF he onept presented in this work pitlized on this previous experiene to desrie how the sme omintion of stress nd mgneti (eld n give rise to ontrolled trnsverse domin wll motion in twoEdomin nnostruturesF purtherD nother key ide in this work is the engineering of the potentil lndspe through vrile widthF hile stndrd miromgneti simultions were rried out nd supported the propoE sition mdeD simplerD ad hoc Ih model le to determine the mgnetiztion distriution of miniml energy ws developed nd presented in hpter PF his hoie ws motivted y two onernsX reduing the omputtionl lod involvedD nd providing to the user ITW full ontrol on the prmeters nd lgorithms usedF es (rst stepD we gve reminE der of the lssil miromgnetism eqution governing the ehvior of mgnetiztion in the generl se derived from vritionl methodsF henD the sme pproh ws folloE wed with more spei( terms re)eting the geometry nd on(gurtion onsideredF he otined eqution ws introdued into relxtion proedure inluding isetionEsed solverD llowing oth fst onvergene nd high ury s on(rmed y the omprison ginst known nlytil solutionsF ith n input geometry with onstnt widthD the numeril proedure desried inde(nite domin wll motionD leving only one domin of lower energyF roweverD it ws shown tht if we onsider vrile width @for instne hourglssEshpedAD the domin wll n reh n equilirium positionD nd returns to its originl position t rest when the stress disppersF prom this reversile ehviorD reltionship etween domin wll position nd stress ws estlishedF yverllD energeti nlyses showed tht the phenomenon is used y the tion of the stti mgneti (eld on two domins under the in)uene of n e'etive nisotropy resulting from the interply etween mgnetoristlline nostropy nd the pplied stressF es suhD the motion n e hrterized s stressEtriggered ut (eldEdrivenF vooking t piezoeletri genertion of stressD we showed tht suh system dissiptes very smll mounts of energyD minly from toule heting dissiption due to the hrge nd dishrge yle of the piezoeletri sustrteF he determintion of equilirium mgnetiztion distriutions does not yield inforE mtion out the pth followed y the mgneti systemF he dynmis were therefore explored in hpter Q through seond ad hoc Ih odeF he outome of the vritionl proedure ws omined with the vnduEvifshitzEqilert eqution to otin the desired dynmil system of equtionsF en impliit iuler lgorithm sed on the xewton method ws used to solve this systemD nd ws vlidted y its suessful preditions on (eldE driven motionF he dynmis re in)uened y the vlues of the pplied stress nd is mgneti (eldD nd it ws shown tht in nnostripe with onstnt widthD stedyEstte motion is rehedF prom the physil point of viewD stressEtriggered motion stnds prt from lssil domin wll motion in tht the orresponding domin wll shpe fetures notle outEofEplne exursions in the viinity of the domin wllF hese exursions wne s the domin rehes its equilirium position in hourglssEshped nnostrutureF he extent of the outEofEplne omponent depends on the reltive strength of ompressive nd tensile stress omponentsD sine they re ssoited with plnr nd unixil nisotropyD respetivelyF his is the reson why with the piezoeletri prmeters onsidered the pplition of negtive nd positive eletri (eld does not result in the sme dynmis IUH nd domin wll shpeF sn ny seD the vlues of veloities mesured @up to VHH mGs nd eyondD depending on prmetersA ompre fvorly to those of more stndrd domin wll motion tehniquesF he plusiility tht the filure of the ode t high (elds hinted t the existene of phenomenon kin to lker rekdown ws disussedF qiven the strong theoretil nd numeril evidene presentedD it seems likely tht the phenomenon desried n e oserved in rel systemF ghpter R overed experimenE tl work rried out during the hhF he hoie of mterils re)ets the requirements of n e0ient mgnetoeletri oupling medited y mehnil stressF hen sujeted to vertil eletri (eldD @HIIAEut @wg 1 Eiy 3 x @wxEA round the morphotropi phse oundry genertes lrge inEplne stress omponents of oppoE site signsF e sw how using rreEerthEsed lloys in sputtered multilyers enles n interesting ompromise etween mgnetiztion t sturtionD mgnetostrition nd niE sotropyF he frition proess of mgnetoelsti nnostrutures on wxE egn y polishing the sustrtes to otin surfes of good qulityF foth dryEething nd liftEo' were testedD lthough only the ltter yielded stisfying resultsF por thtD positive resist lithogrphy ws suessfully developed on wxED so tht mgnetoelsti multilyers ould e deposited y sputtering efore the liftEo' ws performedF es for hrteriztionD piezoeletri mesurements unveiled the strongly nonliner nture of @HIIAEut wxED with its hystereti ehvior tht my prove useful for some pplitionsF he mgneti properties of multilyers were mesured y virting smple mgnetometerF he topoE grphy otined y liftEo' ws proed y tomi fore mirosopyD while mgneti fore mirosopy provided informtion out the lol mgnetiztion of individul elements whih hppened to e mostly monodominF e newly set up uerr mirosope llowed the mgnetoEoptil oservtion of the system under mgneti (eld nd eletrillyEindued stressD nd mgnetoeletri e'et ws notiedF he potentil wys forwrd nd more pplitionEoriented mtters were disussedD inluding e'orts put into the ething of wxE to produe studs filitting the piezoeletri tution of individul deviesF fuilding on this workD there re oth room for improvement nd new lines of reserh to exploreF yn the numeril frontD e'orts ould e put into nlyzing the filure of the dynmil ode further in order to ring support or dismiss the ide tht there is indeed lkerElike rekdown in the system studiedF uh (nding might e signi(nt from the physil point of viewF e(ned nlyses using stndrd miromgneti simultions ould lso help on this mtterF yn the experimentl frontD ollortion with tems possessing omplementry knowEhow on domin wll mnipultion ould very well result in the proper experimentl oservtion of the phenomenon preditedD with the onditions IUI nd geometry onsideredF wye other mterils my prove more suitleF sf suessfully implementedD stressEtriggered (eldEdriven motion my prove oth ompetitive nd onveE nientF purtherD the onept proposed pves the wy for interesting vrintsF sn prtiulrD piezoeletrilly tuted multistle systems my e of interest for some pplitionsF fesidesD the in)uene of vrile width on domin wll motion in generl is lso n interesting led tht my ring new insight on oth fundmentl nd pplied mttersF IUP Appendix A Domain wall with exchange and anisotropy energy e nlyze spei( domin wll on(gurtionD whih is importnt from oth the theE oretil nd numeril points of viewF e onsider the simple se of ferromgneti stripe with only exhnge nd nisotropy energy ontriutionsF he lultion is (rst rried out in the generl se of ounded geometryD nd then extended to n in(nite domin wllF eordinglyD the totl energy n e dedued from iqF @PFPWAD eventully otining

  ηF (arcsin cos Φ, 1) = 0F xowD we use the property QHUD QHV F (ψ, 1) = ln(sec ψ + tan ψ), @eFPHA IUS nd we n esily otin the result

  ∂ w ∂ n sine w = 1F hereforeD in order to impose w × ∂ w ∂ n = 0D it is su0ient to hve ∂ w ∂ n = 0 on ∂V F IVH Appendix C Calculation of the demagnetization eld e provide here tehnique to numerilly evlute the demgnetiztion (eld in our systemF e similr pproh n e found in the literture QHWF o eginD we onsider iqF @PFUA giving the demgnetiztion (eld for n ritrry region V F sn nlogy with the theory of the eletri dipole PTWD we my introdue mgneti slr potentil φ suh tht H d ( r) = -∇φ, @gFIA

Nature

  CommunicationsD UXIQSQPD xovemer PHITF IUI wrtin qjekD wnuel fiesD téphne pusilD urim fouzehouneD tosep pontE uertD egnès frthélémyD nd elert pertF unnel juntions with multiferroi rriersF Nature MaterialsD T@RAXPWT!QHPD epril PHHUF PHR IUP iF F symlD eF qruvermnD F qriD wF fiesD nd eF frthélémyF perroeE letri nd multiferroi tunnel juntionsF MRS BulletinD QU@PAXIQV!IRQD perury PHIPF IUQ ueEei inD wurlikrishn juD eiEtin ruD ioEtun engD ue ouD tun huD ioEqung viD hiEhong hngD nd i viF wultiferroi tunnel juntionsF Frontiers of PhysicsD U@RAXQVH!QVSD eugust PHIPF IUR uF hF felshhenkoD yF hernyshyovD elexey eF uovlevD nd yF eF retikovF wgnetoeletri domin wll dynmis nd its implitions for mgnetoeletri meE moryF Applied Physics LettersD IHV@IQAXIQPRHQD wrh PHITF IUS F mesh nd xiol eF pldinF wultiferroisX progress nd prospets in thin (lmsF Nature MaterialsD T@IAXPI!PWD tnury PHHUF IUT wnfred pieigF evivl of the mgnetoeletri e'etF Journal of Physics D: Applied PhysicsD QV@VAXIPQD PHHSF IUU tFF touleF yn the e'ets of mgnetism upon the dimensions of iron nd steel rsF Philosophical Magazine Series 3D QH@IWWAXUT!VUD perury IVRUF IUV xF gF uoonD gF wF illimsD nd fF xF hsF qint mgnetostrition mterilsF Journal of Magnetism and Magnetic MaterialsD IHH@IAXIUQ!IVSD xovemer IWWIF IUW qF ingdhl nd sF hF wyergoyzF rndook of qint wgnetostritive wterils E Ist iditionF IVH F goehoornF iletroni truture glultions for re irthErnsition wetl gompoundsF sn qry tF vong nd pernnde qrndjenD editorsD Supermagnets, Hard Magnetic MaterialsD numer QQI in xey es eriesD pges IQQ!IUHF pringer xetherlndsD IWWIF IVI © oshin ghikzumiF Physics of magnetismF ileyD IWTRF IVP illim puller frownF Magnetoelastic interactionsD volume WF pringerD IWTTF IVQ vouis xéelF enisotropie mgnétique super(ielle et surstrutures d9orienttionFJournal de Physique et le RadiumD IS@RAXPPS!PQWD epril IWSRF IVR itienne du rémolet de vheisserieD hmien qignouxD nd wihel hlenkerF Magnetism: II-Materials and ApplicationsF pringer iene 8 fusiness wediD heemE er PHIPF PHS IVS s hzyloshinskiiF he prolem of piezomgnetismF Sov. Phys. JETPD TXTPID IWSVF IVT FExF ouD gFEF ngD nd iF nF ymmetry types of the piezoeletri tensor nd their identi(tionF Proc. R. Soc. AD RTW@PISSAXPHIPHUSSD tuly PHIQF IVU gFEF xnD vF viuD xF giD tF hiD F eD F rF vinD vF tF hongD nd gF F iongF e threeEphse mgnetoeletri omposite of piezoeletri ermisD rreEerth iron lloysD nd polymerF Applied Physics LettersD VI@PHAXQVQI!QVQQD xovemer PHHPF IVV geEen xnD wF sF fihurinD huxing hongD hF iehlndD nd qF rinivsnF wultiferroi mgnetoeletri ompositesX ristoril perspetiveD sttusD nd future diretionsF Journal of Applied PhysicsD IHQ@QAXHQIIHID perury PHHVF IVW ting wD timin ruD heng viD nd geEen xnF eent rogress in wultiferE roi wgnetoeletri gompositesX from fulk to hin pilmsF Advanced MaterialsD PQ@WAXIHTP!IHVUD wrh PHIIF IWH F qiordnoD wF qoueygouD xF ierelinD eF liD F ernodD nd F reorzE henskyF wgnetoEeletroEelsti e'etive properties of multilyered rti(il multiE ferrois with ritrry lmintion diretionF International Journal of Engineering ScienceD UVXIQR!ISQD wy PHIRF IWI FEF qu nd FEgF reF gompt losedEform miromehnil expressions for the e'etive unoupled nd oupled liner properties of lyered ompositesF Philosophical MagazineD WS@PSAXPUWQ!PVITD eptemer PHISF IWP tefno qiordnoF ixpliit nonliner homogeniztion for mgnetoEeletroEelsti lminted mterilsF Mechanics Research CommunicationsD SSXIV!PWD tnury PHIRF IWQ F pusilD F qriD eF frthélémyD nd wF fiesF wgnetoeletri hevies for pintronisF Annual Review of Materials ResearchD RR@IAXWI!IITD PHIRF IWR t n uhtelenF rodut propertiesX new pplition of omposite mterilsF Philips Res. RepD PU@IAXPV!QUD IWUPF IWS eF wF tF qF n unD hF F errellD nd tF rF holingF en in situ grown euteti mgnetoeletri omposite mterilF Journal of Materials ScienceD W@IHAXIUIH!IUIRD ytoer IWURF PHT IWT tF vn den foomgrd nd F F tF fornF e sintered mgnetoeletri omposite mteril fiy 3 Exi@goDwnA pe 2 y 4 F Journal of Materials ScienceD IQ@UAXISQV!ISRVD tuly IWUVF IWU rriu lneediD enkteswrlu ennpureddyD hshnk riyD nd tungho yuF ttus nd erspetives of wultiferroi wgnetoeletri gomposite wterils nd epplitionsF ActuatorsD S@IAXWD wrh PHITF IWV tungho yuD elfredo ázquez grzoD uenji hinoD nd ryounEie uimF wgE netoeletri roperties in iezoeletri nd wgnetostritive vminte gompositesF Japanese Journal of Applied PhysicsD RH@VAXRWRVD eugust PHHIF IWW qF rinivsnF wgnetoeletri ilyer nd multilyer strutures of mgnetostritive nd piezoeletri oxidesF Physical Review BD TR@PIAD PHHIF PHH wukesh uumriD ghndr rkshD nd tnml ghtterjeeF oom temperture lrge selfEised mgnetoeletri e'et in nonEled sed piezoeletri nd mgneE tostritive @HEQA prtiulte omposite systemF Journal of Magnetism and Magnetic MaterialsD RPWXTH!TRD wy PHIUF PHI F wF hepleyD eF F ushforthD wF ngD qF furnellD nd F eF wooreF wodi(E tion of perpendiulr mgneti nisotropy nd domin wll veloity in tGgoGt y voltgeEindued strinF Scientic ReportsD SXUWPID tnury PHISF PHP eF F ushforthF oltge ontrol of mgnetorystlline nisotropy in ferromgnetiE semiondutorEpiezoeletri hyrid struturesF Physical Review BD UV@VAD PHHVF PHQ wF eilerD eF frndlmierD F qeprägsD wF elthmmerD wF ypelD gF fihlerD rF ruE elD wF F frndtD F qrossD nd F F fF qoennenweinF oltge ontrolled inversion of mgneti nisotropy in ferromgneti thin (lm t room tempertureF New Journal of PhysicsD II@IAXHIQHPID PHHWF PHR ienEun ghungD qregory F grmnD nd uotekr F wohnhndrF eversile mgneti dominEwll motion under n eletri (eld in mgnetoeletri thin (lmF Applied Physics LettersD WP@IIAXIIPSHWD wrh PHHVF PHS FErF ghuD wF F gruzD gFErF ngD vF F wrtinD FEvF ngD tFEF hngD uF veeD F uD vFEF ghenD nd F meshF homin gontrol in wultiferroi fipey 3 through ustrte iinlityF Advanced MaterialsD IW@IVAXPTTP!PTTTD eptemer PHHUF PHU PHT ingEro ghuD vne F wrtinD wikel fF rolomD wrtin qjekD huEten rnD ing reD xin flkeD ghnEro ngD honkoun veeD ei ruD in hnD eiEving ngD erntx prileEodríguezD endres hollD hn F ngD nd F meshF iletriE(eld ontrol of lol ferromgnetism using mgnetoeletri multiferroiF Nature MaterialsD U@TAXRUV!RVPD tune PHHVF PHU tt ngD q hoD rp inD vf vuoD r hngD t reD nd r vuoF iletri (eld mnipultion of mgnetiztion t room temperture in multiferE roi gope 2 y 4 G@wg 1/3 x 2/3 A 0.7 i 0.3 y 3 heterostruturesF Applied Physics LettersD WR@PIAXPIPSHRD wy PHHWF PHV odd frintlingerD ungErwn vimD uml rF flohD ris elexnderD i iD tohn frryD tohn welngilisD vourdes lmnEiD sF keuhiD nd tohn gumingsF sn itu yservtion of eversile xnomgneti withing sndued y iletri pieldsF Nano LettersD IH@RAXIPIW!IPPQD epril PHIHF PHW uuntl oyD upriyo fndyopdhyyD nd tysimh etulsimhF withing dynE mis of mgnetostritive singleEdomin nnomgnet sujeted to stressF Physical Review BD VQ@PPAXPPRRIPD tune PHIIF PIH o uD elexndre furD uin ongD ing hoD ghristopher F vynhD edrm uhE lili emiriD ung vF ngD nd qregory F grmnF iletril ontrol of reversile nd permnent mgnetiztion reorienttion for mgnetoeletri memory deviesF Applied Physics LettersD WV@PTAXPTPSHRD tune PHIIF PII hi rkesD e gvillD e rindmrhD dleyD p wqeeD g tddonD u idE mondsD gmpionD fv qllgherD nd e ushforthF xonEvoltile voltge onE trol of mgnetiztion nd mgneti domin wlls in mgnetostritive epitxil thin (lmsF Applied Physics LettersD IHI@UAXHUPRHPD eugust PHIPF PIP uemeng rnD vi iD ue viD ioin quoD hong viD hen ngD lu uoD nd hesheng ueF hemonstrtion of mgnetoeletri memory ell in @IIHA @wg 1/2 x 2/3 Ay 3 0.68 Eiy 3 0.32 GuGpego heterostruturesF Applied Physics LettersD IHS@IPAXIPPRHPD eptemer PHIRF PIQ suhiro hirhtD yot hiinD hiego vópez qonzálezD uévin tF eF prnkeD iiji dD witsuru stohD xikoly eF ertsevD estin vn hijkenD nd omoysu niymF iletriE(eld swithing of perpendiulrly mgnetized multilyersF NPG Asia MaterialsD U@UAXeIWVD tuly PHISF PHV PIR rsnin ehmdD tysimh etulsimhD nd upriyo fndyopdhyyF eversile strinEindued mgnetiztion swithing in peq nnomgnetsX thwy to rewriE tleD nonEvoltileD nonEtoggleD extremely low energy strintroni memoryF Scientic ReportsD SD heemer PHISF PIS iF he nieriD F iF oyD hF pngD iF uF ehsthedtD eF gF srvineD hF reissD eF gE sirghiD F F gmpionD fF vF qllgherD F tungwirthD nd tF underlihF iezoeE letri ontrol of the moility of domin wll driven y diti nd nonEditi torquesF Nature MaterialsD IP@WAXVHV!VIRD eptemer PHIQF PIT F hukovD tF wF flnoD wF sptovD nd eF hukovF wgnetoelsti gontriution in hominEll hynmis of wgnetilly fistle wirowiresF IEEE Transactions on MagneticsD RU@IHAXQUVQ!QUVTD ytoer PHIIF PIU F hukovD tF wF flnoD wF sptovD nd eF hukovF wgnetoelsti ontriution in domin wll dynmis of morphous mirowiresF Physica B: Condensed MatterD RHU@WAXIRSH!IRSRD wy PHIPF PIV erdy hukovD tun flnoD wihil sptovD elexnder ghizhikD nd lentin hukovF wnipultion of domin wll dynmis in morphous mirowires through the mgnetoelsti nisotropyF Nanoscale Research LettersD U@IAXPPQD PHIPF PIW wihel poersterD perrn wiàD xhuel ttutoD imone pinizioD elerto rernándezEwínguezD ergi vendínezD ulo F ntosD tosep pontuertD ton wE nel rernàndezD wthis uläuiD nd vui elleF hiret imging of delyed mgnetoEdynmi modes indued y surfe ousti wvesF Nature Communi-cationsD V@IAXRHUD eptemer PHIUF PPH xF eF ertsevF qint mgnetoeletri e'et vi strinEindued spin reorienttion trnsitions in ferromgneti (lmsF Physical Review BD UV@PIAD PHHVF PPI xe ertsevF trinEmedited eletriE(eld ontrol of multiferroi domin strutures in ferromgneti (lmsF Applied Physics LettersD IHP@IIAXIIPRHUD wrh PHIQF PPP eF rerklotzF eversile strin e'et on the mgnetiztion of vgoy 3 (lmsF Physical Review BD UW@WAD PHHWF PPQ uuntl oyD upriyo fndyopdhyyD nd tysimh etulsimhF ryrid spintroE nis nd strintronisX e mgneti tehnology for ultr low energy omputing nd signl proessingF Applied Physics LettersD WW@TAXHTQIHVD eugust PHIIF PHW PPR F xovosdD F ytniD eF yhswD F qF uimD uF pukmihiD tF uoikeD uF wruymD yF uitkmiD nd F himdF xovel mgnetostritive memory devieF Journal of Applied PhysicsD VU@WAXTRHH!TRHPD wy PHHHF PPS wnuel fies nd egnès frthélémyF wultiferroisX owrds mgnetoeletri memoryF Nature MaterialsD U@TAXRPS!RPTD tune PHHVF PPT xiols ierelinD nnik hushD ldimir reorzhenskyD nd hilippe ernodF wgnetoeletri memory using orthogonl mgnetiztion sttes nd mgnetoelsti swithingF Journal of Applied PhysicsD IHW@UAXHUhUPTD epril PHIIF PPU eF uF fiswsD F fndyopdhyyD nd tF etulsimhF inergyEe0ient mgnetoelsE ti nonEvoltile memoryF Applied Physics LettersD IHR@PQAXPQPRHQD tune PHIRF PPV eF sF worosov nd eF F igovF 4vong leg4 mgnetoeletri memoryF Journal of Applied PhysicsD IIS@PPAXPPQWHWD tune PHIRF PPW wohmmd lehi pshmiD uuntl oyD tysimh etulsimhD nd upriyo fnE dyopdhyyF wgnetiztion dynmisD fennett loking nd ssoited energy disE siption in multiferroi logiF NanotechnologyD PP@ISAXISSPHID epril PHIIF PQH xoel h9ouzD tysimh etulsimhD nd upriyo fndyopdhyyF pourEstte nnomgneti logi using multiferroisF Journal of Physics D: Applied PhysicsD RR@PTAXPTSHHID tuly PHIIF PQI xF h9ouzD tF etulsimhD nd upriyo fndyopdhyyF inergyEi0ient fennett gloking heme for pourEtte wultiferroi vogiF IEEE Transactions on Nano-technologyD II@PAXRIV!RPSD wrh PHIPF PQP xoel h9ouzD wohmmd lehi pshmiD upriyo fndyopdhyyD nd tysimh etulsimhF ixperimentl gloking of xnomgnets with trin for ltrlow ower foolen vogiF Nano LettersD IT@PAXIHTW!IHUSD perury PHITF PQQ F rinivsnD eF rkrD fF fehinEeeinD nd F httF ellEpin vogi hevie ith snuilt xonreiproityF IEEE Transactions on MagneticsD RU@IHAXRHPT!RHQPD ytoE er PHIIF PQR xF hrouzD tysimh etulsimhD nd upriyo fndyopdhyyF en ltrfst smge eovery nd eognition ystem smplemented ith xnomgnets ossesE sing fixil wgnetorystlline enisotropyF IEEE Transactions on NanotechnologyD II@SAXVWT!WHID eptemer PHIPF PIH PQS pF vliheD F hoD rF hengD pF truD wF F gruzD FEvF ngD hF roD nd F meshF iletrilly essisted wgneti eording in wultiferroi xnostrutuE resF Nano LettersD U@TAXISVT!ISWHD tune PHHUF PQT uoms rF iF vhtinenD uévin tF eF prnkeD nd estin vn hijkenF iletriE (eld ontrol of mgneti domin wll motion nd lol mgnetiztion reverslF Scientic ReportsD PXPSVD perury PHIPF PQU x veiD hiut hevolderD quillume egnusD sl euertD vurent hnielD tooE on uimD eisheng hoD heodossis rypiniotisD ussell F gowurnD glude ghppertD h(né velosonD nd hilippe veoeurF trinEontrolled mgneti domin wll propgtion in hyrid piezoeletriGferromgneti struturesF Nature CommunicationsD RXIQUVD tnury PHIQF PQV qinrlo gonsolo nd qiovnn lentiF enlytil solution of the strinEontrolled mgneti domin wll motion in ilyer piezoeletriGmgnetostritive nnostruE turesF Journal of Applied PhysicsD IPI@RAXHRQWHQD tnury PHIUF PQW e uwon uimD hniel rillD nd roslv serkovnykF wehnil etution of wgneti hominEll wotionF Physical Review LettersD IIU@PQAXPQUPHID xovemer PHITF PRH tF henD wF F frynD F hre)D nd hF eF ellwoodF tressEsed ontrol of mgneti nnowire domin wlls in rti(il multiferroi systemsF Journal of Applied PhysicsD IHW@PAXHPQWISD tnury PHIIF PRI wF F frynD tF henD nd hF eF ellwoodF hynmis of stressEindued domin wll motionF Physical Review BD VS@IRAXIRRRIID epril PHIPF PRP tiEwin ruD innn ngD usr womeniD ioxing ghengD vei ghenD himing veiD hujun hngD usn rolierEwuinstryD enktrmn qoplnD qregory F grmnD geEen xnD nd vongEing ghenF pst wgneti hominEll wotion in ingEhped xnowire hriven y oltgeF Nano LettersD IT@RAXPQRI!PQRVD epril PHITF PRQ uoms rF iF vhtinenD tussi yF uomiD nd estin vn hijkenF ttern rnsfer nd iletriEpieldEsndued wgneti homin pormtion in wultiferroi reterostruturesF Advanced MaterialsD PQ@PVAXQIVU!QIWID tuly PHIIF PII PRR uévin tF eF prnkeD fen n de ieleD suhiro hirhtD mpo tF rämäläinenD omoysu niymD nd estin vn hijkenF eversile iletriEpieldEhriven wgneti hominEll wotionF Physical Review XD S@IAXHIIHIHD perury PHISF PRS fen n de ieleD vsse vursonD uévin tF eF prnkeD nd estin vn hijkenF iletri (eld driven mgneti domin wll motion in ferromgnetiEferroeletri heE terostruturesF Applied Physics LettersD IHR@IAXHIPRHID tnury PHIRF PRT fen n de ieleD tonthn veliertD uévin tF eF prnkeD nd estin vn hijE kenF iletriE(eldEdriven dynmis of mgneti domin wlls in mgneti nnowires ptterned on ferroeletri dominsF New Journal of PhysicsD IV@QAXHQQHPUD PHITF PRU tiEwin ruD innn ngD tinjun ngD rouing rungD tinxing hngD vongE ing ghenD nd geEen xnF urely iletriEpieldEhriven erpendiulr wgneE tiztion everslF Nano LettersD IS@IAXTIT!TPPD tnury PHISF PRV tF töhrD rF gF iegmnnD eF ushuD nd F tF qmleF wgnetiztion swithing without hrge or spin urrentsF Applied Physics LettersD WR@UAXHUPSHRD perury PHHWF PRW eyn uumr fiswsD rsnin ehmdD tysimh etulsimhD nd upriyo fnE dyopdhyyF ixperimentl hemonstrtion of gomplete IVH • eversl of wgnetizE tion in ssolted go xnomgnets on wx! ustrte with oltge qenerted trinF Nano LettersD wy PHIUF PSH xiols ierelinD nnik hushD hilippe tques ernodD nd ldimir reorE zhenskyF wgnetoeletri memoryD heemer W PHIRF tent VDWHVDRPPF PSI nnik hushF Nano-Système Magnéto-Électro-Mécanique (NMEMS) ultra-basse consommation pour le traitement et le stockage de l'informationF hh thesisD xoE vemer PHIIF PSP xF ierelinD F hushD F qiordnoD eF ulimovD F reorzhenskyD nd F ernodF trin wedited wgnetoeletri wemoryF sn tysimh etulsimh nd upriyo fndyopdhyyD editorsD Nanomagnetic and Spintronic Devices for Energy-Ecient Memory and ComputingD pges PPI!PSUF tohn iley 8 onsD vtdD PHITF PSQ xiols ierelinD nnik hushD elexey ulimovD tefno qiordnoD ldimir reE orzhenskyD nd hilippe ernodF oom temperture mgnetoeletri memory PIP ell using stressEmedited mgnetoelsti swithing in nnostrutured multilyersF Applied Physics LettersD WW@IWAXIWPSHUD xovemer PHIIF PSR nnik hushD xiols ierelinD elexey ulimovD tefno qiordnoD ldimir reE orzhenskyD nd hilippe ernodF tressEmedited mgnetoeletri memory e'et with uniExil go 2 Gpego multilyer on HIIEut wxE ferroeletri relxorF Journal of Applied PhysicsD IIQ@IUAXIUgUIWD wy PHIQF PSS tefno qiordnoD nnik hushD xiols ierelinD hilippe ernodD nd lE dimir reorzhenskyF gomined nnomehnil nd nnomgneti nlysis of mgnetoeletri memoriesF Physical Review BD VS@ISAXISSQPID epril PHIPF PST tefno qiordnoD nnik hushD xiols ierelinD hilippe ernodD nd ldimir reorzhenskyF tohsti mgnetiztion dynmis in single domin prtilesF The European Physical Journal BD VT@TAD tune PHIQF PSU F qiordnoD F hushD xF ierelinD F ernodD nd F reorzhenskyF herE ml e'ets in mgnetoeletri memories with stressEmedited swithingF Journal of Physics D: Applied PhysicsD RT@QPAXQPSHHPD eugust PHIQF PSV nnik hushD syl udenkoD xiols ierelinD tefno qiordnoD ldimir reE orzhenskyD nd hilippe ernodF rystereti mgnetoresistne in stress ontrolled mgneti memory devieF Nanomatériaux et nanostructuresD PXRR!SHD PHIPF PSW elexey ulimovD xiols ierelinD nnik hushD tefno qiordnoD héo wtE hurinD hilippe ernodD ldimir reorzhenskyD enton ghurnovD nd ergei xikitovF wgnetoeletri write nd red opertions in stressEmedited multiferE roi memory ellF Applied Physics LettersD IIH@PPAXPPPRHID wy PHIUF PTH wF rF uryder nd gF F uimF efter rrd hrives ! ht gomes xextc IEEE Transactions on MagneticsD RS@IHAXQRHT!QRIQD ytoer PHHWF PTI nnik hushD xiols ierelinD elexey ulimovD syl udenkoD ury sgntovD mi rgeEeliD hilippe ernodD nd ldimir reorzhenskyF tterned vIHE pet for polriztion of mgneti (lmsF Journal of Applied PhysicsD IHW@UAXHUeUPHD epril PHIIF PTP F pishherD wF prnhinD qF fordignonD nd rF pngohrF e ystemti epproE h to wultiphysis ixtensions of piniteEilementEfsed wiromgneti imultiE onsX xmgF IEEE Transactions on MagneticsD RQ@TAXPVWT!PVWVD tune PHHUF PIQ PTQ ghristophe qeuzine nd tenEprnçois emleF qmshX e QEh (nite element mesh genertor with uiltEin preE nd postEproessing filitiesF International Journal for Numerical Methods in EngineeringD UW@IIAXIQHW!IQQID eptemer PHHWF PTR elexey ulimovD xiols ierelinD ldimir reorzhenskyD nd hilippe erE nodF snhomogeneous spin reorienttion trnsition @srtA in gint mgnetostritive go 2 Gpego multilyersF IEEE transactions on magneticsD RP@IHAXQHWH!QHWPD PHHTF PTS wrtin uruºík nd endres rohlF eent hevelopments in the wodelingD enlysisD nd xumeris of perromgnetismF SIAM ReviewD RV@QAXRQW!RVQD PHHTF PTT héo wthurinD tefno qiordnoD nnik hushD xiols ierelinD hilippe erE nodD nd ldimir reorzhenskyF tressEmedited mgnetoeletri ontrol of ferE romgneti domin wll position in multiferroi heterostruturesF Applied Physics LettersD IHV@VAXHVPRHID perury PHITF PTU héo wthurinD tefno qiordnoD nnik hushD xiols ierelinD hilippe erE nodD nd ldimir reorzhenskyF wehnilly driven domin wll movement in mgnetoelsti nnomgnetsF The European Physical Journal BD VW@UAXITWD tuly PHITF PTV szril woiseevith qelfndD ihrd e ilvermnD et lF Calculus of variationsF gourier gorportionD PHHHF PTW tulius edms trttonF Electromagnetic theory. International series in pure and applied physicsF wqrwErillD xew orkD IWRIF PUH ghristin wiehe nd qutm ithirjF e geometrilly onsistent inrementl vE ritionl formultion for phse (eld models in miromgnetisF Computer Methods in Applied Mechanics and EngineeringD PRS!PRTXQQI!QRUD ytoer PHIPF PUI ssk h wyergoyzD qiorgio fertottiD nd gludio erpioF Nonlinear magnetization dynamics in nanosystemsF ilsevierD PHHWF PUP rerert f uellerF Numerical solution of two point boundary value problemsD voE lume PRF swD IWUTF PUQ el(o urteroniD irdo oD nd pusto leriF Numerical mathematicsD voE lume QUF pringer iene 8 fusiness wediD PHIHF PIR PUR wro feleggiD hkul ndonD imei huD nd wr he qrefF yn the ompuE ttion of the demgnetiztion tensor for prtiles of ritrry shpeF Journal of Magnetism and Magnetic MaterialsD PUPEPUTD upplementXiIIWU!iIIWWD wy PHHRF PUS wF feleggiD F ndonD F huD nd wF he qrefF yn the mgnetostti intertions etween nnoprtiles of ritrry shpeF Journal of Magnetism and Magnetic MaterialsD PUV@IEPAXPUH!PVRD tuly PHHRF PUT ongxin F rung nd ongmei wF tinF hse (eld modeling of mgnetizE tion proesses in growth twinned erfenolEh rystlsF Applied Physics LettersD WQ@IRAXIRPSHRD ytoer PHHVF PUU e uolmogorov nd pominF Elements of the Theory of Functions and Functional Analysis, Moscow, 1974F hoverD xew orkD IWWWF PUV peifei ngD vihui vuoD hn houD ingyong hoD nd rosu vuoF gomplete set of elstiD dieletriD nd piezoeletri onstnts of orthorhomE i HFUI@wg 1/3 x 2/3 Ay 3 EHFPWiy 3 single rystlF Applied Physics LettersD WH@PIAXPIPWHQD wy PHHUF PUW héo wthurinD tefno qiordnoD nnik hushD xiols ierelinD hilippe erE nodD nd ldimir reorzhenskyF hominEwll dynmis in mgnetoelsti nE nostripesF Physical Review BD WS@IRAXIRHRHSD epril PHIUF PVH roslv serkovnykD erne frtsD nd qerrit iF F fuerF inhned qilert hmping in hin perromgneti pilmsF Physical Review LettersD VV@IIAXIIUTHID perury PHHPF PVI eF glrkD F eundiD nd F qillmorF wgnetiztion nd mgneti nisotropy of pe 2 D hype 2 D 0.27 hy 0.73 pe 2 nd mpe 2 F IEEE Transactions on MagneticsD IR@SAXSRP!SRRD eptemer IWUVF PVP iri i pullertonD tF tingD nd F h fderF rrdGsoft mgneti heterostruturesX model exhngeEspring mgnetsF Journal of Magnetism and Magnetic MaterialsD PHH@IEQAXQWP!RHRD ytoer IWWWF PVQ xiols ierelinF Phénomènes dynamiques non linéaires dans les lms minces magnétostrictifs au voisinage de la transition de réorientation de Spin : Application aux micro systèmesF hh thesisD tnury PHHHF PIS PVR o uD ing hoD wingqing foD elexndre furD toshu vF rokelD uin ongD uotekr F wohnhndrD ghristopher F vynhD nd qregory F grmnF homin engineered swithle strin sttes in ferroeletri @HIIA @wg 1/3 x 2/3 Ay 3 (1-x) E iy 3 x @wxED x ≈ 0.32A single rystlsF Journal of Applied PhysicsD IHW@IPAXIPRIHID tune PHIIF PVS F F ghoiD F F hroutD F tF tngD nd eF F fhllF hieletri nd pyroeletri properties in the @wg 1/3 x 2/3 Ay 3 Eiy 3 systemF FerroelectricsD IHH@IAXPW!QVD heemer IWVWF PVT wihel F vngnD xing ngD hinesh gF hueD nd eiEtoo tngF hieletri fehvior of the elxor @wg 1/3 x 2/3 Ay 3 Eiy 3 olidEolution ystem in the wirowve egionF Journal of the American Ceramic SocietyD UP@QAXRVI!RVQD wrh IWVWF PVU ingyong hoD fijun pngD ru goD iping quoD nd rosu vuoF hieletri nd piezoeletri performne of wxE single rystls with ompositions round the wfX in)uene of ompositionD poling (eld nd rystl orienttionF Materials Science and Engineering: BD WT@QAXPSR!PTPD heemer PHHPF PVV iping quoD rosu vuoD inhou reD riqing uD nd hiwen inF homin on(E gurtion nd ferroeletri relted properties of the @IIHA u uts of relxorEsed @wg 1/3 x 2/3 Ay 3 Eiy 3 single rystlsF Japanese journal of applied physicsD RI@QAXIRSID PHHPF PVW uF uF jnD wF hnthiD F F ghngD tF tinD nd vF gF vimF hieletri nd piezoeletri properties of HHI nd HIIEpoled relxor ferroeletri xE nd wxE single rystlsF Sensors and Actuators A: PhysicalD IQQ@IAXIIH!IITD tE nury PHHUF PWH ui hngD fei tingD nd enwu goF ingleEdomin properties of HFTU@wg 1/3 x 2/3 Ay 3 EHFQQiy 3 single rystls under eletri (eld isF Applied Physics LettersD VP@SAXUVU!UVWD perury PHHQF PWI ui hngD fei tingD nd enwu goF ilstiD piezoeletriD nd dieletri proE perties of multidomin HFTU@wg 1/3 x 2/3 Ay 3 EHFQQiy 3 single rystlsF Journal of Applied PhysicsD WH@UAXQRUI!QRUSD ytoer PHHIF PIT PWP wuhtr ehrtD wddury omyzuluD F iF gohenD F qneshD rzemyslw herD roEkwng woD ussell tF remleyD ng enD eter viermnnD nd higng uF yrigin of morphotropi phse oundries in ferroeletrisF NatureD RSI@UIUVAXSRS! SRVD tnury PHHVF PWQ gF ntigteD tF veeD nd eF friF roessing nd properties of @wg 1/3 x 2/3 Ay 3 E iy 3 thin (lms y pulsed lser depositionF Applied Physics LettersD TT@IQAXITII! ITIQD wrh IWWSF PWR omin rerdierD wF hetlleD hvid tenkinsD groline oyerD nd henis emiensF iezoeletri thin (lms for wiw pplitions E e omprtive study of D HFUwxEHFQ nd HFWwxEHFI thin (lms grown on i y p mgnetron sputE teringF Sensors and Actuators A: PhysicalD IRV@IAXIPP!IPVD xovemer PHHVF PWS uF eF fuiEhiD F veoeurD wF hmEhiD nd qF qrryF hse optimistion of wxE thin (lms deposited y ulsed vser heposition on wgy sustrtes nd tEoted silionF sn 2010 IEEE International Symposium on the Applications of Ferroelectrics (ISAF)D pges I!RD eugust PHIHF PWT uim enh fui weurF Réalisation des couches minces PMN-PT dans la technologie MEMS pour les applications hyperfréquencesF hh thesisD niversité ris udEris sD PHIPF PWU i du rémolet de vheisserieF he(nition nd mesurement of the surfe mgE netoelsti oupling oe0ients in thin (lms nd multilyersF Physical review BD SI@PPAXISWPSD IWWSF PWV F hoD rF pujiwrD qF tF wnkeyD gF rouD nd wF unF eonstrution of inE plne mgnetiztion distriutions from mgneti fore mirosope imgesF Journal of Applied PhysicsD VW@IIAXUPQH!UPQPD tune PHHIF PWW eF hivilleD tF wilttD nd tF wF qríF wgneti pore wirosopyX smges of xnostrutures nd gontrst wodelingF sn rofessor rerert ropster nd rofessor rns eter yepenD editorsD Magnetic Microscopy of NanostructuresD xnoiene nd ehnologyD pges PPS!PSIF pringer ferlin reidelergD PHHSF QHH vF eelmnnD eF vn den fosD nd gF vodderF wgneti pore wirosopy " oE wrds righer esolutionF sn rofessor rerert ropster nd rofessor rns eter yepenD editorsD Magnetic Microscopy of NanostructuresD xnoiene nd ehnoE logyD pges PSQ!PVQF pringer ferlin reidelergD PHHSF PIU QHI uF ostvD eF wziewskiD eF tupkiewizD eF wroD vF F fzewskiD F isnoE vskyD nd F mguhiF rnsverse mgnetoEoptil uerr e'et mesured using phse modultionF Journal of the European Optical Society -Rapid publicationsD I@HAD ytoer PHHTF QHP qor veedevF Composites multiferroïques pour dispositifs magnéto-électriques inté-grésF hh thesisD niversité de qrenoleD PHIPF QHQ wrk e wgord nd wihel t ooksF pie hndook of mirolithogrphyD miroE mhining nd mirofritionF sn SPIE, BellinghamD PHHHF QHR wF hemrtin wederD hF hmjnoviD nd xF etterF ved pree iezoeletri wE terilsF Journal of ElectroceramicsD IQ@IEQAXQVS!QWPD tuly PHHRF QHS ghngEryo rongD rwngEill uimD fyungEul ghoiD ryoungEu rnD te ung onD ghng on ehnD nd ook toF vedEfree piezoermis E here to move onc Journal of MateriomicsD P@IAXI!PRD wrh PHITF QHT szril olomonovih qrdshteyn nd sosif woiseevih yzhikF

  of miromgnetism is tht t restD the mgnetiztion distriution is suh tht the orresponding free energy of the system is minimumD t lest lolly in phse speF his quntity is the sum of set of ontriutionsD with sometimes on)iting in)uenesF felow is very short qulittive desription of the origins nd e'ets of the ontriutions relevnt in this workF Zeeman energy wgneti moments re 'eted y surrounding mgneti (eldsF he e'et of mgneti (elds is ounted for through the eemn energyD whih re)ets the tendeny of mgneti moments to lign with pplied (eldsF e mgneti (eld is generlly noted H nd its dimension is the sme s mgnetiztion @mesured in eGmAF sndutionD whih is often noted BD hs the sme dimension s µ 0 H @mesured in eslAF

M = M s w( r, t)D where M s is the mgnetiztion t sturtion nd w = (w x , w y , w z ) is spe dependnt nd time vrying unit vetorF PP enother min spet Demagnetization energy sn ddition to externl (eldsD the mgnetiztion distriuE tion is sujet to the (eld tht it itself genertesD tht will therefter e lled demagnetizing eldF sts e'ets re of longer rnge nd re not lwys esily preditedF husD they dd signi(nt omplexity to the ehvior of mgneti systemsF

riniple of opertion of wivewD where uniform stress n unequivoE lly disriminte etween two sttes e nd fF u(gures A A nd A show shemti of the mgnetoelsti element t rest nd sujet to tensile respF ompressive stressD while

  mterils nnot e piezeoeletriAF he totl numer of symmetry types hs een somewht detedD ut reent works onluded tht there ws IS irreduE ile symmetry types IVTF es wy of turning mehnil signls into eletril signls or the reverseD the piezoeletri e'et is found to hve multiple pplitions todyD suh s sensingD trnsduingD genertion of voltge nd so forthF pigure IFIW ! hemti of the piezoeletri e'et in qurtzF e ompression or tension retes hrges due to the rystlline rrngementF pigure from dev.nsta.orgF he oe0ients d ij whih we will hve use of in the following work re de(ned in this su(gures dA eA nd fA is Qh plot of the orresponding free energy showing the energy lndspe s funtion of the zimuthl φ nd polr ngle θF

	1.3.3 Piezoelectricity sn stressEsed mgnetoeletri ouplingD the most strightforwrd ounterprt of mgE netostrition is piezoeletriityF gontrry to mgneti mterilsD where piezomgnetism is in priniple possile IRU ut in prtie on(ned to ntiferromgnets IVSD it is quite ommon to (nd mterils with liner reltion etween eletri (eld nd polriztionF he origin of this e'et is due to ristllogrphy nd the existene of polr xes in the mteril @morphous wyX D i = d ijk T jk + ij E j , @IFIQA where D is the eletri indutionD T is the lol guhy stress tensor nd is the lol strinF sn prinipleD the piezoeletri tensor d ijk n hve mximum of IV omponentsD lthough this numer is often drmtilly redued depending on the prtiulr symmeE tries exhiited y the mteril onsideredF 1.3.4 Articial magnetoelectric materials yne wy to lift the two onstrints mentioned ove in intrinsi mgnetoeletri mterils" nmelyD wekness of oupling nd low temperture requirements"is to rrnge n inE diret oupling in heterostrutures @usully with two phsesD ut threeEphse systems lso exist IVUD IVVAF ossile wys of hieving the oupling re shown in pigF IFPHF he intermedite etween eletri (elds nd mgnetism is most often mehnilD ltE hough exhngeEis nd hrgeEmedited e'ets re lso studied IVWF he ssoition of piezeoeletri mteril"whih will onvert n eletri (eld in mehnil stress" nd mgnetoelsti mteril"whih will e mgnetilly sensitive to the trnsmitted stress"gives rise to wellEknown heterostruturesF fsillyD piezoeletri phse nd mgnetoelsti phse re elstilly oupledD so tht the strin generted y n pplied eletri (eld in the former will e trnsmitted to the ltterD prompting mgneti e'etF issentillyD this e'et is tntmount to n indued nisotropyF sn prtieD this elsti oupling n for instne e implemented in lmintesD in whih theoretil investigtions of the mgnetoEeletroEelsti oupling in suh systems overed the liner IWHD IWI nd nonliner IWP regimesF por more informtion on the di'erent rti(il mgnetoeletri mterils proposed in the litertureD see reviews IVVD IVWD IWQF ixperimentl reserh on stressEmedited mgnetoeletri e'ets go k to the proE position of twoEphse mterils IWR nd susequent work in the IWUHs with omposite mterils mde of melted IWS nd sintered IWT ferroeletri nd ferromgneti phsesF vterD progress in thin (lm tehnologies mde possile the stking of piezoeletri nd mgnetoelsti lyers to form lmintesF he development of new mterilsD inluding the serh for etter piezoeletri propertiesD ontriuted to the optimiztion of the mgE netoeletri oupling IWUF es resultD the mgnetoeletri ouplingD often mesured in m -1 ye -1 D hs drmtilly inresed over the yers IWV!PHHF everl exmples re shown in pigF IFPIF he si e'et of n indued stress is modi(tion of nisotropy nd hs een oserved repetedlyD inluding in tGgo PHI nd in @qDwnAes PHPF imilr e'ets @inversion of nisotropyA hve een notied in nikel PHQF hese hnges n then result in reversile modi(tions of domin wll on(gurtions PHR s well s mgnetiztion swithing PHS!PIQ t room tempertureD inluding t the nnosle PIRF elsoD sigE ni(nt hnge to domin wll moility nd other kinds of in)uene on domin wll dynmis PHID PIS!PIV hve een mentioned in the literture nd will e further desE ried in susetion IFRFIF hynmi strins ssoited with the propgtion of ousti wves hve lso een shown to in)uene mgneti strutures PIWF wnipultion of domE in wlls in more spei( systems exhiiting mis(t strins etween di'erent lyers hs lso een reported PPHD PPIF enother quite remrkle phenomenon involves hnges in the vlue of mgnetiztion @sturtionAD with the possiility tht the gurie temperture would depend on strin PPPF erti(il mgnetoeletri mterils re thus linked to very dynmi re of reserh of urrent interest ISQF por more omprehensive ount of the tehnologil outlook of mgnetoeletri mterilsD one n onsider severl reviews ISQD IVVD IVWD IWQD IWUF pigure IFPH ! hemti showing the possile rrngements of mgnetoeletri heteE rostrutures nd exmples of their prtil implementsF pigure from IWUF pigure IFPI ! yverview of the vlues of mgnetoeletri oupling otined nd reported in the literture with the ssoited mterils in @A ulk nd @A (lm struturesF pigure from IWUF wuh like domin wllEsed deviesD the integrtion of stressEmedited devies in miE roeletronis hs een proposed in the ontext of severl prtil pplitions in the litertureF ome hve oined nd used the term straintronics to refer to strinEsed systems PIRD PPQF e (rst pplition often onsidered is memoryF wentioned s erly s PHHH PPRD the potentil of mgnetoeletri mterils in memory pplitions hve een repetedly put forwrd PPS!PPVF feside informtion storgeD there re lso posE siilities in informtion proessingD with fennettEloking nd mgneti logi devies in generl PPW!PQP pving the wy for ll spin logi PQQF roponents of mgnetoeletri mterils hve even mentioned their use s reovery devies for nonlssil informtion proessing systems PQRF ytherD more spei( uses hve lso een proposedD suh s eletrilly ssisted mgneti reording PQSF 1.4 Motion induced by uniform stress in magnetoelas-tic materials his setion ontins presenttion of the system proposed nd studied in this thesisD whih dels with domin wll motion driven y uniform stress in mgnetoelsti ferromgnetF fefore it is ddressedD the neessity of reking the mgneti symmetry will e illustrtedF he solution dopted here omes from erlier workD whih will e detiledF pinllyD preliminry studies showing the vlidity of the onept will e disussedF 1.4.1 The need for symmetry breaking spets of improved onveniene of opertion s well s redued energy onsumption PQTF roweverD due to symmetry resons linked to the ft tht the mgnetoelsti energy hs qudrti formD uniform stress typilly nnot disriminte etween two ntiprllel mgnetiztion sttesF st follows tht it nnot indue unidiretionl motion of IVH • wlls @while to e firD it ould do so for WH • domin wlls in prinipleD for instne in mterils with ui nisotropy PPIAF roweverD n pplied stress n ertinly hve n in)uene on the dynmis of motion indued y other mensF his should not ome s surpriseX s mentioned oveD the e'et of the mgnetoelsti intertion is essentilly the retion of new nisotropyF st is quite ovious from iqF @IFWA giving the stedyEstte veloityD tht hnge in the nisoE tropy will result in hnge of veloityF his phenomenon hs een veri(ed experimentlly unidiretionl domin wll motionF roweverD in ring geometryD it n fore domin wll to move towrd given position determined y the prtiulr stress diretionsD in omintion with prtiulr shpes SQ @see pigF IFPQAF purther modi(tions n e rought to the geometry to tully rek the symmetry IIHF rereD prtiulr shpe of the mgnetoelsti element will llow the displement of the domin wllD hieving IVH • 4swithing4 fter eh pplition of the stressF por ontinous domin wll motion in ring geometryD one n lso resort to the use of n rrngement with multiple eletrodes long with the neessry time tution signlsD possiility lredy explored numerilly PRPF e ompletely di'erent pproh is to tke dvntge of the strong pinning tht mgE neti domin wlls n hve on ferroeletri domin wlls PRQF his pinning is gin linked to lol stressEindued nisotropy used y the presene of ferroeletri domin wll etween orthogonl dominsF he result is ptterning of the mgneti mteril pigure IFPQ ! esults of miromgneti simultions showing the ehvior of domin wlls in soElled t eyeEshped mgnetoelsti mgnets sujeted to inEplne stressF he width is vrile ut is typilly round IHH nmF he outer dimeter is SHH nmF pigure from IIHF domin on(gurtion on tht of the ferroeletri on whih it is deposited s illustrE ted in pigF IFPRF he pplition of n eletri (eld will hve the e'et of hnging the domin on(gurtion nd therefore indue domin wll motionD s shown experiE there re high risks tht suh systems would e unrelileF sn dditionD ny imperfetion in shpe or within the mteril will tend to indue is towrd one position or the otherD whih would mke this sheme even more di0ult to implement suessfullyF wore importntlyD this system n only funtion s toggleD whih n e serious limittion in the ontext of memory pplitionsF sndeedD ny stimultion will hnge the stte stress IIHF memory pplitionsD this system hs severl dvntges detiled elowX used to rek the symmetry nd trigger IVH • swithing of mgnetiztion with n pplied fesiility of stressEmedited ontrol of mgnetiztion with nonEvoltile opertionF por energy requirements PPUD PRWF vike with domin wll motionD geometry n lso e ws lso implemented t the nnosle y independent uthors PIRF his shows the ide is to use multiple eletrodesD gin with more omplex opertion ut with redued medited swithing ws rried out t siwx nd showed its nonEvoltilityF his onept destrutive redout in order to know if the system is lredy in the desired stteF enother PSP s well s severl pulitions PPTD PSQ!PSUF ixperimentl vlidtion of stressE regrdless of its previous stteD so tht writing lwys hs to e preeded y nonE pigure IFPS ! wivew hs een the ojet of ptent PSHD hh thesis PSID ook hpter rereD pulse durtionD mgnitude nd orienttion hve to e tightly ontrolledD otherwise presented in the litertureF eginD the interest generlly mentioned here lies in the proE es mentioned in the eginning of this susetionD uniform stress nnot indue desired stte PHWD PPQD PRU @lso desried for intrinsi mgnetoeletri mterils PRVAF egrding domin wll motion indued y n eletri (eldD severl methods hve een with mgneti (eldEdriven domin wll motion in morphous mirowires PIT!PIVF yne n onlude from the experimentl evidene ville tht the in)uene of mehnil stress through the mgnetoelsti intertion hs the generl e'et of deresing domin mentlly PQTD PRRF wiromgneti simultions showed tht distint dynmil regimes existedD nd mentioned the sene of rekdown PRS lthough reserhers lter limed tht depinning ws possile nd somehow linked to the lker rekdown PRTF 1.4.2 Symmetry breaking based on bias magnetic eld • xonEtoggleX the symmetry reking llows for unequivol opertionD whih ensures tht knowledge of the prior mgnetiztion stte of the memory is not required for o rek the symmetry in stressEmedited systemsD n interesting lterntive to wht writingF hs een heretofore mentioned is the use of stti mgneti (eld isF his ide ws wll veloity in the on(gurtions studiedF his result is orroorted y other reports tht the pplition of stress n inrese the oerivity @or propgtion threshold (eldsA implemented in onept ptized wivew euse of its potentil use s mgnetoeE letri memoryF rereD unixil mgnetoelsti element is sujeted to n inEplne stti • pst opertionX studies of its dynmil ehvior showed sunnoseond swithing PSID PSUF nd therefore lok domin wll motion PQUF imilr e'ets hve een oserved in the se of eletri urrent pulses in mterils with perpendiulr nisotropyD with moility vritions up to SHH7 reported y reserhers PISF enlytil tretments hve lso een mgneti (eld H 0 perpendiulr to its esy xisD tilting towrd its diretion the two stle sttes of equl energyF he susequent pplition of unixil stress RS • from its esy • xonEvoltilityX one the stress hs dispperedD the informtion written in the diE retion of mgnetiztion is not lostF fesidesD if pproprite struturl design is xis n energetilly fvor one of these sttesD nd therefore determine unequivolly the proposed in reent works to desrie the in)uene of mehnil stress on the dynmis rried outD signi(nt energy rrier n e rrnged etween the two stle stE (nl stte of mgnetiztion regrdless of its initil stte @see pigF IFPSAF e si underE of spin urrentEdriven motion PQVF enother relted nd intriguing proposition dels with tes so tht the system n retin informtion for hosen @to some extentA period stnding n e otined y n energeti nlysisD tking into ount nisotropy energyD domin wll motion indued y the propgtion of polrized elsti wvesD whih exert of timeF eemn energy nd mgnetoelsti energyF torque nd fore on domin wlls PQWF he fundmentl hurdle of symmetry hs led reserhers to propose lterntives for stressEindued domin wll motionF he most ovious ide is to use nonEuniform stress distriutionsD i.e. stress grdientsF sndeedD the sptil vrition of stress distriution pigure IFPP ! hemti of the systems sed on stressEgrdients onsidered y hen et lF volized stress generted y seprte eletrodes rete potentil wellsF pigure from PRHF • xonEdestrutive redoutX in wivewD reding y eletril mens n e hieved if oupled to gint mgnetoresistne or tunnel mgnetoresistne struture PSVF pigure IFPR ! perroeletri domin pttern n e imprinted on mgneti mterilsF pigure hese tehniques n work with low voltges so s not to disrupt the systemF heE from PQTF reforeD it is possile to red the stte of ell without ltering itF elsoD n ll o'ers the possiility to rete potentil wells in whih domin wll n e trpped @see he sme symmetry prolem n e found in the ontext of mgnetiztion swithing mgnetoeletri nonEdestrutive redout n e dopted to void the use of dditiE pigF IFPPAF eginD this mounts to the retion of lol vritions of nisotropy tht will is to (neEtune the eletri (eld pulse to ensure tht the mgnetiztion does end up in the miromgneti simultions PRIF es onsequeneD vrious solutions hve een put forwrd in the litertureF yne of them urrent PRHF e lter nlysis sed on simpli(ed model gives support to originl the mgnetiztion to hoose etween two stle sttes long two eqully fvorle pthsF n indeed indue domin wll motion in the sene of ny mgneti (eld or eletri xis to stte long the hrd xisD the susequent disppering of the stress will fore pulished in the litertureD miromgneti simultions hve shown tht stress grdients on monodomin mgnet so tht the mgnetiztion goes from one stte long the esy 'et the energy lndspe seen y the domin wllF sn the (rst ount on this ide of monodomin mgneti prtiles using unixil stressF sndeedD if stress is pplied onl strutures llowing mgnetoresistne mesurements PSWF
	RU RV RW SH SI SP SQ SR SS

  PFQPAhe diretion of the e'etive nisotropy thus depends on σF hile it is long the xExis when σ = 0D it n vry nd is prt of the intervl ] -π 4 , + π 4 [F he two ounds orrespond to vlues of σ for whih the mgnetoelsti energy domintes the unixil nisotropy @ 3 2 λ s |σ| K u AF xow it is ovious tht pplying uniform stress hs the e'et of generting tilted nisotropyF he two stle sttes with respet to this nisotropy re not equivlent in terms of eemn energyD hene the symmetryEreking role of the externl mgneti (eld long yF es n sideD it is interesting to notie tht in prinipleD if we pply stress with n ngle loser to π 2 insted of the vlue 3π 4 hosen hereD the eemn energy gp would e even greter nd the system more e0ientF

  his is the min eqution governing the equilirium ehvior of the mgnetiztion orienE ttion within the onsidered strutureF prom mthemtil point of view it is seond UV order integroEdi'erentil eqution with (xed oundry onditions @ H d y,z depends on Φ through n integrl opertionAF he originl hrter of this eqution n e underliE ned y oserving tht it desries oneEdimensionl model utD t the sme timeD its seond term in iqF @PFRHA tkes into ount the tul shpe of the ferromgneti region

	@through the vrile width nd its derivtive AF his point lso represents ruil
	modi(tion introdued with respet to the lssil development summrized in setion
	PFIF hereforeD this pprohD spei(lly elorted for the study of twoEdomin nE
	nostrutureD llows for the nlysis of domin wll on(gurtions with strong redution
	of the neessry omputtionl e'ort with respet toD e.g.D stndrd (nite element miE
	romgneti pprohF he ltterD while llowing to tkle wide vriety of mgneti
	prolemsD is less omputtionlly e0ient for suh simple systemF

rving introdued vrile widthD it is interesting to notie tht it ts through the rtio on one of the exhnge termsF sn order to lerly highlight its in)ueneD let us write the eqution in the sene of ny other ontriutions prt from the exhnge energyD the eqution n e writtenX

  Table of integrals, series, and productsF edemi pressD PHIRF QHU w ermowitz nd se tegunF Handbook of Mathematical FunctionsF hover uE lishersD xew orkD IWUHF QHV prnk t ylverF NIST Handbook of Mathematical Functions Hardback and CD-ROMF gmridge niversity ressD PHIHF QHW omin vud nd quy vemrqundF wgneti pield rodued y rllelepiE pedi wgnet of rious nd niform olriztionF Progress In Electromagnetics ResearchD WVXPHU!PIWD PHHWF PIV Magnetoelectric manipulation of transverse domain walls in magnetoelastic nanostructures

Acknowledgments

Appendix B

Variational procedure he minimiztion of the funtionl U de(ned in iqF @PFTA n e performed s follows

where λ = λ ( r) is vgrnge multiplier introdued to (x the norm of the unit vetor wF hereforeD we hve to minimize

with H l (xedF his funtionl ssumes its extreml vlue when

where the leftEhnd side represents the qâteux derivtive of the funtionl U PUUF sf we de(ne the quntity

elthough in impliit formD this is the eqution giving w( r) in V F et this pointD we n omine it with H l = H 0 + H d F st is importnt to note tht this sustitution nnot e IUU mde in U or U D efore the minimiztionD sine H d is funtion of wF sn henD iqF @fFSA eomes

where we used iqF @PFUA for the demgnetiztion (eldF iquivlentlyD we n lso write