exchange and anisotropy

7/28/2019 Exchange and Anisotropy

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C H A P T E R 5

Exchange and Anisotropy

According to the resul ts of Chapter 4 , in c lass ica l e lec t romagnet ism andthermodynamics magne t ized mat te r i s charac te r ized by the cons t i tu t ivelaw M(H) . A s tudy o f the in te rna l s t ruc tu re o f a g iven medium i s thenneede d , in o rder to iden t i fy the p rocesses con t ro ll ing the onse t o f the m acro-scop ic mag ne t iza t ion M a nd the ensu ing p roper t i e s o f M(H) . In th i s re spect ,two bas ic me chan ism s a re a t the roo t s o f the behav ior o f mag ne t ic m ate ria ls ,exchange and an i so t ropy . Exchange der ives f rom the combina t ion o f theelect ros ta t ic co upl in g betw een e lec t ron orbi ta ls an d the nec ess i ty to sa t is fythe Paul i exclus ion pr inciple . I t resul ts in spin-spin in teract ions that favorlong- range sp in o rd er ing over m acroscop ic d i s tances . Anisotropy i s ins teadm ainly re la ted to in teract ions of e lec t ron orbita ls wi th the poten t ia l createdby the ho st ing la t tice. As the la tt ice sym m etry is ref lec ted in the sym m etryof the po ten t ia l, the resu l t is tha t sp in o r ien ta t ion a long ce r ta in sym m et ryaxes of the ho st ing la tt ice become s ene rget ica l ly favored.The s tu dy of the q ua ntu m or igin of these effects is a fasc inat ing topicin i t se l f , but i t i s outs ide the scope of the approach we are pursuing. Thequan tum t rea tment i s no t needed i f one accep ts the s impl i f i ed pheno-m e n o lo g ic a l f r am e i n t ro d u c e d i n C h a p t e r 3 , w h e r e a m a g n e t i c m e d iu m ism ode le d as a col lec tion o f e leme nta ry m agne t ic mo me nts , an d the magn e t i -z a t io n M , i n t e n d e d a s a n a v e r a g e o v e r v o l u m e s c o n t a in i n g m a n y m o m e n t s ,is the quant i ty of in teres t . To our purposes , the mo lecular ield a p p r o a c h ,discu ssed in Sect ion 5 .1 , wi l l prov e am ply suffic ient . Similar ly , in Sect ion5.2 we are not going to consider the microscopic or ig in of anisotropy. Weshal l jus t accept the existence of aniso tropy as a we l l -prov en e xper im enta lfac t and we sha l l use symmet ry a rguments to d raw conc lus ions abou t thevar ious fo rms in which an i so t ropy can become mani fes t and the va r iousw ays in wh ich i t can a ffec t the behav ior o f M. M any ph enom ena , and f ir s to f a ll the fo rmat ion o f mag ne t ic dom ains , a re gover ned by the sy m m et ryproper t i e s o f the an i so t ropy energy , qu i te independen t ly o f the spec i f i cmic roscop ic m echan ism g iv ing ri se to tha t pa r t i cu la r sym m et ry scheme.

1 2 9


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130 CHAPTER 5 Exc han ge and AnisotropyA perv ad in g aspect , encou nte red severa l t imes in the p reced ing chap-ters , i s the role that spat ia l sca les have in the descr ipt ion of magnet ic

phenomena . The ob jec t ive o f th i s chap te r i s to in t roduce the re levan tenerg y te rms tha t, toge the r wi th m agne tos ta t i c energy , gov ern the fo rma-t ion o f magne t ic domains . The goa l i s thus even tua l ly to t r ea t space-dep en den t p rob lem s , in wh ich M exh ib it s de fin it e spa t ia l va r ia t ions f rompoin t to po in t . As we have d i scussed a t the beg inn ing o f Chap te r 3 , oneaims a t a descr ipt ion involving local quant i t ies that are ac tual ly the resul to f a v e r a g e s o v e r e l e m e n ta r y v o lu m e s & V l a rg e e n o u g h t o c o n ta in m a n ye l e m e n ta r y m o m e n t s , b u t a t th e s a m e t im e s m a l l e n o u g h w i th r e s p e ct tothe typ ica l sca le over which impor tan t magne t iza t ion changes occur . Wesha l l see tha t the th ickness o f the domain wa l l s separa t ing ne ighbor ingdomains can be a good paramete r in th i s r e spec t . In i ron , th i s th icknessi s o f the o rder o f 50 nm. One migh t imagine to choose as e lementa ryvo lumes fo r the desc r ip t ion vo lumes o f l inea r d imens ions o f the o rderof , say, 5 nm. In th is chapter we are in teres ted in the proper t ies of one ofthese e lementa ry vo lumes , a ssuming tha t i t con ta ins enough magne t icm om ents to pe rm i t a r e li ab le use o f s tat is ti c and the rm ody nam ic me thods .In chap te r s 6 and 7 , the mag ne t iza t ion conf igura t ions expec ted in macro-scop ic bod ies wi l l be d i scussed on the bas i s o f the knowledge ga ined onthe p roper t i e s o f the ind iv idua l e lementa ry vo lumes .

5 . 1 E X C H A N G EWe beg in th i s sec t ion wi th a b r ie f d i scuss ion o f param agne t i sm , w h ic h i sthe na tu ra l bas i s fo r the desc r ip t ion o f fe rromag ne t i sm. In fact , a fer rom ag-ne t can be imagined as a pa ramagne t where a s t rong coup l ing , desc r ibedby W eiss m olecular ie ld , ex is ts be tw een the magne t ic m om ents . The molec -ular f ie ld approach is not able to accurate ly reproduce a l l the deta i ls ofthe behav ior o f f e r romagn e ts be low the Cur ie po in t , bu t i t i s o f inva luab leimpo r tance in g iv ing a s imple and a t the same t ime deep phys ica l in te rp re -ta t ion o f the ex i s tence o f spon taneous magne t iza t ion , and in in t roduc ingone o f the bu i ld ing b locks necessa ry to the s tudy o f magne t ic domainsand hysteresis effects.

5 . 1 . 1 P a r a m a g n e t i s mSome aspec t s o f pa ramagne t i sm were b r ie f ly in t roduced in Sec t ion 1 .1 .1and Sec tion 4 .1 .3 . The phys ica l a ssu m pt ions de f in ing an i de a l param a gne t(Fig. 5 .1) can be su m m ariz ed as fo llows.


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5.1 EXCH ANG E 131IIIr"

II

101Y - 1 H aFIGURE 5.1. Magnetic mo men ts in ideal param agnet .

(i) T h e p a r a m a g n e t i s d e sc r i b e d a s a n a s se m b l y o f i d e n t ic a l p e rm a -n e n t m a g n e t ic m o m e n t s m i o f s t reng th m, sub jec t t o the ac t ionof the ex te rna l f i e ld H a. Each m o m en t can be t rea t ed as a c la ss ica le l e m e n t a ry m o m e n t o f t h e t y p e d i s c u s se d i n S e c t i o n 3 .1 .3 , o r a sa q u a n t u m m o m e n t . T h e q u a l i ta t iv e b e h a v i o r is t h e s a m e i n b o t hc a se s , t h o u g h t h e q u a n t i t a t i v e d e t a i l s m a y d i f f e r . W e sh a l l u seq u a n t u m r u l es f o r s p i n - 8 9 m o m e n t s t o c a lc u la t e t h e m a g n e t i z a -t io n , m a i n l y b e c a u s e i n th i s w a y o n e o b t a i n s s i m p l e r m a t h e m a t i -c a l e x p re s s i o n s . T h i s i s a m i n o r e x c e p t i o n t o o u r p ro g ra m o fa v o i d i n g a n y c o n s i d e r a ti o n o f q u a n t u m a s p e ct s , w h i c h , a n y h o w ,wi l l no t a f fec t t he f ina l re su l t s i n any subs tan t i a l way .

(ii) T h e re i s n o i n t e r a c t i o n b e t w e e n t h e m o m e n t s . E v e n t h e m a g n e -t o s t a t i c d i p o l a r c o u p l i n g i s n e g l e c t e d .( i i i ) E a c h m o m e n t h a s a p o t e n t i a l e n e r g y - ~ 0 m i ' H a i n t h e e x t e r n a lf i e ld (Eq . 3 .77 ) . Due to th i s ene rgy , each moment t ends to s t aypara l l e l t o H a

( i v ) T h e r m a l a g i t a t i o n r a n d o m l y s h a k e s e a c h m o m e n t a n d t e n d s t od e s t ro y a n y p re f e r e n t i a l a l i g n m e n t t o t h e f i e l d .

T h e t h e r m o d y n a m i c b e h a v i o r o f a s y s t e m w i t h t h e s e p ro p e r t ie s r e s ul tsf r o m t h e c o m p e t i t io n b e t w e e n t h e fi el d a n d t h e t h e r m a l c o u p li n g . L e t u sf i r s t c o n s i d e r t h e c a se w h e re t h e m o m e n t s a r e c l a s s i c a l v e c t o r s . Be c a u set h e re i s n o i n t e r a ct i o n b e t w e e n t h e m o m e n t s , o n e c a n s t u d y o n e m o m e n ta t a t im e a n d t h e n s u m u p t h e r e s u lt o v e r all m o m e n t s . T h e t h e r m o d y n a m i ci n fo rm a t i o n i s a l l c o n t a i n e d i n t h e s i n g l e -m o m e n t p a r t i t io n f u n c t i o n

Zm(Ha,Z) ~ e x p ( /z 0 H a , m ) f l- "~ as t a t e s ~-B~ = 2r -1 ex p( a cosS) d(cosS) 4r( 5 . 1 )


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132 CHAPTER 5 Exchange and Anisotropywh e re t h e d ime n s io n le s s p a r a me te r a i s g iv e n b y th e e x p re s s io n

/z0m/-/~a = (5.2)k s TIn Eq . (5 .1 ), 8 i s t h e a n g le b e tw e e n th e m o m e n t a n d th e fi eld, a n d th e s u mo v e r a l l m ic ro s t a t e s a m o u n t s t o a n i n t e g ra l o v e r a l l p o s s ib le m o m e n t o r i e n -ta t ions wi th respec t to the f ie ld . The to ta l pa r t i t ion func t ion Z assoc ia tedw i t h a ce r ta i n n u m b e r N o f i n d e p e n d e n t m o m e n t s i s j u st th e p r o d u c t o fthe s in g le - m om ent func t ions , Z = (Z~) N. Ac cord in g to s ta t i st ica l the r m o-d y n a m ic s , t h e f r ee e n e rg y i s g iv e n b y G (Ha ,T ) = - k B T l n Z = - N k B T lnZm,tha t i s,

G ( H a , T ) = - N k B T ( l n sh a - In a + In 4rThe average magne t iza t ion M is g iven by Eq . (4 .43) ,

(5.3)

M(Ha,T) = /Zo-aVL-d-~aaJ (5.4)w h e r e AV i s t h e v o lu me o c c u p ie d b y th e mo me n t s . B y c a r ry in g o u t t h ef ie ld de r iva t ive , one f inds

M (H a ,T ) = M o ( c o t h a - ~ ) = M o L (a ) (5.5)w h e r e

XmM 0 = ZkV (5.6)is t h e m a x i m u m p o s s i bl e m a g n e t i z at i o n , o b ta i n e d w h e n t h e m o m e n t s a r ea l l pe r fec t ly a l igned . The func t ion L ( a) is ca l led the L a n g e v i n fu n c t i o n . Itsb e h a v io r a s a f u n c t io n o f H a / T i s shown in F ig . 5 .2 .

L e t u s n o w c o n s id e r t h e q u a n t u m d e s c r ip t i o n o f t h e p ro b l em . P a r a -m a g n e t i s m i s a p h e n o m e n o n i n w h i c h t h e q u a n t u m n a t u r e o f m a g n e t i cm o m e n t s b e c o me s c l ea r ly ma n i f e s t, a n d i t i s o n ly a f te r q u a n tu m c o n s id e r a -t i o n s a r e i n t ro d u c e d th a t a s a t i s f a c to ry a g re e me n t w i th e x p e r ime n t s i sf o u n d . N o n e t h e le s s , w e w i ll a b a n d o n q u a n t u m i d e a s s o o n a ft er t hi s se c-t i o n . On c e a s u i t a b l e e x p re s s io n fo r t h e a v e ra g e ma g n e t i z a t i o n M o f t h es y s t e m i s d e r iv e d , we wi l l f o rg e t a b o u t t h e c l a s s i c a l o r q u a n tu m o r ig inof tha t re su l t , and we wi l l use i t a s the macroscop ic s ta r t ing po in t fo r thed iscuss ion o f molecu la r f ie ld e f fec ts .

L e t u s t h e n c o n s i d e r t h e c a s e w h e r e t h e m o m e n t s a r e q u a n t u m m o -m ents , a ssoc ia ted w i th sp in 1A. O nly tw o s ta tes a re then poss ib le , in wh ich


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5.1 EXCH ANG E 133

1.00

0.75o

0.50

0.25

0.00

, , , , , , , " ' " ' " ' " ' "~s

1 2 4

FIGURE 5.2. Mag netization of ideal param agne t. Continuous line: Q uan tum spin-1A m om ents (Eq. 5.9). Da shed line: Classical m om ents (Eq. 5.5).

t he magne t i c momen t componen t a l ong t he f i e l d i s e i t he r + m o r - m .Equat ion (5.1) through Eq. (5 .5) are modif ied as fol lows:

[[zomHa~ ( # 0 m H ~ ) = 2 c h aZm(Ha,Z) = exp ~ kB~ j + ex p kBT ( 5 . 7 )G(/-/~,T) = - N k B T ( l n ch a + In 2) ( 5 . 8 )

M (H a ,T ) = M o t h a = M 0 th \ kBT ] ( 5 . 9 )The behavior of Eq. (5 .9) i s shown in Fig. 5 .2 . Because th(a) ~- a w h e n ai s smal l , we see tha t , a t low f ie lds , the magnet iza t ion i s p ropor t iona l tothe f i e ld , M = xH, , and the suscept ib i l i ty i s g iven by

# o m M o 1 C- ( 5 . 1 0 )X = kB T TT he i nve r s e p r opo r t i ona l i t y w i t h t emper a t u r e exp r e s s e s t he C u r i e l a w . T h eclass ical resu l t (Eq. 5 .5) i s qua l i ta t ively s imilar , bu t th e lo w f ield susce pt ibi l -i ty is di f ferent , beca use L ( a ) ~ a / 3 for smal l a. No te , howev er , tha t the quan-t it y m app ea r i n g i n the t w o exp r e s s i ons has a d if f er en t phys i ca l mean i ng ,becau se in the class ica l case i t r epresen t s the m od ulu s of the magn et ic m o-men t , w he r eas i n t he quan t um cas e i t r ep r e s en t s i t s componen t a l ong t hef ie ld . Sa tura t ion se ts in wh en a ll m om en ts b eg in to be subs tan t i a l ly a l igned


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134 CHAPTER 5 Exchange and Anisot ropy

to t he f i e ld . Th i s t akes p l ace a t f i e lds H, --- kB T / /x om . I t i s i n t e re s t i ng t om a k e s o m e o r d e r - o f - m a g n i t u d e e s t i m a te . T h e n a t u r a l a t o m i c u n i t of m a g -n e ti c m o m e n t s is th e B o h r m a g n e t o n , / z B = h q e / 4 - a ' m e ~- 9 .27 10 -24 A m 2(see Sec t ion 3 .2 .1) . By taking m --~/zB an d / z0 M 0 --- 2 T , w e f i nd , a t ro omt e m p e r a t u r e , w h e r e k B T ~ 4 10 -21 J, th a t ,t' "" 5 10 -3 a n d k B T / t z o m --- 4 108A m -~. T h e r m a l a g i t a t i o n i s v e r y e f f ec t iv e a t r o o m t e m p e r a t u r e a n d s u b s t a n -ti al m o m e n t a l ig n m e n t c a n be o b t a i n e d o n l y b y a p p l y i n g e x t r em e l y h i g hf ie lds .

A n a c c o u n t of t h e q u a n t u m t h e o r y o f p a r a m a g n e t i s m is o u t s id e t h escop e of t h i s book . W e s im ply reca l l t ha t , fo r sp in s J g re a t e r t h an 1A, i ns t e ado f E q . (5 .9 ) t h e m o r e c o m p l i c a t e d e x p r e s s i o n M = M o B l (a ) is f o u n d , w h e r eBl(a ) i s the B r i l lo u i n f u n c t i o n

Bl(a ) = 2J +________!11o th (2J + 1)a 1 c o th a (5.11)21 2J 2J 21Fo r J = 1A, Eq. (5.9) is reco ver ed. Co nv erse ly , th e c lass ica l expr ess io n, Eq.(5.5) , i s ap pr oa ch ed i n t he l im i t J --4 co. Eq ua t ion (5.9) wi l l be a s s um ed int h e n e x t s e c t i o n a s t h e s t a r t i n g p o i n t f o r t h e m a c r o s c o p i c d e s c r i p t i o n o fs p o n t a n e o u s m a g n e t i z a t i o n .

5 . 1 . 2 W e i s s m o l e c u l a r f i e l dW e h a v e s e e n t h a t e n o r m o u s f ie ld s , o f th e o r d e r o f l 0 s A m -1, a r e r e q u i r e dt o p r o d u c e s u b s t a n t i a l m o m e n t a l i g n m e n t i n a p a r a m a g n e t . T h i s i s i ns t r i k i n g c o n t r a d i c t i o n t o t h e o b s e r v a t i o n t h a t c e r t a i n m a t e r i a l s , w h i c h a r ep a r a m a g n e t i c a t h i g h t e m p e r a t u r e s , a t r o o m t e m p e r a t u r e c a n ex h ib i t am a g n e t i z a t i o n o f t h e o r d e r o f 1 T u n d e r f i el d s a s l o w a s a f e w t e n s o fa m p 6 r e s p e r m e t e r . W e i s s m o l e c u l a r f i e l d h y p o t h e s i s r e s o l v e s t h i s d i f f i -c u l ty b y d e s c r i b in g a f e r r o m a g n e t a s a p a r a m a g n e t w h e r e o n e o f t h ef o u r a s s u m p t i o n s l i s t e d i n t h e p r e v i o u s s e c t i o n n o l o n g e r h o l d s . I n af e r r o m a g n e t , th e m o m e n t s a r e b y n o m e a n s u n c o u p l e d . T h e r e e x is ts as t r o n g i n t e r a c t i o n f i e ld , w h i c h f a v o r s c o l le c t iv e a l i g n m e n t o f t h e m o m e n t sa l o n g a c o m m o n d ir ec ti on . A s th e m e a n d e g r e e o f m o m e n t a l i g n m e n t i sm e a s u r e d b y t h e m a g n e t i z a t i o n M , i t s e e m s r e a s o n a b l e , a s a f ir st a p p r o x i -m a t i o n , t o p o s t u l a t e t h a t t h e i n t e r a c t i o n f i e l d s h o u l d b e p r o p o r t i o n a l t oM i t se l f . Th i s i n t e rac t i on f i e ld , named m o l e c u l a r f i e l d , t h u s t a k e s t h e f o r m

H w = N w M (5.12)w h e r e t h e d i m e n s i o n l e s s c o n s t a n t N w m e a s u r e s t h e s t r e n g t h o f t h e in t e r a c -t io n . U n d e r t h e p r e s e n c e o f H w , e a c h m o m e n t e x p e r i e n c e s a n e f fe c ti v e


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5.1 EXCHANGE 1 3 5f ie ld that i s the sum of the appl ied f ie ld and of the molecular f ie ld . Thisis descr ib ed b y m odif ying Eq. (5.9) as fo llows:

MoM = th[~BTlZ~ (Ha + NwM)] (5 .13)According to Eq. (5 .13) , the presence of the molecular f ie ld in t roduces apos i t ive feedback mec han ism, becau se a nonz ero m agne t iza t ion g ives r iseto a f ie ld that favors a fur ther increase of the magnet iza t ion i tse l f . Wecan u se E q. (5.13) to calculate, as before, the lo w field su sceptibil ity. Byexpanding th(x) to f i rs t order , th(x) = x , we again obta in M = x H a, b u tn o w

# o m M o 1 C= (5.14)X = kB T - T c T - T cwhere the charac te r i s t i c t empera tu re

T c = I ~ ~ 1 7 6 (5.15)kBis the C u r i e t e m p e r a t u r e . Equa t ion (5.14) i s kn ow n as the C u r i e - W e i s s l a w .The com par i son wi th Eq. (5 .10) show s tha t the on ly d i f fe rence caused bythe molecu la r f i e ld i s the appearance o f the add i t iona l t empera tu re T c i nt h e d en o m in a to r . Wh e n T - ~ T c f rom h ighe r t emp era tu res , the suscep t ib i l-i ty tends to inf in i ty . In addi t ion, the law cer ta inly fa i ls for T < T c as alarge negat ive suscept ib i l i ty is a sure indicat ion of ins tabi l i ty . Somethingimportant occurs a t T = T~.To ana lyze the p rob lem in more de ta i l , l e t us s impl i fy the no ta t ionby in t roduc ing appropr ia te d imens ion less va r iab les :

M T - T ch a - H a x - - - t = (5.16)N w M o M o TcEqua t ion (5.13) becom es

x = t h +This equa t ion expresses in impl ic i t fo rm the eq ua t ion o f s ta te x (h a, t ) forthe sys tem. Le t us f i r s t d i scuss i t s mean ing when no app l ied f i e ld i sp resen t . The magne t iza t ion then fo l lows the l aw

x = t h 1


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1 3 6 CHAPTER 5 Exchange and Anisot ropyT h e s o l u t i o n o f t h i s e q u a t i o n f o r x a s a f u n c t i o n o f t d e s c r i b e s t h e s t a t eo f t h e s y s t e m u n d e r n o e x t e r n a l f i e l d . I n o r d e r t o v i s u a l i z e t h e s o l u t i o n so f E q . (5 .1 8 ), o n e c a n s e p a r a t e l y p l o t t h e t w o m e m b e r s o f t h e e q u a t i o na n d c o n s i d e r t h e p o i n t s o f i n t e rs e c t io n . T h e r e s u l t i s s h o w n i n F i g . 5 .3 .W h en t > 0 , i .e ., T > T , t he re i s on ly on e so lu t i on a t x = 0 . Con ve rse ly ,t h ree so lu t i ons , a t x = 0 and a t x = + x s ( t ) ( i.e. , M = +M s(T)) , exis t for t< 0 (i.e., T < T c ) . A s we sha l l s ee shor t l y , t he so lu t i on x = 0 i s uns t ab l ea n d s h o u l d n o t b e c o n s i d e r e d . T h e o t h e r t w o s o l u t io n s r e p r e s e n t s t a te sw h e r e t h e s y s t e m s p o n t a n e o u s l y a c q u i r e s t h e n o n z e r o m a g n e t i z a t i o nM s ( T ) e v e n if n o f i el d i s a p p l i e d . Tc i s j u s t t h e t e m p e r a t u r e a t w h i c h t h ep o s i t i v e f e e d b a c k o p e r a t e d b y t h e m o l e c u l a r f i e l d o v e r c o m e s t h e r m a la g i t a t i o n a n d d r i v e s t h e s y s t e m t o a g l o b a l l y o r d e r e d s t a t e . M s (T ) i s ca l led

1.00.50.0

-0.5-1.0

t = l

_ s s S S t

s i A" I

s

~ s S S S

L s S S S

s t a b l e N , ,

sss

i i I i

1.00.50.0

-0.5-1.0

- t = - 0 . 5

, /- s t ab l e \ ~

s

. 6 - 0 . 8

th[x/(l+t)~]~ _/ \stable

\ x , i t lunstableo o ' 0 1 8 '

FIGURE 5.3 . G raphical solut ion of Eq. (5.18), show ing onset of spon taneou sma gnet izat ion x = +Xs(t wh en t < 0.


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5.1 EXCH ANG E 137s p o n t a n e o u s m a g n e t i z a t i o n or s a t u r a t i o n m a g n e t i z a t i o n . I t s t e m p e r a t u r e d e -pendence , as p red ic ted by Eq . (5 .18) , i s shown in F ig . 5 .4 .

T h e m o l e c u l a r f i e l d a p p r o a c h o p e r a t e s o n m a c r o s c o p i c , p h e n o m e n o -l o g i c a l g r o u n d s . T h e a v e r a g e m a g n e t i z a t i o n i s t h e i m p o r t a n t p a r a m e t e r ,t he ex i s t ence o f t he mo l ecu l a r f i e l d i s pos t u l a t ed , and now her e i n t het heo r y i s t he r e d i r ec t re f e rence t o t he mi c r os cop i c con f i gu r a t i ons a t t a i nedb y m a g n e t i c m o m e n t s . A m i c ro s c o p ic i n t e r p r e t a ti o n o f t h e m o l e c u l a r fi el dc a n b e w o r k e d o u t o n l y i n t h e f r a m e w o r k o f q u a n t u m m e c h a n i c s , i nt e r ms o f t he s o - ca l l ed e x c h a n g e i n t e r a c t i o n . T h e idea i s tha t the overa l lan t i s ymmet r i c cha r ac t e r o f e l ec t r on i c s t a t e s g i ves r i s e t o a coup l i ng be -t w e e n t h e s p i n m o m e n t u m a n d t h e w a v e f u n c t i o n i n r e a l s p a c e . I n o t h e rw o r ds , i f t w o e l ec t rons ca r r y pa r a ll e l s p in s ( w h i ch co r r e s p ond s t o a s ym-me t r i c s p i n w ave f unc t i on ) , t hey canno t s t ay c l o s e t o each o t he r ( w h i chi s t h e t y p i c a l p r o p e r t y o f a n a n t i s y m m e t r i c t w o - e l e c t r o n w a v e f u n c t i o ni n r ea l s pace ) . T he f ac t t ha t t hey neve r ge t c l o s e r educes t he ave r ageenergy as soc ia ted wi th the i r e lec t ros ta t i c in te rac t ion . I t i s th i s r educ t iont ha t f avo r s t he pa r a l l e l s p i n con f i gu r a t i on . T hus , i n t he end , exchangei n t e r ac t i on , t he bas i s o f f e r r omagne t i s m, t u r ns ou t t o be an e l ec t r o s t a t i ce f f ec t . T h i s s chema t i c p i c t u r e i s no t h i ng mor e t han a s t a r t i ng po i n t . T her ea l s it u a t io n i s m u c h m o r e c o m p l e x . O n e h a s t o d e a l n o t w i t h t w o , b u tw i t h a m a c r o s c o p i c n u m b e r o f e le c tr o n s, a n d o n e h a s t o d e c id e w h e t h e rt he e lec t r ons con t r i bu t i ng t o the m agn e t i za t i on a r e loca l i zed a r oun d s i ng leions or i t ineran t over l a rge d i s tances . We sha l l no t go in to the de ta i l s o f

1 .~0.

O. 0.5 1.0 1.5 2.0TGFIGURE 5.4 . Thick solid line: Zero-field spontane ous m agn etization versus tem-pera ture pred icted by Eq. (5.18). Thin lines: Solutions of Eq. (5.17) un de r nonz erofield ha - 0.005, 0.02 , 0.05, 0.1. h a = H a / N w M o (see Eq. 5.16).


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138 CHAPTER 5 Exchange and Anisotropyt hes e des c r i p t i ons . W e on l y r eca l l t he f ac t t ha t t he s e p r ob l ems a r e o f t endes c r i bed i n t e r ms o f t he H e i s e n b e r g H a m i l t o n i a n

H - - - ~ , J i j S i " S j (5.19)i,jw h e r e S i is t h e s p i n a n g u l a r m o m e n t u m o f t h e i o n l o c a te d a t t h e i th s it eo f s ome l a t t i c e , and t he exchange i n t eg r a l Ji j ( pos i t i ve i n a f e r r omagne t )m e a s u r e s t h e s tr e n g t h o f t h e e x c h a n g e c o u p l i n g b e t w e e n t h e m o m e n t s ian d j. E xcha nge is a s ho r t - r ang e i n t e r ac ti on , w h i ch f al ls o ff r ap i d l y w i t hd i s t ance , and i t i s o f t en a pe r f ec t l y accep t ab l e app r ox i ma t i on t o t r t mca t et he s ummat i on i n E q . ( 5 . 19 ) t o nea r e s t ne i ghbor s . W hen Ji j > 0 , the min i -m u m e n e r g y c o n f i g u r a ti o n is th e o n e w h e r e a ll sp i n s a r e a l i g n e d t o e a c ho th e r. T h e e n e r g y i n c r e a se s a b o v e t h is g r o u n d l e ve l w h e n e v e r s o m e m i s -a l i g m n e n t o f n e i g h b o r i n g s p i n s o c c ur s.

A f i n a l p o i n t s h o u l d b e m e n t i o n e d , w h i c h i s n o t e v i d e n t f r o m t h es i mp l i f i ed des c r i p t i on j u s t g i ven . S p i n o r de r i ng a t l ow t emper a t u r e s doesno t neces s a r i l y r educe t o pa r a l l e l a l i gnmen t . D epend i ng on t he l a t t i c es y n ~ n e t r y a n d t h e v a l u e o f t h e e x c h a n g e i n te g r a l, m o r e c o m p l e x o r d e r e ds t r uc t u r e s may be f o r med , w he r e one can i den t i f y s eve r a l s ub l a t t i c e s ,s uch t ha t t he s p i n s be l ong i ng t o t he s ame s ub l a t t i c e a r e a l i gned , bu t t hem agn e t i za t i on o f d i f fe r en t s ub l a tt i c e s po i n t s a l ong d i f fe r en t d i rec t i ons. I nt h is con t ex t , t he w o r d f e r r o m a g n e t i s m des c r i bes t he ca se w he r e M s i s duet o the pa r a l l e l a l i gnm en t o f i den t i ca l m om en t s on one l a tt ic e , and f e r r i mag -n e t i s m t h e o n e w h e r e M s r e su l ts f r o m t h e v e c t o r s u m o f c o m p e t i n g p a r t i a lcon t r i bu t i ons comi ng f r om s eve r a l s ub l a t t i c e s . T h i s compe t i t i on can l eadt o pe r f ec t cance l l a t i on o f t he pa r t i a l con t r i bu t i ons . O ne t h en ha s a n t i f e r r o -m a g n e t i s m , c h a r a c t e r iz e d b y l o n g - r a n g e s p i n o r d e r i n g b u t n o s p o n t a n e o u smagne t i za t i on . A s a l r eady men t i oned i n S ec t i on 4 . 1 . 3 , i n t h i s book w egene r i ca l l y u s e t he exp r e s s i on m a g n e t i c m a t e r i a l t o r e f e r t o any s ys t emd i s p l a y i n g s p o n t a n e o u s m a g n e t i z a t i o n a n d h y s t e r e s i s , w i t h n o f u r t h e rd i s t inc t ion .

5.1.3 Energetic aspectsWe said th at , of the thre e solu t ion s of Eq. (5 .18) for T < Tr th e on e gi vin gx = 0 i s ac t ua l l y uns t ab l e and s hou l d no t be cons i de r ed . T h i s a s pec t i sb e t t e r c la r if ie d b y a n a l y z i n g t h e p r o b l e m f r o m t h e e n e r g y v i e w p o i n t . L e tus r eco ns id er Eq . (5 .17). By inv er t in g the hyp erbo l ic t ang ent , on e canrewr i te i t in the form

h a 1 /= 2 1 - - x (5.20)


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5.1 EXCHANGE 1 3 9A c c o rd i n g t o th e a p p ro a c h p re se n t e d i n S ec t io n 2 .1 .4 , t h e s t a b le o r m e t a s t a -b l e s t a t e s o f a sy s t e m a re lo c a l m i n i m a o f i ts L a n d a u f r e e e n e r g y G L , sa t i s fy i n gt h e c o n d i t i o n O G L / O X = 0 . C o n s i d e r i n g t h a t G L = F ( X ) - H X , t h i s cond i -t i o n c a n a lso b e w r i t t e n a s H = 3 F / O X . T h i s is j u s t t h e fo rm t a k e n b y E q .(5.20), with H ~ ha , X - -~ x . By in teg ra t ing Eq . (5 .20) wi th re spec t t o x ,w e c a n t h u s o b t a i n t h e d i m e n s i o n l e s s e n e rg y o f t h e sy s t e m . W e sh a l ld e n o t e t h is e n e r g y b y g E x t o r e m i n d u s o f it s e x c h a n g e o r ig i n . T h e r e l a t io nof g E x t o t h e e n e rg y G E X e x p re s se d i n p h y s i c a l u n i t s i s g E X = G E X //z0NwM02AW. The in te gra t ion of Eq. (5.20) w i th resp ect to x g ives[ ] x 2g E x ( X ; h a , t ) = 1 + t (1 + x)ln(1 + x) + (1 -- x)ln(1 - x ) - h a x2 2 - - (5.21)The f i r s t t e rm , p ro por t iona l t o 1 + t = T / T c , desc r ibes the e f fect o f the rm alag i t a t ion . I t i s a func t ion o f x wi th pos i t ive cu rva tu re , favor ing the ze ro -m a g n e t i z a t i o n s t a t e x = 0. C o n v e r se l y , th e s e c o n d t e rm , - x 2 / 2 , h a s n e g a -t ive cu rva tu re and favors s t a t e s where x : / : 0 . Th i s i s t he e f fec t o f them o l e c u l a r f i e l d . W e c a n a p p re c i a t e t h e c o n se q u e n c e s o f t h i s c o m p e t i t i o nby mak ing a Tay lo r expans ion o f Eq . (5 .21) wi th re spec t t o x , and byk e e p i n g t h e l o w e s t - o r d e r t e r m s . O n e f i n d s

g E x ( X ; h a , t ) = 1 + t X4 4 - t X 212 ~ - h a x + . . . (5.22)W e r e c o g n i z e t h e c u s p c a t a s t r o p h e discussed in Section 2.2.2. t = 0, i .e . T= T c , i s t he c r i t i ca l po in t where a qua l i t a t ive change in the po ten t i a l t akesp lace , a s shown in F ig . 5 .5 . A t the Cur i e po in t , t he x = 0 s t a t e becomesu n s t a b l e a n d t h e s y s t e m s p o n t a n e o u s l y a c q u ir e s a n o n z e r o m a g n e t i z a t i o n .

T h e b e h a v i o r o f g E x a round the Cur i e po in t g iven by Eq . (5 .22) i s a l soin te res t in g in an o th e r re spec t . F igure 5 .4 sh ow s the so lu t ions o f Eq . (5 .17)u n d e r n o n z e ro f ie ld . T h e p re se n c e o f t h e fi e ld d e s t ro y s t h e m a g n e t i z a t i o ns i n g u l a r i t y a t th e C u r i e t e m p e ra t u re . T h i s r a i se s t h e p ro b l e m o f h o w o n esh o u l d m e a su re t h e C u r i e t e m p e ra t u re i n p r a c t i c e . I n f a c t , T c m a r k s aq u a l it a ti v e c h a n g e i n th e s p o n t a n e o u s m a g n e t i z a t i o n M s (T ) b u t i n o r d e rt o m e a s u r e M s o n e n e e d s t o a p p l y m a g n e t i c f ie ld s l ar g e e n o u g h t o s w e e pa w a y m a g n e t i c d o m a i n s a n d t h is j u s t d e s t r o y s t h e s in g u l a r i t y t h at o n ea i m s a t d e t e r m i n i n g . A p o s s i b l e s o l u t i o n i s t o c a r r y o u t m e a s u r e m e n t su n d e r d i f f e r e n t f i el d s, a n d t h e n t o a t t e m p t so m e e x t r a p o l a t i o n o f t h eresul t s to zero f ie ld . Here i s where Eq. (5 .22) comes in to p lay . In fac t , thee q u i l i b r i u m c o n d i t i o n O g E x / O X = 0 , w h e n a pp l ie d to Eq. (5 .22) , g ives

h a = t x + 1 + t x 3 4 - . . . (5.23)


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0.02

CHAPTER 5 Exchange and Anisot ropy

0 .00

- 0 .02

0.02

0.00o )

- 0 .02!

t = - 0 . 2

90 .02

o.ooo ~

-0.02

_ 1

140

.0 -0.5 0.0 0.5 1.0

FIGURE 5.5 . Q ualitative chan ge in zero-field free energ y (Eq. 5.21) at the Curiepoint t = 0. Com pare w ith Fig. 2.9.

B y d i v i d i n g b o t h m e m b e r s b y x a n d b y e x p r e s s i n g t h e r e s u l t i n p h y s i c a lu n i t s t h r o u g h E q . ( 5 . 1 6 ) , o n e f i n d sHa

Ms(H~,T) = N w T - T + N w T M 2 ( H a , T ) (5.24)Tc 3TcM 2E q u a t i o n (5 .2 4) s h o w s t h a t, i f w e m e a s u r e t h e s a t u r a t i o n m a g n e t i z a t i o nMs(Ha,T ) u n d e r d i f f e r e n t f i e l d s a n d w e r e p r e s e n t H a / M s a s a f u n c t i o n o fM 2, w e sh ou ld ge t a s t r a igh t l i ne . In add i t i on , t he i n t e r sec t i on of t h i s li new i t h t h e f i e ld a x i s s h o u l d b e p o s i t i v e i f T > Tc, a n d n e g a t i v e i n t h e o p p o s i t ec a s e . T h r o u g h t h i s c h a n g e o f s i g n , o n e h a s a s e n s i t i v e w a y t o e s t i m a t ew h e r e t h e m e a s u r e m e n t t e m p e r a t u r e is w i t h r e s p e c t t o t h e C u r i e p o i n t.R e p r e s e n t a t i o n s b a s e d o n t h i s i d e a a r e k n o w n a s Arrot t p lo t s , a n d a r e o fg r e a t v a l u e i n C u r i e p o i n t s t u d i e s .


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5.1 EXCHANGE 1 4 1T h e e n e r g y p e r u n i t v o l u m e U w / A V a s s o c i a t e d w i t h t h e m o l e c u l a rf ie ld is r e p r e s e n te d b y th e t e rm -x 2 / 2 o f Eq . (5 .2 1 ). B y g o in g b a c k to

phys ica l quan t i t i e s , one f indsUw - - /z ~ Nw M2 (5.25)AV 2

A n e n e rg y t e rm a n a lo g o u s t o Eq . (5 .2 5 ) w a s e n c o u n te r e d in S e c tio n 4 .1 .1(Eq . 4 .5 ) , when we d iscussed magne tos ta t ic ene rgy . The molecu la r f ie lde n e rg y i s g r e a te r t h a n th e m a g n e to s t a t i c o n e b y th e f a c to r 3 N w . W e c a ne s t ima te N w f ro m th e e x p e r ime n ta l k n o w le d g e o f t h e Cu r i e p o in t . F ro mEq. (5 .15) , we have tha t Nw = kBTc/~ omMo 9 In iron, where T ~- 103 K,by tak ing m - - - / zB --- 10 -23 A m 2 and /~0 M 0 --- 2 T, w e ob ta in 3N w --- 2 103.Th e m o le c u la r f i e ld i n t e r ac t io n is t h r e e o rd e r s o f m a g n i tu d e g r e a t e r t h a nthe m agn e tos ta t ic one . Th is con f i rms th e fac t tha t th e M 2 te rm of Eq .(4 .5 ) i s in fac t o f neg l ig ib le impor tance when fe r romagne t ic ma te r ia l s a rec o n s id e r e d .The f ree energy o f F ig . 5 .5 has the ty p ica l two -w el l s t ruc tu re d i scusse din Ch a p te r 2 . Th e p r e s e n c e o f two e q u iv a l e n t m in ima p o in t s a t t h e e x i s -t en c e o f t w o p h a s e s , a " m a g n e t i z a t i o n u p " a n d a " m a g n e t i z a t i o n d o w n "phase . Al l the cons ide ra t ion s m ad e in Sec t ion 2 .2 .3 abou t pha se coex is tencec a n b e a p p l i e d t o t h e p r e s e n t c a s e . B u t o n e s h o u ld b y n o me a n s j u mp toth e c o n c lu s io n th a t we h a v e a l r e a d y a t h a n d a re a li st ic f r a m e w o rk fo r t h ein t e rp r e t a t i o n o f ma g n e t i c d o m a in s . I n t h is r es p e c t, t h e a p p ro a c h th a t weh a v e d i s c u s s e d s o f a r i s i n a d e q u a te a n d u n re a l i s t i c . Th e p o in t i s t h a t H aa n d M h a v e b e e n t r ea t e d e v e r y w h e r e ju s t as n u m b e r s , w h e r e a s t h e y a rev e c to r s . A t t h e b e g in n in g , t h e r e a s o n fo r t h i s wa s t h a t w e w e re c o n s id e r in gth e ma g n e t i z a t i o n c o mp o n e n t a lo n g th e f i e ld d i r e c t io n . Ho we v e r , l a t e rw e fo c u s e d a t t e n t io n o n th e c a s e w h e re n o f ie ld is p r e se n t , a n d th is r a is e st he q u e s ti o n of w h a t m a g n e t i z a ti o n " u p " o r " d o w n " m i g h t m e a n u n d e rsuch c i rcumstance s . The tw o-w el l po ten t ia l o f F ig . 5 .5 desc r ibes a s i tua t ionwh e re , d u e to s o me a n i s o t ro p y o f u n s p e c i f i e d o r ig in , t h e m a g n e t i z a t i o nis fo rced to l i e a long a ce r ta in f ixed d i rec t ion , and we a re es t ima t ing thee n e r g y i n v o l v e d i n p a s s i n g f r o m m a g n e t i z a t i o n " u p " t o " d o w n " a l o n gth a t d i r e ct io n . Th i s e n e rg y i s h u g e . I f w e a p p ly t h e a n a ly s i s o f two -w e l lp o t e n t i a l s p r e s e n t e d in Ch a p te r 2 t o t h e p r e s e n t c a s e , we c o me to t h ec o n c lu s io n th a t th e f ie ld t h a t we m u s t a p p ly t o re v e r s e t h e ma g n e t i z a t i o nf ro m " u p " t o " d o w n " i s o f t h e o rd e r o f t h e mo le c u la r f ie ld , H w = NwM.W ith N w --- 103 an d/z 0M ~- 1 T, th is g ives f ie lds o f the ord er of 109 A m -1 .Cer ta in ly , i t i s a p red ic t io n g ross ly ou t o f sca le w i th respec t to the hys te res i slo o p s o b s e rv e d in re al it y. Th e p o in t i s t h a t t h e a n i s o t ro p y ju s t me n t io n e dis an a r t i fac t o f the desc r ip t ion , use fu l to keep the d iscuss ion s impleb u t u n p h y s i c a l . Ex c h a n g e in t e r a c t io n s , a s d e s c r ib e d fo r i n s t a n c e b y th e


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1 4 2 CHAPT ER 5 Exchange and An isot ropy

H e i s e n b e r g H a m i l t o n i a n , E q . (5 .1 9 ), a r e i s o t r o p i c i n sp a c e . T h e r e i s n o ap r i o r i fa v o r e d d i r e c ti o n f or t h e s p o n t a n e o u s m a g n e t i z a ti o n , b u t o n l y t h a tp a r t i c u l a r d i r e c t i o n s e l e c t e d b y t h e s y s t e m w h e n t h e s p o n t a n e o u s m a g -n e t i z a t i o n s e t s in . To il l u s t r a t e t h e c o n s e q u e n c e s o f th i s f a ct , i t i s s u f f i c i e n tt o p a s s f r o m o n e t o t w o d i m e n s i o n s . T h e e q u a t i o n a n a l o g o u s t o E q .(5.21) isgEx(X;ha ,t) l + t [ ] x22 (1 + x) ln(1 + x) + (1 - x) ln(1 - x) - ~ - - h a .x

(5.26)w h e r e h a a n d x a r e n o w t w o - d i m e n s i o n a l v e c t o r s a n d x = Ixl. T h e b e h a v i o ro f t h e p o t e n t i a l u n d e r z e r o f ie ld i s s h o w n i n th e u p p e r p a r t o f F ig . 5 .6 .

FIGURE 5.6 . Spon taneous m agnet iza t ion in the presence of cont inuous sym metry.Top: h a = 0. Bottom : h~ ~ 0.


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5.1 EXCH ANG E 143T her e i s a con t i nuo us r i ng o f equ i v a l en t g r ou nd s t a te s ( it w o u l d be as phe r i ca l s he l l i n t h r ee d i mens i ons ) . T h i s re f lec ts t he r o t a t i ona l s y m m et r yo f t h e p r o b l e m . O n e s a y s t h a t t h e s y m m e t r y i s c o n t i n u o u s , b e c a u s e o n ec a n m o v e t h r o u g h t h e w h o l e s e t of g r o u n d s ta t es w i t h c o n t in u it y . W h e nt h e s y s t e m s p o n t a n e o u s l y a c q u i r e s a m a g n e t i z a t i o n i n t h e w a y s h o w n i nFig . 5 .6 , i t s e lec t s one among the ava i l ab le ground s ta tes , thus des t roy ingt he o r i g i na l s y m m et r y o f t he p r ob l em . I n f ie l d and pha s e t r ans i t i on t heo ry ,t h is me ch an i s m i s o f t en re f e r r ed t o a s s p o n t a n e o u s s y m m e t r y b r e a k i n g . W h e na nonze r o ex t e r na l f i e l d i s app l i ed , t he ene r gy s u r f ace i s mod i f i ed a ss ho w n i n the l ow er pa r t o f F ig . 5 .6 . T he ex i s tence o f con t i nuou s s y m m et r yin the zero- f i e ld po ten t ia l p lays a c ruc ia l ro le , because i t permi t s thes ys t em t o ad j u s t i t s s t a t e t o t he f i e l d w i t hou t hav i ng t o ove r come anyene r gy ba r ri e r. T he pa t h " a r ou nd t he h i ll " s ho w n i n the f i gu re i s j u s tc o h e r e n t a n d s i m u l t a n e o u s r o t a t i o n o f a l l m a g n e t i c m o m e n t s t o w a r d t h ef ie l d d ir ec t ion . T h i s does no t i m p l y an y chan ge i n t he mo du l u s o f spon t a -neous magne t i za t i on and i n exchange i n t e r ac t i ons , and i t c an t ake p l aceu n d e r f i e l d s n o m a t t e r h o w s m a l l . O n t h e c o n t r a r y , t h e p a t h " a b o v e t h eh i l l , " which was the on ly one ava i l ab le in one d imens ion (F ig . 5 .5 ) , i s ap a t h w h e r e t h e m a g n e t i z a t i o n m o d u l u s m u s t c h a n g e w i t h c o n t in u i ty f ro mi t s in i t i a l va lue to the oppos i t e o f i t . Thi s can on ly be ob ta ined by f ine lym i x i n g m a g n e t i c m o m e n t s p o i n t i n g a l o n g d i f fe r en t d i r ec t io n s , a p ro c e s st ha t en t a i ls a hu ge am ou n t o f exchan ge ene r gy . T h i s pa t h i s nev e r f o l l ow ed .

T he cu r va t u r e o f t he ene r gy p r o f i l e i n t he r ad i a l M d i r ec t i on , w he r et h e s t re n g t h o f th e s p o n t a n e o u s m a g n e t i z a t i o n c h a n g e s , p l a y s a r ol e w h e no n e w a n t s t o e s t i m a t e t h e j o i n t f i e l d - t e m p e r a t u r e d e p e n d e n c e M s ( H a , T ) .Le t u s cons i de r t he s pon t aneous magne t i za t i on M s ( 0 , T ) unde r ze r o f i e l d .x = Xs(0,T = M s ( O , T ) / M o i s then the so lu t ion of Eq . (5 .18). N ow le tu s a pp lya fi e ld pa r a l l e l to t he mag ne t i za t i on . T he f i e ld f avo r s f u r t he r a l i gnm en t o ft h e m a g n e t i c m o m e n t s a n d t h u s p r o d u c e s a f o r c e d i n c r e a s e o f M s. T h i si nc r ease i s eva l ua t e d b y e qua t i ng t he ex t e r na l f i e ld f o rce t o t he s p r i ng l i kecounter force a r i s ing f rom the curva ture of the po ten t ia l o f Eq . (5 .21)a r o und t he m i n i m um a t x = X s(0 ,T ). W e have

I c~2~EX]cgxi x = I x s ( h a ' t ) - x s ( O ' t ) l ' - h aha = O (0't) (5.27)The curva ture , ca lcu la ted f rom Eq. (5 .21) , i s

OX2 Jx = xs(O,t)ha = 0x s + t X s (5.28)2 - - x2 -} - t ch2 1 ~ t1 - x s

w h e r e w e h a v e t a k e n i n t o a c c o u n t t h a t X s ( O , t ) sat is f ies Eq. (5 .18). At te m pe r-2a t u r e s T < < T c , X s - 1 an d X s + t ~ - 1 + t = T / T c . B y us i ng t h i s app r ox i -


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144 CHAPTE R 5 Exchange and Anisotropym at io n w he n Eq . (5 .28) i s inse r te d in to Eq . (5 .27) , and by tak in g in toacc ou nt E q. (5 .16) , one f inds

Ms(G ,T) M s(0,T) + ch 2( )(529,N w TAs T < < Tc, the second te rm of Eq . (5 .29) i s expo nen t ia l ly sm al l and thefu n c t io n M s ( H a , T ) i s t h u s a v e r y w e a k f u n c t i o n o f H a. In t h e fo l l o win gc h a p te r s , we s h a l l u s u a l ly n e g le c t th i s d e p e n d e n c e , b y a s s u m in g th a t t h e r ee x i s t s a s p o n t a n e o u s m a g n e t i z a t i o n o f m o d u l u s M s (T ) i n d e p e n d e n t o ff ie ld. Th is is a pe r fec t ly accep tab le app rox im at io n in m os t cases, p ro v id edth e t e mp e ra tu r e i s n o t t o o c lo s e t o t h e Cu r i e p o in t .

5 . 2 ANIS O TRO P YI n t h e p r e v i o u s s e ct io n , w e h a v e d i s c u s se d h o w t h e s p o n t a n e o u s m a g n e t i -z a t i o n M s ( T ) c h a ra c t e r i z in g a f e r ro ma g n e t c a n b e d e t e rm in e d in t h e mo le c -u l a r f i e ld a p p ro x ima t io n . W e h a v e s e e n th a t e x c h a n g e in t e r a c t io n s a r ei s o t ro p ic i n s p a c e , wh ic h me a n s t h a t t h e e x c h a n g e e n e rg y o f a g iv e nv o l u m e AV i s t h e s a me fo r a n y o r i e n t a t i o n o f t h e ma g n e t i z a t i o n v e c to r ,p rov ided tha t i t s s t reng th remains the same . In rea l i ty , th i s ro ta t iona lin v a r i a n c e i s a lwa y s b ro k e n b y a n i s o t r o p y e ff ec ts , wh ic h m a k e p a r t i c u l a rspa t ia l d i rec t ions energe t ica l ly favored . In th i s sec t ion we d iscuss thero u te s l e a d in g to a b r e a k in g o f t h e e x c h a n g e s p h e r i c a l s y mme t ry . Th ea p p r o a c h w i l l b e b a s e d o n s y m m e t r y a r g u m e n t s , a n d w i l l m a k e n o d i s ti n c-t i o n b e twe e n th e v a r io u s mic ro s c o p ic me c h a n i s ms , d i s c u s s e d in S e c t io n5 .3 , eve n tu a l ly respons ib le fo r an iso t ropy . In fac t, th i s d i s t inc t ion i s o f teni r r e l e v a n t . Th e c o n s e q u e n c e s o f a n i s o t ro p y o n th e ma c ro s c o p ic b e h a v io ro f a ma g n e t i c ma te r i a l a r e ma in ly d i c t a t e d b y th e s p a t i a l s y mme t ry o fth e a n i s o t ro p y e n e rg y , r a th e r t h a n b y th e p a r t i c u l a r me c h a n i s m g iv in gr i se to the an iso t ropy i t se l f .

5 . 2 . 1 S y m m et ry b rea k i ngL e t u s c o n s id e r a c e r t a in v o lu me AV w i t h u n i f o r m m a g n e t i z a ti o n M . W eare in te res ted in the de pen den ce o f the AV f ree energ y FAN(m) on theo r i e n t a t i o n o f M . W e a s s u m e a g iv e n f ix e d t e mp e ra tu r e , w h e re IM I = M s .T h e d e p e n d e n c e o n T is n o t i m p o r t a n t a n d w i l l b e u n d e r s t o o d . U n d e rth e s e c o n d i t i o n s , t h e s ta t e o f t h e s y s t e m i s d e s c r ib e d b y th e m a g n e t i z a t i o nu n i t v e c to r m = M / M s. W e wi l l i d e n t i f y m b y i t s Ca r t e s i a n c o mp o n e n t s


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5.2 AN ISOTROPY 145

rex, my, m z or by the sph eric a l ang les 8, ~b (F ig . 5.7) . The re la t ion shi pb e t w e e n t h e s e q u a n t i t i e s i s

mx = sin 8 cos~bm v = si n8 si nm z = cos8

(5.30)

T h e f re e e n e rg y d e n s i t y fA N (m) = F A N ( m ) / A V c a n b e g ra p h i c a l l y r e p re -s e n t e d b y p l o t t i n g i t a s a su r f a c e i n sp a c e , w h e re t h e d i s t a n c e f ro m t h eor ig in o f the po i n t o f the su r fa ce ly in g a long the d i rec t ion m i s j u s t fAN(m)(see F ig . 5 .8 ) . In th i s rep resen ta t ion , i so t rop ic exchange g ives r i se to a

mLmz q,.

m

/ " "

my

F I G U R E 5.7. M agnetization un it vector m (Iml = 1), w ith definition of Cartesiancomponents and spherical angle coordinates.

FIGURE 5.8. Broken spherical sym me try with formation of easy magn etizat ionaxis.


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146 CHAPTE R 5 Exchange and Anisotropys p h e re . Th e a b s o lu t e v a lu e o f t h e e x c h a n g e e n e rg y p l a y s n o ro l e a n d ,a c c o rd in g ly , we s h a l l a s s u me th a t f A N ( m ) i s de f ined bu t fo r a cons tan t ,i n d e p e n d e n t o f m . Ch o o s in g d i f f e r e n t v a lu e s fo r t h e c o n s t a n t w i l l g iv ee n e rg y s u r f a c e s d if f e rin g in q u a n t i t a t i v e d e t a i ls , b u t c o n ta in in g th e s a m ein fo rm a t io n o n s y m m e t ry b r e a k in g . I n p a r t ic u l a r, t h e p r e s e n c e o f d e p re s -s io n s i n t h e e n e rg y s u r f a c e imme d ia t e ly s h o ws th e s p a c e d i r e c t io n s t h a ta re energe t ica l ly favored . These d i rec t ions a re ca l led e a s y m a g n e t i z a t i o na x e s . Th e y r e p re s e n t t h e d i r e c t io n s a lo n g wh ic h th e ma g n e t i z a t i o n n a tu -r a l ly t e n d s t o p o in t , i n o rd e r t o m in im iz e t h e s y s t e m f re e en e rg y .

Th e m a in p ro p e r t i e s o f t h e a n i s o t ro p y e n e rg y s u r fa c e ar e d e t e rm in e db y th e s e t o f m d i r e ct io n s s a t is fy in g th e e q u i l i b r iu m c o n d i t i o n0fAN(m) = 0 (5.31)3m

under the cons t ra in t Iml = 1 . The so lu t ions o f th i s equa t ion represen tlo c a l m in ima , s a d d le p o in t s , o r l o c a l ma x ima o f t h e e n e rg y s u r f a c e . Alo c a l m in imu m id e n t i f i e s a n e a s y a x i s . On th e o th e r h a n d , t h e t e rmsm e d i u m - h a r d a x i s a n d h a r d a x i s a re s o me t ime s u s e d to r e f e r t o a s a d d lep o in t o r t o a l o c a l ma x imu m, r e s p e c t iv e ly . De p e n d in g o n th e s y mme t ryo f t h e e n e rg y s u r f a c e , o n e ma y e n c o u n te r s imp le s i t u a t io n s wh e re j u s to n e e a s y a x i s i s p r e s e n t , o r mo re c o m p le x s i t u a t io n s wh e re m u l t i p l e a x e sa n d s a d d le p o in t s a r e i n v o lv e d . W e s h a ll d i s c u s s t h e v a r io u s p o s s ib i li t ie scase by case . To th i s end , i t i s appropr ia te to cons ide r the deve lopmento f t h e a n i s o t ro p y e n e rg y in p o we r s o f m c o m p o n e n t s , fAN(m) i s i n g e n e ra li n v a r i a n t u n d e r m i n v er s io n , s o t h e d e v e l o p m e n t w i ll c o nt a in e v e n p o w e r so n ly . Th e v a r io u s c o mb in a t io n s c o m p a t ib l e w i th t h i s r e q u i r e m e n t y i e ld ac o n v e n ie n t wa y to d e s c r ib e a n d c l a s s i fy d i f f e r e n t k in d s o f s y mme t ryb re a k in g . I n t h i s r e s p e c t , a f u n d a me n ta l r o l e i s p l a y e d b y u n i a x i a l a n dc u b i c an iso t ropy .

5 .2 .2 Uniax ia l an i so tropyL e t u s c o n s id e r t h e c a s e w h e re j u s t o n e p r iv i l e g e d s p a c e d i r e c t io n c o n t ro l sthe an iso t ropy energy . We wi l l iden t i fy th i s d i rec t ion by the z ax is . Thea n i s o t ro p y e n e rg y i s i n v a r i a n t w i th r e s p e c t to r o t a ti o n s a ro u n d th e a n i so t -ro p y a x is , a n d d e p e n d s o n ly o n th e r el a ti v e o r i e n t a ti o n o f m w i th r e s p e c tto t h e a x i s . Un d e r t h e s e c o n d i t i o n s , t h e a n i s o t ro p y e n e rg y i s a n e v e nf u n c t io n o f t h e m a g n e t i z a t i o n c o m p o n e n t a l o n g z , m z = cosS. It i s co m m onto use m x2 j r _ my2 = 1 _ mz2 = 1 - - C O 8 2 8 - s i n2 ~ , i n s t e a d o f COS2~, as th ee x p a n s io n v a r ia b l e . Th u s t h e e n e rg y d e n s i ty fAN (m ) w i l l h a v e th e g e n e ra le x p a n s i o n


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5 .2 AN ISOTROPY 147

fAN(m) = K0 + K1 s i n 2 ~ q - K2 s i n 4 ~ if- K3 s i n 6 ~ if- 9 9 9 (5.32)w h e r e t h e anisotropy constants K1, K2, K3 h a v e t h e d i m e n s i o n s o f e n e r g yp e r u n i t v o l u m e . F o r t h e m o m e n t , l e t u s l i m i t o u r c o n s i d e r a t i o n s t o t h ec a s e w h e r e t h e s e r i e s c a n b e t r u n c a t e d a f t e r th e s i n 2 8 t e r m . T h e n f AN ( m )e x h i b i t s t h e b e h a v i o r s h o w n i n F i g . 5 . 9 ( K 1 > 0 ) a n d F ig . 5 .10 (K1 < 0).W h e n K ~ > 0 , w e h a v e t w o e n e r g y m i n i m a a t 8 = 0 a n d 8 = or, t h a t is ,w h e n t h e m a g n e t i z a t i o n l i e s a l o n g t h e a n i s o t r o p y a x i s i n t h e p o s i t i v e o rn e g a t i v e d i r e c t i o n . T h e a n i s o t r o p y a x i s i s a n e a s y a x i s f o r m , w i t h n op r e f e r e n t i a l o r i e n t a t i o n , a c c o r d i n g t o t h e f a c t t h a t fA N d e p e n d s o n e v e np o w e r s o f m z. T h i s t y p e o f a n i s o t r o p y is k n o w n a s easy-axis anisotropy.

0.4

-0.4

-1 -0.5 0 0.5 1FIGURE 5 .9 . Un iax ia l an i so tropy wi th K1 > 0 . Top : ene rgy surface associatedwi th Eq . (5 .32), w he n K0 = 0.1 , K1 = 1, K2 = K3 = 0. Bottom: Vertical cut of theenergy surface. The z axis i s an easy magnet izat ion axis .


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1 4 8 C H A P T E R 5 E x c h a n g e a n d A n i s o t r o p y

lJ0.5

0

-0.5

-1

-0.4 0 0.4F IG U R E 5 .1 0 . U n i a x i a l a n i s o t r o p y w i t h K 1 < 0 . T o p : E n e r g y s u r f a c e a s s o c i a t e dwi th Eq . ( 5 .32), wh en K 0 = 1 .1 , K 1 = -1 , K 2 = K 3 = 0 . Bo t t om: Ver t i ca l cu t o f t hee n e r g y s u r f a c e . T h e x-y p l a n e i s a n e a s y m a g n e t i z a t i o n p l a n e .


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5.2 AN ISOTROPY 149

C o n v e r s e l y , w h e n K 1 K 0 , t h e e n e r g y is a t a m i n i m u m f o r t? = r w h i c hc o r r e s p o n d s t o m p o i n t i n g a n y w h e r e i n t h e x - y p l a n e . T h i s s i t u a t i o n i sd e s c r i b e d b y t h e t e r m e a s y - p l a n e a n i s o t ro p y . I n t h i s c a s e , t h e r e i s n o l o n g e ra s i n g le p r e f e r r e d a x is , b u t a c o n t i n u o u s d e g e n e r a c y o f e q u i v a l e n t o r i e n t a -t i ons .

L e t u s c o n s i d e r t h e c a s e w h e r e K 1 ~ 0 a n d m l ie s a l o n g t h e e a s y a x i s .L e t u s t a k e t h e e n e r g y o f t h is s t a te a s z e r o e n e r g y l ev e l. F o r s m a l l d e v i a t i o n so f t h e m a g n e t i z a t i o n v e c t o r f r o m t h e e q u i l i b r i u m p o s it io n , t h e a n i s o t r o p ye n e r g y d e n s i t y c a n b e a p p r o x i m a t e d , t o s e c o n d o r d e r i n 8 , a s

f A N ( m ) ~ K 182 ~ 2K1 - 2K1 cost? (5.33). 2K 1= 2K 1 - ]zolVIslz~VI cos t? = 2K 1 - ~0 M 9 HAN

T h e a n g u l a r d e p e n d e n c e o f t h e e n e r g y i s t h e s a m e a s if t h e r e w a s a f ie ldo f s t r e n g t h

H A N = 2K1 (5 .34 )

a c t i n g a l o n g t h e e a s y a x i s . T h e a n i s o t r o p y f i e l d H A N g i v es a n a t u r a l m e a s u r eo f t h e s t r e n g t h o f t h e a n i s o t r o p y e ff ec t a n d o f t h e t o r q u e n e c e s s a r y t o t a k et h e m a g n e t i z a t i o n a w a y f r o m t h e e a s y ax is . H A N w i l l o f t e n a p p e a r i nt r e a t m e n t s o f m a g n e t i c f r e e e n e r gy .

5 . 2 . 3 C u b i c a n i s o t r o p y

W e n o w c o n s i d e r t h e c a se w h e r e t h e a n i s o t r o p y e n e r g y h a s c u b ic s y m m e -t r y . T h i s i s t y p i c a l l y t h e c a s e f o r t h e a n i s o t r o p y o r i g i n a t i n g f r o m s p i n -l a tt ic e c o u p l i n g i n c u b i c c r y s ta l s. T h i s s ~ e t r y i m p l i e s t h e e x i s te n c e o ft h r e e p r i v i l e g e d d i r e c t i o n s , w h i c h w e t a k e a s t h e x - y - z a x e s . T h e l o w e s t -o r d e r c o m b i n a t i o n s o f m c o m p o n e n t s c o n s i st e n t w i t h t h e r e q u i r e d s y m m e -

2 2t r y a r e t h e f o u r t h - o r d e r c o m b i n a t i o n m x m l + y 2 2 + mz2 mx2 a n d t h eY 4s i x t h - o r d e r o n e , m 2 m ~ m z . T h e c o m b i n a t i o n , / a l s o z a d m i s s i b l e , m x +4 4 "m y + m z i s n o t i n d e p e n d e n t f r o m t h e p r e v i o u s o n e s , a s

4 4 if_ 4 2(m 2 m y m y m z + m z m yx + m y m z q_ 2 q_ 2 2 2 m2x)= (m2x + 2 _}_ m z2 )2 = 1 (5.35)T h e d e v e l o p m e n t o f t h e a n i s o t r o p y e n e r g y t a k e s th e f o r m

2 + 2 2 2m2x) + K 2 2 2 2f A N ( m ) = Ko + K l (m 2 m y m y m z + m z m x m y m z + . . . (5.36)


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150 CHAPTER 5 Exchange and AnisotropyT h e e q u i v a l e n t e x p re s s i o n i n t e rm s o f sp h e r i c a l c o o rd i n a t e s i s

fAN(m) = K~ + K1(sina S sina 2r + c~ (5.37)s in 2 2 r . .+ K 2 1 6

L e t u s d i s c u s s t h e a p p r o x i m a t i o n w h e r e o n l y t h e f o u r t h - o r d e r t e r m i simpor tan t ( i . e . , K2 = 0 ) . The behav io r of fAN(m ) i s shown in F ig . 5 .11 (K 1> 0) and Fig . 5 .12 (K1 < 0) . When K1 > 0 , there are s ix equivalent energym i n i m a w h e n t h e m a g n e t i z a t i o n p o i n t s a l o n g t h e x , y , o r z a x e s , i n t h ep o s i t i v e o r n e g a t i v e d i r ec t io n . T h e se t h r e e d i r e c t i o n s i d e n t i fy e a sy m a g n e -t i z a t i o n a x e s . I n t h e n o t a t i o n e m p l o y e d i n c ry s t a l l o g ra p h y , t h e e a sy a x e sa re axes . We see f rom F ig . 5 .11 tha t d i rec t ions a re sa dd lep o i n t s o f t h e e n e rg y su r f a c e (m e d i u m -h a rd a x e s) , w h e re a s t h e < 11 1> d i re c -t i o n s a r e l o ca l m a x i m a (h a rd a x e s) . If o n e c o n s i d e r s t h e e n e rg y b e h a v i o rfo r sm a l l d e v i a t i o n s a ro u n d o n e o f t h e e a sy a x e s , o n e f i n d s t h a t , a s fo rt h e u n i a x ia l c a se , th e e n e r g y p r o p e r t i e s c a n be s u m m a r i z e d b y a n a n i s ot -ro p y f i e l d HAN of s t reng th iden t i ca l t o Eq . (5 .34) . The energy p ic tu recha nge s com ple te ly w h en K1 < 0 (F ig . 5 .12) . The ro le o f the d i rec t ionsj u s t m e n t i o n e d i s i n t e r c h a n g e d . W e n o w h a v e e i g h t e q u i v a l e n t e n e r g ym i n i m a w h e n t h e m a g n e t i z a t i o n p o i n t s a l o n g t h e < 1 1 1 > d i r e c t i o n s . T h e< 1 1 0 > d i r e c t i o n s a r e m e d i u m -h a rd a x e s , t h e < 1 0 0 > d i r e c t i o n s a r e h a rda x es , a n d t h e a n i s o t r o p y f ie ld g o v e r n i n g t h e e n e r g y b e h a v i o r a r o u n d t h ee a s y a x e s b e c o m e s

4 IKII (5.38)HAN = 3 /~oMs

5 . 2 . 4 H i g h e r - o r d e r a n i s o t r o p y c o n s t a n t sNeglec t ing h igher -o rde r t e rms in Eq . (5 .32) and Eq . (5 .36) may be no ta l w a y s a c c e p ta b l e . W h e n h i g h e r -o rd e r t e rm s g i v e a n a p p re c i a b le c o n t r i b u -t i o n , e a sy a x e s m a y b e t r a n s fo rm e d i n t o h a rd a x e s o r c o m p l e x e n e rg yp ro f i le s m a y s e t i n. L e t u s s e e w h i c h a r e th e m a i n c o n se q u e n c e s fo r u n i a x i a la n d c u b i c sy s t e m s .U niaxial anisotropy. W h e n u n i a x i a l a n i so t ro p y i s d e sc r i b e d u p t o s i x t horder , a s in Eq . (5.32), t he poss ib le eq u i l ib r ium pos i t ion s fo r the ma gn e t i z a -t ion (Eq. (5 .31)) are so lu t ions of the equat ion

O/AN=382[K1 s i n S + 2 K2 sin28 + 3K3 s in 3 8 ] = 0 (5.39)


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5 .2 A N I S O T R O P Y 1 5 1

0 .30 .20.1

-0.1.- 0 . 2

i /I

t !Ii

IIII

,/- 0 . 3 . . . . . . . . . .

-0.4 -0.2 0 0.2 0.4

FIGURE 5 .11. Cub ic a n iso t ropy w i th K1 > 0 . Top: Ene rgy su r f ace assoc ia ted w i thEq. (5.36), w h e n K0 = 0.1, K 1 = 1, K2 = 0 . Bottom: Ver t ical cu t of the energysu r face a long (100) p lane ( con t inuous l ine) and (110) p lane (b roken l ine) . d i r e c t i o n s a r e e a s y m a g n e t i z a t i o n a x e s .


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1 5 2 C H A P T E R 5 E x c h a ng e a n d A n i s o t r o p y

, - . . . , . . . . . . . . . . . . . . , .

0.4

0.2 , ,

0 ~ . . . . . . . . . .

- 0 . 2

- 0 . 4-0.4 -0.2 0 0.2 0.4

F IG U R E 5.1 2. C u b i c a n i s o t r o p y w i t h K 1 < 0. T o p: E n e r g y s u r f a c e a s s o c i a t e d w i t hEq . (5 .36) , when K 0 = 0 .4 , K 1 = -1 , K2 = 0 . Bo t tom: Ver t i ca l cu t o f the ene rgysur face a long (110) p lane (con t inuous l ine ) and (100) p lane (b roken l ine ) . d i r e c t i o n s a r e e a s y m a g n e t i z a t i o n a x e s .


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5.2 ANISOTROPY 1 5 3I n add i t i on t o 8 = 0 and 8 = or, w h i ch a r e a l w a ys s o l u t i ons , one f i ndst ha t t w o add i t i ona l s o l u t i ons may ex i s t f o r ce r t a i n combi na t i ons o f t hean i s o t r opy cons t an t s :

si n8 c = ~ - x -+- ~ /x 2 - 3y3y (5.40)w h e r e

x K2 K3= K--~ y = ~ (5.41)T he s o l u t i on co r r e s pond i ng t o t he + s i gn i n E q . ( 5 . 40 ) i s an ene r gym a x i m u m w i t h r e s p e c t t o 8 , a n d t h e o t h e r o n e i s a m i n i m u m . B e c a u s et h e e n e r g y is i n d e p e n d e n t o f t he a z i m u t h r t hi s m i n i m u m c o r r e sp o n d st o a s ta t e w h e r e m m a y l ie a n y w h e r e o n t h e c o n e o f a p e r t u r e 8c a r o u n dz. This i s cal led an e a s y - c o n e s ta t e . T he p r e s en ce o f add i t i ona l equ i l i b r i ump o i n t s l e a d s t o m o r e c o m p l i c a t e d e n e r g y p r o f i l e s , w h e r e m e t a s t a b l es t a t e s can ex i s t i n add i t i on t o t he g l oba l mi n i mum. T he na t u r e o f t heg r o u n d s t a t e c a n c h a n g e f r o m e a s y - a x i s o r e a s y - p l a n e t o e a s y - c o n et y p e , d e p e n d i n g o n t h e v a l u e s o f t h e r a ti o s K 2 / K 1 a n d K 3 / K 1. T h es i tua t ion i s summar ized by F ig . 5 .13 and F ig . 5 .14 . In F ig . 5 .14 , typ ica le n e r g y p r o f i l e s a s s o c i a t e d w i t h p a r t i c u l a r p o i n t s o f t h e p h a s e d i a g r a m sof F ig . 5 .13 a re shown. F igure 5 .14 i l lus t r a tes wel l the ex i s tence ofm e t a s t a b l e m o r ie n t a ti o n s , w h i c h a r e lo c al b u t n o t g l o b a l e n e r g y m i n i m a .I f t he s y s t em i n i t i a l l y occup i e s one o f t he s e s t a t e s , i t c an becomeuns t ab l e unde r t he ac t i on o f t he ex t e r na l f i e l d , i n a w ay s i mi l a r t ot ha t d i s cus s ed i n S ec t i on 2 . 2 . T h i s g i ves r i s e t o a B a r khaus en j ump , i nw h i c h t h e m a g n e t i z a t i o n s u d d e n l y r o t a t e s t o w a r d a n e w o r i e n t a t i o no f l o w e r e n e r g y . T h i s p h e n o m e n o n i s a c t u a l l y o b s e r v e d a n d w i l l b ed i scussed in Sec t ion 8 .2 .3 .W e p r ev i ous l y i n t r oduced t he concep t o f an i s o t r opy f i e l d i n r e l a t i onto easy-ax i s un iax ia l an i so t ropy (Eq . (5 .34) ) . HA N w a s d e f i n e d a s t h ef ie ld t h a t m i m i c s th e r e s t o r in g t o r q u e a c t i n g o n t h e m a g n e t i z a t i o n w h e nm l i e s i n t he v i c i n i t y o f t he ea s y ax i s . A no t he r de f i n i t i on o f an i s o t r opyf i e l d i s pos s i b l e , w h i ch co i nc i des w i t h t he p r ev i ous one on l y w henh i ghe r - o r de r cons t an t s a r e neg l i g i b l e . I n t he new de f i n i t i on , t he an i s o t -r o p y f i e l d i s t h e m i n i m u m f i e l d s t r e n g t h H K a p p l i e d p e r p e n d i c u l a r l yt o t he ea s y ax i s , t ha t i s ab l e t o f o r ce t he magne t i za t i on t o becomep e r p e n d i c u l a r t o t h e e a s y a x i s . H K c a n b e c a l c u l a t e d b y c o n s i d e r i n gt ha t t he t o r que 3 f A N / 3 8 i s b a l a n c e d b y t h e t o r q u e ]zo H a Ms s i n 8 exe r t edby t he ex t e r na l f i e l d . B y t ak i ng i n t o accoun t t he exp r e s s i on f o r OfA N / 0 8give n by Eq. (5.39) an d by no t ing tha t 8 = r tha t i s, s in8 = 1 , w h en


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154 CHAPTER 5 Exchange and Anisotropy6

42

T. -

-2-46

2

-2-4

. / e a s y/ X x c o n er " .2B

- e a s yp l a n e!

e a s ya x i s

/ A

K I > O

w

i , , ~ 1 I

K I < Oe a s yp l a n e

A

B ~e a s yc o n e i t | I

' ' o 2 4

K2/K ~

FIGURE 5.13. Phase diagra m sho win g natu re of gro und state for uniaxial anisot-rop y energy, Eq. (5.32), as a function of higher-orde r anisotropy constants. A,B,Cm ark states wh ose energ y profile is show n in Fig. 5.14 (after Ref. 5.1).

m i s p e rp e n d i c u l a r t o t h e e a sy a x i s , o n e f i n d s t h a t t h e f i e l d s t r e n g t h t ob e a p p l i e d i s2Il K- (K 1 + 2K2 + 3K3) (5.42)~0Ms

t o be co m pa red wi th Eq . (5 .34) . H K = /- /AN on ly w h en K2 an d K3 areneg l ig ib le .

F i na ll y, w e r e m a rk t h a t i n cry s t a ls w h e re t h e a n i so t ro p y i s of m a g n e t o -c rys t a l l i ne o r ig in , a s wi l l be d i scussed in the nex t sec t ion , t he p resence o fh i g h e r -o r d e r t e rm s i s r e l a t e d t o t h e sy m m e t r y o f t h e c ry s ta l . F o r e x a m p l e ,a n i so t ro p y i n t e t r a g o n a l c ry s t a ls m a y e x h i b i t a fo u r t h -o rd e r b i a x i al c o n t r i-2 2 ~ s in48 s in 2 2~. A fter re-u t i o n i n t h e x -y p l a n e , p ro p o r t i o n a l t o m x my


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5.2 ANISOTROPY 155

-1

/

-2 -1 0 1 2

-1K 1 < 0

-2 -1 0 1 2

FIGURE 5.14. Ve rticalcuts, calculated from Eq. (5.32), of energ y surfa ce assoc iatedwith points A , B , C of the phase diag ram s of Fig. 5.13 (compare w ith Fig. 5.9 andFig. 5.10). Top: K0 = 1, K1 = 1. Bottom: K0 = 1, K1 = - 1. Values of K 2 / K 1 a n dK3/K 1 a re given by the coordinates of points A , B , C of F ig. 5.13.

c o m b i n a t i o n w i t h t h e s in 4 8 t e rm a l r e a d y p re se n t i n E q. (5.3 2), t h i s g iv e sa t e rm p ro p o r t i o n a l t o s in 4 8 c o s4 r In h e x a g o n a l c ry s ta l s , t h e s ix - fol dsy m m e t ry a ro u n d t h e c - a x i s g i v e s r i s e t o a t e rm p ro p o r t i o n a l t o s i n 4 8cos6r

C u b i c a n i s o t r o p y . I n sy s t e m s w i t h c u b i c a n i so t ro p y , t h e s i t u a t i o n i s m o rec o m p l i c a t e d t o r e p re se n t , g i v e n t h e i n h e re n t t h r e e -d i m e n s i o n a l c h a ra c t e ro f t h e sy n m l e t ry . W e j u s t re c a ll th a t , d e p e n d i n g o n t h e r e l a t iv e v a l u e a n dsign of the c on stan ts K1 an d K2 of Eq. (5.36) , the qua l i ta t iv e fea tu res o ft h e e n e rg y su r f a c e m a y c h a n g e a n d t h e c h a ra c t e r o f < 1 0 0 > , < 1 1 0 > , a n d< 11 1> d i r e c t io n s m a y c h a n g e f ro m e a sy -a x i s t o h a rd -a x i s t y p e . T h e s i tu a -t i o n i s s u m m a r i z e d b y t h e f o l l o w i n g p r o s p e c t .


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156 CHAPTE R 5 Exchange and Anisotropy

K 1 > 0x - - K 2 / K 1 o o < x < - 9< 1 0 0 > M e d i u m - h a r d Hard EasyK 1 < 0x = K 2 / l g l [ oo < x < 9/4< 1 0 0 > Ha rd< 1 1 0 > M e d i u m - h a r d Easy

- 9 < x < - 9 / 4Ea s yH a r dM e d i u m - h a r d

9 / 4 < x < 9H a r dEa s yM e d i u m - h a r d

- 9 / 4 < x < + o oEa s yM e d i u m - h a r dH a r d

9 < x < + o oM e d i u m - h a r dEa s yH a r d

5.3 A N I S O T R O P Y M E C H A N I S M STh e re ma y b e s e v e ra l c a u s e s b r e a k in g th e ro t a t i o n a l s y mme t ry o fi so t rop ic exchange . Exchange i t se l f can con ta in an iso t rop ic co r rec t ionsd e p e n d i n g o n t h e s y m m e t r y o f t h e a t o m i c s p i n c o n f i g u r a t i o n , a n da to mic s p in s c a n b e c o u p le d to t h e h o s t in g l a t t i c e . Th e s e me c h a n i s msa r e s u m m a r i z e d u n d e r t h e t e r m m a g n e t o c r y s t a l l i n e a n i s o t r o p y . For s imi la rr e a s o n s , t h e e n e rg y o f a f e r ro m a g n e t c a n b e a f f e ct e d b y e l as ti c d e fo rm a -t io n s o f t h e h o s t in g me d iu m, wh ic h g iv e s r i s e t o m a g n e t o s t r i c t i o n effectsa n d s t r e s s a n i s o t r o p y . On a c o mp le t e ly d i f f e r e n t s c a l e , ma g n e to s t a t i ce n e r g y c a n p r o d u c e a n i s o t r o p y w h e n e v e r t h e m a g n e t i c b o d y i s n o t o fs p h e r i c a l s h a p e . F in a l ly , a n i s o t ro p y c a n b e i n d u c e d b y a to mic r e -a r r a n g e m e n t s , m a d e p o s s i b l e b y b r i n g i n g t h e s y s t e m t o t e m p e r a t u r e sh igh enough to induce a tomic d i f fus ion . In th i s sec t ion , we sha l l b r ie f lyd i s c u s s s o m e a s p e c t s o f t h e s e p h e n o m e n a .

5 . 3 . 1 M a g n e t o c r y s t a l l i n e a n i s o t r o p yIn a ma g n e t i c c ry s t a l , t h e d i s p o s i t i o n o f t h e ma g n e t i c mo me n t s r e f l e c t sth e s y m m e t r y o f t h e h o s t in g la tt ic e. I n t e ra c t io n s o f t h e mo m e n t s b e tw e e nth e ms e lv e s o r w i th t h e l a t t i c e a r e t h u s a f f e c t e d b y th e s y mme t ry o f


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5.3 ANISOTROPY MECHAN ISMS 157t he c r y s t a l and g i ve r i s e t o an i s o t r op i c ene r gy con t r i bu t i ons , s ummar i zedu n d e r t h e te r m o f magnetocrystalline anisotropy. The d o m i n a n t m e c h a n i s mg i v i ng r i s e t o magne t oc r ys t a l l i ne an i s o t r opy i s spin-orbit coupling. D u eto th i s coupl ing , e lec t ron ic orb i t a l s a re to some ex ten t t i ed to thee l e c t r o n i c s p i n a n d t e n d t o r i g i d l y f o l l o w t h e s p i n w h e n t h e m a g n e t i z a -t i on changes o r i en t a t i on i n s pace . T h i s has t w o cons equences : i t mayl e a d t o v a r i a t i o n s o f i n t e r a c t i o n e n e r g y b e t w e e n t h e m o m e n t - c a r r y i n gi on and t he hos t i ng l a t t i c e , o r i t may l ead t o an i s o t r op i c co r r ec t i onsto exchange i t s e l f .

T he magne t i c behav i o r o f a s i ng l e i on i n t he l a t t i c e depends onexchange , on t he e l ec t r i c po t en t i a l c r ea t ed a t t he i on pos i t i on bys u r r ound i ng a t oms ( c r y s t a l f i e l d ) , and on s p i n - o r b i t coup l i ng . T he r ea r e t w o ma i n pos s i b i l i t i e s , depend i ng on t he r e l a t i ve s t r eng t h o f t hecrys ta l f i e ld and sp in-orb i t e f f ec t s . On the one hand , the e lec t ron icorb i t a l s a re coupled to the l a t t i ce v ia the c rys ta l f i e ld , and on the o therh a n d t h e y a r e c o u p l e d t o t h e m a g n e t i z a t i o n v i a t h e s p i n - o r b i t c o u p l i n g .W h e n t h e o r b i t - l a t t i c e c o u p l i n g d o m i n a t e s a n d t h e o r b i t a l a n g u l a rmomen t um L i s r i g i d l y t i ed t o t he l a t t i c e , t he an i s o t r opy ene r gy i sde t e r mi ned by t he va r i a t i on o f t he s p i n - o r b i t ene r gy , p r opo r t i ona l t oL 9 S , w h e r e t h e o r b i ta l m o m e n t u m L is f ix e d a n d t h e s p i n m o m e n t u mS r o t a t e s w i t h M . T he oppos i t e ca s e i s w hen t he s p i n - o r b i t coup l i ngdomi na t e s . I n t h i s ca s e , t he o r b i t a l angu l a r momen t um L r i g i d l y f o l l ow sS , and t he an i s o t r opy ene r gy de r i ves f r om t he va r i a t i ons o f t he o r i en t a -t ion of L wi th r espec t to the l a t t i ce , i . e . , f rom the var ia t ion of theene r gy o f t he e l ec t r on i c w ave f unc t i on i n t he c r y s t a l f i e l d . A n i s o t r op i cco r r ec t i ons t o exchange a r i s e f r om va r i a t i ons i n t he deg r ee o f ove r l apo f e l e c t r o n w a v e f u n c t i o n s . W h e n t h e m a g n e t i z a t i o n p o i n t s a l o n gd i f f e r en t s pace d i r ec t i ons , t he o r i en t a t i on o f e l ec t r on i c w ave f unc t i onsi s a l so d i f f e ren t , because of sp in-orb i t coupl ing . This may g ive r i s e toa p s e u d o - d i p o l a r e n e r g y c o n t r i b u t i o n h a v i n g t h e s a m e s y m m e t r y o fd ipo lar f i e lds .

A quan t i t a t i ve p r ed i c t i on o f t he va l ues o f t he an i s o t r opy cons t an t sf r om quan t um mechan i c s i s d i f f i cu l t . M agne t oc r ys t a l l i ne an i s o t r opy i su s ua l l y des c r i bed no t by t heo r e t i ca l va l ues o f t he an i s o t r opy cons t an t s ,b u t b y m e a s u r e d v a l u e s o b t a i n e d f r o m s u i t a b l e e x p e r i m e n t s . S o m ed a t a o n m a g n e t o c r y s t a l l i n e a n i s o t r o p y c o n s t a n t s a n d c o r r e s p o n d i n ga n i s o t r o p y f i e l d s a t r o o m t e m p e r a t u r e a r e g i v e n i n A p p e n d i x D . I nF e , magne t oc r ys t a l l i ne an i s o t r opy has cub i c s ymmet r y , w i t h pos i t i veK 1 . T hus d i r ec t i ons a r e ea s y magne t i za t i on axes . N i a l s o hascubic an i so t ropy , bu t w i th nega t ive K1, so tha t the d i rec t ionsa r e ea s y axes . C o h as he xago na l s y m m et r y w i t h K 1 > 0 , and i s t huscha r ac t e r i zed by un i ax i a l e a s y - ax i s an i s o t r opy .


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158 CHAPTER 5 Ex chan ge and Anisotropy5 . 3 . 2 M a g n e t o s t r i c t i o n a n d s t r e s s a n i s o t r o p yThe mechanisms respons ib le for magnetocrys ta l l ine an iso t ropy a l so g iver i se to energy var ia t ions when the re la t ive pos i t ion of the magnet ic ions


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5.3 ANISOTRO PY MECHAN ISMS 1 5 9of the fo r m of Eq . (5 .43), a c rys ta l o f g iven m agn e t iz a t io n na tu ra l ly te ndsto d e fo r m in a wa y th a t d e c r e as e s i t s t o ta l fr ee e n erg y. Th e s p o n ta n e o u sd e fo rma t io n th u s a c q u i r e d i s c a l c u l a t e d b y min imiz in g th e s u mfM E + f E Lwi th r e s p e c t t o e ij u n d e r f i x e d m i. When th is ca lcu la t ion i s ca r r ied ou t , i ti s f o u n d th a t t h e s p o n ta n e o u s r e l a t iv e c h a n g e o f l e n g th 8 l / l of the c rys ta li n t h e d i r e c t io n n wh e n th e ma g n e t i z a t i o n p o in t s i n t h e d i r e c t io n m i sg iv e n b y th e e x p re s s io n

3 ( m 2 2T -- 2 /~100 Y/i -- 3,~111 ~ . j m i m j n i n j (5 .44)i irTh e c o n s t a n t s / ~ 1 0 0 a n d ' ~ 1 1 1 a r e cal led m a g n e t o s t r i c t i o n c o n s t a n t s a n d a r er e l a t e d t o t h e ma g n e to e l a s t i c c o u p l in g c o n s t a n t s p r e v io u s ly i n t ro d u c e dand to the crysta l e las t ic s t i f fness constants Cll c12 c44 conta ined in f E L ,t h ro u g h th e r e l a t i o n s

2 B1 1 B2'~100 = --- - ~111 = (5.45)3 Cll - c12 3 c44Eq ua t ion (5 .44) ra i ses the p rob lem of the re fe rence s ta te wi th respec t tow h i c h t h e e l o n g a ti o n s h o u l d b e e s t im a t e d . C o m m o n l y , o n e a s s u m e s t h a tth e s y s t e m i s i n i t i a l l y i n a mu l t i d o ma in s t a t e i n wh ic h m t a k e s , f r o mdomain to domain , a l l the poss ib le easy-ax is d i rec t ions o f the c rys ta l ( s ixfo r a cub ic c rys ta l wi th K1 > 0 , e igh t when K 1 < 0 ), in such a wa y tha tthe avera ge e lo nga t ion ca lcu la ted f ro m Eq . (5.44) van ishes . Ho we ver , them os t re l iab le ap pro ach i s ce r ta in ly to use Eq . (5 .44) to e s t im a te d i f fe rencesin e lo n g a t io n wh e n a c ry s t a l i s ma g n e t i c a l l y s a tu r a t e d a lo n g d i f f e r e n tdirec t ions .

Of p a r t i c u l a r i n t e r e s t i s t h e c a s e wh e re t h e e lo n g a t io n c o n s id e r e d i sin t h e s a me d i r e c t io n a s t h e ma g n e t i z a t i o n (n ~ m) . W h e n n = m , Eq .(5 .44) becomes8/- T - - '~100 if- 3('~111 -- '~ 1 0 0 ) ~ m2 m~ (5.46)ir

Eq ua t io n (5 .46) sho w s tha t/~100 an d ' ~ 1 1 1 rep resen t the re la t ive e longa t iono f t h e c ry s t a l w h e n m p o in t s a lo n g th e [10 0 ] d i r e c tio n ( m x = 1, my = m z= 0) or a lo ng the [111] one ( m x = m y = m z = 1 / V '3 ) .

Eq u a t io n (5. 43 ) c a n b e a p p l i e d t o t h e c a se wh e re t h e d e fo rma t io n i sp ro d u c e d b y th e a p p l i c a t i o n o f a g iv e n s t r e s s t e n s o r o i j . Let us cons ide r ,


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1 6 0 CHAPTER 5 Exchange and Anisot ropy

in pa r t i cu l a r , a un i ax i a l s t r e s s Or ap p l i e d a lon g t he d i rec t i on ~t. A p os i t i veo r n e g a t i v e v a l u e o f Or c o r r e s p o n d s t o a te n s i le o r c o m p r e s s i v e s tr e ss . B ye x p r e s s i n g t h e d e f o r m a t i o n t e n s o r o f E q . (5 .4 3) in t e r m s o f t h e s t r e ss a n dby t ak ing i n to account Eq . (5 .45) , one f i nds t ha t Eq . (5 .43) i s t r ans formedin to

3f r = -~ ,t 100or ~ m 2~ / - 3 , ~ 1 1 1 O r~ , m i m j Y i~i i~j

(5.47)

W e s e e t h a t , f o r g i v e n s t r e s s , t h e e n e r g y o f t h e c r y s t a l d e p e n d s o n t h ed i r e c t i o n o f m . T h u s w e h a v e a n a n i s o t r o p y e f f e c t s i m i l a r t o t h e o n e sd i s c u s s e d i n S e c ti o n 5 .2 . T h e s y m m e t r y i n v o l v e d h a s a m o r e c o m p l i c a te ds t r u c tu r e , t h o u g h . T h is c a n b e il l u s tr a t e d b y c o n s i d e r i n g s o m e p a r t i c u l a rc a se s . In t h e f o l l o w i n g e q u a t i o n s , 8 w i l l a l w a y s r e p r e s e n t t h e a n g l e b e -t w e e n m a n d t h e s t r e s s a x i s .St ress ap pl ied a long the [001] direct ion. Th is m e a n s t h at Yx = ) 'y = 0, Yz =1 . Equa t ion (5 .47) r educes t o

3 2 3f o " = --2~100or m z = const . + 2 ~ 1 0 0 o r sin28 (5.48)St ress a pp l ied a long the [111] direct ion. In th is ca se, Yx = ~'yT h e n = Yz = l /X /3 .

3f r = c o n s t . - 2 ~ 1 1 1 o r ( m . ~ /) 2 3= const . + ~ A 1 1 1 O r sin26~ (5.49)as m 9 "y - cos #.Iso trop ic m agnetos t r ic t ion , where ~ 1 0 0 - -- ~ 1 1 1 - - A s " T h is s o m e w h a t u n r e a li s ti cc a s e i s o f t e n c o n s i d e r e d a s a u s e f u l s i m p l i f y i n g h y p o t h e s i s . B y m a k i n gthe subs t i t u t i ons i n to Eq . (5 .47) , one f i nds

3f o " - - - - 2 ~ s o r ( m " ~ ) 2 3= const. + -,~s~r s i n 2 ~2 (5.5o)In t e re s t i ng ly enough , a l l t h ree ca se s de sc r ibed by Eq . (5 .48) , Eq . (5 .49 ) ,a n d E q . ( 5 . 5 0 ) l e a d t o e n e r g y e x p r e s s i o n s w i t h t h e s a m e s t r u c t u r e ,

3f r = ~,~or si n 2~9 (5.51)w h e r e # is th e a n g l e b e t w e e n t h e m a g n e t i z a t i o n a n d t h e s tr e ss d i r e ct io n ,a n d ,~ is t h e a p p r o p r i a t e m a g n e t o s t r i c t i o n c o n s t a n t . E q u a t i o n ( 5.5 1) d e -sc r ibe s un i ax i a l an i so t r op y ( see Eq . (5 .32)) . The s t r e s s c rea t e s an an i so t ro py


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5.3 ANISOTRO PY MECHAN ISMS 161ax is a long the d i rec t ion o f s t re ss app l ica t ion . The assoc ia ted an iso t ropyc o n s t a n t i s

3K r = K,~r (5.52)ZTh e a n i s o t ro p y i s o f t h e e a s y -a x i s o r e a s y -p l a n e ty p e , d e p e n d in g o nw he th er ,~cr > 0 or ,~cr < 0 . As an exa m ple , i f one a pp l ies a tensi le s t ressto an i ron crysta l for wh ich , a t ro om tem pe rat ure , ,~100 ~ - ,~111 ~ 2 10 -5 ,o n e o b ta in s e a s y -a x i s a n i s o t ro p y w h e n th e s t r es s is a p p l i e d a lo n g < 1 0 0>d i r e c t io n s , a n d e a s y -p l a n e a n i s o t ro p y wh e n th e s t r e s s i s a lo n g < 1 1 1 >direct ions .

5.3 .3 Shape anisotropyThe o r ig in o f shap e an iso tropy i s magne tos ta t ic ene rgy . S t r ic t ly speak ing ,th e t e rm d e s c r ib e s n o n e w p h y s i c a l me c h a n i s m. Th e c o n c e p t s n e c e s s a ryto exp la in i t hav e b een g iven in Sec t ion 3 .2 .3 an d Sec t ion 4 .1 .1 . Non e the -less , i t i s impor tan t to inc lude i t among the causes o f an iso t ropy , becausei t c a n b e a s imp o r t a n t a s ma g n e to c ry s t a l l i n e a n i s o t ro p y in d r iv in g th em a g n e t i z a t i o n p ro c e s s u n d e r m a n y c i r c u ms ta n c e s. W e h a v e s e e n in Se c tio n4 . 1 . 1 t h a t a b o d y o f e l l i p s o id a l s h a p e , w i th u n i fo rm ma g n e t i z a t i o n M s ,p o s s e s s e s a ma g n e to s t a t i c e n e rg y FM t h a t is a q u a d ra t i c f o rm in t h e ma g n e -t i z a t io n c o mp o n e n t s . W h e n w e t a k e th e p r in c ip a l ax e s a,b,c of the e l l ipso ida s Ca r t e s i a n a x e s , t h e ma g n e to s t a t i c e n e rg y d e n s i ty fM = F M / A V ( A V isin th i s case the e l l ipso id vo lume) takes the fo rm (see Eq . (4 .10) )

/ ~ 2 (Nam2x + N b m ~ + Ncm2z) (5.53)f M - - 2w h e r e N a, N b, a n d N c a re th e d e m a g n e t i z in g f a c to r s p e r t a in in g to t h e t h re epr inc ipa l axes , a n d N a + N b + N c = 1 . The energy su r face i s thus i t se l fa n e l l i p s o id , o f s h a p e c o mp le me n ta ry t o t h e o n e o f t h e r e a l b o d y , i n t h es e n s e t h a t , t h e l o n g e r t h e p r in c ip a l a x i s , t h e l o we r t h e c o r r e s p o n d in gd e m a g n e t i z i n g f a c to r a n d t h e l o w e r t he e n e r g y w h e n m p o i n t s a lo n g t h a td i rec t ion . In the case o f a sphero id , where two p r inc ipa l axes a re equa la n d th e b o d y h a s ro t a t io n a l s y m m e t ry a ro u n d th e t h ird , Eq. (5 .5 3) b e c o m e s

fM -" /Z ~ 2S [N m x m y 2 + N ,m z] /z 0 M s2 ( N s i n 2 ~ if- N Il c O s 2 ~ )(5 .54)


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162 CHAPTER 5 Exchange and Anisotropywh ere Nit and N~ are the dem agn et i z ing fac tors a long the sy m me t ry ax isand perpendicular to i t , respect ively. Equat ion (5.54) can be rewri t ten inthe form /M - 2 NIJ + 2 N N li sin28 (5.55)Equat ion (5 .55) has the symmet ry charac te r i s t i c of un iax ia l an i so t ropy .The t e rm shape an i so t ropy i s us ed to refer to i t, as i t i s the me re c ons equ enc eof the geomet r i ca l shape of the body. The corresponding an i so t ropy con-stant i s

1 - 3 / 5 5 6 ,K M - 2 4where we have made use of the fac t tha t 2N1 + N, = 1 . In a pro la t esph eroid , wh ere N > N, an d Nil < 89 we have easy-axi s an i so t ropy a longthe symmet ry ax i s of the sphero id . Converse ly , in an obla te sphero id ,w he re N , < Nil an d N, > 8 9 w e have ea sy -p l ane an i so t ropy i n t he p l anepe rpend i cu l a r t o t he sym m et ry ax i s .

5 .4 B I B L I O G R A P H I C A L N O T E SExchange and an i so t ropy are d i scussed in prac t i ca l ly a l l t ex t s on magne-t ism an d m ag ne tic ma terials [e.g. , B.55, B.67, B.69, B.71, B.77, B.81, B.91].Spec i f i c aspec t s re l a t ed to mater i a l p roper t i es a re addressed in a numberof rece nt school pro cee ding s [B.101, B.105, B.106, B.107, B.108]. W eissmolecular f i e ld theory , o r the equiva len t formula t ion in t e rms of Landaufree energy, i s a lso discussed in many texts of s ta t i s t ical mechanics andthe rm od yn am ics [e.g. , B.32, B.34, B.37, B.40, B.52]. For the qu an tu m aspectsof exch ang e an d anisot ropy, see [B.69, B.77, B.84, B.99].

A va luable d i scuss ion of the ro le of h igh er-o rder an i so t rop y co ns tan t sis given in Ref. 5.1.Magnetos t r i c t ion has only been superf i c i a l ly touched upon. The pre-sentat ion fol lowed in Sect ion 5.3.2 i s the his torical one, and refers onlyto s ingle crystals w i th cubic anisot ropy. Info rm at ion on the case of uniax ialan i so t rop y can be found in [B .96] , wh ere a thoro ug h account of the subjecti s g i ven , w i t h pa r t i cu l a r em phas i s on t he u se o f app rop r i a t e o r t hogona lbases fo r t he expans i on o f an i so t ropy and m agne t oe l a s t i c ene rgy w i t hdi f fe ren t symmet r i es .

exchange and anisotropy

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