J Exchange energy. Fe spin resolved

Transcription

1 ac popete agetm, why bothe? Up to ow oly tebly wea ect: χ -4 << ollectve couplg of p lead to tog magetc momet that ofte ca be detected dectly but eve moe commoly lead to hdde ode: atfeomaget ac popete The xchage teacto etal fo udetadg magetc teacto old Ae fom oulomb electotatc teacto ad the Paul excluo pcple eomaget Atfeomaget emaget lemet: e, o,, Gd Alloy: Pemalloy e lemet:, Oxde: O, O, HTSc Oxde: e O, Gd e 5 O 4 /T: o 5, Dy e 4 oulomb epulo-eegy oulomb epulo-eegy hgh loweed e 8 U J 4 πε ( 5 K!) ac popete xchage eegy Wave fucto of two electo mut be atymmetc fo exchage of patcle (, :, ) (, :, ) hace that two electo wth ame p ae at the ame place zeo. Paul pcple tae cae that paallel p avod each othe. lecto dety aoud each electo fee electo ga (bach ad Lüth) xchage coelato gap adu: / -Å ac popete xchage eegy oequece fom exchage coelato: Atpaallel p (glet tate) have lowe oulomb eegy tha paallel p (tplet tate) Δ exchage J xchage eegy J S A t J S See bach&luth Tplet Sglet 4 ac popete Heebeg Hamltoa ac popete e p eolved Heebeg Hamltoa fo lattce: ove all atom H J S S ove the eghbo atom eomagetc couplg fo potve J Atfeomagetc couplg fo egatve J What doe th mea? lecto o ame atom pefeg paallel p (Hud ule/atomc coelato): J> dg ofte eult atpaallel p o eghbo atom: J< ( eao why ot o may feomagetc mateal ext) Heebeg Hamltoa a good tatg pot fo theoe fo magetc mateal wth pedomat eghbo teacto (e.g. olato) 5 Schematc: agetc momet: μ

2 ac popete Popete of feomaget ot oly whole o half tege momet ac popete Heebeg Hamltoa Heebeg model Hamltoa H J S S eomagetc couplg fo potve J Atfeomagetc fo egatve J t S A J S Tplet Sglet lecto o the ame atom ca have paallel p (Hud ule/atomc coelato) odg ted to lead to atpaallel momet o eghbog atom (oe eao why thee ae ot may magetc compoud) ut f J potve eghbog atom momet couple paallel Heebeg Hamltoa good tatg pot fo may theoe o magetm fo ytem wth oly pa we teacto 7 8 ac popete &L8. xchage teacto betwee ee lecto ee electo wavefucto ( ) e Two fee electo pa wavefucto ( ) e e pace atymmetc: (, ) e e e e ( ) ( )( ) ( ) e e ac popete xchage teacto betwee ee lecto Pobablty of fdg electo d ad electo d : ( ) d d ( ) ( )( ) ( ) e e [ co( )( )] d d Two electo wth ame p caot be at ame poto oc chage of a p-up electo ot ceeed by othe p-up electo. Th lowe the eegy ad lead to a collectve exchage teacto wth potve g 9 ac popete xchage hole xchage teacto caue each p to puh away othe p al chage dety modfed: xchage hole up electo oly: ρ () e ex 9 All electo ρ () e ex ( co ) ( ) 9 ( co ) ( ) adu / -Å Ue th dety Schödge equato: Hatee-oc appoxmato. ull theoy Dety uctoal theoy Local Dety Appoxmato (D-LDA) ac popete ad (Stoe) odel of feomagetm Heebeg model doe ot completely expla feomagetm metal. S ( ( S ( ( Stoe paamete ad decbe eegy educto due to electo p coelato, dety of up, dow p

3 The ac popete ad (Stoe) odel (p exce) μ ( ( / S ( ( / Sp exce gve by em tattc: f ( f ( f [ exp( ( m / T )] /, S ( ) ( ( ac popete ad (Stoe) odel wth Δx Let be mall, ue Taylo expao: Δx g( x Δx / ) g( x Δx / ) g'( x) Δx g'''( x)...! ( ) f f ( ( ) ( )... ( 4 ( f f d ( ) π D( ) f() D.O.S.: dety of tate at em level ( π ) d( δ ( )) (at T) 4 ac popete ad (Stoe) odel D D( The D ( ) ) () ( ) O ( D O Whe >? < ( ) () ( ) ) Dety of tate pe atom pe p Thd ode tem D o D( ) > o e, o, th codto tue Doe t wo fo ae eath, though Stoe odto fo eomagetm 5 ac popete g() ) ad Δ fo a d feomaget. DOS at em level geate a d T: d 4 badwdth of d bad malle (d-d ovelap malle tha -) bad alo cota moe electo (ca tae total, v. fo 4 bad) ac popete Sp eolved DOS: d feomaget. Δ fom algmet wth (o agat) geate a d T: ac popete ad tuctue plu: eve a mall chage eegy, Δ, lead to a elatvely lage o. of d electo chagg p tate Δ 4 xchage teacto plt p degeeated bad Ẽ ( 7 8

4 Scadum ac popete ac popete Suceptblty of feomaget oppe d: t g d:e g 4p 4 t g eg p d bad alzed electo 4p bad fee electo How ca we coect the mcocopc Heebeg Hamltoa wth macocopc popete? ema: agetzato geeate o-ale ect elewhee: eld at the atom te the appled feld μ H the feld geeated by all the atom of the magetc obect. μ H H ) o( appled oal 9 ac popete We molecula feld Pee We: xchage teacto of eghbog momet poduce molecula feld popotoal to magetzato λμo We feld ehace We λμ o H What coecto betwee macocopc λ ad mcocopc exchage teacto J? J S S ac popete ea feld theoy appoxmate opeato Hamltoa wth t mea (aveage) value eplace S by mea value <S> H J S S H S J < S > <S> ca be expeed magetzato: om pevou chapte: μ -g L μ S Thu magetzato of atom Afte ome opeato ee gay box p. We λμo λ & poblem 8. gμ We S J μg μ ac popete ea eld Theoy fo dodeed phae T>T c gμ J ( T gj, ) μ J T μ ( T, ) μ tah T llou fucto wth Smple cae: L, JS/, g Hgh tempeatue: x<< : tah x x μ (, T ) T ecaue mall: μ H ue-we law χm ( T ) H T T μ μ /, ue tempeatue: T λ μ ( λμο ) T χ m (T ) ue-we T c T ue-we law μ μ /, ac popete tmato of exchage teacto J zj λ z eghbo μ μ g μ J me g μ χm ( T ) H T T ue tempeatuu: T λ o e: 9x 8 m - K T 4 K λ 4 4

5 ac popete ea eld Theoy fo odeed phae T<Tc T<T c : Spotaeou magetzato (alo zeo feld H) μ μ tah T y μ Aume (feld geeated by et of ample mall) Tc λμo λμo x μ y T T tah x ac popete ea eld Theoy fo odeed phae T<Tc Tc y tah x x μ y μ T T T c 7K Two multaeou equato fo y. a oly be olved gaphcally. 5 ac popete alue of ea eld T>T c ac popete alue of ea eld T<T c ea feld theoe exclude fluctuato fal aoud phae tato: ctcal behavo χ ( T T ) γ T c 7K l χ χm ( T ) T T ue ( T T ) β See SSS g T-T c (K) 7 8 ac popete eyod mea feld ac popete alue of ea eld T At low T: Spwave T c 7K ago ea feld γ ad β

6 ac popete Sp wave ac popete Sp wave Smla to phoo lea cha Low exctato leaze cha egy ( S S ) JS S Sˆ z σ ffectve feld J ( S S ) gμ ds quato of moto h μ JS S S dt ( ) quato of moto ds h JSˆ z dt JSˆ z ( σ σ ) ( σ σ σ ) 4JSσ ˆz ac popete Sp wave ac popete Sp wave Wave le oluto ( awt ) σ Ae Dpeo elato hω JS 4JS[ co( a) ] a a ( e e ) ε hω JSa egetc of exctato umbe of tate ( d d π g Specfc heat ad magetzato fom exctato d dt AT g T μ Sp wave the atfeomaget peovte LaO: A euto-catteg tudy ", Phy. ev 54, 549 (99) 4 H o ac popete H o ac popete Tght bdg evted gle electo h e e e H Δ m 4πε 4πε 4πε Ty lea combato of atomc obtal c φ c φ A A Petubato theoy gve mmze w..t. c A & c A ' H d d A - ' H Ogal eegy tegal teacto eegy tegal Ovelap tegal d d H AA ± H ± ± S A Tght bdg evted H AA φahφad e S ϕ A ϕd 4πε H φ Hφ d A H AA Atbodg odg H 5

7 H molecule ac popete H molecule ac popete xchage teacto ad odg Symmety equemet of wavefucto two electo H eglect H t ad ty poduct of atomc wavefucto: (, ) H ( ) H ( ) H (, ) t (,) [ ϕ () ϕ () ][ ϕ ( ) ϕ ( ) ] A A A - - (,) ϕ ( ) ϕ ( ) ϕ ( ) ϕ ( ) A A Space pat ymmetc fo exchage of () ad () p pat atymmetc: p glet S >- > (,) ϕ ( ) ϕ ( ) ϕ ( ) ϕ ( ) A A eaoable mplfcato: eglect tem wth both electo o ame atom Hetle Lodo wavefucto: (,) ϕ () ϕ () ϕ () ϕ () A A Space pat aymmetc fo exchage of () ad () p pat ymmetc: p tplet S > > > > 7 8 H ac popete molecule ac popete S: Ovelap tegal e S ϕa () ϕa () ϕ () ϕ () dd 4πε xchage teacto Petubato theoy gve Tplet ± A fo glet ± S - fo tplet Sglet : oulomb tegal e ϕa () ϕ ( ) dd 4πε A A A: xchage tegal e A ϕa ( ) ϕa ( ) ϕ ( ) ϕ ( ) dd 4πε A A mple molecule bodg lead to glet goud tate ± A ± S Sze of plttg defed a exchage cotat J xchage teacto ad odg fo glet - fo tplet S A J S t Geealzato to may atom: Heebeg Hamltoa eomagetc couplg fo potve J Atfeomagetc fo egatve J H Tplet Sglet J S S 9 4 7