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The compensation behavior of a mixed spin-2 and spin-52 Heisenberg ferrimagnetic system on a honeycomb lattice
ARTICLE IN PRESS
Physica B 348 (2004) 79–88
The compensation behavior of a mixed spin-2 and spin-52 Heisenberg ferrimagnetic system on a honeycomb lattice Jun Lia,b,*, An Dua, Guozhu Weia b
a College of sciences, Northeastern University, Shenyang 110006, China Department of Physics, Northeastren University, Shenyang 110044, China
Received 15 December 2002; received in revised form 4 August 2003; accepted 11 November 2003
Abstract The magnetic properties of a mixed spin-2 and spin-52 Heisenberg ferrimagnetic system on a layered honeycomb lattice, both of whose sublattices have interlayer interactions and single-ion anisotropy were investigated theoretically by a multisublattice Green-function technique which takes into account the quantum nature of Heisenberg spins. We studied the spin-wave spectra of the ground state. We found three types of the magnetization curves predicted by Neel; in particularly; we investigated the effect of single-ion anisotropies and interlayer interactions on the transition and compensation phenomenon. r 2003 Elsevier B.V. All rights reserved. PACS: 75.10.Jm; 75.30.Ds; 75.40.Gb Keywords: Heisenberg ferrimagnet; Mixed spin system; Sublattice magnetization; Transition temperature; Compensation temperature; Green function
1. Introduction Molecular-based magnetic materials have been a subject of growing interest in recent years [1–5]. Such materials are of interest in experiment and theory because of their properties such as low density, transparency, electrical insulation, and low-temperature fabrication and photoresponsiveness. The search for materials that order at or above room temperature is a major driving force moving the field. Experimentally, a number of molecular-based magnetic materials have been synthesized. Among these materials, ferrimagnets in which two kinds of magnetic atoms regularly alternate antiferromagnetically seem to play an important role. A group compounds synthesized by Mathoniere et al. [6] in 1996 with spins residing only in d orbits connected via covalent bonds AFeIIFeIII(C2O4)3(A=N(nCnH2n+1)4, n ¼ 325) have critical temperatures between 35 and 48 K and some of them have compensation temperature near 30 K. Another recently developed amorphous V (TCNE)x y order ferrimagnetically at as high as 400 K [6].
*Corresponding author. Tel.: 62268995; fax: +24315275642. E-mail address: [email protected] (J. Li). 0921-4526/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2003.11.074
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Theoretically, mixed Ising systems, which provide simple models that can show ferrimagnetic ordering and may have compensation temperatures, have been used to describe the above materials. The magnetic properties of these models have been studied by several methods such as a mean-field [7] and an effectivefield theory [8], a cluster variational theory [9], Monte Carlo simulations [10,11], and so on. In the mixed Ising model used to study the properties of the molecular-based magnetic materials [11], the spins are scalar quantities. In fact, the spins should be vectors in the molecular-based magnetic materials AfeIIFeIII(C2O4)3(A=N(n-CnH2n+1)4, n ¼ 325) [6], the magnetic properties of the system may be affected by the quantum fluctuation of the spins. Otherwise, the mixed Ising model mentioned above does not take into account the interlayer interaction and the single-ion anisotropy of FeIII and, we think it is necessary and interesting to take them into account. So, in this paper, we will study the mixed quantum Heisenberg system, whose both sublattices have interlayer interactions and single-ion anisotropy, with the method of the double-time–temperature Green function, to investigate the magnetic behavior of the molecular-based magnet. In Section 2, we start with a mixed spin-2 and spin-52 ferrimagnetic Heisenberg model that takes the interlayer interaction and the single-ion anisotropy of FeIII into account and then give the fundamental equations. In Section 3, we give the numerical results of sublattice magnetizations in the whole range of temperatures, and illustrate the effects of single-ion anisotropies and interlayer interactions. Section 4 is devoted to summary.
2. Model and fundamental equations We consider a mixed Heisenberg ferrimagnetic model on a layered honeycomb lattice with spin-2 and spin-52 spins that represent FeII and FeIII atoms, respectively, as show in Fig. 1. For convenience, the system is divided into two sublattices A and B with each sublattice being a honeycomb lattice, and the Hamiltonian we adopt takes the form X X X X X H ¼2 Jij Si Sj D1 ðSiz Þ2 D2 ðSjz Þ2 Jii0 Si Si0 Jjj 0 Sj Sj 0 ; ð1Þ ðijÞjj
i
j
ðii0 Þ>
ðjj 0 Þ>
where the summations are taken over all nearest–neighbor spin pairs, iA the sublattice A, Sa=2(FeII), jA the sublattice B, Sb=52(FeIII). In addition, we define Jij ¼ Jð> 0Þ denotes an antiferromagnetic exchange interaction between spins Si and Sj in the layer, Jii0 ¼ J1 ð> 0Þ and Jjj 0 ¼ J2 ð> 0Þ denote a ferromagnetic interlayer interaction between spins in the sublattices A and B in the perpendicular direction, respectively. D1 and D2 denote the single-ion anisotropy of sublattices A and B, respectively.
Fig. 1. The structure of the stacked honeycomb layers, the big balls represent FeIII, the small balls represent FeII.
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To analyze the layered Heisenberg ferrimagnets, we introduce the following double-time Green function according to Callen [12]: GilR ðt; 0Þ ¼ //Siþ ðtÞ; Bl ð0ÞSS; GjlR ðt; 0Þ ¼ //Sjþ ðt; 0Þ;
ð2aÞ
Bl ð0ÞSS;
z eaSl ð0Þ Sl ð0Þ;
ð2bÞ Siþ ðtÞ
Sl ð0Þ
a is a parameter, and and are Heisenberg spin operators defined where Bl ð0Þ ¼ from the x and y spin components. The equations of motion for Green functions read X d i //Siþ ðtÞ; Bl ð0ÞSS ¼ dðtÞ/½Siþ ðtÞ; Bl ð0Þ S þ 2 Jij //ðSiz Sjþ Siþ Sjz Þ; Bl ð0ÞSSÞ dt jðiÞ== X Jii0 //ðSiz Siþ0 Siþ Siz0 Þ; Bl ð0ÞSS þ D1 //ðSiþ Siz þ Siz Siþ Þ; Bl ð0ÞSS; ð3aÞ i0 ðiÞ>
i
X d //Sjþ ðtÞ; Bl ð0ÞSS ¼ dðtÞ/½Sjþ ðtÞ; Bl ð0Þ S þ 2 Jij //ðSjz Siþ Sjþ Siz Þ; Bl ð0ÞSS dt iðjÞjj X z þ þ z Jjj 0 //ðSj Sj 0 Sj Sj 0 Þ; Bl ð0ÞSS þ D2 //ðSjþ Sjz þ Sjz Sjþ Þ; Bj ð0ÞSS; j 0 ðjÞ>
ð3bÞ where dðtÞ is the Dirac d function, /?S means thermal average, and _ ¼ 1 has been used. The higher-order Green’s functions are decoupled as follows. The Green functions coming from the exchange term such as //ðSiz Sjþ Siþ Sjz Þ; Bl SS are decoupled by using the so-called Tyablikov decoupling scheme [13], for example //Siz Sjþ ; Bl SSD/Siz S//Sjþ ; Bl SS:
ð4Þ //ðSiþ Siz
The Green functions coming from the single-ion anisotropy term such as treated by using the Anderson and Callen’s decoupling scheme [14], for example //ðSiþ Siz þ Siz Siþ Þ; Bl SSDti //sþ i ; Bl SS; where ti ¼
2
þ
Siz Siþ Þ;
Bl SS are ð5Þ
1 z 2 S ðS þ 1Þ /ðS Þ S /Saz S: a a a Sa2
ð6Þ
We obtain the Fourier components of the Green function, YA ðaÞ o C oC GA ðo; KÞ ¼ þ ; E E o Eþ o E YA ðgÞ oA oA GB ðo; KÞ ¼ þ ; E E o Eþ o E
ð7aÞ ð7bÞ
where z
YA ðaÞ ¼ /½Siþ ; eaSi Si S;
z
YB ðgÞ ¼ /½Sjþ ; egSj Sj S;
E 7 ¼ 12fðA þ CÞ7½ðA CÞ2 þ 4BD 1=2 g; A ¼ D1 ti 2JZjj /Sbz S þ J1 Z> ð1 cosðkz CÞÞ/Saz S;
ð8Þ ð9Þ ð10aÞ
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B ¼ 2JZjj /Saz Sg;
ð10bÞ
C ¼ D2 tj 2JZjj /Saz S þ J2 Z> ð1 cosðkz CÞÞ/Sbz S;
ð10cÞ
D ¼ 2JZjj /Sbz Sg ; " # pffiffiffi ! 3 1 iky a iky a=2 g¼ kx a e e þ 2 cos : Zjj 2
ð10dÞ ð10eÞ
After using the spectral theorem and Callen’s technique [12], we finally obtain the magnetization of each sublattice as /Saz S ¼ /Sbz S ¼
ðSa Fa Þð1 þ Fa Þ2Sa þ1 þ ðSa þ 1 þ Fa ÞFa2Sa þ1 ; ð1 þ Fa Þ2Sa þ1 Fa2Sa þ1 ðSb Fb Þð1 þ Fb Þ2Sb þ1 þ ðSb þ 1 þ Fb ÞFb2Sb þ1 ð1 þ Fb Þ2Sb þ1 Fb2Sb þ1
ð11aÞ
ð11bÞ
;
/ðSaz Þ2 S ¼ ð2Fa þ 1Þð1 þ Fa Þ
Sa2 Fa2Sa þ1 ð1 þ Sa Þ2 ð1 þ Fa Þ2Sa þ1 þ 2 Sa Fa2Sa þ1 þ ð1 þ Fa Þ2Sa þ1 ð1 þ Sa Þ ð1 þ Fa Þ ð1 þ Fa Þ2Sa þ1 Fa2Sa þ1
; ð11cÞ
/ðSbz Þ2 S ¼ ð2Fb þ 1Þð1 þ Fb Þ
h i Sb2 Fb2Sb þ1 ð1 þ Sb Þ2 ð1 þ Fb Þ2Sb þ1 þ 2 Sb Fb2Sb þ1 þ ð1 þ Fb Þ2Sb þ1 ð1 þ Sb Þ ð1 þ Fb Þ ð1 þ Fb Þ2Sb þ1 Fb2Sb þ1
where Fa and Fb are the auxiliary functions, þ 2X 1 E C E C Fa ¼ ; N K E þ E ebE þ 1 ebE 1 þ 2X 1 E A E A ; Fb ¼ N K E þ E ebE þ 1 ebE 1
; ð11dÞ
ð12aÞ
ð12bÞ
and the total magnetization of the system is given by /S z S ¼ /Saz S þ /Sbz S:
ð13Þ
Eqs. (7)–(12) are the fundamental equations of the sublattice magnetizations, they are complicated and must be solved self-consistently.
3. Results and discussion We now discuss the behaviors of the sublattice magnetizations in whole temperature regimes, the effects of single-ion anisotropies, and the effects of interlayer interactions. 3.1. The spin-wave spectrum of the ground state The first Brillouin zone in the XY plane is shown in Fig. 2.
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kya R
P 2π/3
1/2
(2(3) π /9,2 π/3)
M 0
1/2
4(3) π /9
kxa
Fig. 2. The first Brillouin zone in XY plane.
6 6
+
E
E
4
+
4 2
2
0
0
-2
-2 -4
-4
-6
D1/J=D2/J=0.2
+
E ,E
+
E ,E
-
D1/J=D2/J=0
E
-6
-
E
-
-8 -8
(a)
θπ/2 k xa
θπ (b)
θπ/2 k xa
θπ
Fig. 3. The spin-wave spectra of the ground state with J1 =J ¼ J2 =J ¼ 1:0 (solid line: along OP in The first Buliyoun region in the XY plane, dash line: along OR in The first Buliyoun region in the XY plane for kz ¼ 0) for different values of D1 and D2 (a) D1 =J ¼ pffiffiffi D2 =J ¼ 0; (b) D1 =J ¼ D2 =J ¼ 0:2 ðy ¼ 4 3p=9Þ:
Figs. 3 and 4 show the spin-wave spectra of the ground state for some representative values of D1 ; D2 ; J1 and J2 : In Fig. 3, the solid lines are the case of ky ¼ kz ¼ 0 corresponding to pffiffithe ffi diagonal line (OM) of the first Brillouin zone along the kx axis, and the dot lines are the case of ky ¼ 3kx ; kz ¼ 0; corresponding to the diagonal line (OR) of the first Brillouin zone not along the kx axis. As shown in Fig. 3a, the spin wave has two branches that in E is the acoustic branch and E þ is the optical branch when the anisotropy D1 and D2 are equal to zero. When the anisotropy D1 and D2 are unequal to zero, both branches of spin-wave spectrum are optical branches and an energy gap appears, which is just the effect of the anisotropies. In both cases, the branch of spin-wave spectrum E is always negative. The negative energy can be understood
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20 60
20
-40
0
-20
(a)
-20
+
E ,E
+
E ,E
-
0
-
40
J2/J=1.0 J2/J=4.0 J2/J=6.0
π/2 kz c
-60
π
(b)
J1/J=1.0 J1/J=4.0 J1/J=6.0
π/2 kzc
π
pffiffiffi Fig. 4. The spin-wave spectra of the ground state with D1 =J ¼ D2 =J ¼ 0; kx a ¼ 2 3p=9; ky a ¼ 2p=3 for different values of J1 and J2 (a) J1 =J ¼ 1:0 and J2 =J ¼ 1:0; 4.0, 6.0, (b) J2 =J ¼ 1:0 and J1 =J ¼ 1:0; 4.0, 6.0.
in the following manner. As elementary excitations, we consider the magnon vacuum as the ground state. Therefore, magnons excited out of the filled sea constitute the two branches with energy E and E þ [15]. As shown in Fig. 4(a), the negative energy branch E is insensitive to the change of J2 as J1 is fixed, and the positive energy branch E þ is sensitive to change of J2 : As shown in Fig. 4(b), the positive energy branch E þ is insensitive to the change of J1 as J2 is fixed, and the negative energy branch E is sensitive to change of J1 : 3.2. Magnetization curves In Fig. 5, we plot total magnetization /S z S as a function of the temperature for several values of D1 and D2 : From this figure, we can clearly recognize the existence of a compensation point for D1 ¼ 4:0 and 8.0 and the compensation points vary with the value of D1 : As shown in the figure, when D1 is equal to 4.0 or 8.0, the magnetization curves (c) and (d) behave as N-type, when D1 is equal to 0.0 or 2.0, the magnetization curves (a) and (b) behave as Q-type, and (e) and (f) behave as P-type for D2 ¼ 2:0; or 4.0 and D1 ¼ 2:0: In AFeIIFeIII(C2O4)3 (n ¼ 325), A=(C6H5)3 PNP(C6H5)3, the experiment result of the temperature dependence of the magnetization is similar to the curves of (c) and (d), indicating the importance of the single-ion anisotropy of sublattice A. 3.3. The effects of the interlayer interactions and the single-ion anisotropy on the compensation temperature TC and the transition temperature TN The compensation temperature TC is the temperature where the resultant magnetization vanishes below the critical temperature. That is, when total magnetization oSz > is equal to zero but the sublattice magnetization oSaz > and oSbz > are not equal to zero, we can determine the compensation temperature TC :
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1.0 0.8 0.6
(f)
z
/J
0.4
(e) 0.2
(a)
(b)
0.0
(c) (d)
-0.2 -0.4 0
4
8
12
16
20
24
K BT/J Fig. 5. Total magnetization as a function of temperature for several group values of D1 and D2 when J ¼ 1:0; J1 ¼ 0:2; J2 ¼ 0:0: (a) D1 ¼ D2 ¼ 0:0 (b) D1 ¼ 2:0; D2 ¼ 0:0; (c) D1 ¼ 4:0; D2 ¼ 0:0 (d) D1 ¼ 8:0; D2 ¼ 0:0; (e) D1 ¼ 2:0; D2 ¼ 2:0; (f) D1 ¼ 2:0; D2 ¼ 4:0:
The transition temperature TN is the temperature where the resultant magnetization and the sublattice magnetizations all vanish. At this temperature, the sublattice magnetization/Saz S; /Sbz S-0; E 7 -0; FA; ; FB -0; then we have 0 Sa ðSa þ 1Þ 2X C ¼ TN b ; ð14aÞ 3 N K A0 C 0 B0 D0 Sb ðSb þ 1Þ 2X A0 ¼ TN ; 0 0 3 N K A C B0 D0
ð14bÞ
where b ¼ /Saz S=/Sbz S; and A0 ¼ A=/Sbz S; C 0 ¼ C=/Sbz S B0 ¼ B=/Sbz S; D0 ¼ D=/Sbz S:
ð15Þ
We obtain
P Sa ðSa þ 1Þð2=NÞ K A0 =A0 C 0 B0 D0 P b¼ ; Sb ðSb þ 1Þð2=NÞ K C 0 =A0 C 0 B0 D0 TN ¼
3ð2=NÞ
S ðS þ 1Þ Pb b0 0 0 : 0 0 K A =A C B D
ð16aÞ ð16bÞ
Figs. 6–9 gives the variations of the compensation temperature TC and the transition temperature TN with the interlayer interactions and the single-ion anisotropies. In every figure, the dashed lines are the transition curves, the solid lines below them are compensation curves, and each transition curve intersects with the corresponding two compensation curve. Figs. 6 and 7 show the variations of the compensation temperature TC and the transition temperature TN as a function of D1 =J for some values of D2 and J2 ; we can see from them that the critical temperature (TN ) increases and compensation temperature (TC ) (if exists) decreases as D1 increases, and TN and TC increase with D2 or J2 increasing, and the cross point moves towards right. Figs. 8 and 9 show the variations of the compensation temperature TC and the
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86 28
24
20
kBT/J
16
D2=2 D2=1
12
D2=0
8
4
0 0
2
4
6
8
10
D1/J Fig. 6. The variations of the compensation temperature TC and the transition temperature TN as a function of D1 =J for some values of D2 at J1 =J ¼ J2 =J ¼ 0: Dashed line: transition temperature TN ; solid line: compensation temperature TC :
28
24
20
J2=2
kBT/J
16
J2=1
12
J2=0
8
4
0 0
2
4
6
8
10
D1/J Fig. 7. The variations of the compensation temperature TC and the transition temperature TN as a function of D1 =J for some values of J2 at J1 =J ¼ D2 =J ¼ 0: Dashed line: transition temperature TN ; solid line: compensation temperature TC :
transition temperature TN as a function of J1 =J for some values of D2 and J2 ; the situation is similar to the above. These phenomenona can be explained as follow. Firstly, the compensation point is the temperature where the sublattice magnetizations of the ferrimagnetic system have the same value but toward the opposite directions. The maxima of the magnetizations of sublattices A and B are 2 and 52 separately, the magnetization of sublattices A must decrease slower and that of sublattice B decrease faster as the temperature increases in order to make the compensation point. Secondly, the interlayer interaction J1 and
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24
20
D2=2
kBT/J
16
D2=1
12
D2=0
8
4
0
2
4
6
8
10
J1/J Fig. 8. The variations of the compensation temperature TC and the transition temperature TN as a function of J1 =J for some values of D2 at D1 =J ¼ J2 =J ¼ 0: Dashed line: transition temperature TN ; solid line: compensation temperature TC :
J2=2
20
16
J2=1
kBT/J
12
J2=0
8
4
0 0
2
4
6
8
10
J1/J Fig. 9. The variations of the compensation temperature TC and the transition temperature TN as a function of J1 =J for some values of J2 at D1 =J ¼ D2 =J ¼ 0: Dashed line: transition temperature TN ; solid line: compensation temperature TC :
the single-ion anisotropy D1 of sublattice A make the spin-52 sublattice remain ordered at high temperature, the interlayer interaction J2 and the single-ion anisotropy D2 of sublattice B make the spin-52 sublattice remain ordered at high temperature. So, the higher D1 and J1 ; the more easily the compensation point appears; the higher D2 and J2 ; the more difficult the compensation point appears. In experiment, it is found that the transition temperature increases with increasing interlayer separation in the series N(n-CnH2n+1)4FeIIFeIII(C2H4)3, (n ¼ 325), meanwhile the lattice constant within the layer is almost unchangeable. According to our calculation result, we think the effective interlayer superexchange interaction may increase although the distance between the layers increases [6].
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4. Conclusion We have investigated a mixed spin-2 and spin-52 Heisenberg model on a layered honeycomb lattice model which both of whose sublattices have interlayer interactions and single-ion anisotropy by means of a double-time–temperature spin Green function. We have given the spin-wave spectra of the ground state. In particular, we have studied the effects of a single-ion anisotropy and interlayer interactions on the compensation point and transition point and given the spin-wave spectrum of the ground state. As seen from Fig. 5, there are three types of magnetization curves that Neel predicted [16]. When D1 ¼ 0:0 and 2.0, the spin-2 ordering resulting from the single-ion anisotropy of sublattice of A is not stronger to generate a compensation point. When D1 ¼ 4:0 and 8.0, the single-ion anisotropy of sublattice of A makes it possible for spin-2 sublattice to remain more ordered at higher temperature, then a compensation point appears. When the single-ion anisotropies D1 ¼ 2:0; D2 ¼ 2:0 and D1 ¼ 2:0; D2 ¼ 4:0; the single-ion anisotropy of sublattice A makes the spin-2 sublattice less ordering and the single-ion anisotropy of sublattice B makes the spin-52 sublattice more ordering, so the difference of magnetization between the two sublattices will be larger than the classical value 0.5 at certain temperature. Our results suggest that both the single-ion anisotropies and the interlayer interactions are the causes for the compensation phenomenon. The compensation temperature TC decreases as D1 (or J1 ) increases and seems to become insensitive to the change of D1 (or J1 ) as D1 (or J1 ) becomes large. The higher D1 and J1 ; the more easily the compensation point appears; the higher D2 and J2 ; the more difficult it is for the compensation point to appear.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
M. Kinoshita, Jpn. J. Appl. Phys. 33 (1994) 5718. In: D. Gatteschi, O. Kahn, J.S. Miller, F. Palacio (Eds.), Magnetic Molecular Materials, Kluwer Academic, Dordrecht, 1991. A. Ganeschi, D. Gatteschi. Prog. Inorg. Chem. 37 (1991) 331. K. Ionue, H. Iwamura, J. Am. Chem. Soc. 116 (1994) 3173. K. Ionue, T. Hayamizu, H. Iwamura, D. Hashizume, Y. Ohashi, J. Am. Chem. Soc. 118 (1996) 1803. C. Mathoni"ere, C.J. Nuttall, S.G. Carling, P. Day, Inorng. Chem. 35 (1996) 1201. T. Kaneyoshi, J.C. Chen, J. Magn. Magn. Mater. 98 (1991) 201. T. Kaneyoshi, Solid State Commun. 70 (1989) 975. J.W. Tucker, J. Magn. Magn. Mater. 195 (1999) 733. Y. Nakamura, J. Phys. : Condens. Matter 12 (2000) 4067. Y. Nakamura, Phys. Rev. B 62 (2000) 11742. H.B. Callen, Phys. Rev. 130 (1963) 890. R.A. Tahir-Kheli, D. ter Haar, Phys. Rev. 127 (1962) 88. F.B. Anderson, H.B. Callen, Phys. Rev. 136 (1964) 1068. D.L. Lin, H. Zheng, Phys. Rev. 37 (1988) 5394. L. Neel, Ann. Phys. 3 (1948) 137 (Paris).
Physica B 348 (2004) 79–88
The compensation behavior of a mixed spin-2 and spin-52 Heisenberg ferrimagnetic system on a honeycomb lattice Jun Lia,b,*, An Dua, Guozhu Weia b
a College of sciences, Northeastern University, Shenyang 110006, China Department of Physics, Northeastren University, Shenyang 110044, China
Received 15 December 2002; received in revised form 4 August 2003; accepted 11 November 2003
Abstract The magnetic properties of a mixed spin-2 and spin-52 Heisenberg ferrimagnetic system on a layered honeycomb lattice, both of whose sublattices have interlayer interactions and single-ion anisotropy were investigated theoretically by a multisublattice Green-function technique which takes into account the quantum nature of Heisenberg spins. We studied the spin-wave spectra of the ground state. We found three types of the magnetization curves predicted by Neel; in particularly; we investigated the effect of single-ion anisotropies and interlayer interactions on the transition and compensation phenomenon. r 2003 Elsevier B.V. All rights reserved. PACS: 75.10.Jm; 75.30.Ds; 75.40.Gb Keywords: Heisenberg ferrimagnet; Mixed spin system; Sublattice magnetization; Transition temperature; Compensation temperature; Green function
1. Introduction Molecular-based magnetic materials have been a subject of growing interest in recent years [1–5]. Such materials are of interest in experiment and theory because of their properties such as low density, transparency, electrical insulation, and low-temperature fabrication and photoresponsiveness. The search for materials that order at or above room temperature is a major driving force moving the field. Experimentally, a number of molecular-based magnetic materials have been synthesized. Among these materials, ferrimagnets in which two kinds of magnetic atoms regularly alternate antiferromagnetically seem to play an important role. A group compounds synthesized by Mathoniere et al. [6] in 1996 with spins residing only in d orbits connected via covalent bonds AFeIIFeIII(C2O4)3(A=N(nCnH2n+1)4, n ¼ 325) have critical temperatures between 35 and 48 K and some of them have compensation temperature near 30 K. Another recently developed amorphous V (TCNE)x y order ferrimagnetically at as high as 400 K [6].
*Corresponding author. Tel.: 62268995; fax: +24315275642. E-mail address: [email protected] (J. Li). 0921-4526/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2003.11.074
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Theoretically, mixed Ising systems, which provide simple models that can show ferrimagnetic ordering and may have compensation temperatures, have been used to describe the above materials. The magnetic properties of these models have been studied by several methods such as a mean-field [7] and an effectivefield theory [8], a cluster variational theory [9], Monte Carlo simulations [10,11], and so on. In the mixed Ising model used to study the properties of the molecular-based magnetic materials [11], the spins are scalar quantities. In fact, the spins should be vectors in the molecular-based magnetic materials AfeIIFeIII(C2O4)3(A=N(n-CnH2n+1)4, n ¼ 325) [6], the magnetic properties of the system may be affected by the quantum fluctuation of the spins. Otherwise, the mixed Ising model mentioned above does not take into account the interlayer interaction and the single-ion anisotropy of FeIII and, we think it is necessary and interesting to take them into account. So, in this paper, we will study the mixed quantum Heisenberg system, whose both sublattices have interlayer interactions and single-ion anisotropy, with the method of the double-time–temperature Green function, to investigate the magnetic behavior of the molecular-based magnet. In Section 2, we start with a mixed spin-2 and spin-52 ferrimagnetic Heisenberg model that takes the interlayer interaction and the single-ion anisotropy of FeIII into account and then give the fundamental equations. In Section 3, we give the numerical results of sublattice magnetizations in the whole range of temperatures, and illustrate the effects of single-ion anisotropies and interlayer interactions. Section 4 is devoted to summary.
2. Model and fundamental equations We consider a mixed Heisenberg ferrimagnetic model on a layered honeycomb lattice with spin-2 and spin-52 spins that represent FeII and FeIII atoms, respectively, as show in Fig. 1. For convenience, the system is divided into two sublattices A and B with each sublattice being a honeycomb lattice, and the Hamiltonian we adopt takes the form X X X X X H ¼2 Jij Si Sj D1 ðSiz Þ2 D2 ðSjz Þ2 Jii0 Si Si0 Jjj 0 Sj Sj 0 ; ð1Þ ðijÞjj
i
j
ðii0 Þ>
ðjj 0 Þ>
where the summations are taken over all nearest–neighbor spin pairs, iA the sublattice A, Sa=2(FeII), jA the sublattice B, Sb=52(FeIII). In addition, we define Jij ¼ Jð> 0Þ denotes an antiferromagnetic exchange interaction between spins Si and Sj in the layer, Jii0 ¼ J1 ð> 0Þ and Jjj 0 ¼ J2 ð> 0Þ denote a ferromagnetic interlayer interaction between spins in the sublattices A and B in the perpendicular direction, respectively. D1 and D2 denote the single-ion anisotropy of sublattices A and B, respectively.
Fig. 1. The structure of the stacked honeycomb layers, the big balls represent FeIII, the small balls represent FeII.
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To analyze the layered Heisenberg ferrimagnets, we introduce the following double-time Green function according to Callen [12]: GilR ðt; 0Þ ¼ //Siþ ðtÞ; Bl ð0ÞSS; GjlR ðt; 0Þ ¼ //Sjþ ðt; 0Þ;
ð2aÞ
Bl ð0ÞSS;
z eaSl ð0Þ Sl ð0Þ;
ð2bÞ Siþ ðtÞ
Sl ð0Þ
a is a parameter, and and are Heisenberg spin operators defined where Bl ð0Þ ¼ from the x and y spin components. The equations of motion for Green functions read X d i //Siþ ðtÞ; Bl ð0ÞSS ¼ dðtÞ/½Siþ ðtÞ; Bl ð0Þ S þ 2 Jij //ðSiz Sjþ Siþ Sjz Þ; Bl ð0ÞSSÞ dt jðiÞ== X Jii0 //ðSiz Siþ0 Siþ Siz0 Þ; Bl ð0ÞSS þ D1 //ðSiþ Siz þ Siz Siþ Þ; Bl ð0ÞSS; ð3aÞ i0 ðiÞ>
i
X d //Sjþ ðtÞ; Bl ð0ÞSS ¼ dðtÞ/½Sjþ ðtÞ; Bl ð0Þ S þ 2 Jij //ðSjz Siþ Sjþ Siz Þ; Bl ð0ÞSS dt iðjÞjj X z þ þ z Jjj 0 //ðSj Sj 0 Sj Sj 0 Þ; Bl ð0ÞSS þ D2 //ðSjþ Sjz þ Sjz Sjþ Þ; Bj ð0ÞSS; j 0 ðjÞ>
ð3bÞ where dðtÞ is the Dirac d function, /?S means thermal average, and _ ¼ 1 has been used. The higher-order Green’s functions are decoupled as follows. The Green functions coming from the exchange term such as //ðSiz Sjþ Siþ Sjz Þ; Bl SS are decoupled by using the so-called Tyablikov decoupling scheme [13], for example //Siz Sjþ ; Bl SSD/Siz S//Sjþ ; Bl SS:
ð4Þ //ðSiþ Siz
The Green functions coming from the single-ion anisotropy term such as treated by using the Anderson and Callen’s decoupling scheme [14], for example //ðSiþ Siz þ Siz Siþ Þ; Bl SSDti //sþ i ; Bl SS; where ti ¼
2
þ
Siz Siþ Þ;
Bl SS are ð5Þ
1 z 2 S ðS þ 1Þ /ðS Þ S /Saz S: a a a Sa2
ð6Þ
We obtain the Fourier components of the Green function, YA ðaÞ o C oC GA ðo; KÞ ¼ þ ; E E o Eþ o E YA ðgÞ oA oA GB ðo; KÞ ¼ þ ; E E o Eþ o E
ð7aÞ ð7bÞ
where z
YA ðaÞ ¼ /½Siþ ; eaSi Si S;
z
YB ðgÞ ¼ /½Sjþ ; egSj Sj S;
E 7 ¼ 12fðA þ CÞ7½ðA CÞ2 þ 4BD 1=2 g; A ¼ D1 ti 2JZjj /Sbz S þ J1 Z> ð1 cosðkz CÞÞ/Saz S;
ð8Þ ð9Þ ð10aÞ
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B ¼ 2JZjj /Saz Sg;
ð10bÞ
C ¼ D2 tj 2JZjj /Saz S þ J2 Z> ð1 cosðkz CÞÞ/Sbz S;
ð10cÞ
D ¼ 2JZjj /Sbz Sg ; " # pffiffiffi ! 3 1 iky a iky a=2 g¼ kx a e e þ 2 cos : Zjj 2
ð10dÞ ð10eÞ
After using the spectral theorem and Callen’s technique [12], we finally obtain the magnetization of each sublattice as /Saz S ¼ /Sbz S ¼
ðSa Fa Þð1 þ Fa Þ2Sa þ1 þ ðSa þ 1 þ Fa ÞFa2Sa þ1 ; ð1 þ Fa Þ2Sa þ1 Fa2Sa þ1 ðSb Fb Þð1 þ Fb Þ2Sb þ1 þ ðSb þ 1 þ Fb ÞFb2Sb þ1 ð1 þ Fb Þ2Sb þ1 Fb2Sb þ1
ð11aÞ
ð11bÞ
;
/ðSaz Þ2 S ¼ ð2Fa þ 1Þð1 þ Fa Þ
Sa2 Fa2Sa þ1 ð1 þ Sa Þ2 ð1 þ Fa Þ2Sa þ1 þ 2 Sa Fa2Sa þ1 þ ð1 þ Fa Þ2Sa þ1 ð1 þ Sa Þ ð1 þ Fa Þ ð1 þ Fa Þ2Sa þ1 Fa2Sa þ1
; ð11cÞ
/ðSbz Þ2 S ¼ ð2Fb þ 1Þð1 þ Fb Þ
h i Sb2 Fb2Sb þ1 ð1 þ Sb Þ2 ð1 þ Fb Þ2Sb þ1 þ 2 Sb Fb2Sb þ1 þ ð1 þ Fb Þ2Sb þ1 ð1 þ Sb Þ ð1 þ Fb Þ ð1 þ Fb Þ2Sb þ1 Fb2Sb þ1
where Fa and Fb are the auxiliary functions, þ 2X 1 E C E C Fa ¼ ; N K E þ E ebE þ 1 ebE 1 þ 2X 1 E A E A ; Fb ¼ N K E þ E ebE þ 1 ebE 1
; ð11dÞ
ð12aÞ
ð12bÞ
and the total magnetization of the system is given by /S z S ¼ /Saz S þ /Sbz S:
ð13Þ
Eqs. (7)–(12) are the fundamental equations of the sublattice magnetizations, they are complicated and must be solved self-consistently.
3. Results and discussion We now discuss the behaviors of the sublattice magnetizations in whole temperature regimes, the effects of single-ion anisotropies, and the effects of interlayer interactions. 3.1. The spin-wave spectrum of the ground state The first Brillouin zone in the XY plane is shown in Fig. 2.
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kya R
P 2π/3
1/2
(2(3) π /9,2 π/3)
M 0
1/2
4(3) π /9
kxa
Fig. 2. The first Brillouin zone in XY plane.
6 6
+
E
E
4
+
4 2
2
0
0
-2
-2 -4
-4
-6
D1/J=D2/J=0.2
+
E ,E
+
E ,E
-
D1/J=D2/J=0
E
-6
-
E
-
-8 -8
(a)
θπ/2 k xa
θπ (b)
θπ/2 k xa
θπ
Fig. 3. The spin-wave spectra of the ground state with J1 =J ¼ J2 =J ¼ 1:0 (solid line: along OP in The first Buliyoun region in the XY plane, dash line: along OR in The first Buliyoun region in the XY plane for kz ¼ 0) for different values of D1 and D2 (a) D1 =J ¼ pffiffiffi D2 =J ¼ 0; (b) D1 =J ¼ D2 =J ¼ 0:2 ðy ¼ 4 3p=9Þ:
Figs. 3 and 4 show the spin-wave spectra of the ground state for some representative values of D1 ; D2 ; J1 and J2 : In Fig. 3, the solid lines are the case of ky ¼ kz ¼ 0 corresponding to pffiffithe ffi diagonal line (OM) of the first Brillouin zone along the kx axis, and the dot lines are the case of ky ¼ 3kx ; kz ¼ 0; corresponding to the diagonal line (OR) of the first Brillouin zone not along the kx axis. As shown in Fig. 3a, the spin wave has two branches that in E is the acoustic branch and E þ is the optical branch when the anisotropy D1 and D2 are equal to zero. When the anisotropy D1 and D2 are unequal to zero, both branches of spin-wave spectrum are optical branches and an energy gap appears, which is just the effect of the anisotropies. In both cases, the branch of spin-wave spectrum E is always negative. The negative energy can be understood
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20 60
20
-40
0
-20
(a)
-20
+
E ,E
+
E ,E
-
0
-
40
J2/J=1.0 J2/J=4.0 J2/J=6.0
π/2 kz c
-60
π
(b)
J1/J=1.0 J1/J=4.0 J1/J=6.0
π/2 kzc
π
pffiffiffi Fig. 4. The spin-wave spectra of the ground state with D1 =J ¼ D2 =J ¼ 0; kx a ¼ 2 3p=9; ky a ¼ 2p=3 for different values of J1 and J2 (a) J1 =J ¼ 1:0 and J2 =J ¼ 1:0; 4.0, 6.0, (b) J2 =J ¼ 1:0 and J1 =J ¼ 1:0; 4.0, 6.0.
in the following manner. As elementary excitations, we consider the magnon vacuum as the ground state. Therefore, magnons excited out of the filled sea constitute the two branches with energy E and E þ [15]. As shown in Fig. 4(a), the negative energy branch E is insensitive to the change of J2 as J1 is fixed, and the positive energy branch E þ is sensitive to change of J2 : As shown in Fig. 4(b), the positive energy branch E þ is insensitive to the change of J1 as J2 is fixed, and the negative energy branch E is sensitive to change of J1 : 3.2. Magnetization curves In Fig. 5, we plot total magnetization /S z S as a function of the temperature for several values of D1 and D2 : From this figure, we can clearly recognize the existence of a compensation point for D1 ¼ 4:0 and 8.0 and the compensation points vary with the value of D1 : As shown in the figure, when D1 is equal to 4.0 or 8.0, the magnetization curves (c) and (d) behave as N-type, when D1 is equal to 0.0 or 2.0, the magnetization curves (a) and (b) behave as Q-type, and (e) and (f) behave as P-type for D2 ¼ 2:0; or 4.0 and D1 ¼ 2:0: In AFeIIFeIII(C2O4)3 (n ¼ 325), A=(C6H5)3 PNP(C6H5)3, the experiment result of the temperature dependence of the magnetization is similar to the curves of (c) and (d), indicating the importance of the single-ion anisotropy of sublattice A. 3.3. The effects of the interlayer interactions and the single-ion anisotropy on the compensation temperature TC and the transition temperature TN The compensation temperature TC is the temperature where the resultant magnetization vanishes below the critical temperature. That is, when total magnetization oSz > is equal to zero but the sublattice magnetization oSaz > and oSbz > are not equal to zero, we can determine the compensation temperature TC :
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1.0 0.8 0.6
(f)
z
0.4
(e) 0.2
(a)
(b)
0.0
(c) (d)
-0.2 -0.4 0
4
8
12
16
20
24
K BT/J Fig. 5. Total magnetization as a function of temperature for several group values of D1 and D2 when J ¼ 1:0; J1 ¼ 0:2; J2 ¼ 0:0: (a) D1 ¼ D2 ¼ 0:0 (b) D1 ¼ 2:0; D2 ¼ 0:0; (c) D1 ¼ 4:0; D2 ¼ 0:0 (d) D1 ¼ 8:0; D2 ¼ 0:0; (e) D1 ¼ 2:0; D2 ¼ 2:0; (f) D1 ¼ 2:0; D2 ¼ 4:0:
The transition temperature TN is the temperature where the resultant magnetization and the sublattice magnetizations all vanish. At this temperature, the sublattice magnetization/Saz S; /Sbz S-0; E 7 -0; FA; ; FB -0; then we have 0 Sa ðSa þ 1Þ 2X C ¼ TN b ; ð14aÞ 3 N K A0 C 0 B0 D0 Sb ðSb þ 1Þ 2X A0 ¼ TN ; 0 0 3 N K A C B0 D0
ð14bÞ
where b ¼ /Saz S=/Sbz S; and A0 ¼ A=/Sbz S; C 0 ¼ C=/Sbz S B0 ¼ B=/Sbz S; D0 ¼ D=/Sbz S:
ð15Þ
We obtain
P Sa ðSa þ 1Þð2=NÞ K A0 =A0 C 0 B0 D0 P b¼ ; Sb ðSb þ 1Þð2=NÞ K C 0 =A0 C 0 B0 D0 TN ¼
3ð2=NÞ
S ðS þ 1Þ Pb b0 0 0 : 0 0 K A =A C B D
ð16aÞ ð16bÞ
Figs. 6–9 gives the variations of the compensation temperature TC and the transition temperature TN with the interlayer interactions and the single-ion anisotropies. In every figure, the dashed lines are the transition curves, the solid lines below them are compensation curves, and each transition curve intersects with the corresponding two compensation curve. Figs. 6 and 7 show the variations of the compensation temperature TC and the transition temperature TN as a function of D1 =J for some values of D2 and J2 ; we can see from them that the critical temperature (TN ) increases and compensation temperature (TC ) (if exists) decreases as D1 increases, and TN and TC increase with D2 or J2 increasing, and the cross point moves towards right. Figs. 8 and 9 show the variations of the compensation temperature TC and the
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24
20
kBT/J
16
D2=2 D2=1
12
D2=0
8
4
0 0
2
4
6
8
10
D1/J Fig. 6. The variations of the compensation temperature TC and the transition temperature TN as a function of D1 =J for some values of D2 at J1 =J ¼ J2 =J ¼ 0: Dashed line: transition temperature TN ; solid line: compensation temperature TC :
28
24
20
J2=2
kBT/J
16
J2=1
12
J2=0
8
4
0 0
2
4
6
8
10
D1/J Fig. 7. The variations of the compensation temperature TC and the transition temperature TN as a function of D1 =J for some values of J2 at J1 =J ¼ D2 =J ¼ 0: Dashed line: transition temperature TN ; solid line: compensation temperature TC :
transition temperature TN as a function of J1 =J for some values of D2 and J2 ; the situation is similar to the above. These phenomenona can be explained as follow. Firstly, the compensation point is the temperature where the sublattice magnetizations of the ferrimagnetic system have the same value but toward the opposite directions. The maxima of the magnetizations of sublattices A and B are 2 and 52 separately, the magnetization of sublattices A must decrease slower and that of sublattice B decrease faster as the temperature increases in order to make the compensation point. Secondly, the interlayer interaction J1 and
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24
20
D2=2
kBT/J
16
D2=1
12
D2=0
8
4
0
2
4
6
8
10
J1/J Fig. 8. The variations of the compensation temperature TC and the transition temperature TN as a function of J1 =J for some values of D2 at D1 =J ¼ J2 =J ¼ 0: Dashed line: transition temperature TN ; solid line: compensation temperature TC :
J2=2
20
16
J2=1
kBT/J
12
J2=0
8
4
0 0
2
4
6
8
10
J1/J Fig. 9. The variations of the compensation temperature TC and the transition temperature TN as a function of J1 =J for some values of J2 at D1 =J ¼ D2 =J ¼ 0: Dashed line: transition temperature TN ; solid line: compensation temperature TC :
the single-ion anisotropy D1 of sublattice A make the spin-52 sublattice remain ordered at high temperature, the interlayer interaction J2 and the single-ion anisotropy D2 of sublattice B make the spin-52 sublattice remain ordered at high temperature. So, the higher D1 and J1 ; the more easily the compensation point appears; the higher D2 and J2 ; the more difficult the compensation point appears. In experiment, it is found that the transition temperature increases with increasing interlayer separation in the series N(n-CnH2n+1)4FeIIFeIII(C2H4)3, (n ¼ 325), meanwhile the lattice constant within the layer is almost unchangeable. According to our calculation result, we think the effective interlayer superexchange interaction may increase although the distance between the layers increases [6].
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4. Conclusion We have investigated a mixed spin-2 and spin-52 Heisenberg model on a layered honeycomb lattice model which both of whose sublattices have interlayer interactions and single-ion anisotropy by means of a double-time–temperature spin Green function. We have given the spin-wave spectra of the ground state. In particular, we have studied the effects of a single-ion anisotropy and interlayer interactions on the compensation point and transition point and given the spin-wave spectrum of the ground state. As seen from Fig. 5, there are three types of magnetization curves that Neel predicted [16]. When D1 ¼ 0:0 and 2.0, the spin-2 ordering resulting from the single-ion anisotropy of sublattice of A is not stronger to generate a compensation point. When D1 ¼ 4:0 and 8.0, the single-ion anisotropy of sublattice of A makes it possible for spin-2 sublattice to remain more ordered at higher temperature, then a compensation point appears. When the single-ion anisotropies D1 ¼ 2:0; D2 ¼ 2:0 and D1 ¼ 2:0; D2 ¼ 4:0; the single-ion anisotropy of sublattice A makes the spin-2 sublattice less ordering and the single-ion anisotropy of sublattice B makes the spin-52 sublattice more ordering, so the difference of magnetization between the two sublattices will be larger than the classical value 0.5 at certain temperature. Our results suggest that both the single-ion anisotropies and the interlayer interactions are the causes for the compensation phenomenon. The compensation temperature TC decreases as D1 (or J1 ) increases and seems to become insensitive to the change of D1 (or J1 ) as D1 (or J1 ) becomes large. The higher D1 and J1 ; the more easily the compensation point appears; the higher D2 and J2 ; the more difficult it is for the compensation point to appear.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
M. Kinoshita, Jpn. J. Appl. Phys. 33 (1994) 5718. In: D. Gatteschi, O. Kahn, J.S. Miller, F. Palacio (Eds.), Magnetic Molecular Materials, Kluwer Academic, Dordrecht, 1991. A. Ganeschi, D. Gatteschi. Prog. Inorg. Chem. 37 (1991) 331. K. Ionue, H. Iwamura, J. Am. Chem. Soc. 116 (1994) 3173. K. Ionue, T. Hayamizu, H. Iwamura, D. Hashizume, Y. Ohashi, J. Am. Chem. Soc. 118 (1996) 1803. C. Mathoni"ere, C.J. Nuttall, S.G. Carling, P. Day, Inorng. Chem. 35 (1996) 1201. T. Kaneyoshi, J.C. Chen, J. Magn. Magn. Mater. 98 (1991) 201. T. Kaneyoshi, Solid State Commun. 70 (1989) 975. J.W. Tucker, J. Magn. Magn. Mater. 195 (1999) 733. Y. Nakamura, J. Phys. : Condens. Matter 12 (2000) 4067. Y. Nakamura, Phys. Rev. B 62 (2000) 11742. H.B. Callen, Phys. Rev. 130 (1963) 890. R.A. Tahir-Kheli, D. ter Haar, Phys. Rev. 127 (1962) 88. F.B. Anderson, H.B. Callen, Phys. Rev. 136 (1964) 1068. D.L. Lin, H. Zheng, Phys. Rev. 37 (1988) 5394. L. Neel, Ann. Phys. 3 (1948) 137 (Paris).