Magnetic Behavior of Ferri-ferromagnetic Alloy in Presence of External Magnetic Field

Transcription

1 Commun. Theor. Phys. (Bejng, Chna) 49 (2008) pp c Chnese Physcal Socety Vol. 49, No. 4, Aprl 15, 2008 Magnetc Behavor of Ferr-ferromagnetc Alloy n Presence of External Magnetc Feld HU Hong-Lang, XIN Z-Hua, and LIU We-Je epartment of Physcs, Shangha Unversty, Shangha , Chna (Receved Aprl 10, 2007; Revsed May 25, 2007) Abstract The magnetc behavor of the mxed ferro-ferrmagnetc alloy wth (A a B b C c ) y structure composed of Isng spns S A = 1, S B = 5/2, S C = 2, and S = 3/2 n the presence of the external magnetc feld s nvestgated by the use of the effectve feld theory. The role of concentraton b s dscussed n ths system n detal. Results show that for a = 0.4, only when the concentraton b s n the regon 0.60 b > 0.34 can the ferrmagnetc behavor be seen. Otherwse, the alloy shows ferromagnetc behavor. PACS numbers: Hk, K, Gg Key words: magnetc susceptblty, magnetaton, Prussan blue analogs, effectve feld theory 1 Introducton Prussan blue analogs often exhbt noble magnetc propertes such as photonduced magnetaton and hgh Cure temperature. 1 8] Moreover, among these compounds, partcular attenton has been pad to mult-metal Prussan blue analogs such as (A p Mn 1 p ) 1.5 Cr(CN) 6 ] H 2 O or (N p Mn q Fe r ) 1.5 Cr(CN) 6 ] H 2 O, 9 11] whch have ncorporate both ferromagnetc and antferromagnetc exchange nteractons between dfferent spns. In partcular, because varous types of metal ons can be ncorporated there as spn centers, moreover, the fcc structure of the Prussan blue analogs s mantaned even when the metal ons are substtuted, the ratonal desgn of ths new magnetc property s relatvely easy. 10] However, because of the complexty of the structure, lttle work has been done n the theoretcal descrpton of these molecular-based magnets n the aspect of magnetc behavor. In recent works, some ferromagnetc systems were dscussed n detal by P.B. He et al. 12,13] T. Kaneyosh et al. studed some ferrmagnetc systems wth two or three magnetc ons n theory ] S. Ohkosh et al. dscussed the temperature dependence of magnetaton n a mxed ferro-ferrmagnetc system wth four magnetc ons based on molecular feld theory. 11] In ths paper, we study magnetc behavor for the mxed ferro-ferrmagnets composed of Prussan blue analogs (A a B b C c ) y wth four magnetc ons n detal. Ths alloy has both ferromagnetc (J A > 0, J C > 0) and ferrmagnetc (J B < 0) exchange nteractons wth dfferent spns S A = 1, S B = 5/2, S C = 2, and S = 3/2, respectvely. For the mxed ferro-ferrmagnets, the superexchange nteractons between the second-nearest-neghbour metal ons are neglected because of ther relatvely long dstances. 10] Therefore, we only need to take nto account the superexchange nteractons between the nearest neghbor metal ons. The effectve-feld theory has been appled successfully to varous physcal problems due to ncludng some effects of spn-spn correlatons. So we wll study ths ferro-ferrmagnetc system based on the effectve feld theory wth correlatons. Ths paper s organed as follows. In Sec. 2, a precse descrpton of the model s gven and relevant expressons n the effectve feld theory are derved. Secton 3 s devoted to the dscusson of the numercal results concernng the phase dagrams. Fnally, some conclusons are presented n Sec Isng Model and Its Effectve Feld Theory To be consstent wth a structure of the Prussan blue analogs such as (N II 0.22 Mn II 0.60 Fe II 0.18) 1.5 Cr III (CN) 6 ], the, and A (or B, C) ons, whch are randomly dstrbuted n the lattce, are lnked n an alternatng fashon. The Hamltonan of the system s gven by H = 2J A ξ A SAS +δ,,+δ 2J B ξ B SBS +δ,,+δ 2J C ξ C SCS +δ,,+δ ( h ξ A (SA) + ξ B (SB) + ξ C (SC) + ) (S), (1) where J A, J B, and J C are the nearest-neghbor exchange nteractons, and h s an external longtudnal magnetc feld, all n energy unts. SA = 0 and ±1 for A ons, SB = ±1/2, ±3/2 and ±5/2 for B ons, S C = 0, ±1 and The project supported by Shangha Leadng Academc scplne Project under Grant No. T0104

2 1060 HU Hong-Lang, XIN Z-Hua, and LIU We-Je Vol. 49 ±2 for C ons, S = ±1/2 and ±3/2 for ons. The summaton s carred out only over nearest-neghbor pars of spns on dfferent sublattces. ξ A, ξ B or ξ C s a set of ndependent, unformly dstrbuted random varables whch take the value of unt or ero, dependng on whether the ste s occuped by an on of the type A, B or C. The dstrbuton functons of ξ A, ξ B and ξ C should satsfy ξ A + ξ B + ξ C = 1, a = ξ A, b = ξ B, and c = ξ C, (2) where denotes the random confguratonal average. a, b and c represent the concentratons of A, B, and C ons, respectvely, whch satsfy the relaton a + b + c = 1. The startng pont for the statstcs of our spn system for any operator A at ste s gven by 22] A = Tr () A exp( βh ) Tr () exp( βh ) 0, β = 1 k B T, (3) where k B s Boltmann constant, Tr () denotes a partal trace for the ste, and 0 denotes a thermal average over the ensemble defned by H. Wthn the effectve-feld theory, the average sublattce magnetatons for the lattce are gven by m A ξ A SA 0 = a cosh(2j A η ) + m ] ZAFA snh(2j A η ) (x + h) x=0, (4) η m B ξ B SB 0 = b cosh(2j B η ) + m ] ZBFB snh(2j B η (x + h) x=0, (5) η m C ξ C SC 0 = c cosh(2j C η ) + m ] ZCFC snh(2j C η (x + h) x=0, (6) η m S 0 = a cosh(2j A η A ) + m A snh(2j A η A ) + 1 a η A b cosh(2j B η B ) + m B snh(2j B η B ) + 1 b η B c cosh(2j C η C ) ] ZA ] ZB + m ] ZC C snh(2j C η C ) + 1 c η C F (x + h) x=0, (7) where Z KL s the numbers of the nearest-neghbor L-ste ons surroundng a K-ste on. = / x s the dfferental operator, and η α (α = A, B, C) are expressed as η 2 A ξ A SA 2 0 = a cosh(2j A η ) + m ] ZAGA snh(2j A η ) (x + h) x=0, (8) η η 2 B ξ B SB 2 0 = b cosh(2j B η ) + m ] ZBGB snh(2j B η (x + h) x=0, (9) η ηc 2 ξ C SC 2 0 = c cosh(2j C η ) + m ] ZCGC snh(2j C η (x + h) x=0, (10) η η 2 S 2 0 = a cosh(2j A η A ) + m A snh(2j A η A ) + 1 a η A b cosh(2j B η B ) + m B snh(2j B η B ) + 1 b η B c cosh(2j C η C ) ] ZA ] ZB + m ] ZC C snh(2j C η C ) + 1 c η C G (x + h) x=0. (11) The functons F k (x) and G k (x) (k = A, B, C, or ) are defned as F A (x) = 2 snh(βx) 2 cosh(βx) + 1, (12) F B (x) = 1 5 snh( 5 2 βx) + 3 snh(3 2 βx) + snh(3 2 βx) 2 cosh( 5 2 βx) + 2 cosh(3 2 βx) +, (13) cosh(1 2βx) 4 snh(2βx) + 2 snh(βx) F C (x) = 2 cosh(2βx) + 2 cosh(βx) + 1, (14) F (x) = 1 3 snh(3βx/2) + snh(βx/2) 2 cosh(3βx/2) + 2 cosh(βx/2), (15) G A (x) = 2 snh(βx) 2 cosh(βx) + 1, (16) G B (x) = 1 25 cosh( 5 2 βx)+9 cosh(3 2 βx) + cosh(1 2 βx) 4 cosh( 5 2 βx)+cosh(3 2 βx) + cosh(1 2 βx), (17) G C (x) = 8 cosh(2βx) + 2 cosh(βx) 2 cosh(2βx) + 2 cosh(βx) + 1, (18) G (x) = 1 9 cosh(3βx/2) + cosh(βx/2) 4 cosh(3βx/2) + cosh(βx/2). (19) Then the averaged total magnetaton can be obtaned from the relaton M = M A + M B + M C + M = g(ym A + ym B + ym C + m ). (20) On the other hand, the total susceptblty per ste s gven by χ = M h, (21)

3 No. 4 Magnetc Behavor of Ferr-ferromagnetc Alloy n Presence of External Magnetc Feld 1061 especally, the ntal susceptblty can be obtaned from the equaton M χ = lm h 0 h. (22) 3 Numercal Results and scussons In ths secton, we shall dscuss magnetc behavor wth the external magnetc feld by solvng Eqs. (4) (11). The magnetc feld takes cm 1 as ts unt. To smulate compounds (N II a Mn II b Fe II c ) 1.5 Cr III (CN) 6 ], 11] we choose y = 1.5, J A = 5.6 cm 1, J B = 2.5 cm 1, J C = 0.9 cm 1, Z A = Z B = Z C = 4, Z A = Z B = Z C = 6. For ferromagnets, accordng to the Cure Wess law, the magnetc susceptblty χ decreases wth the ncrease of T above T c and the χ 1 value decreases lnearly wth the decrease of temperature, and become ero at T c. The χt value ncreases monotonously, approachng nfnty at T c wth decreasng temperature. Otherwse, for ferrmagnets, the χ 1 versus T curve devates from the lnear lne and fnally drops to ero at T c. The χt versus T plots have a mnmum value and then approach nfnty at T c. 23,24] The theory mentoned above s for pure ferromagnet and pure ferrmagnet. What wll the magnetc behavor be for ths mxed ferro-ferrmagnetc alloy? Fg. 1 The temperature dependence of magnetc susceptblty χ at a = 0.40 and h = 0.0,0.5,1.0,2.0. The dotted curves denote the poston of the peak. (a) b = 0.05, c = 0.55; (b) b = 0.35, c = 0.25; (c) b = 0.50, c = Fg. 2 Temperature dependence of the nverse magnetc susceptblty for the system wth a = 0.40 and h = 0,0.5,1.0,2.0. (a) b = 0.05, c = 0.55; (b) b = 0.35, c = 0.25; (c) b = 0.50, c = 0.10.

4 1062 HU Hong-Lang, XIN Z-Hua, and LIU We-Je Vol. 49 The magnetc susceptblty of ths mxed ferroferrmagnetc (A a B b C c ) y alloy s gven n Fgs The peak whch corresponds to the second-order transton pont decreases wth ncreasng h n Fg. 1. It s the result of competton between the exchange couplng and the external magnetc feld. For a defnte h, fgure 1 shows the transton temperature ncreases wth the ncrease of b at a = 0.40, whch s consstent wth the result gven n Fg. 1(a). From Fg. 1, one can also see that the ntal susceptblty gets almost nfnte at T c. Fgure 2 shows that χ 1 ncreases wth the decrease of T and gets to nfnte as T approaches ero at T < T c. At T > T c, χ 1 ncreases wth the ncrease of T. But they ncrease at the dfferent speeds for dfferent b values. By comparng three fgures, one can fnd that the smaller the value of b s, the smaller the curvature s n the paramagnetc regon. It s because the smaller b s, such as 0.05, the less the ferrmagnetc component s n ths alloy. Ths makes the mxed ferro-ferrmagnetc behave more lke a ferromagnet. Otherwse, when b takes a bgger value, such as 0.50, ths alloy behaves more lke a ferrmagnet. Our numercal results also show that at b = 0, the χ 1 changes lnearly wth the change of T above T c. Fgure 3 gves the temperature dependence of χt at a = 0.4. It shows that χt approaches nfnte at T = T c, and decreases monotonously when b takes smaller values, such as b = 0.00, When b takes bgger value, such as b = 0.50, 0.60, there s a mnmum n the χt-t plane. That also ndcates that the bgger b s, the more lke a ferrmagnet ths mxed ferro-ferrmagnetc alloy behaves. Our numercal results show that the real threshold for the appearance of mnmum of χt s about b = It means that the effect between the ferrmagnetc component and ferromagnetc component keep balance and the Wess temperature s ero. Fg. 3 The χt-t curves of the (A a B b C c ) y system at a = 0.40 and h = 0. On the other hand, the magnetaton s also gven n Fg. 4 under the external feld. At T = 0 K, a sharp rse s found n M when the external magnetc feld h s about 30 cm 1 and then magnetaton does not vary nearly wth the ncrease of the external magnetc feld h. Contrarly, at T = 15 K, magnetatons always ncrease smoothly wth ncreasng h. It ndcates that the effect of temperature becomes more pronounced. Fg. 4 Magnetaton of the mxed (A a B b C c ) y system wth the external magnetc feld. (a) T = 0 K; (b) T = 15 K. 4 scusson In the presence of the external magnetc feld, magnetc susceptblty ncludng the ntal magnetc susceptblty also shows the maxmum at T c. Moreover, ts maxmum decreases wth ncrease of external magnetc feld due to the competton between the exchange couplng and the external magnetc feld. Partcularly, the maxmum value of the ntal magnetc susceptblty s nearly nfnte. The magnetc susceptblty curves also show that when b takes smaller values, ths mxed ferro-ferrmagnet behaves more lke a ferrmagnet, and otherwse t behaves more lke a ferromagnet. In addton, n spte of the change of a, a

5 No. 4 Magnetc Behavor of Ferr-ferromagnetc Alloy n Presence of External Magnetc Feld 1063 sharp ncrease appears n magnetaton at the same external magnetc feld at T = 0 K. Conversely, magnetatons ncrease smoothly when the external magnetc feld ncreases at T = 15 K due to the effect of temperature. References 1] S. Ferlay, T. Mallah, R. Ouahés, P. Vellet, and M. Verdaguer, Nature (London) 378 (1995) ] M. Verdaguer, Scence 272 (1996) ] O. Sato, T. Iyoda, A. Fujshma, and K. Hashmoto, Scence 271 (1996) 49. 4] O. Sato, T. Iyoda, A. Fujshma, and K. Hashmoto, Scence 272 (1996) ] O. Sato, Y. Enaga, T. Iyoda, A. Fujshma, and K. Hashmoto, J. Electrochem. Soc. 144 (1997) L11. 6] S. Ohkosh, M. Muno, G.J. Hung, and K. Hashmoto, J. Phys. Chem. 104 (2000) ] Y. Mortomo, M. Hanawa, Y. Ohsh, K. Kato, M. Takata, A. Kurk, E. Nshbor, M. Sakata, S. Ohkosh, H. Tokoro, and K. Hashmoto, Phys. Rev. B 68 (2003) ] H.W. Lu, K. Matsuda, Z.Z. Gu, K. Takahash, A.L. Cu, R. Nakajma, A. Fujshma, and O. Sato, Phys. Rev. Lett. 90 (2003) ] S. Ohkosh, S. Yorou, O. Sato, T. Iyoda, A. Fujshma, and K. Hashmoto, Appl. Phys. Lett. 70 (1997) ] S. Ohkosh, T. Houm, and K. Hashmoto, Phys. Rev. B 64 (1997) ] S. Ohkosh, Y. Abe, A. Fujshma, and K. Hashmoto, Phys. Rev. Lett. 82 (1999) ] P.B. He and W.M. Lu, Phys. Rev. B 72 (2005) ] P.B. He, X.C. Xe, and W.M. Lu, Phys. Rev. B 72 (2005) ] T. Kaneyosh and Y. Nakamura, J. Phys.: Condens. Matter 10 (1998) ] T. Kaneyosh, Y. Nakamura, and S. Shn, J. Phys.: Condens. Matter 10 (1998) ] T. Kaneyosh, J. Phys.: Condens. Matter 14 (2002) ] O.F. Abubrg,. Horváth, A. Bobák, and M. Jaščur, Physca A 296 (2001) ] A. Bobák and M. Jaščur, Phys. Rev. B 51 (1995) ] G.M. Buendıa and R. Cardona, Phys. Rev. B 59 (1999) ] M. Godoy, V.S. Lete, and W. Fgueredo, Phys. Rev. B 69 (2004) ] G.M. Buendıa and J.E. Vllarroel, J. Magn. Magn. Mater. 310 (2007) e495 e ] T. Kaneyosh, Acta Phys. Pol. A 83 (1993) ] L. Néel, Ann. Phys. (Lepg) 3 (1948) ] P. Wess, J. de Physque 6 (1907) 661.