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Paramagnetic-ferromagnetic transition in a double-exchange model Eugene Kogan1 and Mark Auslender2 1 Jack and Pearl Resnick Institute of Advanced Technology, Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel 2 Department of Electrical and Computer Engineering, Ben-Gurion University of the Negev, P.O.B. 653, Beer-Sheva, 84105 3 Israel 0 (February 1, 2008) 0 2 Westudyparamagnetic -ferromagnetic transition duetoexchangeinteraction between classical n localized magnetic moments and conduction electrons. We formulate the Dynamical Mean Field a Approximation equations for arbitrary electron dispersion law, concentration and relation between J exchangecouplingandtheelectronbandwidth. Solvingtheseequationswefindexplicitformulafor 8 thetransition temperatureTc. Wepresent theresults ofcalculations of the Tc for thesemi-circular electron density of states. ] h PACS numbers: 75.10.Hk, 75.30.Mb, 75.30.Vn c e m I. INTRODUCTION II. DMFA EQUATIONS AND TC - t a t The double-exchange (DE) model [1–3] is one of the Spins being considered as classical vectors Sn = mn s (with the normalization m2 =1), the problembecomes t. basic ones in the theory of magnetism. Magnetic order- | | a ing appears in this model due to exchange coupling be- a single electron one. The DE Hamiltonian in a single m electron representation can be presented as tween the core spins and the conduction electrons. The - Hamiltonian of the model is d Hnn′ =tn−n′ Jmn σδnn′. (2) n − · o H = tn−n′c†nαcn′α−J Sn·σαβc†nαcnβ, (1) Let us introduce Green’s function c nn′α nαβ [ X X Gˆ(E)=(E H)−1 (3) 2 wherecandc†aretheelectronsannihilationandcreation − v operators, Sn is the operator of a core spin, tn−n′ is the and local Green’s function 0 electron hopping, J is the exchange coupling between a 45 core spin and n electrons, σˆ is the vector of the Pauli Gˆloc(E)= Gˆnn(E) . (4) 2 matrices, and α,β are spin indices. D E 1 The model (the core spins being treated as classical InthelastEquation,theaveragingiswithrespecttoran- 2 vectors) began to be studied several decades ago [1–3]. domconfigurationsofthecorespins. Intheframeworkof 0 Because of a recent general interest in manganites, it the DMFA approach to the problem (see [8,10] and ref- / t reappeared in the focus of attention (see reviews [4–7] erences therein) the local Green’s function is expressed a m and references therein). In particular, application of the through the the local self-energy Σˆ by the equation Dynamical Mean Field Approximation (DMFA) [8] sub- - d stantially advanced our understanding of the properties Gˆ (E)=g E Σˆ(E) , (5) loc 0 n of the DE model [9,10]. Most of the papers dealing with − (cid:16) (cid:17) o themodel,startingfromclassicalpaperbyDeGennes[3], where c : considered the DE Hamiltonian with infinite exchange Xiv (cahnadngwe,itwhhtihcehaidsdcirtuiocnialoffotrhethaenteixfeprlraonmataigonneotifcmsuapgenreetxic- g0(E)= N1 k (E−tk)−1 (6) X r properties of manganites). However lately many proper- a ties of the model for arbitrary strength of the exchange is the bare (in the absence of the exchange interaction) were analyzed [11]. localGreen’s function. The self-energysatisfies equation In this paper we calculate the temperature of a paramagnetic-ferromagnetic transition Tc in a double- Gˆ (E)= 1 , (7) exchange model for arbitrary electron dispersion law, loc *Gˆ−1(E)+Σˆ(E)+Jm σˆ+ loc · concentration and relation between the exchange coupling and the electron band width by formulating and where X(m) X(m)P(m),andP(m)is aprobabil- h i≡ solving the DMFA equations. We treat the core spins as ityofagivenspinorientation(one-siteprobability). The classical vectors. quantities Gˆ andRΣˆ are 2 2 matrices in spin space. × 1 The difference between Eq. (7) and similar equation muchlessthechemicalpotential,sowecanconsiderelec- appearingintheCoherentPotentialApproximation[12], tron gas as degenerate. In this case is due to the fact that in our problem the disorder is an- ∞ µ nealed (and not quenched). That is the probability of f(E)...dE ...dE, (14) a given (random) core spin configuration is determined Z−∞ ≡Z−∞ bythe energyofelectronsubsysteminteractingwiththis where the chemical potential (in paramagnetic phase) is configuration. found from the equation To understandthe DMFA assumptionallowingto find the probability P(m) self-consistently with the solution 2 µ of Eq. (7), notice that Eq. (7) reduces the problem of n= Img dE, (15) −π electronscatteringbymanyspins, toscatteringby asin- Z−∞ glespinwiththeeffectivepotentialV = Jm σ Σˆ. The and n is the number of electrons per site. − · − spin is embedded in an effective medium, described by Forthe T , afterstraightforward,thoughlengthyalge- the Hamiltoniantk+Σˆ, and,hence,by the localGreen’s bra, we obtacin functionGˆ . Considerthechangeinthenumberofelec- loc tron states in such system due to a single spin. We get 2J2 µ g [12,13] Tc = 3π Im (Σg′−g)(1+Σg) 2J2g dE, (16) Z−∞ " g′+g2 − 3 # 1 ∆D(E,m)= Imlndet 1+ Jmσˆ+Σˆ Gˆ , (8) −π loc where Σ and g are determined by the properties of the h (cid:16) (cid:17) i system in the PM phase; they are the solutions of Eqs. wherethe argumentofbothG andΣ isE+i0. Sothe loc (5) and (11); g′ dg0(ε) . Eq. (16) is the main change in thermodynamic potential is [14,11,15] ≡ dε ε=E−Σ result of our paper. (cid:12) (cid:12) ∆Ω(m)= f(E)∆D(E,m)dE, (9) Inthecaseofweakexch(cid:12)angeJ 0Eq. (16)takesthe → form Z where f(E) is the Fermi function. The DMFA approxi- 2J2 1 µ mation for the one-site probability P(m) is: Tc = 3 N0(µ)− π Img02(E) dE. (17) (cid:20) Z−∞ (cid:21) P(m) exp[ β∆Ω(m)]. (10) In fact, this equation is the MF approximation [16] for ∝ − theRKKYHamiltonian[17],towhichtheoriginalHamil- Eqs. (7) and (10) are the system of non-linear (inte- tonian (1) can be reduced in the case J W. gral) equations. However, the system is grossly simpli- ≪ In the opposite case J we have to consider only fiedinparamagnetic(PM)phase,wherethemacroscopic →∞ one spin sub-band. (Because of the symmetry of the magnetization M = 0. In this case P(m) = const, the problem with respect to transformation n 2 n, ev- averagingin Eq. (7) canbe performed explicitly, and we → − erywhere further on we consider only the case n 1.) obtain ≤ After we shift E, Σ and µ by J, we obtain 1 1 g(E)= 2 g−1(E)+Σ(E) J, (11) 2 µ g′+g2 T = Im dE, (18) X(±) ± c π g′ 2g2 Z−∞ (cid:20) − (cid:21) where Σˆ =Σˆ1, Gˆ =gˆ1=g (E Σ)ˆ1, where ˆ1 is a unity 0 where g is found from the equation − matrix. In the ferromagnetic (FM) phase near the Curie tem- 1 1 g(E)= . (19) perature, Eqs. (7) and (10) can be linearized [9,11] with 2 g−1(E)+Σ(E) respectto M. Thus we reduce the DMFA equations to a traditional MF equation [16] P(m) exp( 3βTcM m). (12) III. TC FOR SEMI-CIRCULAR DOS ∝ − · The parameter T is formally introduced as a coefficient c LetusapplyEq. (16)tothe caseofsemi-circular(SC) in the expansion of ∆Ω(m) with respect to M. Non- bare density of states (DOS) trivial solution of the MF equation M= m (13) 2 E 2 h i N0(ε)= 1. (20) Ws W − (cid:18) (cid:19) can exist only for T < T , hence T is the ferromagnetic c c transition temperature. In all cases T turns out to be Then c 2 2 0.05 N (ε)dε 2 E E 0 g (E)= = 1 . (21) 0 E ε W W −s W −  Z − (cid:18) (cid:19) 0.04   From Eq. (5) follows 0.03 Σ=E W2g/4 g−1, (22) W − − T/c 0.02 and Eq. (11) can be written as 1 1 0.01 g = . (23) 2 E W2g/4 J X(±) − ± 0 0 0.2 0.4 0.6 0.8 1 Eq. (16) takes the form n FIG.1. Tc asafunctionofelectronconcentrationnfordif- ferent relative strengthsof the exchange: J/W =.25 (dotted J2W2 µ T = line), J/W = 1 (dash-dotted line), J/W = 2 (dashed line), c 6π Z−∞ and J/W =20 (solid line). g2 Im dE. (24)  E W2g E W2g J2W2g2 IV. NONFERROMAGNETISM − 4 − 2 − 6 (cid:16) (cid:17)(cid:16) (cid:17)  Notethatthatourmainresult(equationfortheT )in- In the case of weak exchange J/W n1/3, we obtain c ≪ dicatesit’s ownlimitsofvalidity. InpartoftheJ/W n − plane,Eq. (24)givesT <0. NegativevalueofT means, 4J2 c c T = (4y2 1) 1 y2, (25) that at any temperature, including T = 0, the para- c 3π2W − − magnetic phase is stable with respect to appearance of p small spontaneous magnetic moment, and strongly sug- where geststhatthegroundstateinthispartofthephaseplane 2 isnonferromagnetic[18]. The mechanismsleadingtothe n=n0(y) 1+ sin−1y+y 1 y2 . (26) destruction of ferromagnetism can be easily understood ≡ π − (cid:16) p (cid:17) in the extreme cases of weak and strong exchange. In the case of strong exchange J/W n1/3, Eq. (18) It is well known, that for J → 0 the DE Hamiltonian ≫ canbereducedtototheRKKYHamiltonian(seereview takes the form [10,15] [17] and references to the original papers therein) Tc = W6π2 Z−µ∞Im"1−gW262g2#dE HRKKY =Xnn′ Inn′mn·mn′. (30) Thus weak exchange between the electrons and the core W√2 1 = 1 y2 tan−1 3(1 y2) , (27) spins generates effective long-range and sign oscillating 4π − − √3 − (cid:20) (cid:21) effective exchange interaction between the core spins. p p In the opposite case J , we can reduce the DE where n=n (y)/2. If, in addition, n 1, we obtain → ∞ 0 Hamiltonian (2) to effective (classic) t J model (see ≪ − review [19] and references therein). To do that, first di- 3 T = Wn .53Wx. (28) agonalize the exchange part of the DE Hamiltonian by c 4√2 ≈ choosing local spin quantization axis on each site in the direction of m, the latter being determined by polar an- Notice, that virtual crystal approximation (for J ) gle θ and azimuthal angle φ. In this representation the → ∞ [3] gives Hamiltonian is Tc = 125Wx≈.13Wx, (29) Hnn′ =−Jσzδnn′ +tnn′(cid:18)ab∗nn′nn′ abnnn′n′ (cid:19), (31) where thuFsorbaeirnbgitrimarpyreecxicsheabnygea,tfahcetionrteogfr4a.linEq. (24)canbe ann′ =cosθn cosθn′ +sinθn sinθn′ei(φn′−φn) 2 2 2 2 ctiaolncualaretepdreosnelnytendumonerFicIaGll.y.1.The results of such calcula- bnn′ =sinθn cosθn′e−iφn cosθn sinθn′e−iφn′. (32) 2 2 − 2 2 3 The next step is to apply a canonical transformation describing a single spin in aneffective field, proportional to the macroscopic magnetization, and we obtained ex- 1 H H˜ =eSHe−S =H +[S,H]+ [S[S,H]]+... (33) plicit formula for the Tc. For the semi-circular electron → 2 density of states we explicitly calculated the transition whichexcludesallband-to-bandtransitions. Thiscanbe temperature as a function of the exchange interaction achieved if we chose the operator S in the form and electron concentration. Snn′ =−t2nJn′ (cid:18)−abn∗nn′′n abnnn′n′ (cid:19). (34) FoTunhdisatrieosneaardcmhinwiastserseudppboyrttehde IbsyraethleAcIasrdaeemliySocfieSnccie- ences and Humanities. We have [S,Hex]nn′ =−tnn′(cid:18)b∗n0′n bn0n′ (cid:19) [S,[S,Hex]]nn =2Xn′′ Inn′′(cid:18)−|bn0n′′|2 |bnn0′′|2 (cid:19), (35) [1] C. Zener, Phys.Rev. 82, 403 (1951). whereInn′′ = tnn′′ 2/(2J). Keepingtermsuptothesec- [2] P. W. Anderson and H. Hasegawa, Phys. Rev. 100, 675 | | (1955). ond order with respect to t (and only site-diagonal part [3] P. G. De Gennes, Phys. Rev. 118, 141 (1960). of the second order terms) we obtain [4] J. M. D. Coey, M. Viret, S. von Molnar, Adv.Phys. 48, Hñn′ =−Jσzδnn′ +tnn′(cid:18)an0n′ an0′n (cid:19) [5] Y1464u7,.1(A10.999I(z92y)0.u0m1)o.v and Yu. N. Skryabin, Physics-Uspekhi, + Inn′′(mn mn′′ 1)σzδnn′, (36) [6] M. Ziese, Rep.Prog. Phys. 65, 143 (2002). · − n′′ [7] D. M. Edwards, Adv.Phys. 51, 1259 (2002). X [8] A.Georges,G.Kotliar,W.Krauth,andM.J.Rozenberg, InthesecondquantizationtheHamiltonianobtainedhas Rev. Mod. Phys. 68, 13 (1996). the form (ignoring the constant term) [9] N. Furukawa,cond-mat/9812066. [10] N. Furukawa: in Physics of Manganites, ed. T. Kaplan H˜ = tnn′ cosθn cosθn′ and S.Mahanti (Plenum Publishing, New York,1999). 2 2 [11] A. Chattopadhyay and A. J. Millis Phys. Rev. B 64, nn′ (cid:20) X 024424 (2001); A. Chattopadhyay, S. Das Sarma, and +sinθ2n sinθ2n′ei(φn′−φn) d†ndn′ A. J. Millis Phys.Rev. Lett. 87, 227202 (2001). (cid:21) [12] J. M. Ziman, Models of Disorder (Cambridge University + Inn′mn·mn′d†ndn, (37) [13] APr.eCss.,HCeawmsborni,dgTeh,e19K7o9n).do Problem to Heavy Fermions, nn′ X (Cambridge University Press, Cambridge, England, whered†(d)istheoperatorofcreation(annihilation)ofa 1993). spinlesselectron. Thusstrongexchangebetweentheelec- [14] S. Doniach and E. H. Sondheimer, Green’s functions for trons and the core spins generates effective short-range solid state physicists (Imperial College Press, London, 1998). antiferromagnetic exchange between the core spins. [15] M. Auslender and E. Kogan, Phys. Rev. B 65, 012408 In conclusion, we formulated the Dynamical Mean (2002); Physica A 302, 345 (2001). Field Approximation equations for the double-exchange [16] M. Auslender and E. Kogan, EuroPhys. Lett. 59, 277 model with classical spins for arbitrary electron disper- (2002). sion law, concentration and relation between the ex- [17] J. H.Van Vleck, Rev.Mod. Phys. 34, 681 (1962). changeandtheelectronbandwidth. Inthevicinityofthe [18] A.Chattopadhyay,A.J.Millis, andS.DasSarmaPhys. paramagnetic-ferromagnetic transition critical tempera- Rev. B 64, 012416 (2001). ture Tc, these equations were reducedto a MF equation, [19] Yu.A. Izyumov,Physics-Uspekhi, 40, 445 (1997). 4