Susceptibility and Inverted Hysteresis Loop of Prussian Blue Analogs with Orthorhombic Structure

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1 Commun. Theor. Phys. 58 (202) Vol. 58, No. 5, November 5, 202 Susceptblty and Inverted Hysteress Loop of Prussan Blue Analogs wth Orthorhombc Structure GUO An-Bang (ÁËǑ) and JIANG We ( å) School of Scences, Shenyang Unversty of Technology, Shenyang 0870, Chna (Receved July 30, 202) Abstract The magnetc susceptblty of ternary metal Prussan blue analogues wth orthorhombc structure s studed usng Isng model. Wthn the frame work of effectve-feld theory wth correlatons, the roles of the mole fracton y, unaxal magnetc ansotropy, transverse and longtudnal magnetc feld are dscussed n detals. The temperature dependence of the magnetc susceptblty s also nvestgated. The nterestng phenomenon of the nverted magnetc hysteress loop has been found. The results can help to understand the expermental work of the molecule-based ferrferrmagnet. PACS numbers: 75.0.Hk, 75.0.Dg Key words: prussan blue analogues, magnetc susceptblty, nverted hysteress loop Introducton Prussan blue analogues become one of the most attractve classes of molecule-based magnets because the face-centered cubc (fcc) structure s mantaned even when metal on substtuton s carred out. 2] They present several attrbutes unavalable n conventonal metal alloys and metal oxde magnets, such as electrcal nsulaton, low densty, transparency, and low temperature fabrcaton. By varyng the composton, dfferent magnetc propertes, such as saturaton magnetzaton, coercve feld, Cure temperature, and compensaton temperature, can be controlled. By consderng only the nearestneghbor superexchange nteractons and neglectng the second nearest neghbor stes exchange nteractons due to relatvely longer dstances between metal ons ( 0 Å), 3] desgn of novel Prussan blue analogues magnets s possble. Based on the mean-feld and effectve-feld theory, mxed-spn Isng model has been successfully used to explan the magnetc propertes of exstng molecule-based ferro-ferrmagnet (N II a Mn II b FeII c ).5 Cr III (CN 6 )] zh 2 O and (N II x Mn II x).5 Cr III (CN) 6 ] zh 2 O. 4 5] However, all these works only deal wth the face-centered cubc (fcc) structure of Prussan blue analogues. The standard mean feld approxmaton s used to nvestgate the magnetc propertes, especally the compensaton pont of bnaryalloy Isng ferrmagnets. 6 7] In our prevous works, 8 9] we have studed the magnetc propertes of the mxed Isng model. Ohkosh et al. have successfully desgned and prepared molecule-based ferr-ferrmagnet Sm III y Gd III ycr III (CN) 6 ] 4H 2 O system based on the molecular feld theory. 0] Ths new Prussan blue analogue has the orthorhombc structure, dfferent from the molecule-based magnets wth the face-centered cubc (fcc) structure mentoned above. They have also found that the novel magnet exhbts nverted magnetc hysteress loops by controllng varous magnetc parameters such as magnetzaton, molecular feld, and unaxal magnetc ansotropy. The effect of an external magnetc feld on the classcal ferromagnet, nanowre and doman-well n thn flm wth two sngle-on ansotropes has been dscussed. 3] The works on the orthorhombc system by the effectve-feld theory wth self-spn correlatons have not been reported. In ths paper, we focus on the effects of the mole fracton y, the unaxal magnetc ansotropy, the transverse and longtudnal magnetc feld on the magnetc susceptblty of the mxed ferr-ferrmagnetc orthorhombc crystal, under a threesublattce Isng system. In a magnet composed of three metal ons A III y B yc III III, the magnetc ons A and B are randomly ncorporated n the orthorhombc lattce and the letter y denotes the molar fracton. The exchange couplng constant between nearest neghborng magnetc ons A and C (or B and C) s ferrmagnetc such that J AC < 0 (or J BC <0). The exchange couplng constant between magnetc ons A and B s neglected due to C and ether A or B s lnked randomly. 0] The Hamltonan of the mxed ferr-ferrmagnets composed of Prussan blue analogs A III y B yc III III s gven by H = 2J AC ξ A SAS z jc z 2J BC ξ B SBS z jc z,j D ξ A (SA) z 2 ( h ξ A (SA) z +,j ξ B (S z B) + j ) (SjC) z Supported by the Excellent Talents Program of the unversty of Laonng Provnce of Chna under Grant No. LR2003 Correspondng author, E-mal: jang we63@yahoo.com.cn

2 No. 5 Communcatons n Theoretcal Physcs 773 ( Ω ξ A (SA) x + ξ B (S x B) + j ) (SjC) x, () where the summaton s over all nearest-neghbor pars. D s a unaxal magnetc ansotropy from the Sm III on. 0] h and Ω are an external longtudnal and transverse magnetc feld. The unformly dstrbuted random varables occupaton number ξ A (or ξ B ) = 0, depend on whether the -th ste s occuped by an on of type A or B. They should satsfy the relatons ξ A + ξ B =, y = ξ A C, y = ξ B C, (2) where C s the random confguraton average. M A = y cosh(2j AC ) + M C M B = ( y) cosh(2j BC ) + M C M C = y cosh(2j CA η A ) + M A The paper s organzed as follows. We wll present the model Hamltonan of ths three sublattce Isng model and the formulaton n the effectve-feld theory wth A III y BIII y CIII. In the followng secton, numercal results for the sublattce and total magnetzatons are gven. Fnally, we shall gve the summary n Sec Formulatons We denote A = Sm III, B = Gd III, and C = Cr III, then the parameters should be S A = 5/2, S B = 7/2, S C = 3/2. Wthn the effectve-feld theory, the average sublattce magnetzatons for the lattce are gven by ] 6FA snh(2j AC ) (x) x=0, (3) ] 6FB snh(2j BC ) (x) x=0, (4) snh(2j CA η A ) + y η A ( y)cosh(2j CB η B ) + M B snh(2j CB η B ) + y η B ] 6 ] 6FC (x) x=0, (5) where = / x s the dfferental operator. η k (k = A, B, C) are gven by η A = y cosh(2j AC ) + M ] 6GA C snh(2j AC ) (x) x=0, (6) η B = ( y) cosh(2j BC ) + M ] 6GB C snh(2j BC ) (x) x=0, (7) = y cosh(2j CA η A ) + M ] 6 A snh(2j CA η A ) + y η A ( y)cosh(2j CB η B ) + M ] 6GC B snh(2j CB η B ) + y (x) x=0. (8) η B The functons F k and G k (k = A, B, or C) are defned as F A (x) = F B (x) = F C (x) = G A (x) = G B (x) = G C (x) = 6 m= n= exp(βλ m ) { 6 m= φ m SA φ z m exp(βλ m ), (9) { 8 φ n SB 8 z φ n exp(βλ n ), (0) exp(βλ n ) n= { 4 φ 4 p SjC φ z p exp(βλ p ), () exp(βλ p ) p= p= 6 m= n= exp(βλ m ) { 6 m= φ m (SA) z 2 φ m exp(βλ m ), (2) { 8 φ n (SB 8 z )2 φ n exp(βλ n ), (3) exp(βλ n ) n= { 4 φ 4 p (SjC) z 2 φ p exp(βλ p ), (4) exp(βλ p ) p= p=

3 774 Communcatons n Theoretcal Physcs Vol. 58 where β = /k B T, k B s the Boltzmann constant and T s the absolute temperature. The ntal susceptblty and the average magnetzaton per ste can be obtaned by the followng relaton χ = M, (5) h h=0 M = k g k M k, (6) where g k (k = A, B, C) are the Lande factors. 3 Numercal Results and Dscussons We wll dscuss the effects of mole fracton y, unaxal magnetc ansotropy, transverse and longtudnal magnetc feld on the magnetc susceptblty of ternary metal Prussan blue analogues wth orthorhombc structure A III y BIII y CIII. To smulate the compound (Sm III y GdIII y )CrIII (CN) 6 ] 4H 2 O, 0] we choose J AC = J CA = 0.32 cm, J BC = J CB = 0.23 cm. What s the magnetc susceptblty for ths mxed ferrferrmagnetc alloy? From Eq. (5), the magnetc susceptblty s a functon of temperature, unaxal magnetc ansotropy, mole fracton y, transverse and longtudnal magnetc feld. The numercal results of magnetc susceptblty are plotted n Fgs. 3. Fg. Temperature dependence of the magnetc susceptblty for ferr-ferrmagnets. The dotted curves denote the poston of the peak. The number besde each curve s the value of the A-on mole fracton y. (a) D =.0; (b) D = 3.0. Fg. 2 Temperature dependence of the magnetc susceptblty for ferr-ferrmagnets. The number besde each curve s the value of the transverse magnetc feld. (a) y = 0.4; (b) y = 0.6. The dotted lnes n these fgures locate peak postons,.e. the phase transton ponts. The magnetc susceptblty versus temperature curves exhbt dfferent behavors, dependng on the A-on mole fracton y as n Fgs. (a) (b) wth h = 0.5 and Ω = 0. For example, the curves for y = 0.3 show two maxma n the magnetc susceptblty wth ncreasng T, whle the curves for other y values exhbt only one maxma. The peak value whch corresponds to the second-order transton pont decreases wth the ncreasng the A-on mole fracton y. From Fgs. (a) (b), t s also revealed that the curves ncrease the ncreasng temperature when T < Tc. But the ncreasng rates at dfferent y values are dfferent. For T > Tc, all curves decrease on further ncreasng T. Furthermore, the magnetc susceptblty s larger for smaller A-on mole fracton y at the same temperature. These results are attrbuted

4 No. 5 Communcatons n Theoretcal Physcs 775 to the opposte magnetzatons between sublattce C and sublattces A and B. Ther temperature dependence on the A-on mole fracton y s also dfferent. The effects of unaxal magnetc ansotropy on the magnetc susceptblty can be obtaned by comparng Fgs. (a) and (b). It s revealed from these curves that the transton temperature ncreases wth decreasng A-on mole fracton y and the unaxal magnetc ansotropy D. In Fgs. 2(a) 2(b), we plot the numercal results of the susceptblty for the ferrferrmagnets when the unaxal magnetc ansotropy and the longtudnal magnetc feld are selected as D = 0.5 and h = 0.5, respectvely. The numerc label denotes the value of transverse magnetc feld. It s noted that the susceptblty rapdly ncreases and exhbts one peak at the transton temperature n Fg. 2(a) when y = 0.4. The stronger the transverse magnetc feld, the larger the susceptblty s at T < Tc. On a contrary, the reverse trend occurs for T > Tc: the susceptblty decreases wth the transverse magnetc feld. However, two peaks on all the curves are found n Fg. 2(b) when y = 0.6, whch correspond to the postons of the compensaton and the transton temperatures. The dstance of these two peaks becomes larger wth ncreasng value of Ω. On the other hand, accordng to (5), the ntal susceptblty s also gven n Fg. 3. Fg. 3 Temperature dependence of the ntal susceptblty for ferr-ferrmagnets wth D = 3.0, y = 0.35, 0.4, 0.5, From Fg. 3, t can be seen that the curves of the ntal susceptblty rapdly ncreases and attans almost nfnte at the transton temperature T c. It then rapdly decreases on ncreasng temperature above T c. Fgure 4 shows the hysteress loops for y = 0.5 and D = 3 n the appled magnetc feld between 4 and +4 at temperature T = 2. The sold and dotted curves wth crcles descrbe the ncreasng and decreasng parts of the magnetc hysteress loops, respectvely. The nterestng phenomenon of the nverted magnetc hysteress loop has been found. The magnetzaton became negatve n the decreasng part when the appled feld was stll postve, whle the magnetzaton became postve n the ncreasng part when the appled feld was stll negatve. The smlar behavor has been observed expermentally n Ref. 0]. Fg. 4 Inverted magnetc hysteress loops for ferrferrmagnets. 4 Conclusons Wthn the framework of the effectve-feld theory based on the dfferental technque, we have developed the method to study the magnetc susceptblty and the hysteress loop of mxed ferr-ferrmagnets composed of Prussan blue analogs wth orthorhombc structure. The results show that the magnetc susceptblty ncludng the ntal magnetc susceptblty of the novel magnet Sm III y Gd III ycr III (CN) 6 ] 4H 2 O can be controlled by varous parameters such as unaxal magnetc ansotropy, on mole fracton, transverse and the longtudnal feld. All magnetc susceptblty curves have the maxma at T c. The peak decreases wth ncreasng the A-on mole fracton (or the transverse magnetc feld) due to the competton among these parameters. Moreover, the maxmum value of the ntal magnetc susceptblty approaches nfnte. The nterestng phenomenon of the nverted magnetc hysteress loop has been found. It should be emphaszed that the present method by consderng the selfspn correlatons s essentally dfferent from the meanfeld method. Ths method s helpful for desgn and understandng the related expermental results of Prussan blue analogs composed of rear-earth alloys and metal oxdes.

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