Magnetic transitions in metallic compounds determined by RKKY interactions

Transcription

1 JOURAL O OPTOELECTROICS AD ADVACED MATERIALS Vol. 9, o. 4, Aprl 7, p Magnetc transtons n metallc compounds determned by RKKY nteractons E. IRSA *, C. CADI a, M. RACUCIU Physcs Department, Unversty Lucan laga, str. Dr I.Ratu, o.5-7, 554, Sbu, Romana a aculty of Engneerng Hermann Oberth, Unversty Lucan laga of Sbu, Romana We study the role of RKKY (Ruderman-Kttel-Kasuya-Yoa) nteractons n magnetc transtons observed n varous metallc compounds. These transtons were found to play an mportant role n problems nvolvng the nteracton of localzed moments n a metal, va polarzaton of conducton electrons. We show the exstence of the antferromagnetsmferromagnetsm transton n metallc compounds generated by RKKY nteractons. We study the role of ntersttals mpurtes n ths process and we provde the nfluence of exchange couplng constant. Indrect exchange couples moments over relatvely large dstances. It s the domnant exchange nteracton n metals where there s lttle or no drect overlap between neghborng magnetc electrons. It therefore acts through an ntermedary, whch n metals are the conducton electrons (tnerant electrons). The RKKY exchange coeffcent oscllates from postve to negatve as the separaton of the ons changes and has the damped oscllatory nature. Therefore, dependng upon the separaton between a par of ons ther magnetc couplng can be ferromagnetc or antferromagnetc. A magnetc on nduces a spn polarzaton n the conducton electrons. Ths spn polarzaton n the tnerant electron system s felt by the moments of other magnetc ons wthn range, leadng to an ndrect couplng. (Receved ovember, 6; accepted ebruary 8, 7) Keywords: Itnerant ferromagnetsm, RKKY nteractons, Magnetc transtons. Introducton The RKKY nteracton was found to play an mportant role n varous problems nvolvng the nteracton of localzed moments n a metal va polarzaton of conducton electrons [], []. It was demonstrated that at large dstances r the nteracton decays as /r and has the k (k been the erm wave-vector) oscllaton wth the erm momentum k that can be the premse of the magnetc transton bandferromagnetsm (M) band-antferromagnetsm (AM) and reverse. At ntermedate temperatures, when the wdth of the flled part of the band s comparable to kt( k s the oltzmann constant), the ampltude of the oscllaton s dmnshed. At hgh temperature, for the oltzmann gas, the magnetzaton has a Gaussan shape, where the recprocal wave number of the electron wth energy kt.e. mkt, s the decay length []. Thus we take n account the erm Drac dstrbuton law [4]: f D ( ε ) = ε μ exp kt At hgh temperatures, there are many unoccuped states wtch excted electrons can occupy. Thus, the Paul Excluson Prncple and erm Drac statstcs are useful n descrbng low temperature behavor (dfferent fromt = K ). As the temperature ncreases, only () electrons wth ktenergy closed to ε wll be thermally excted [5]. The total energy of the system n ths case s: total ( ) ( ) U = ε ε f ε dε () RKKY theory explans that a magnetc on s able to spn polarze surroundng conducton electrons wth λ [6]. In turn, these spn polarzed electrons can couple to the spn of a nearby on, thus creatng a cooperatve nteracton between dstant magnetc ons. The oscllatory nature of the polarzaton at large dstance s of the form: D cos( k a) f ( a) = () a where a s the dstance from the local moment, thus, there are regons n whch the spns are polarzed successvely n the up and down confguratons wth respect to the magnetc on. Whether ferromagnetc or antferromagnetc behavor s favored s dependent on the dstance between the conducton electron and the magnetc on.. The physcal model We propose a model that nclude the RKKY nteracton, the Hubbard model and the double exchange nteracton, n fact a hybrd model between the tnerant and locals moments magnetc behavor. We study ths model n a numercal way (Monte Carlo smulaton) and we fnd the nfluence of the RKKY nteracton n the stablty of the condensed state.

2 9 E. rsan, C. Candn, M. Racucu. The Hubbard model The Hubbard model s the smplest band model that proposes an electronc correlated approach and the Hamltonan nteracton s gven by [7]: H = tc c U n n (4) Hubbard,, σ σ σ where, t s a general hoppng matrx element between stes and. n ths Hamltonan c σ, c σ are the creaton and annhlaton operators, respectvely and U s the Coulomb nteracton. Also, n, n are the number partcle operators wth spn up and spn down. Thus, the Hubbard nteracton s nterplay between the knetc and the Coulomb nteracton terms... The double exchange (DE) model The double exchange model s descrbed by the followng Hamltonan [8]: where: c( ) and c ( ) HDE = Hhoppng Hnteracton (5), ( μδ ) H hoppng = t c () c( ) (6) H = J s S nteracton () () are the annhlaton and creaton operators for electrons at ste R n the spnor notaton c () = ( c (), c () ), μ s the chemcal potental, t s the hoppng ntegral, s ( ) s the spn densty operator of the conducton electrons and s gven by: s c c () = () σ () σ denotes the vector of Paul matrces and S( ) localzed spn, J s the ndrect exchange ntegral... The RKKY nteracton (7) (8) s a The RKKY nteracton s a well know nstrument for magnetc mpurtes propertes development. Ths magnetc nteracton s a basc model n the physcs of magnetsm []. The dagram formalsm s descrbed n some good monograph [4], [5]. The s-d ndrect exchange nteracton s determned by the Hamltonan: Hˆ = J s R S s R S ( ˆ( ) ( ) ) (9) and s conered lke a small perturbaton. Here R and R are the space vectors of the mpurtes, S and S are the spns mpurtes and sˆ ( R ) and sˆ ( R ) are electron spn operators n the correspondng space ponts. Usng the smple matrx equaton: Tr σ S σ S = S S () ( ) we obtan the space dependence of the ndrect exchange: RKKY where R = R R, R = R ( R ) SS Hˆ = J () Here ( R) J means the RKKY exchange ntegral and s gven by [9]: ( ) J R = d k d k ε( K) μ ε( h ) μ J a =. cos m (( k k ') R) m τ * k k * () where π and μ = s the erm * m energy and k s the erm momentum [6]. The expresson of the exchange ntegral s: where: and k ( ) ( ) ( ) J R = J A k R () ak A= 6 8 π μ e R / λ (4) k k τ λ = = (5) * q m s the electron mean path [6]. We see that the functon that descrbes the RKKY nteracton s: wth x cos x sn x ( x) = (6) 4 x x = (7) k R The exchange nteracton decrease lke / R n R lmt and oscllate wth / k perodc dstance. Thus the polarzaton of the conducton electrons nduced by the local magnetc spns s not unform and decrease wth dstance. Thus the nteracton between the nearest neghbor may be ferromagnetc or antferromagnetc and the sgn of ths nteracton s gven by the RKKY functon (x) [], [9].

3 Magnetc transtons n metallc compounds determned by RKKY nteractons 9.4. The hybrd model We propose a hybrd model that arses from these three models: RKKY, Hubbard, and DE. Our Hamltonan s gven by: H = HHubbard HRKKY HDE (8) The latest two terms n ths Hamltonan are correlated by J (the ndrect exchange nteracton). We apply the Monte Carlo algorthm to descrbe ths smulaton magnetc model for a fcc lattce wth parameter a and for a perturbated lattce wth a centered ntersttal atom. We fnd that the magnetc propertes are changng n ths way. The condensed phase s changng and s fnd a magnetc transton A M (antferromagnetsm ferromagnetsm). The generalzed nfnte dmensonal band fcc lattce, wth hoppng scaled as []: t = (9) Z (where Z s the number of neghbors) has a densty of states (DOS), gven by []: ( ε ) ( ( ε ) ) π ( ε) exp / = () that presents a square root sngularty at the lower band edge. t of Our parameters of work are the matrx ( ) hoppng, the Coulomb nteracton ( U ), the constant of the lattce ( a ), the chemcal potental μ, the erm wave vector k, the ndrect exchange ntegral J. Thus, we can study the role of nter ste hybrd nteractons n decdng ferromagnetc (or antferromagnetc) state n the tnerant electron (narrow band) systems lke the transton metals. We have conered Hubbard lke tght bndng model along wth exchange and hybrd nteractons. All these nteractons have been treated wthn mean feld approxmaton []. It s well known that the onset of ferromagnetsm n tnerant electron sold need not be due to the relatve shft of the poston of maorty and mnorty spn bands as a result of large ntra atomc Coulomb nteractons. The nter atomc exchange nteractons may play crucal roles for the onset of ferromagnetc state. It has been argued that the ferromagnetc state may be realzed wth nter atomc exchange nteractons alone. The onset of ferromagnetsm s manfested through the narrowng and wdenng of, respectvely, (maorty) up and (mnorty) down spn electrons.. Monte Carlo algorthm The Monte Carlo (MC) smulatons are done an cubc lattce wth perodc boundary condtons. Although the smulaton MC s a sem-quantum concept (we use the standard Metropols algorthm), for the localzed spns we use the classc coneratons and the tnerant electrons are conered n an quantum pcture. The standard Metropols algorthm that was use s n the fact the follow the spn random dstrbuton s conered n start of algorthm and the random reorentaton on the ste (for example), wll nduce the energy change E. If the quantty exp ( Δ E / kt ) s smaller than a random number between and, the change s allowed, otherwse t s reected. In ths mode we can determne dfferent physcs parameters of the conered lattce by the calculaton of the statstcal average of those parameters: A exp( β E) A = exp( β E ) () where β = s the oltzmann factor and A s a kt physcal quantty A, n mcrostate. An obvous physcal studed quantty s the magnetzaton gven m by: M = s and the magnetzaton per spn m =. = Usually we are nterested n the average values M and the fluctuatons M M. The zero feld magnetc susceptblty χ s an example of a lnear response functon, because t measures the ablty of a spn to respond to a change n the external magnetc feld. χ s related to the fluctuatons of the magnetzaton: M M χ = () kt Thus we can determne the magnetc propertes of the smulated model and next we can fnd the others characterstcs parameters. 4. Results a) Magnetc transton Take n account the oscllatory behavor of the RKKY nteracton we explan the magnetc transtons that appears n fcc Mn as a result of ntersttal mpurtes ncluson ( for example). Thus, Mn 4 present band ferromagnetsm whle the pure Mn s antferromagnetc. Ths s due to the dlataton of the crystallne lattce and thereafter, the ncrement of the dstance between Mn ons. The lattce parameter n Mn 4 (sold soluton) s.86 Å and x s 4.6. Thus the RKKY ntegral can change the sgn and the magnetc transton s possble.

4 94 E. rsan, C. Candn, M. Racucu (x) x b) Magnetzaton g.. RKKY oscllatng functon. We studed the magnetzaton n the MC smulaton for our hybrd model for dfferent values of * J = J t (where t s the hoppng factor).we fnd that the condensed phase s nfluenced by the couplng factor. The crtcal temperature s nfluenced by the ndrect exchange couplng (see n g. ): x.5 χ rel J * =5 J * = J * =5 T (K) g.. Quadratc temperature dependence of the magnetc relatve susceptblty (U=5). d) Magnetc phase dagram The phase dagram dependence by the ndrect exchange ntegral s plotted n fg.4. The ferromagnetc doman s enhanced wth the ndrect exchange ncreasng (n s the fllng level wth tnerant electrons) m J * =5 J * = J * = T(K) Para J * =5 J * = J * =5.. erro.. T(K) g.. The relatve magnetzaton n fcc lattce wth mpurtes (U=5). c) The magnetc relatve susceptblty The magnetc relatve susceptblty n a fcc lattce wth ntersttal mpurtes s nvestgated by our model n obect to fnd the nature of the magnetc state. It s mportant to see (g..) that the relatve susceptblty ( χ χ ) s enhanced by the exstence of the mpurtes. ( χ s the correspondng susceptblty wthout mpurtes). The lnear dependence of susceptblty (versus quadratc temperature) denotes a tnerant orgn of the ferromagnetc state n g.4. T vs. n phase dagram of the hybrdmodel for a fcc lattce, for dfferent J (U=5). 5. Conclusons The MC smulaton s used to test the hybrd model that ncludes a composte between tnerant and localzed standponts. We nvestgated the nfluence of the mpurtes n the observed magnetc transton (band antferromagnetsm-band ferromagnetsm). We take n account n our smulaton the exstence of non-magnetc ntersttals mpurtes by the factor of hoppng (n ths way the lattce s transformed n a smple cubc lattce). y the magnetcally pont of vew, the fcc structure s modfed (the spn arrangement s parallel), thus the new ordered state s ferromagnetc. We verfed the tnerant behavor of the condensate state and the dependence of the model by the ndrect exchange couplng. We found the phase dagrams (crtcal temperature versus the fllng level).

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