STABILITY OF METALLIC FERROMAGNETISM: CORRELATED HOPPING OF ELECTRONS IN Mn 4 N

Transcription

1 STABILITY OF METALLIC FERROMAGNETISM: CORRELATED HOPPING OF ELECTRONS IN Mn 4 N EUGEN BIRSAN 1, COSMIN CANDIN 2 1 Physcs Department, Unversty Lucan Blaga, Dr. I. Ratu str., No. 5 7, , Sbu, Romana, beneugen10@yahoo.com 2 Faculty of Engneerng, Unversty Lucan Blaga of Sbu, Romana Receved September 26, 2006 We study the ferromagnetc phase stablty n Hubbard model n Mn 4 N sold soluton. It s beleved that strong ferromagnetc orders n some solds are generated by subtle nterplay between quantum many-body effects and spn-ndependent Coulomb nteractons between electrons. Here we descrbe our numercal approach to ferromagnetsm n the Hubbard model, whch s a standard dealzed model for strongly nteractng electrons n a sold. We consder the correlated hoppng of electrons n two-dmensonal Hubbard model, to nvestgate the stablty of ferromagnetsm n Mn 4 N sold soluton wth a Monte Carlo smulaton. The ferromagnetc transton n double-correlated hoppng model s studed by the Monte Carlo method and we determned the crtcal temperature nvolved n ths process. 1. INTRODUCTION The theoretcal model of mangante materals, based on double-exchange (DE) nteracton, was gven n 1951 by Zener and has long been regarded as satsfactory. The ferromagnetc stablty n Mn 4 N sold soluton s studed by the nterference between the localzed magnetc moment of Mn ons and the tnerant pcture of the electrons. The correlated nteracton of the band electrons was gven by a Hubbard Hamltonan lke [1]: (1) H = t cˆ cˆ + U nˆ nˆ j + j j,, The pure Mn has an FCC type lattce and an antferromagnetc algnment of the Mn spns. The antferromagnetc behavor of the FCC Mn s gven by the correlated nteractons of the electrons [6]. By the exstence of ntersttals N atoms t s created a brdge between the electronc hoppngs (the N been magnetc nert). Ths s a DE (Zener) nteracton where are nvolvng the two Paper presented at the Natonal Conference on Appled Physcs, June 9 10, 2006, Galaþ, Romana Rom. Journ. Phys., Vol. 53, Nos. 1 2, P , Bucharest, 2008

2 262 Eugen Brsan, Cosmn Candn 2 types of atoms Mn (wth a magnetc nfluence) and N (wthout a magnetc behavor). The Hubbard nteracton U represents the Coulomb repulson of electrons n the same orbtal at a gven lattce ste. It s gven by the matrx elements [2]: U = V r r (2) ( ) of the Coulomb potental, and s typcally on the order of a few ev [3]. The t j matrx determnes the nter-ste hoppng of tnerant electrons and s generally smaller than U, but may not be neglgble small [4]. The operators c ˆ +, c ˆ ndcate the creaton and the annhlaton operators of an electron and, ˆ, n nˆ are the number of partcle operators (wth spn up and down). Ths s the classcal Hubbard Hamltonan whch s a good nstrument of the band ferromagnetsm theory n the pure magnetc systems. 2. PHYSICS MODELS To explan the magneto-conductve propertes of mangantes, n whch the manganese s present n at least two dfferent valence states (Mn 3+ and Mn 4+ ), Zener proposed the mechansm of double exchange (DE). The Mn 3d levels are splt nto two subsets: t 2g (threefold degenerate) and e g (twofold degenerate). The t 2g electrons, lower n energy, are more localzed. The outermost e g electrons are more delocalzed, capable of hoppng from ste to ste provded that the t 2g spns on adjacent manganese atoms are parallel. Accordng to the Jahn Teller effect, as soon as an electron hops nto an empty e g orbtal, a dstorton of the cage of O atoms lowers the symmetry and further splts e g and t 2g levels. The electron s now more tghtly bound, formng a so-called lattce polaron. Accordng to DE, the algnment of adjacent localzed t 2g, spn on manganese atoms rules the dynamcs of tnerant e g carrers, whch hop from one atom to the next to yeld electrcal conductvty [12]. If adjacent t 2g spns are parallel (the ferromagnetc state), conducton s favored; f they are randomly algned (the paramagnetc hgh-temperature state), conductvty drops dramatcally. We proposed a Double Correlated Hoppng (DCH) model n Mn 4 N, where the ntersttals N can play the role of a brdge hoppng of the tnerant electrons. The hoppng of the electrons s helped by the exstence of N and thus the adjacent Mn spns are correlated wth N partcpaton. The double-correlated hoppng model s descrbed by the followng Hamltonan: H t cˆ+ = cˆ() j + U nˆ nˆ J s S (3) () () () j H j,,,

3 3 Stablty of metallc ferromagnetsm = where u s the Coulomb nteracton, t j s the hoppng ntegral, c () = ( c + (), c + () ) s the creaton operators and c () c (), c () annhlaton operators. Also, s () c+ () c() ( ) = s the 1 = s the spn densty operator 2 of the conducton electrons and s the vector of Paul matrces, S () s the localzed spn and J H s the ferromagnetc Hund couplng (0 < J H < ). 3. MONTE CARLO ALGORITHM The Monte Carlo (MC) smulatons are done n N 3 cubc lattces wth perodc boundary condtons. Although the localzed spns are consdered classcal, the knetc energy of the conducton electrons s calculatng by dagonalzng the DE Hamltonan [11]. The standard Metropols algorthm was used n the MC smulatons. The spn random reorentaton on the ste, wll nduce the energy change ΔE. If the quantty exp ( Δ E/ KT) s smaller than a random number between 0 and 1, the change s allowed, otherwse t s rejected [10]. The average of the magnetzaton, m, are [13]: m= 1 S 3 N (4) where denotes statstcal average. Due to the fnte sze of the unt cells used n the smulatons, m s dfferent from zero at any temperature, but we defne the crtcal temperature T C, as the pont where the second dervatve of m, change the sgn. 4. RESULTS In Fg. 1 we plot the magnetzaton versus temperature for the FCC lattce of Mn 4 N, wth ntersttals N. Thus we can approxmate the crtcal temperature T c, for dfferent Hund s rule couplng and we can provde the nfluence of ths parameter by the stablty of ferromagnetsm. In Fg. 2 we show the crtcal temperature versus J H (Hund couplng). Ths fgure descrbes the phase dagram and provdes a stablzaton of ferromagnetc state for J H > 2. It s evdent that the ncrease of the Hund s couplng leads of the stablty of the magnetc ordered states (ferromagnetc algnments of magnetc spns).

4 264 Eugen Brsan, Cosmn Candn 4 Fg. 1 Magnetzaton versus T, at N = 20, wth n = 1.25 and J H = 6, U= 6. In Fg. 3 we plot the dependence of crtcal temperature versus the electron denstes n. The ferromagnetc phase s found n a broad range of fllngs. The maxmum of the crtcal temperature s observed around n = Closed n = 2 (half fllng) s observed the antferromagnetc phase, because the ant-ferromagnetc Hesenberg exchange suppresses the ferromagnetc order. We can say that the N atom play the role of an electronc brdge that produce a ferromagnetc adjacent algnment of the Mn atoms. Ths s a hybrd exchange model, between the Hubbard model and the Double-Exchange (DE) model. The exstence of ferromagnetc state s condtoned by the range of tnerant electronc densty. Around n = 1.25, the ferromagnetc state s characterzed by a crtcal temperature around 800 K, very closed by the expermental results reports by I. Pop [14, 15]. Fg. 2 Cure temperature versus Hund s rule couplng, for n = 1.25, U = 6.

5 5 Stablty of metallc ferromagnetsm 265 Fg. 3 Magnetc T-n phase dagram for U = 6 and J H = CONCLUSIONS We used a Monte Carlo Algorthm to descrbe the ferromagnetc state n Mn 4 N sold soluton. The physcs model s a hybrd model between the Hubbard model and Zener model. In our double correlated hoppng model the N atom s a brdge atom, magnetcally nert that play the role of an ntermedate hoppng poston. Thus, we can fnd the phase dagram wth two type parameter: Hund s couplng and tnerant electronc densty. Also we can approxmate the crtcal temperature of Mn 4 N n a metallc state (n = 1.25) wth an ntermedate electronc nteracton (J H = 6 and U = 6). REFERENCES 1. J. Hubbard, Proc. Roy. Soc. London A 276, 238 (1963). 2. A. Schller, Phys. Rev. B 60, (1990). 3. M. Ulmke, Eur. J. Phys B 1, 301 (1998). 4. D. Vollhardt, Rev. Mod. Phys 56, 99 (1984). 5. D. Vollhardt n Correlated Electron Systems, edted by V. J. Emery (World Scentfc, Sngapore, 1993). 6. P. Fulde, Electron Correlatons n Molecules and Solds, Sprnger, Berln Hedelberg, J. E. Hrsch, Phys. Rev. B 56, (1997). 8. M. J. Razenberg, Phys. Rev. B 55, R 4855 (1997). 9. J. Kamamor, Prog. Theor. Phys. 30, 275 (1963). 10. Monte Carlo Methods n Statstcal Physcs, (Sprnger Verlag, Berln, 1979). 11. K. Held, D. Vollhardt, Phys. Rev. Lett 84, 5168 (2000). 12. D. C. Matts, The Theory of Magnetsm I (Sprnger, Berln, 1988). 13. V. L. Moruzz, J. F. Janack, A.R. Wllams, Calculated Electronc Propertes of Metals (Pergamon Press, New York 1978).

6 266 Eugen Brsan, Cosmn Candn I. Pop, R. Munteanu, O. Pop, Materals letters, 28, 155 (1996). 15. I. Pop, M. Andrcuţ, I. Burda, R. Munteanu, H. Crveanu, Materals chemstry and Physcs, 372, 52 (1994). 16. T. Morya, Spn fluctuatons n Itnerant Electron Magnetsm, Sprnger, Berln Hedelberg, 1985.