Dynamics of magnetization coupled to a thermal bath of elastic modes

Transcription

1 PHYSICAL REVIEW B 7, Dyamcs of magezao coupled o a hermal bah of elasc modes Erco Ross, Olle G. Heoe, ad A. H. MacDoald Deparme of Physcs, Uversy of Texas a Aus, Aus, Texas 787, USA Seagae Techology, 780 Compuer Ave., Souh, Bloomgo, Mesoa 55435, USA Receved 8 May 005; revsed mauscrp receved 9 Augus 005; publshed 0 November 005 We sudy he dyamcs of magezao coupled o a hermal bah of elasc modes usg a sysem plus reservor approach wh realsc mageoelasc couplg. Afer egrag ou he elasc modes we oba a self-coaed equao for he dyamcs of he magezao. We fd explc expressos for he memory frco kerel ad hece, va he flucuao-dsspao heorem, for he specral desy of he magezao hermal flucuaos. For magec samples whch he sgle doma approxmao s vald, we derve a equao for he dyamcs of he uform mode. Fally we apply hs equao o sudy he dyamcs of he uform magezao mode sulag ferromagec h flms. As expermeal cosequeces we fd ha he flucuao correlao me s of he order of he rao bewee he flm hckess, h, ad he speed of soud he mage ad ha he lewdh of he ferromagec resoace peak should scale as B h where B s he mageoelasc couplg cosa. DOI: 0.03/PhysRevB PACS umbers: 76.0.q, , q, Gb I. INTRODUCTION Thermally duced flucuaos of he magezao are resposble for oe fudameal lm o he sgal-o-ose rao of small mageoressve sesors. The ose scales versely wh he volume of he sesors ad peaks a frequeces,3 ha are ow close o he ever creasg daa rae of magec sorage devces. The crease of daa raes combed wh he coug decrease of he dmesos of he sesors makes magec ose evable ad movaes work amed a achevg a dealed udersadg of s characer. The sadard approach oward modelg of magezao flucuaos s o sar from he Ladau-Lfshz-Glber- Brow equao 4 = E + h +, where s he gyromagec rao, =M/ s he magezao dreco, M s he magezao, M S s he magude of he saurao magezao, E s he free eergy, ad h s a radom magec feld. Ths equao assumes ha he characersc me scale of he magezao dyamcs s loger ha he ypcal me scale of he evrome ha s resposble for he dsspave erm proporoal o. I pracce he use of hs equao s parally cosse, resulg some praccal lmaos o s applcao. 5,6 The source of he problem s ha he dsspao s local me. Because of he flucuaos-dsspao heorem, hs mplcly requres he radom feld o have whe ose properes,.e., o have zero auocorrelao me. Sce he corbuo of he radom feld o he magezao dyamcs h depeds o, Eq. exhbs whe mulplcave ose. 7 I follows ha order o egrae Eq. relably we eed o rack he evoluo of o very shor me scales for whch he whe ose approxmao for h s lkely o be uphyscal. I hs paper we address he physcs ha deermes he correlao me of he radom feld. We sar Sec. II by cosderg a formal model of a magec sysem coupled o a evrome ad specalze Sec. III o a evrome cossg of elasc modes. I Sec. IV we cosder he case whch a sgle magec mode correspodg o cohere evoluo of he magezao a small sgle-doma sysem s coupled o he elasc evrome. I Sec. V we cosder a h flm geomery whch he magezao s coupled o elasc modes of he sysem ad s subsrae. Fally Sec. VI we coclude wh a dscusso of he possble role of oher sources of dsspao, parcular dsspao due o parcle-hole excaos he case of meallc ferromages. II. GENERIC RESERVOIR Callg q he degrees of freedom of he reservor, we cosder he followg form for he oal Lagraga: L = L S x, x + L R q,q + L I x,q Lx, where L S x, x s he Lagraga ha descrbes he dyamcs of he magezao whe o coupled o exeral degrees of freedom, L R q,q s he Lagraga for he reservor ad L I x,q s he eraco Lagraga ha couples he magezao o he reservor degrees of freedom. The erm Lx s a couer erm ha depeds o ad he parameers of he reservor bu o o he dyamc varables of he reservor. 8,9 Ths erm s roduced o compesae a reormalzao of he eergy of he sysem caused by s couplg o he reservor. 8 The Ladau-Lfshz equaos for he decoupled sysem magezao follow from he magec Lagraga, 098-0/005/77/744/$ The Amerca Physcal Socey

2 ROSSI, HEINONEN, AND MACDONALD L S =V M A E s dx, where A s a vecor feld defed by he equao A= ad E S s he magec free eergy fucoal ad V M he volume of he ferromage. We model he reservor as a se of classcal degrees of freedom, L R = 3 m q E R q. 4 The Euler-Lagrage equaos for he oal Lagraga yeld he followg coupled dyamcal equaos: m q = q L R q,q + L I,q = E S, L I,q + L. 6 Whe L I s lear he coordaes of he bah, we ca formally egrae 5 o ge q as a fuco oly of he al codos ad ad he ser he resul 6 o elmae he reservor coordaes from he dyamcal equaos for, egrag ou he reservor degrees of freedom. A example of he applcao of hs procedure for a quaum mechacal model of he eraco bewee magezao ad reservor degrees of freedom ca be foud Ref. 0. III. MAGNETIZATION COUPLED TO ELASTIC MODES: GENERAL If we cosder oly log wavelegh vbraos we ca rea he lace as a couous medum ad use resuls from elascy heory. The poeal eergy fucoal, E R,ofhe elasc medum ca he be expressed erms of he sra esor u,j, u j u + u j x j x, where u s he dsplaceme vecor feld. We wa o sudy he dyamcs of he magezao whe coupled o elasc deformaos of he sysem. We wll be eresed applyg our resuls o polycrysalle elasc meda whch ca be reaed as soropc o a good approxmao. I s que sraghforward, albe que edous, o exed our resuls o he case of osoropc meda wh specfc lace symmeres. For soropc elasc meda follows from geeral symmery cosderaos ha, o lowes order, we ca express he mageoelasc eergy he form, 3 E I = B j u j dx, V M,j= where B s he mageoelasc couplg cosa. For he case of sof ferromage h flms, he ma corbuo o 5 7 he mageoelasc eergy wll be gve by he mageosac eergy depedece o sra. Ths corbuo o E I s ormally referred o as he form effec. 3 The cosa B ca be exraced from mageosrco daa. For a soropc elasc medum wh soropc mageosrco,, we have ha B = 3 E, 8 where E s he Youg s modulus ad he Posso s rao. The Lagraga for a elasc reservor L R s L R = V u dx E R, 9 where s he mass desy, V he oal volume of he elasc medum magec flm plus subsrae ad E R s gve by 4 E E R =V 3 u E + j + 3 u,j= + dx. = 0 The equao of moo for he dsplaceme wll he be u = ux E Ru + E I,u. I wll prove useful o expad u erms of he elasc ormal modes f, u = q f x, where he fucos f sasfy he boudary codos approprae for u ad sasfy E R f f x = f x, N, 3 MV f x f m xdx = m, where M s he oal mass, M V dx. I erms of he degrees of freedom, q, we have wh PHYSICAL REVIEW B 7, L I = E I = B q j f,j V j dx M f j f x j + f j. x 4 5 We he see ha he eraco Lagraga s lear he coordaes q, wh couplg cosas c j f,j V j dx. 6 M Ths propery wll allow us o egrae ou he reservor degrees of freedom o oba a equao for he dyamcs of he magezao erm of aloe. 744-

3 DYNAMICS OF MAGNETIZATION COUPLED TO A Le us frs dscuss he dyamcs of he reservor degrees of freedom q. Usg Eqs. 4 we fd he dyamcal equaos q = q B M c. Iegrag 7 we fd q = q 0 cos + q 0 s B M 0 s c d, 7 8 where q 0 ad q 0 are he al values of q ad q, respecvely. The couplg of he magezao o he reservor wll cause dampg ad frequecy reormalzao. I order o be able o separae he wo effecs s useful o egrae he las erm o he rgh-had sde of 8 by pars obag q = q 0 cos + q 0 s B M c + B M c 0cos + B M 0 V M c dcos x, dx. x 9 Usg he expresso of he eraco Lagraga gve by 5 ad he defo of he couplg cosas c we have L I = B q c. 0 Combg Eqs. 6, 9, ad 0 for he dyamcs of he magezao we fd = E S + L + x, B c q 0 cos + q 0 s B M c c x, + B M c 0cos c x, + B dx cos M c 0 dvm x, x c x,. The couer erm L of he oal Lagraga s defed o cacel he frequecy reormalzg erm B, M c c I follows from Eq. 6 ha c l = f l x x, + f. x l. 3 To smplfy ad exrac he physcal coe from hese cumbersome equaos, we defy he memory frco kerel esor jm, jm,,x,x c mx, c jx, B M cos, 4 where s he Heavsde fuco. We also recogze he radom feld h, hx, B s c. q 0 cos + q 0 5 Assumg ha he dsrbuo of al posos of he evrome degrees of freedom follows he caocal classcal equlbrum desy for he uperurbed reservor we fd ha hx, =0, 6 h j x,h m x, = K BT jm,,x,x. 7 I erms of jm ad h he dyamcal equao for akes he form l = jl E S j + jl h j + jl 0 + jl, dx jm dv M,,x,x m m x B M c 0cos c j. The fal erm s a arfac of he assumpo ha he al sae he reservor was decoupled from he sysem. 9,5 Droppg hs erm, he dyamcal equaos for magezao coupled o a hermal bah of elasc modes s l = jl E S j + jl h j PHYSICAL REVIEW B 7, jl 0 dx jm dv M,,x,x m m x 8 wh jm defed by 4 ad h a radom feld wh sascal 744-3

4 ROSSI, HEINONEN, AND MACDONALD properes gve by 6 ad 7. Equao 8 s que geeral. I parcular oce ha o oba 8 we dd o perform ay expaso. As a cosequece, as log as we keep he exac form for E S, Eq. 8 cludes also he effecs of sp wave eracos. I prcple we could also clude E S a erm o ake o accou he scaerg of sp waves due o dsorder. Equao 8 does o, however, ake o accou he couplg bewee he magezao ad parcle-hole excaos. As we dscuss Sec. VI, hs couplg appears o be of crcal mporace may meallc ferromages. Equao 8 s very dffere from he sadard sochasc Ladau-Lfshz-Glber s-llg equao, Eq.. Because he mageoelasc eergy, E I, 7, s olear he magezao, 8 boh he dampg kerel ad he radom feld deped o he magezao ad herefore are sae depede. Ths s coras wh he s-llg equao for whch boh he dampg kerel,, ad he radom feld are depede of. Aoher dfferece bewee Eq. 8 ad he s-llg equao s ha he dampg kerel, jm, s geeral a esor. The esor characer of he dampg has bee suggesed prevously o pheomeologcal grouds. 6 Sarg from he physcal couplg 7, our approach he esor characer of jm appears aurally as a cosequece of a he oleary of he mageoelasc couplg 7, b he asoropy of he elasc modes due o he boudary codos ad/or asoropy of he elasc properes. For small oscllaos of aroud s equlbrum up o quadrac order, he kerel jm ca be assumed o be depede of. Eve hs learzed case, he dampg kerel ha appears 8 wll sll have a esor form due o he asoropy of he elasc modes. As meoed above, he sadard s-llg dampg kerel s smply,.e., he dampg s frequecy depede. As a cosequece, from he flucuao-dsspao heorem, we have ha he specrum of he radom feld ha appears s also frequecy depede. Ths dffers from Eq. 8 for whch he dampg kerel, ad herefore he specrum of he radom feld, s frequecy depede. Gve he geomery ad he maeral properes of he sysem we ca fd he elasc modes, f, ad he egrae Eq. 8 usg a mcromagec approach. The egrao of Eq. 8 could gve sgh parcular o he dampg of he uform magezao mode for dffere geomeres ad show he rage of valdy of he classc pcure 6 of a wo sage dampg process whch he moo of he cohere magezao duces ouform magec modes o shor me scales ha he decay o lace vbraos. IV. MAGNETIZATION COUPLED TO ELASTIC MODES: UNIFORM MAGNETIZATION We ow sudy he dyamcs of he uform magec mode he case whe we ca eglec s eraco wh sp waves ad he oly couplg o exeral degrees of freedom s mageoelasc. Projecg Eq. 8 o he uform mode we fd ha d l d = jl V M V M E S j dx + jl V M V M h j dx + jl V M 0 dx jm,,x,x d m dv M dxv M m d. Le us defe he space averaged error feld h hx,dx, V M V M he dampg kerel jm, dx jm,,x,x, V M V M dxv M ad he coeffces c c l dx. V M l Usg he fac ha s uform we oba c l = V M f l + f dx. x x l I erms of he coeffces c l we ca he wre h l = B V M c l q 0 cos + q 0 s 9 30 ad B jm = MV M c j c m cos. 3 The uform magezao dyamcs ca he be expressed erms of he spaally averaged radom feld h ad memory frco kerel jl, wh ad d l d PHYSICAL REVIEW B 7, = jl V M V M E S j dx + jl h j + jl 0 d jm, d m m d h =0 h jh m = K BT V M jm,. V. THIN FILM UNIFORM MAGNETIZATION DYNAMICS We ow apply Eq. 3 o sudy he dyamcs of he uform magezao a h ferromagec flm placed o 744-4

5 DYNAMICS OF MAGNETIZATION COUPLED TO A PHYSICAL REVIEW B 7, TABLE I. Elasc properes. c,c l are he rasverse ad logudal speed of soud respecvely. Magec flm Subsrae/cappg layer E 00 Gpa 80 Gpa g/cm g/cm 3 c 4.0 km/s.0 km/s c l 5.0 km/s 4. km/s FIG.. Geomery cosdered for he case of a h ferromagec flm o a omagec subsrae. op of a omagec subsrae ad covered by a omagec cappg layer, as llusraed Fg.. We assume ha all meda are polycrysalle ad rea hem as soropc. We wll assume he laeral sze, L s, Fg., o be much bgger ha he flm hckess h. Noce ha f we ake L s bgger ha he doma wall wdh our assumpo ha he ouform magec modes are queched would o be vald aymore. We wll cosder oly oscllaos of he magezao aroud a equlbrum poso parallel o he x 3 axs so ha we ca calculae he dampg kerel esor jm assumg he elasc modes o deped oly o x 3. Oherwse, o fd he correc dampg kerel, we would have o ake o accou he fac ha he laeral sze, L s, s fe ad solve he full hree-dmesoal 3D elascy problem for he elasc modes. A. Dampg kerel ad radom feld To fd he dyamcs of he magezao usg Eq. 3 we eed o evaluae he memory frco kerel jm. The frs sep hs calculao s he deermao of he elasc ormal modes f whch sasfy he followg equao: f = E + f E + f. 35 We allow he flm, he subsrae, ad he cappg layer o have dffere elasc properes ad solve Eq. 35 separaely he dffere subsysems usg he approprae elasc cosas. We assume for he sake of defeess ha he subsrae ad cappg layer maeral s decal. We he mach soluos by mposg he couy of dsplaceme ad sresses a he erfaces x 3 =0, ad x 3 =h. As boudary codos we assume he op surface of he cappg layer o be free ad o dsplaceme a he boom of he subsrae. Because our case he elasc modes oly deped o x 3, Eq. 30 smplfes o wh c l = L s f l 3 + 3l f f h f 0. The spaally averaged dampg coeffces have a smple expresso erms of he f, jl = L sb f cos Mh j 3 + 3j l 3 + 3l. 36 Equao 36 follows from he compleeess relao of he polarzao vecors. Oce we kow he coeffces f, Eqs. 33, 34, ad 36 compleely specfy he dyamcal equao 3 for he magezao. As a example we cosder he case of a polycrysalle ferromagec h flm, lke YIG, placed o a subsrae of a polycrysalle paramage lke aalum, Ta. As ypcal values we ake 7 he oes lsed Table I. For he mageosrco we assume =0 6. Usg Eq. 8, we fd ha B =40 6 ergs/cm 3. Gve he elasc modes mpled by hese parameer values, we ca calculae he coeffces f. Oce we kow he coeffces f we have all he elemes o compleely specfy Eq. 3. We geerae a sochasc feld h wh he correc sascal properes by usg s Fourer represeao. To oba yy = G 37 we choose 8 yy = G, 38 where ad y = G = ye d Ge d. I our case we have from Eq. 3, ha he memory frco kerel jl depeds separaely o ad. As a cosequece, hrough 34, we have ha he average hh does o deped oly o he me dfferece =. The 744-5

6 ROSSI, HEINONEN, AND MACDONALD PHYSICAL REVIEW B 7, radom feld h herefore does o defe a ergodc process ad parcular we cao use Eq. 38. For hs reaso s covee o defe he auxlary radom varables, x f q 0 cos + q 0 s ad he auxlary kerels g so ha we have f cos x x j = K BT M g j. The radom varables x herefore descrbe a ergodc process ad we ca use Eq. 38 o geerae hem. I erms of x ad g we have h l = B h x l 3 + 3l, jm = L sb Mh g j 3 + 3j m 3 + 3m. 39 To geerae he radom feld ad calculae jl we he mus calculae he quaes g ad her Fourer rasforms g. Fgures a, b, 3a, ad 3b show some ypcal profles for g ad g usg for he mechacal properes he values of Table I. We fd ha geeral g does o deped o he hckess of he cappg layer L. I he lm whch we ca learze he mageoelasc eraco wh respec o, we have jm = B L s Mh g j jm. 40 The dampg kerel s dagoal wh compoes equal, apar from a overall cosa, o g j, coras o he s-llg equao for whch we have jl = jl. The power specrum of he radom feld compoe, h j, s he proporoal o g j, coras o he s-llg equao for whch he power specrum of each compoe h j s smply a cosa. Noce ha eve hs lm jm preserves s esor form due o he asoropy of he elasc modes. I our specfc case we have g =g g 3 due o he dfferece bewee he rasverse ad logudal speeds of soud. From Fgs. a ad b, we see ha g goes o zero for mes loger ha D 50 0 =5h/c, where cc,m s he rasverse speed of soud he mage. For a flm 0 m hck we he fd D 0 ps. Whe he releva frequeces of are much lower ha / D, we ca replace he dampg kerel gve by 40 wh he smple kerel wh jeff gve by jm = jeff jm FIG.. Color ole Profles of ĝ g c /hl+h+l a ad ĝ 3 g 3 c /hl+h+l b for he case of a h magec flm o a aalum subsrae; 0 L/c,M. For he sadard s-llg equao g would smply be a Drac dela ceered a =0. jeff = B L s g j d. 4 Mh0 I hs lm we recover a dampg kerel of he same form as he oe ha appears he s-llg equao. Here jeff s he equvale o. From he resuls show Fgs. a ad b we see ha we have ad he 0 g j d h L + h + L c 3 jeff = B h c 3. 4 We fd ha he dampg of magec modes h flms s proporoal o B h. Assumg he values gve Table II we fd eff = eff

7 DYNAMICS OF MAGNETIZATION COUPLED TO A PHYSICAL REVIEW B 7, B. Iegrao Afer geerag he radom feld h he way descrbed above we ca proceed egrag Eq. 3. We assume E S /= V M H eff wh H eff =0,0,H eff ad H eff smply a cosa. Le us defe he dmesoless quaes ˆ H eff, Ĥ eff H eff H eff, ĥ h H eff, ˆ jm jm, Tˆ K BT, H eff H eff V M he dmesoless form Eq. 3 akes he form d l wh dˆ = jl Ĥ effj + jl ĥ j + jl 0 ˆ dˆ ˆ jm ˆ,ˆ d m m dˆ 43 ĥ j ˆ =0, ĥ j ˆĥ m ˆ = Tˆ ˆ jm ˆ,ˆ. 44 Smlarly, for E S /= V M H eff, he sadard s-llg equao,, for he uform mode, akes he dmesoless form wh d dˆ = Ĥ eff + Ĥ + d dˆ 45 FIG. 3. Color ole Values of Reĝ Reg c /hl+h+l a ad Reĝ 3 Reg 3 c /hl+h+l b a he elasc modes frequeces for he case of a h magec flm o a aalum subsrae. Show are he values for h=0.0l, damods, ad h=0.0l, crcles. For ay Reĝ s uque eve hough hs s o compleely evde from he fgure because order o show he behavor of he auxlary kerels over a wde frequecy rage, he resoluo s o hgh eough o always show he separao bewee he sgle pos. For he sadard s-llg equao g would smply be a cosa. TABLE II. Magec properes ad dmesos for he sysem suded. Quay Value s G B 40 6 ergs/cm 3 L h 50 G m 0 m ĥ j ˆ =0, ĥ j ˆĥ m ˆ = Tˆ ˆ ˆ. 46 Usg for jm he expresso 39 ad for g,g he resuls show Fgs. a, b, 3a, ad 3b ad assumg Tˆ =0 ad he values gve Table II we egrae Eq. 43. We used he sochasc Heu scheme ha esures covergece o he Sraoovch soluo eve he lm of zero auocorrelao me for he radom feld. 7 The resuls of he egrao are show Fgs. 4a, 4b, ad 5a. As al codo we ook =0.6,0,0.8, d/dˆ=0. We he egraed Eq. 45 seg = eff wh eff calculaed usg 4. The resuls of he egrao are show Fgs. 4a, 4b, ad 5b. From Fgs. 4a, 4b, 5a, ad 5b we see ha o average Eqs. 43 ad 45 gve very smlar resuls. Ths s expeced because for he al codos chose we are he lm of small oscllaos aroud he equlbrum poso ad herefore he depedece of ˆ jm o s eglgble. The ma dffereces, for he case cosdered, bewee he resuls obaed usg 43 ad 45 are he radom flucuaos of. Ths s a cosequece of he dffere correlao me of he radom feld h used 43 ad 45. For example, we oce ha Eq. 43 seems o gve a less osy dyamcs ha 45 eve hough for boh smulaos ĥ s of he same order of magude. If we zoom o a shor me erval, Fg. 4b, as a maer of fac, we see ha o very shor me scales he amplude of he radom flucuaos for he wo smulaos s very smlar. However for 45 flucuaos wh he same sg are much more lkely ha for 744-7

8 ROSSI, HEINONEN, AND MACDONALD PHYSICAL REVIEW B 7, FIG. 4. Color ole 3 as a fuco of me obaed egrag he sadard s-llg equao, 45, ad Eq. 43. Ia he race obaed usg Eq. 43 has bee offse up by for clary. I b he race of 3 s show o a shor me scale. 43. Ths s due o he dffere specral desy of he radom feld. For 45 we smply have h j =T, whereas for 43 h j s equal o g j cosderg ha for our smulao, o a good approxmao, we ca eglec he depedece of he radom feld o. I parcular for 43 h j has a low frequecy cuoff a = 0 c,m /L, where c,m s he rasverse speed of soud he mage. Ths mples ha for 43 we have a much lower probably ha for 45 o have cosecuve flucuaos of he radom feld wh he same sg wh he resul ha he dyamcs appears less osy. VI. DISCUSSION AND CONCLUSIONS I hs paper we derved he equao for he dyamcs of he magezao akg o accou s couplg o he lace vbraos. The equao ha we oba, 8, s que FIG. 5. Evelope curves of he race of me as obaed egrag Eq. 43 a ad Eq. 45 b. oscllaes bewee he maxmum ad mmum value gve by he evelope curves wh frequecy H eff, equal o he dmesoless us used. geeral. Equao 8 wll have he same form also f we clude sp-sp ad sp-dsorder eracos. To ake o accou hese pheomea s ecessary oly o add he approprae erms o he eergy fucoal E S. From he geeral equao we derved he equao, 3, for he dyamcs of he uform magec mode a h magec flm whe ouform magec modes ca be assumed froze ou. We fd ha geeral he radom feld ha appears he dyamcal equao for he magezao has a correlao me, D, of he order of he rao bewee he flm hckess, h, ad he soud velocy c. Whe he me scale for he dyamcs of he magezao s much loger ha D, we recover he sochasc LLG equao. I hs lm we calculaed he value of he effecve Glber dampg cosa,. For ypcal ferromagec sulaors, lke YIG, we fd 0 4, good agreeme wh he values measured expermes. 6,9 We ca he coclude ha for magec sulaors mageoelasc couplg s he ma source of magezao dampg

9 DYNAMICS OF MAGNETIZATION COUPLED TO A Our work predcs ha magec resoace expermes o ferromagec sulaors should be able o observe he asoropy of he dampg ad as a cosequece of he correlao of he hermal flucuaos. Wh our heory s possble o exacly calculae he specral desy of he hermal flucuaos. The specral deses for small samples wll be dffere from he oe observed bulk expermes because of he dscreeess of he elasc modes. I would be very eresg o es hese resuls wh ew expermes o small ferromagec sulaors samples. I parcular for h flms oe expermeal cosequece of our work s ha he correlao me of he magec flucuaos wll be of he order of h/c where h s he hckess of he ferromagec flm ad c he speed of soud he mage. We also foud ha he lm whe he magezao evolves o me scales much bgger ha h/c he dampg of he magec modes s drecly proporoal o B h. The lewdh of he ferromagec resoace peak sulag ferromagec h flms should herefore scale as B h, whch, prcple, ca be cofrmed expermeally. For ferromagec meals, lke permalloy, we also fd 0 4. Ths value s abou wo orders of magude smaller ha he value observed expermeally. 0 The reaso s ha ferromagec meals he elecroc degrees of freedom are he ma source of dsspao for he magezao., Sarg from a model of localzed d sps exchage coupled o he s-bad elecro, he eraco Lagraga wll be L I = J sd dxx sx, where J sd s he exchage couplg cosa ad s s he coduco elecros sp desy sx = a,b a x ab b x, where are he s-bad carrer feld operaors ad ab he represeao of he sp operaor erms of Paul marces. By egrag ou he s-bad degrees of freedom, he lear respose approxmao Sova e al., 3 for he dampg of he uform magec mode fd g B J = lm sd 0 d3 k 3 a k + b k a,b d A a,ka b,k + f f +, 47 where A a,k ad A b,k are he specral fucos for s-bad quasparcles ad f s he Ferm-Drac dsrbuo. Equao 47 gves zero dampg uless here s a fe-measure Ferm surface area wh sp degeeracy or here s a broadeg of he specral fuco due o dsorder. 4 Characerzg he quasparcle broadeg by a smple umber / s, where s s he quasparcle lfeme, we ca assume PHYSICAL REVIEW B 7, A a,k = a,k + /4. 48 Iserg hs expresso for he specral fucos 47 we fd as a fuco of he pheomeologcal scaerg rae. Noce ha 47 cludes he corbuo boh of rabad, ad erbad 5 7 quasparcles scaerg eves. The rabad corbuo s due o sp-flp scaerg wh a sp-spl bad ad s ozero oly whe rsc sp-orb couplg s prese. From Eq. 47, usg he expresso for A a,k gve 48, we see ha he lm of weak dsorder, small, he rabad corbuo o s proporoal o /, agreeme wh expermeal resuls for clea ferromagec meals wh srog sp-orb couplg 8 3 ad prevous heorecal work. 6,7,3 35 Smlarly from 47 we see ha he erbad corbuo o s proporoal o. Ths resul agrees wh he expermeal resuls for ferromagec meals wh srog dsorder 36 ad prevous heorecal work. 5 7 Noce ha Eq. 47 mplcly also cludes he corbuo due o he so-called sp-pumpg effec 37 4 whch sps are rasferred from he ferromagec flm o adjace ormal meal layers as a cosequece of he precesso of he magezao. I order o calculae hs effec frs approxmao we smply mus subsue 47 he coduco bad quasparcle saes,, calculaed akg o accou he heerogeey of he sample. Assumg for he scaerg rae, / s, ypcal values esmaed by raspor expermes, from Eq. 47 we fd values of good agreeme wh expermes. I summary we have suded deal he effec of he mageoelasc couplg o he dyamcs of he magezao. Sarg from a realsc form for he mageoelasc couplg we have foud he expresso for he dampg kerel, jm. We fd ha geeral jm s a odagoal esor olocal me ad space. The kowledge of he exac expresso of jm allows us o correcly ake o accou he auocorrelao of he ose erm overcomg he zero correlao approxmao of he sochasc Ladau-Lfshz- Glber equao. We fd ha for h flms for whch he sgle doma approxmao s vald, boh he dampg ad he flucuaos correlao me are proporoal o he flm hckess. Our resuls apply o sysems for whch he drec couplg of he magezao o he lace vbraos s he ma source of he magezao relaxao. We have show ha hs s he case for ferromagec sulaors whereas for ferromagec meals he magezao relaxao s maly due o he s-d exchage couplg. ACKNOWLEDGMENTS I s a pleasure o hak Harry Suhl, Thomas J. Slva, Alvaro S. Núñez, ad Joaquí Ferádez-Rosser for helpful dscussos. Ths work was suppored by he Welch Foudao, by he Naoal Scece Foudao uder Gra Nos. DMR ad DMR-00383, ad by a gra from Seagae Corporao. APPENDIX: SIMPLE ESTIMATE OF eff Le us sar from he defo of jm Eq. 3, 744-9

10 ROSSI, HEINONEN, AND MACDONALD jm = A c j c m cos, A where A B / MV M. For he case of h flm we foud c l = L s f l 3 + 3l, where f f h f 0. Noce ha, by defo, f are dmesoless ad so are he quaes f. Assumg ha a equlbrum s =0,0, ad keepg oly he leadg erms he expresso for c l we have c l = L s f l. Le us ow expad he collecve dex s compoes, k,s where s s he polarzao dex of he elasc modes. The, usg he compleeess of he polarzao vecors ad he fac ha he polarzao drecos are parallel o he axs x,x,x 3 we have jm = A = A L s4 k,s = A L s4 k Now oe ha c j c m cos f j k,s k,s f m k,s cos k,s f k j f k m jm cos k,j. k,j M = L s L+ĥ + Lˆ, V M = L s h, where ĥh/l, LˆL/L. The we ca wre B jm = L+ĥ + Lˆh f k j jm cos k,j. k k,j For small eough h/l we ca assume f k j kh wh a cuoff for k D such ha k D h=. We ca he defe he cuoff frequecy D ck D =c/h. Wh hs approxmao we have k f j k cos k,j k,j = k + cos k,j D k,j = k,j D 0 cos d + D D D 0 0 cos d + D = D e D, D 0 where 0 c/l. I hs approxmao we ca he wre B jm D e D. L+ĥ + Lˆh D 0 Iegrag hs expresso bewee =0 ad = we fd B eff = B = h L L+ĥ + Lˆh D 0 L+ĥ + Lˆh c c B = h +ĥ + Lˆ c 3. A To be more accurae le us defe he fucos c f ĝ j k j ĥ+ĥ + Lˆ L cos k,j k k,j so ha we ca wre jm = B L L c ĝ j. The fucos ĝ j are ploed Fg.. Iegrag ĝ j bewee 0 ad we fd ad he fally PHYSICAL REVIEW B 7, D ĝ j d h c eff = B h c c, aalogously o wha we foud prevously A. N. Smh, Appl. Phys. Le. 78, O. Heoe, IEEE Tras. Mag. 38, O. Heoe ad H. S. Cho, IEEE Tras. Mag. 40, W. F. Brow, Phys. Rev. 30, D. A. Gara, Phys. Rev. B 55, N. Smh, J. Appl. Phys. 90, J. L. García-Palacos ad F. J. Lazaro, Phys. Rev. B 58, A. O. Caldera ad A. J. Legge, A. Phys. N.Y. 49, U. Wess, Quaum Dsspave Sysems World Scefc, Sgapore,

11 DYNAMICS OF MAGNETIZATION COUPLED TO A 0 A. Rebe ad G. J. Parker, Phys. Rev. B 67, H. Suhl, IEEE Tras. Mag. 34, C. Kel, Rev. Mod. Phys., E. W. Lee, Rep. Prog. Phys. 8, L. D. Ladau ad E. M. Lfshz, Theory of Elascy Pergamo, New York, P. Hagg, Sochasc Dyamcs, eded by L. Schmasky- Geer ad T. Pöschel, Lec. Noes Phys. Vol. 484 Sprger, New York, M. Sparks, Ferromagec Relaxao Theory McGraw-Hll, New York, R. M. Bozorh, Ferromagesm Va Nosrad, New York, C. W. Garder, Hadbook of Sochasc Mehods for Physcs, Chemsry, ad he Naural Sceces Sprger-Verlag, New York, H. Che, P. De Gaspers, ad R. Marcell, IEEE Tras. Mag. 9, M. Covgo, T. M. Crawford, ad G. J. Parker, Phys. Rev. Le. 89, C. Kel ad A. H. Mchell, Phys. Rev. 0, E. A. Turov, Ferromagec Resoace, eded by S. V. Vosovsk Pergamo, Oxford, U.K., J. Sova, T. Jugwrh, X. Lu, Y. Sasak, J. K. Furdya, W. A. Akso, ad A. H. MacDoald, Phys. Rev. B 69, PHYSICAL REVIEW B 7, Y. Tserkovyak, G. A. Fee, ad B. I. Halper, Appl. Phys. Le. 84, B. Herch, D. Fraová, ad V. Kamberský, Phys. Saus Sold 3, V. Kamberský, Ca. J. Phys. 48, L. Berger, J. Phys. Chem. Solds 38, S. M. Bhaga ad L. L. Hrs, Phys. Rev. 5, S. M. Bhaga ad P. Lubz, Phys. Rev. B 0, B. Herch, D. J. Meredh, ad J. F. Cochra, J. Appl. Phys. 50, J. M. Rudd, J. F. Myrle, J. F. Cochra, ad B. Herch, J. Appl. Phys. 57, V. Korema ad R. E. Prage, Phys. Rev. B 6, V. Korema, Phys. Rev. B 9, J. Kues ad V. Kamberský, Phys. Rev. B 65, J. Kues ad V. Kamberský, Phys. Rev. B 68, S. Igvarsso, L. Rche, X. Y. Lu, G. Xao, J. C. Sloczewsk, P. L. Troulloud, ad R. H. Koch, Phys. Rev. B 66, L. Berger, Phys. Rev. B 54, Y. Tserkovyak, A. Braaas, ad G. E. W. Bauer, Phys. Rev. Le. 88, E. Smaek ad B. Herch, Phys. Rev. B 67, E. Smaek, Phys. Rev. B 68, A. Rebe ad M. Smoao, cod-ma/0450 upublshed. 744-