4. Collective magnetism

Transcription

1 4. Collectve agnets In ths chapter we wll nvestgate the effect of the nteracton between partcles on agnets. Magnetc phase transtons cannot be explaned wthout ncludng nteractons. We wll show that the Coulob repulson s fundaentally portant for understandng collectve agnets. 4.1 Dpole-dpole nteracton Let us frst consder the agnetostatc energy between two agnetc dpoles µ 1 and µ 2 : U dpole dpole = 1 r 3 [ µ 1 µ 2 3 µ n µ 2 n] where r n s the poston vector between the two agnetc dpoles. Approxately Wth U dpole dpole µ 1µ 2 r 3. µ 1 µ 2 gµ e h and r 2 Å one obtans a 0 = h 2 e s the ohr radus: 2 U gµ 2 r 3 e 2 hc 2 a0 3 e ev 1 K. r a 0 Ths nteracton s too weak to explan agnetc orderng! 4.2 Exchange nteracton between localzed spns We consder now the exchange nteracton. Ths nteracton s a quantu echancal effect steng fro the fact that the electrons are ferons. In order to calculate the exchange nteracton let us consder a hydrogen 79 Abbldung 4.1: The hydrogen odecule. olecule wth nucle A and Fg The Haltonan descrbng ths syste s gven by H = H 1 + H 12 H 1 = h e2 e2 r 1A r 1 H 12 = e2 + e2 r 12 R A and the olecule s wavefuncton s gven by Ψ 12 = [ϕ A 1 + ϕ 1][ϕ A 2 + ϕ 2] spns = [ϕ A1ϕ A 2 + ϕ 1ϕ 2 +ϕ A 1ϕ 2 + ϕ A 2ϕ 1] spns. Snce e2 r 12 s strong the states wth two electrons on one on can be neglected. Ths s the Hetler-London approxaton. The wavefuncton of the syste s then Ψ 12 = 1 2 [ϕ A1ϕ 2 ± ϕ 1ϕ A 2] snglet trplet where the requreent that the total wavefuncton be antsyetrc s fulflled. The energy of the syste can be expressed as E = Ψ 12 H Ψ 12 Ψ 12 Ψ 12 = 2E 1 + C ± A 1 ± S 80

2 where E 1 = d 3 r 1 ϕ A1 { h 2 } e2 ϕ A 1 < 0 r 1A s the one-partcle energy { 1 C = e 2 d 3 r 1 d 3 r } ϕ A 1 2 ϕ 2 2 < 0 R A r 12 r 2A r 1 s the Coulob ntegral Hartree ter { 1 A = e 2 d 3 r 1 d 3 r } ϕ R A r 12 r 2A r A1ϕ A 2ϕ 1ϕ 2 1 s the exchange ntegral and S = d 3 r 1 d 3 r 2 ϕ A1ϕ A 2ϕ 1ϕ 2 0 < S < 1 s the overlap atrx. E 1 C A S R. We calculate now the dfference n energy between the trplet and snglet states the exchange energy: { J = E t E s = 2E 1 + C A } { 2E 1 + C + A } 1 S 1 + S = C A 1 S C + A 1 + S J = 2 A SC 1 S 2. Snce the overlap S 1 and the exchange ntegral A > 0 J 2A > 0. Wth ths we have derved the effectve spn-spn Haltonan for two ons: H = 2J S 1 S 2 whch s called the Hesenberg Haltonan. It favors antparallel spns snce E t = J 4 / 2 E s = 3J In ost cases agnetc nteractons are antferroagnetc. We can generalze the Hesenberg odel: H = 2J j S S j. Ths s an effectve Haltonan that can be wrtten out of the Coulob nteracton between electrons and the Paul prncple. Apart fro the exchange nteracton there are other possble nteractons that are also portant for agnets lke the so-called RKKY nteracton. Ths nteracton acts between localzed agnetc oents through conducton electrons naely a localzed spn couples to the spns of conducton electrons and the second localzed spn sees the nduced spn-polarzaton of the conducton electrons. 4.3 Hesenberg odel and related lattce odels for collectve agnets We consder a ravas lattce descrbed by vectors { R }. At each lattce ste we have a spn S and a total oentu J = L + S. Ths stuaton s characterstc to systes wth partally flled f-shells lke Gd EuO etc. or partally flled d-shells lke MnO TOCl etc. nsulators and seconductors. Ths does not apply to Fe Co etc. snce n those systes the 3d electrons are not localzed and the collectve agnets s dscussed n ters of band agnets. As was shown n the prevous secton localzed spns are descrbed by the Haltonan H = J j S S j + gµ S. j For J > 0 the syste favors ferroagnets and for J < 0 antferroagnets. In one denson 1D there s no phase transton to an ordered agnetc state due to quantu fluctuatons. In one and two densons 2D at fnte teperatures the syste doesn t show long range order Mern-Wagner theore. 82

3 concepts lke frustraton and spn glass behavor can be descrbed by ths Haltonan. Let us consder specal cases of the odel. 1 Isng odel: H = j J j S z S z j + gµ Ths odel can be solved exactly n 1D and 2D. There s no phase transton n 1D at fnte T but the syste undergoes a phase transton to an ordered state n 2D. S z. wth the pecularty that contans the operators S j. In order to be able to treat the spns as non-nteractng we substtute the operators S j by ther average values S j so that wth H eff = eff = j gµ eff S J j gµ S j. The dea behnd ths approxaton s to consder the effect of nteractng spns wth a gven spn as a ean-feld effect Fg Instead of havng to 2 XY-odel: H = j J j S x S x j + S y Sy j + gµ S. The Hesenberg odel and related odels cannot be exactly solved n any cases. It s therefore useful to consder soe approxatons. 4.4 Mean-feld theory for the Hesenberg odel We consder the ferroagnetc Hesenberg odel n the presence of a agnetc feld: H = j Forally t has the for H = gµ S J j S j + gµ S. whch corresponds to a syste of non-nteractng spns n the presence of a agnetc feld gven by = j J j S j gµ + 83 Abbldung 4.2: Mean-feld approxaton for nteractng spns. deal wth a coplex proble of two-body nteractons nteractng spns we consder the case of a spn nteractng wth a ean feld created by the nteracton wth the other spns. The effectve agnetc feld has to be found self-consstently. We defne the agnetzaton M as the expectaton value of the spn operator: M = µ = gµ S. We consder = 0 0 and as a specal case S = 1 2. Then see prevous chapter M = µ tanhβµ eff = µ tanh βµ j = µ tanh [βµ + J j S jz gµ ZJ gµ 2M 84 ]

4 wth Z beng the nuber of nearest neghbors as we assued that spnspn nteractons take place only between nearest neghbors. Then the agnetzaton per ste s = M = µ tanh [ ] µ + A k T Ths s a self-consstent equaton wth A = ZJ g 2 µ 2. Wthout an external agnetc feld the equaton reduces to = µ tanh µ A k T. We can solve ths equaton graphcally as shown n Fg We consder. For a 1 there s only one soluton = 0 but for a < 1 there are three solutons. Fg. 4.3 shows that non-trval solutons are only possble f the dervatve of fx at x = 0 s bgger than 1: f 0 > 1 µ 2 A k T 1 cosh 2 µ 2 A k T x x=0 = µ 2 A k T. Ths gves a crtcal teperature between the solutons = 0 and 0: k T c = µ 2 A = ZJ g 2. T c s called the Cure teperature. For T < T c 0 even though the external agnetc feld s zero. The syste orders spontaneously whch s accopaned by a spontaneous breakng of rotatonal syetry. The Hesenberg odel s rotatonally nvarant as S S j s a scalar product. The soluton 0 breaks the rotatonal syetry snce the syste orders n a gven drecton n our case the z-drecton. M s the order paraeter of the ferroagnetc phase transton. The prevous calculaton was done for spn In the case of a syste of spns S the crtcal teperature s gven by k T c = 1 ZJSS Abbldung 4.3: The self-consstent equaton for agnetzaton solved graphcally. the functon fx = tanh µ2 A k T x and ntroduce 1 a = µ2 A k T. 85 ehavor near the phase transton We consder the Taylor expanson of µ A = µ tanh k T for sall : Then tanh x = x 1 3 x x 1. µ 4 A3 = µ2 A k T 1 3 k T

5 For T < T c 0 and T c = µ2 A k 1 = µ2 A k T 1 µ 2 A3 3 k T = T c T Tc 2 T µ Ths equaton can be rewrtten as 2 = 3µ 2 T 3 2 Tc T Tc 3 T 1 3µ 1 2 TTc T c whch defnes the behavor of the order paraeter near the phase transton: 3µ 2 Tc T. T c Abbldung 4.4: Magnetzaton as a functon of teperature T. Ths shows that the crtcal exponent for the agnetzaton α defned as s T c T α α = 1 2. Ths value s coonly obtaned n the ean-feld approxaton. We wll show that the order paraeter n the CS superconductvty has a slar behavor. 87 ehavor at sall teperatures T T c Let us wrte tanh n the followng way: Tc Tc 1 e 2 T = tanh = Tc µ T µ 2 T 1 + e Then for T T c 1 e 2 µ Tc T µ 1 e Tc 2 T. µ. µ Ths result devates fro observatons. At sall teperatures the nteractons aong spns are portant and cannot be accounted for n ters of a ean feld. Susceptblty n the ean-feld approxaton y defnton χ = M = M eff eff = 1 µ2 β cosh βµ eff ZJ M gµ 2. For T > T c M = 0 f 0. Therefore eff = 0 and cosh 2 βµ eff = 1. Then χ = µ2 1 + k T c k T µ 2 χ χ = 0 = µ2 k 1 T T c whch s the Cure-Wess law. It should be copared wth the Cure law for ndependent localzed spns derved n the prevous chapter. The antferroagnetc case was dscussed n the presentaton n class. 4.5 Spn wave exctatons: agnons We saw n the prevous secton that the behavor of agnetzaton at low teperatures s wrongly descrbed wthn a ean-feld approxaton. In order to descrbe agnetc exctatons of a syste of nteractng spns at low teperatures we shall ntroduce n ths secton the spn-wave theory. 88

6 We wll consder here the ferroagnetc case. Our startng Haltonan s the Hesenberg odel wth nearest neghbor couplngs n the presence of a agnetc feld: H = J S S +δ + gµ S δ where J > 0 and δ denotes the nearest neghbors. The wavefuncton for the -spn syste can be descrbed as Ψ = S 1... S... S wth S = S S S. The ground state T = 0 at = 0 s the ferroagnetc state: Ψ 0 = S S... S... S = Ths state s not anyore an egenstate of the Hesenberg Haltonan snce the ters S + j S j+δ propagate the local exctaton further. Ψ j s only an egenstate of the Isng odel but not of the Hesenberg odel. In order to take nto account the propagaton of the local spn exctaton n the Hesenberg odel we ntroduce the plane wave state Ψ = 1 e R j Ψ j j also called the one-agnon state. We wll show that ths s an egenstate of the ferroagnetc Hesenberg Haltonan. We frst show that [ ] H Ψ j = J ZS 2 2ZS Ψ j + 2S δ Ψ j+δ Ψ 0 = for spn The energy of the ground state s snce E 0 = ZJS 2 H = J δ S + S +δ Ψ 0 = 0 S S +δ = J δ S z S z +δ Ψ 0 = 1 4 Ψ 0 and therefore H Ψ 0 = 1 4 JZ Ψ 0 = E 0 Ψ 0. {S zs z+δ + 12 S+ S +δ + S S++δ } We consder now a local spn exctaton on the ground state: j whch for a general spn S corresponds to a wavefuncton Ψ j = S S... S S}{{ 1} S... S. j 89 whch follows fro the followng consderatons: S z S +δ S S... S 1... S = [ Z 1ZS 2 + ZZ 1S 2 + 2ZSS 1 ] Ψ j = ZS 2 2ZS Ψ j δ { S + S +δ + S S+ +δ} S S... S 1... S = [SS + 1 SS 1] δ = 4S δ where we used then Ψ j+δ Ψ j δ + [SS + 1 SS 1] δ S S = SS S 1 S + S = SS S + 1 ; H Ψ j = J [ ZS 2 2ZS Ψ j + 2S δ 90 Ψ j+δ ] Ψ j δ

7 and [ H Ψ = 1 e R j JZS 2 2ZS Ψ j 2SJ ] Ψ j+δ j δ = JZS 2 2ZS Ψ 2SJ 1 e R j + δ Ψ j+δ e δ [ = JZS 2 2ZS 2SZ δ jδ e δ ] Ψ. The dsadvantage of workng n spn space s connected wth the spn coutaton relatons: [S α S β ] = ǫ αβγ S γ. We can substtute the spn operators by ose operators such that all coutaton relatons are preserved. Forally the Hesenberg spn Haltonan wll be reforulated n ters of a Halton operator of nteractng bosons. Ψ are ndeed egenstates of H: H Ψ = E 0 + E Ψ wth E = 2S ZJ J δ e δ. Instead of beng descrbed by ts z coponent the spn state on a ste can be descrbed by the quantu nuber : = S... +S. Here s the azuthal quantu nuber and not the agnetzaton. We ntroduce the occupaton nuber n: n = S The one-agnon spectru E does not have an energy gap: E q 0 E 0. Ths s a specal case of the Goldstone theore whch states that all systes wth a broken contnuous syetry have gapless exctatons the so-called Goldstone odes. The contnuous syetry broken here s the spn rotatonal syetry. such that = +S corresponds to n = 0 and n general n = S. Then S + n = S + S = SS S + 1 = S 2 + S n 2 + 2Sn S 2 + n S S + 1 = 2S + 1 n n n 1 and Abbldung 4.5: Representaton of a spn wave. For phononc exctatons breakng of translatonal nvarance creates gapless exctatons whch are acoustc phonons. 4.6 Quantzaton of spn waves In order to characterze the agnon exctatons we can ether rean n the spn space or choose a dfferent representaton for these exctatons. 91 S n = 2S n n + 1 n + 1 S z n = S = S n S = S n n n. Holsten-Prakoff transforaton We defne the bosonc creaton and annhlaton operators va a n = n n 1 a n = n + 1 n + 1 a a n = n n 92

8 wth [a a ] = 1 and consder the canoncal transforaton S + S S z = 2S = 2Sa = S a a 1 a a 2S a 1 a a 2S Ths transforaton s canoncal snce the spn coutaton relatons are preserved: [S + S ] = 2Sz [S Sz ] = S [S+ Sz ] = S +. For nstance the frst relaton can be confred n the followng way: [S + S ] n = S + S n S S + n = 2S n n 2S z n. Then n ters of the newly ntroduced operators the Hesenberg Haltonan s wrtten as H = J [ δ Here a and a obey S 1 a a 2S a a +δ +a 1 a a 2S [a a j ] = δ j [a a j ] = [a a j ] = 0. 1 a +δ a +δ 2S 1 a +δ a +δ 2S a +δ +S a a S a +δ a +δ ] 4.1 These ose operators descrbe spn wave exctatons that are quantzed called agnons. The ground state of 4.1 corresponds to the state wthout agnons and the ground state energy s E 0 = JZS 2. Magnons are quaspartcles that descrbe exctatons of a spn syste. They do not fulfll the partcle conservaton prncple. We obtaned a Haltonan that s not spler than the ntal spn Haltonan due to the square roots n the defnton of a and a. Therefore we consder an approxaton where all ters that are quadratc or hgher n the partcle nuber operators are neglected. Wth that we get a lnearzed agnon Haltonan operator: H J Sa a +δ + a +δ a a +δ a +δ a a JZS 2. δ Ths approxaton neglects the agnon-agnon nteracton and therefore one deals here wth non-nteractng bosons. Through ths approxaton the Hlbert space of possble states changes: the spn Haltonan operator has 2S+1 states per ste whle the lnearzed agnon Haltonan operator has n states per ste there s no lt for exctatons. That s why the lnearzaton akes sense only at sall teperatures. The Haltonan 4.1 can be dagonalzed by consderng the Fourer transforaton a = 1 e R a Then a = 1 e R a. 4.2 H E 0 = J S 1 e R e R + δ δ + e R + δ e R + δ e R a a = JS e δ + e δ 2 a a = δ E a a 94

9 wth E = 2JS Z cos δ. δ The dagonalzed Haltonan 4.1 looks very slar to the dagonalzed Haltonan for phonons but the agnon dsperson for a ferroagnetc syste s quadratc n δ: cos δ 1 δ 2 2 E JS δ 2 = 2JSa 2 q δ wth a beng the lattce constant. Please note the dfference wth respect to the correspondng result for an antferroagnetc syste where the dependence s lnear. ow we can calculate the therodynac propertes of the syste. The nternal energy s gven by U = E E V k e T 2π h 1 3 d 3 q 2JSa2 q 2. e 2JSa2 q 2 k T 1 Out of ths ntegral one obtans the low-teperature teperature dependence of U U T 5/2 and of the specfc heat U T T 3/2. For the agnetzaton we get M = gµ S z = gµ a a S = gµ S a a = gµ S a a Then a a q 2 dq e E /k T 1 T 3/2. µ = M = gs AT 3/2 = 1 AT 3/2 whch agrees wth experental observatons at sall teperatures. The descrpton of exctatons n an antferroagnetc spn syste s consdered n the presentaton. We shall consder now tnerant agnets. 4.7 Itnerant agnets Itnerant agnets occurs n 3d systes systes that contan atos wth partally flled 3d electronc shells lke Fe Co or. There collectve agnets of tnerant electrons s possble due to the Coulob nteracton and Paul prncple. As a odel for tnerant electrons wth Coulob repulson we consder the Hubbard odel where and H = σ = σ ε k c σ c σ + U n.n. δ tc σ c +δσ + U c c c c c c c c c σ = 1 e R c kσ c σ = 1 e R c σ ε k = t n.n. δ e δ = 2tcos k x a + cos k y a + cos k z a for a cubc lattce. In the presence of a agnetc feld the Haltonan becoes H = σ ε k + gµ σc σ c σ + U c c c c

10 In order to treat the proble we can consder the followng ean-feld approxaton: H eff = ε k µ + U c c c c + ε k + µ + U c c c c U c c c c The derved result deonstrates an enhanceent of susceptblty due to correlatons whch s known as Stoner enhanceent. There s ferroagnetc nstablty for U 2µ 2 χ 0 1 U c ρ 0 E F 1 whch results n the exchange splttng of the two bands. = µ c c c c = µ c c c c { = µ dε fε ρ 0 ε + µ + U c c = µ dε fερ 0ε } ρ 0 ε µ + U c c { 2µ U c c c c } χt = 0 = = µ dε = 2µ 2 ρ 0 ε F f ε { ρ 0 ε { 1 + U 2µ 2 χt = 0 2µ + U } χt = 0 µ }. 4.4 The frst ter n the above susceptblty expresson s the Paul susceptblty for non-nteractng electrons n a agnetc feld and the second ter s present only for an nteractng syste. Fnally χt = 0 = χ 0 1 U χ 2µ 2 0 wth χ 0 = 2µ 2 1 f = 2µ 2 dε ρ 0 ε f T 0 = 2µ 2 ε k ε ρ 0 E F