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The effect of magnetic field on the susceptibility maximum in the spatially anisotropic Heisenberg antiferromagnet
Solid State Communications 148 (2008) 369–373
Contents lists available at ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
The effect of magnetic field on the susceptibility maximum in the spatially anisotropic Heisenberg antiferromagnet Ying Xiang a,∗ , Yuan Chen b , Qi-Zhou Chen c , Jun Zhang a , Yi-Kun Liu c a
School of Information Engineering, Guangdong University of Technology, Guangzhou, 510006, PR China
b
Department of Physics, Guangzhou University, Guangzhou, 510006, PR China
c
Department of Physics, Sun Yat-sen University, Guangzhou 510275, PR China
article
info
Article history: Received 29 May 2008 Accepted 20 September 2008 by B.-F. Zhu Available online 26 September 2008 PACS: 75.40.Cx Keywords: A. Magnetically ordered materials D. Thermodynamic properties
a b s t r a c t The effect of magnetic field h on the longitudinal susceptibility in a spin S = 1/2 exchange anisotropic three-dimensional Heisenberg antiferromagnet, is studied by the double-time Green’s function method within Tyablikov approximation. The calculation results indicated that the height χ (Tm ) and position Tm of the maximum of the longitudinal susceptibility display different behaviors related to the magnetic fields and exchange anisotropic parameters. These behaviors are very different from that in the exchange anisotropic Heisenberg ferromagnet in the magnetic field. The results are: (1) When the field h is weak, in a antiferromagnet, the height χ (Tm ) is a constant χ0 which is independent of field and exchange anisotropy, but the position Tm is only a function of the exchange anisotropy. While in a ferromagnet, both χ (Tm ) and Tm are a function of field and the exchange anisotropy. (2) When the field h is strong, in a antiferromagnet, χ (Tm ) becomes dependent of field and the exchange anisotropy, and χ (Tm ) and Tm are fitted satisfactory to power laws: χ (Tm ) − χ0 ∝ hd and TN − Tm ∝ hc , respectively. Here TN is the Neel 0 temperature. On the contrary, in a ferromagnet, χ (Tm ) and Tm are fitted to power laws: χ(Tm ) ∝ h−d 0 and Tm − Tc ∝ hc , where Tc is the Curie temperature. The above results are very useful in studying the magnetic property of coordination polymers. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction Recently, the magnetic field dependence on the susceptibility maximum χ (Tm ) as well as the dependence on the corresponding temperature Tm in ferromagnets have attracted increasing attention [1,2]. It can be connected with the experiments on La0.91 Mn0.95 O3 , which showed a shift of the maximum electrical resistivity ρ(Tm ) at an applied field h according to h2/3 . Assuming this maximum of ρ(Tm ) coincides with the maximum of magnetic susceptibility χ (Tm ) due to spin scattering, within the mean-field Landau’s theory, the susceptibility has a maximum χ(Tm ) at the point Tm , and under magnetic field h, this point is shifted according to h2/3 . This 2/3 power law is claimed to be independent on a spin model. Afterward many researchers [3–5] addressed this issue, they indicated that χ (Tm ) and Tm are fitted to 0 0 power laws: χ (Tm ) ∝ h−d and Tm − Tc ∝ hc , where Tc is the Curie temperature. However, the exponents c’ and d’ of power laws are found to be strongly dependent on exchange anisotropy and disagree with this 2/3 power law. On the other hand, in contrast with the wealth of literature and experiments on the power laws of ferromagnets, there has
∗
Corresponding author. Tel.: +86 20 34773543; fax: +86 20 34773543. E-mail address: [email protected] (Y. Xiang).
0038-1098/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2008.09.040
been comparatively little research concerning that of antiferromagnets [6], although these two magnetic materials are somewhat related. This phenomenon is attributed to the strong exchange coupling in the traditional oxide antiferromagnets, because χ (Tm ) occurs at the spin-flop transition, and the strong exchange coupling means that very strong magnetic fields are needed to achieve this transition. Therefore, the research and application of χ (Tm ) are limited. However, the new emergence of coordination polymers have revived this issue [7–10]. In these materials, ferromagnetic coupling exists in intra-layer, while antiferromagnetic coupling exists between the inter-layer. Because the antiferromagnetic coupling gets very weak, only the medium magnetic field is needed to achieve the spin-flop transition. In fact, it is very difficult to identify the magnetic phase of polymers in experiments. It is because the polymers with the same composition may possess different magnetic properties, such as ferromagnetic, antiferromagnetic or paramagnetic phase. Furthermore, the polymers easily undergo phase transition (such as spin-flop transition) under different external conditions, which makes identification more difficult. Here we present a method based on the measurement of the behaviors of χ (Tm ) and Tm to investigate the magnetic properties of coordination polymers. The effect of magnetic field h on the longitudinal susceptibility in a spin S = 1/2 exchange anisotropic three-dimensional
370
Y. Xiang et al. / Solid State Communications 148 (2008) 369–373
Heisenberg antiferromagnet, is studied by the double-time Green’s function method within Tyablikov approximation. The calculation results indicated that the height χ (Tm ) and position Tm of the maximum of the longitudinal susceptibility display different behaviors related to the magnetic fields and exchange anisotropic parameters. These behaviors are very different from that in the exchange anisotropic Heisenberg ferromagnet in the magnetic field. The results are: (1) When the field h is weak, in a antiferromagnet, the height χ (Tm ) is a constant χ0 which is independent of field and exchange anisotropy, but the position Tm is only a function of the exchange anisotropy. While in a ferromagnet, both χ (Tm ) and Tm are a function of field and the exchange anisotropy. (2) When the field h is strong, in a antiferromagnet, χ (Tm ) becomes dependent of field and the exchange anisotropy, and χ (Tm ) and Tm are fitted satisfactory to power laws: χ (Tm ) − χ0 ∝ hd and TN − Tm ∝ hc , respectively. Here TN is the Neel temperature. On the contrary, in a ferromagnet, 0 χ(Tm ) and Tm are fitted to power laws: χ (Tm ) ∝ h−d and Tm − Tc ∝ c0 h , where Tc is the Curie temperature. In short, the investigation of χ(Tm ) and Tm in antiferromagnets are both meaningful in academic field and applications. The paper is organized as follows. In Section 2, the formalisms of Green’s function theory are described, and the basic self-consistent equations for the S = 1/2 exchange anisotropic antiferromagnet are obtained. In Section 3, the numerical results of the χ (Tm ) and Tm under weak h are presented. In Section 4, the numerical results of the χ(Tm ) and Tm under strong h are presented. In Section 5, a brief conclusion is given.
The equation of motion for G+− aa (t ) is given by i
+ − d
+ − Sai (t ); Sai Sai (t ), H ; Sai 0 (0) = 2ma · δ(t )δii0 + 0 (0) dt
+ = 2ma · δ(t )δii0 + h Sai (t ); Sai−0 (0) + J · η · ma ·
ρ
− J · mb ·
+ Sbj (t ); Sai−0 (0)
X
+ Sai (t ); Sai−0 (0) .
ρ
(3)
z Here the Tyablikov decoupling [15] was applied as Sai = ma
z
P
and Sbj = mb , and ρ denote a sum over the nearest-neighbor sites. After Fourier transforming the Eq. (3) with respect to the space and time variables, we obtain a algebraic equation that are +− easily for
+ solved the transformed Green’s function Gaa (k, ω) of − Sai (t ); Sai0 (0) , which can be evaluated as: +− +− ωG+− aa (k, ω) = 2ma + hGaa (k, ω) + J · η · ma · γk · Gba (k, ω)
− J · mb · γ0 · G+− aa (k, ω)
(4) * * i k ·ρ
where γ0 = Z is the coordination number and γk = . ρe Hereafter we consider the case of simple cubic lattice, with rk = 2(cos kx + cos ky + cos kz ) and r0 = 6. Similar to the G+− aa (t ), the equation of motion for G+− ( t ) is given by: ba
P
i 2. Method and discussion
+ d
+ − − Sbj (t ); Sai 0 (0) = h Sbj (t ); Sai0 (0)
dt
+ J · η · mb ·
We consider the S = 1/2 three-dimensional Heisenberg antiferromagnet (AFM) model with anisotropic spin exchange. Its Hamiltonian is given by [11]
X
+ Sai (t ); Sai−0 (0)
ρ
− J · ma ·
X η H = J [Saiz · Sbjz + (Sai+ · Sbj− + Sai− · Sbj+ )]
X
ρ
+ (t ); Sai−0 (0) . Sbj
(5)
+ And the transformed Green’s function G+− ba (k, ω) of hhSbj (t );
2
hiji
X
Sai0 (0)ii, which can be evaluated as: −
−h
X i
z Sai −h
X
z Sbj .
(1)
j
Here η denotes the exchange anisotropic parameters with 0 < η < 1, and hiji denotes a sum over the nearest-neighbor sites. J is the exchange coupling constant between neighboring spins and for AFM, J > 0. In order to keep the translational invariance, the AFM lattices are divided into two sublattices A and B, where ai refers to the lattice site i in the sublattice A, so does the bj. It is assumed that the spontaneous magnetization ma of sublattice A is in the +z direction, but mb of sublattice B is in the – z direction. h is the x applied longitudinal magnetic field, which is along the z axis. Sai , y z Sai and Sai represent the three components of the spin S = 1/2 ± operator for a spin at site i in sublattice A. Sai = Saix ± iSaiy are the spin raising and lowing operators, which satisfy the commutation + − relation [Sai , Sbj ] = 2Saiz δij δab and [Sai± , Sbjz ] = ∓Sai± δij δab . In order to calculate the magnetic properties of this model,
we introduce retarded Green’s function Aai (t ); Bbj (0) =
−iθ(t ) Aai (t ), Bbj (0) , where θ (t ) is the step function. The equation of motion for Green’s function follows in a straight forward fashion, which is given by [12–14] i
d
Aai (t ); Bbj (0) = δ(t )δij δab [Aai , Bbj ] dt
+ [Aai (t ), H ]; Bbj (0) .
− J · ma · γ0 · G+− ba (k, ω).
(6)
From Eqs. (4), (6), Gaa (k, ω) can be written as: +−
G+− aa (k, ω) =
2ma
ω1 − ω2
·
ω1 − Ea ω2 − Ea − ω − ω1 ω − ω2
,
(7)
where Ea = h − J · γ0 · ma , Eb = h − J · γ0 · mb ,
ω1,2 =
(Ea + Eb ) ±
p (Ea − Eb )2 + 4ma · mb · (J · η · γk )2 2
.
The correlation function of G+− aa (t ) is found to be
1 1 − + Sai − Saz = − ma . 0 (0)Sai (t ) i0 =i,t =0 = 2 2
With the help of the spectral theorem, we obtain the following expression for the magnetization of sublattice A: 1
1
3 Z
π
Z
π
Z
π
1
= 2π ω1 − ω2 −π −π −π ω1 × (ω1 − Ea ) · coth
2ma (2)
Now, we choose different operators A and B to form Green functions as follow:
+ − + − G+− G+− aa (t ) = Sai (t ); Sai0 (0) , ba (t ) = Sbj (t ); Sai0 (0) , DD EE DD EE + + G+− Sbj (t ); Sbj−0 (0) and G+− Sai (t ); Sbj−0 (0) . bb (t ) = ab (t ) =
+− +− ωG+− ba (k, ω) = hGba (k, ω) + J · η · mb · γk · Gaa (k, ω)
2kB T
− (ω2 − Ea ) · coth
ω2 2kB T
*
dk.
(8)
Following the same way, we obtain the following expression for the magnetization of sublattice B:
Y. Xiang et al. / Solid State Communications 148 (2008) 369–373
371
Fig. 1. Longitudinal susceptibilities χ (T , h) of AFM and FM vs. magnetic field h and temperature T with the same exchange anisotropy η = 0.5.
Fig. 3. The magnetic field h dependence on Tm of AFM and FM with different exchange anisotropy η.
Fig. 2. Longitudinal susceptibilities χ(T , h) of AFM and FM vs. exchange anisotropy η and temperature T under the same magnetic field h = 0.05.
Fig. 4. The magnetic field h dependence of m(Tm ) of AFM and FM with the different exchange anisotropy η.
3 Z π Z π Z π 1 1 = 2mb 2π ω1 − ω2 −π −π −π ω1 × (ω1 − Eb ) · coth
For the field h fixed, in the AFM, χ (Tm ) is found to be independent of the exchange anisotropy η. But its Tm shifts and gets smaller with the increasing η. However, in the FM, as η increases, χ (Tm ) increases, and Tm shifts and decreases, as shown in Figs. 2 and 3. In the end, when h → 0 and η → 1, the Tm of AFM will approach the Neel temperature TN ≈ 0.99, and the Tm of FM will approach the Curie temperature Tc ≈ 1.98. The magnetic field h dependence of the magnetization m(Tm ) of AFM and FM are plotted in Fig. 4 with different exchange anisotropy η. In AFM, the magnetization m(Tm ) at temperature Tm is linear with h, and is independence of η. The slope of the line is found to be χ0 . But in FM, m(Tm ) is nonlinear with h and η. Let us consider the Eqs. (8) and (9), when under the condition h → 0 and T → TN , where TN is the Neel temperature. By using ω1 → 0, ω2T2 → 0 and coth x|x→0 ≈ 1x , the Eq. (8) can be written 2T as:
1
2kB T
− (ω2 − Eb ) · coth
ω2 2kB T
*
dk.
(9)
From Eqs. (8) and (9), we can obtain the magnetization of AFM m = 12 (ma + mb ) and the longitudinal susceptibility χ = ∂∂mh . In the following, for the convenience we let J = kB = 1. When under different conditions h and η, the longitudinal susceptibility χ possess maximum values χ (Tm ) at the temperature location Tm , and the effects of h and η on the χ (Tm ) and Tm are described as follow. 3. Under weak magnetic field conditions When the field h is weak, in antiferromagnets, the height χ (Tm ) is a constant χ0 which is independent of field and exchange anisotropy, but the position Tm is only a function of the exchange anisotropy. While in ferromagnets, both χ (Tm ) and Tm are a function of field and the exchange anisotropy. These different behaviors between antiferromagnet (AFM) and ferromagnet (FM) are shown in Figs. 1–4. If the exchange anisotropy is given, Fig. 1 shows that in AFM, the height χ (Tm ) and position Tm of the longitudinal susceptibilities χ are independent of magnetic field h. Whereas in FM, as h increases, χ(Tm ) always decreases, but Tm increases.
h ma
=
1
3 Z
2π
π −π
Z
π −π
Z
π
4 · h · T · Ea
ω1 · ω2
−π
*
dk.
Then the following equations can be obtained: 4 · T · χa · (1 − γ0 · χa ) 1
= 1 2π
3 R π R π R π −π
−π
−π
1 1−γ0 ·(χa +χb )+χa ·χb ·(γ02 −η2 ·γk2 )
(10)
*
dk
4 · T · χb · (1 − γ0 · χb )
=
1
Rπ Rπ Rπ
1 3 2π
−π
−π
1
−π 1−γ0 ·(χa +χb )+χa ·χb ·(γ 2 −η2 ·γ 2 ) d k 0
*
k
.
(11)
372
Y. Xiang et al. / Solid State Communications 148 (2008) 369–373
Fig. 6. log–log plot of the magnetic field h dependence on Tm .
Fig. 5. Longitudinal susceptibility χ(h, T ) of AFM vs. the magnetic field h and temperature T .
From the Eqs. (10) and (11), we can have:
γ0 · (χa + χb ) = 1. Therefore, the longitudinal susceptibility maximum of AFM can be written as:
χa + χb
χ(Tm ) = χ0 =
2
=
1 2γ0
≈ 0.083.
(12)
The analytic result of Eq. (12) coincides well with the numerical calculation, and shows that χ (Tm ) is a constant which is independence of h and η. If we introduce the Eq. (12) into (10), we can obtain the position temperature is given by TN = Tm =
γ0 4 · C (η)
1
,
3 Z
Fig. 7. log–log plot of the magnetic field h dependence on χ(Tm ).
π
Z
π
Z
π
1
−π
1 − (η · γk /γ0 )
*
The analytic result of Eq. (13) shows that Tm decreases as the η increases.
Furthermore, our calculation results for the longitudinal susceptibility maximum χ (Tm ) and temperature position Tm of the AFM as a function of h are plotted logarithmically in Figs. 6 and 7, respectively. It is shown that χ (Tm ) and Tm is fitted satisfactory to power laws:
4. Under strong magnetic field conditions
χ (Tm ) − χ0 ∝ hd
When under strong magnetic field h, χ (Tm ) and Tm of AFM are the function of h and η, and their behaviors are far different from that in small h. Fig. 5 shows that in AFM, as the field h increases, χ (Tm ) gets higher, and Tm gets smaller. However, in FM, χ (Tm ) gets lower, and Tm gets bigger. These kinds of behaviors of χ (Tm ) and Tm can be used to investigate the magnetic property of coordination polymers. There exist two problems in this area. The first one is the synthesis of coordination polymers. Under different synthetic conditions, coordination polymers which possess the same chemical composition will display different magnetic properties which can’t be predicted before synthesis. Therefore, a method is needed to identify the different magnetic phases of polymers, such as ferromagnetic, antiferromagnetic and metamagnetic phase. The second problem is the phase transition of polymers. Under different conditions (h, T and η), the phases of polymers are easily changed to other phases, such as changing from the antiferromagnetic phase to the spinflop phase under magnetic field h, or changing from the antiferromagnetic phase to the paramagnetic phase under temperature T . Therefore, the way is needed to find this phase transition. Hence, the investigation of χ (Tm ) and Tm is an easy and reliable solution to overcome these two problems.
and
with C (η) =
2π
−π
−π
2
dk.
(13)
T N − T m ∝ hc ,
(14)
(15)
respectively. As mentioned in the introduction, the power laws in FM have obtained much concern in theory as well as experiment, and the corresponding power laws are expressed as follow: 0
χ ( T m ) ∝ h−d
(16)
and 0
T m − T c ∝ hc ,
(17)
where Tc is the Curie temperature. Comparing the Eqs. (14) and (15) with Eqs. (16) and (17), we can conclude that the power laws in AFM are different from those in FM. However, there is little study on the power laws in AFM. The discussion in this paper is a useful supplement in this field. As for the Tm , Tm > Tc exists in FM, while Tm < TN exists in AFM; As for the χ (Tm ), in FM, the power exponent −d0 < 0 (where d0 > 0), while in AFM, the power exponent d > 0, and term χ (Tm ) − χ0 replaces term χ (Tm ), which is present in FM.
Y. Xiang et al. / Solid State Communications 148 (2008) 369–373
In addition, the power exponents are found to be dependent on the exchange anisotropy. If the values of the exchange anisotropy are η = 0.25, 0.50, 0.75, the power exponents c = 2.036, 2.014, 2.006, and d = 2.013, 2.034, 2.076, respectively. From the above data, the exponents c and d possess opposite tendencies with varying η. Let us consider the analytical results of Eqs. (8) and (9). According to the calculation, when χ (T ) → χ (Tm ), there will be ma ≈ mb , and Ea = Eb = E0 . Applying this condition and the approximation coth x ≈ 1x + 3x , the Eqs. (8) and (9) can be simplified as: 1 m
= 4 · T · E0 · E0
+
3T
1
3 Z
2π
π −π
Z
π −π
Z
*
π
dk
E02 − (η · m · γk )2
−π
.
(18)
373
5. Conclusion The effect of magnetic field h on the longitudinal susceptibility in a spin S = 1/2 exchange anisotropic three-dimensional Heisenberg AFM, is studied by the double-time Green’s function method within Tyablikov approximation. The calculation results indicated that the height χ (Tm ) and position Tm of the maximum of longitudinal susceptibility display different behaviors related to the magnetic fields and exchange anisotropic parameters. These behaviors are very different from that in the exchange anisotropic Heisenberg FM in the magnetic field. Considering that there is currently little research on the power laws of AFM, this paper is a useful supplement in this field. Furthermore, the above theory results can be developed as an experimental method to investigate the magnetic property of coordination polymers, where the χ (Tm ) and Tm can be measured easily and accurately.
Let us introduce the following approximations into Eq. (18)
χ(Tm ) = χ0 + 1χ ,
Tm = TN − 1T ,
where 1χ χ0 = 2γ1 , 1T TN = 0 From Eq. (18) we can have 1≈
TN − 1T TN
+
E0 · m 3TN
Acknowledgements
m(Tm ) = m0 + 1m, γ0
4C (η)
, and 1m m0 = 2γh . 0
.
(19)
In addition, applying the derivative ∂∂h of Eq. (18), we can have, 1
γ
2 0
·
1 m
−
h·χ m2
· C 0 (η) =
m + χ · (h − 2γ0 m)
Rπ Rπ Rπ
6T 2
(20)
*
where C 0 (η) = ( 21π )3 −π −π −π d k [ (1−η·γ1 /γ )2 + (1+η·γ1 /γ )2 ]. k 0 k 0 After straightforward and approximately deduction, we have the analytical results
1T ≈
h2
(21)
12γ0
and
1χ ≈
h2 48 · γ0 · C 0 (η) · TN2
(22)
which yield the power laws TN − Tm ∝ hc , χ (Tm ) − χ0 ∝ hd , with the corresponding exponent c = 2.0, d = 2.0.
This work was supported by the National Natural Science Foundation of China (10674033) and the Science and Technology Program of Guangdong Province of China (2007B010600060). References [1] V. Markovich, E. Rozenberg, G. Gorodetsky, B. Revsin, J. Pelleg, I. Felner, Phys. Rev. B 62 (2000) 14186. [2] I. Junger, D. Ihle, J. Richter, A. KlÄumper, Phys. Rev. B 70 (2004) 104419. [3] J. Sznajd, Phys. Rev. B 64 (2001) 052401. [4] Ai-Yuan Hu, Yuan Chen, Li-Jun Peng, Physica B 393 (2007) 368. [5] Ai-Yuan Hu, Yuan Chen, Li-Jun Peng, J. Magn. Magn. Mater. 313 (2007) 366. [6] L. Siurakshina, D. Ihle, Phys. Rev. B 61 (2000) 14601. [7] Michelle L. Taliaferro, Fernando Palaciob, Joel S. Miller, J. Mater. Chem. 16 (2006) 2677. [8] Hong-Peng Jia, Wei Li, Zhan-Feng Ju, Jie Zhang, Dalton Trans. (2007) 3699. [9] Souhei Kaneko, Yoshihide Tsunobuchi, Shunsuke Sakurai, Shin-ichi Ohkoshi, Chem. Phys. Lett. 446 (2007) 292. [10] Long Jiang, Xiao-Long Feng, Tong-Bu Lu, Song Gao, Inorg. Chem. 45 (2006) 5018. [11] Huai-Yu Wang, Phys. Rev. B 69 (2004) 174431. [12] Jian-Hua Cai, et al., The Green’s Function Theory of Quantum statistics, Science Press, Beijing, 1982. [13] Norberto Majlis, The Quantum Theory of Magnetism, World Scientific Publishing Co. Pte. Td, Singapore, 2000. [14] Huai-Yu Wang, The Green’s Function Theory in Condensed Matter Physics, Science Press, Beijing, 2008. [15] S.V. Tyablikov, Method in Quantum Theory of Magnetism, Plenum, New York, 1967.
Contents lists available at ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
The effect of magnetic field on the susceptibility maximum in the spatially anisotropic Heisenberg antiferromagnet Ying Xiang a,∗ , Yuan Chen b , Qi-Zhou Chen c , Jun Zhang a , Yi-Kun Liu c a
School of Information Engineering, Guangdong University of Technology, Guangzhou, 510006, PR China
b
Department of Physics, Guangzhou University, Guangzhou, 510006, PR China
c
Department of Physics, Sun Yat-sen University, Guangzhou 510275, PR China
article
info
Article history: Received 29 May 2008 Accepted 20 September 2008 by B.-F. Zhu Available online 26 September 2008 PACS: 75.40.Cx Keywords: A. Magnetically ordered materials D. Thermodynamic properties
a b s t r a c t The effect of magnetic field h on the longitudinal susceptibility in a spin S = 1/2 exchange anisotropic three-dimensional Heisenberg antiferromagnet, is studied by the double-time Green’s function method within Tyablikov approximation. The calculation results indicated that the height χ (Tm ) and position Tm of the maximum of the longitudinal susceptibility display different behaviors related to the magnetic fields and exchange anisotropic parameters. These behaviors are very different from that in the exchange anisotropic Heisenberg ferromagnet in the magnetic field. The results are: (1) When the field h is weak, in a antiferromagnet, the height χ (Tm ) is a constant χ0 which is independent of field and exchange anisotropy, but the position Tm is only a function of the exchange anisotropy. While in a ferromagnet, both χ (Tm ) and Tm are a function of field and the exchange anisotropy. (2) When the field h is strong, in a antiferromagnet, χ (Tm ) becomes dependent of field and the exchange anisotropy, and χ (Tm ) and Tm are fitted satisfactory to power laws: χ (Tm ) − χ0 ∝ hd and TN − Tm ∝ hc , respectively. Here TN is the Neel 0 temperature. On the contrary, in a ferromagnet, χ (Tm ) and Tm are fitted to power laws: χ(Tm ) ∝ h−d 0 and Tm − Tc ∝ hc , where Tc is the Curie temperature. The above results are very useful in studying the magnetic property of coordination polymers. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction Recently, the magnetic field dependence on the susceptibility maximum χ (Tm ) as well as the dependence on the corresponding temperature Tm in ferromagnets have attracted increasing attention [1,2]. It can be connected with the experiments on La0.91 Mn0.95 O3 , which showed a shift of the maximum electrical resistivity ρ(Tm ) at an applied field h according to h2/3 . Assuming this maximum of ρ(Tm ) coincides with the maximum of magnetic susceptibility χ (Tm ) due to spin scattering, within the mean-field Landau’s theory, the susceptibility has a maximum χ(Tm ) at the point Tm , and under magnetic field h, this point is shifted according to h2/3 . This 2/3 power law is claimed to be independent on a spin model. Afterward many researchers [3–5] addressed this issue, they indicated that χ (Tm ) and Tm are fitted to 0 0 power laws: χ (Tm ) ∝ h−d and Tm − Tc ∝ hc , where Tc is the Curie temperature. However, the exponents c’ and d’ of power laws are found to be strongly dependent on exchange anisotropy and disagree with this 2/3 power law. On the other hand, in contrast with the wealth of literature and experiments on the power laws of ferromagnets, there has
∗
Corresponding author. Tel.: +86 20 34773543; fax: +86 20 34773543. E-mail address: [email protected] (Y. Xiang).
0038-1098/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2008.09.040
been comparatively little research concerning that of antiferromagnets [6], although these two magnetic materials are somewhat related. This phenomenon is attributed to the strong exchange coupling in the traditional oxide antiferromagnets, because χ (Tm ) occurs at the spin-flop transition, and the strong exchange coupling means that very strong magnetic fields are needed to achieve this transition. Therefore, the research and application of χ (Tm ) are limited. However, the new emergence of coordination polymers have revived this issue [7–10]. In these materials, ferromagnetic coupling exists in intra-layer, while antiferromagnetic coupling exists between the inter-layer. Because the antiferromagnetic coupling gets very weak, only the medium magnetic field is needed to achieve the spin-flop transition. In fact, it is very difficult to identify the magnetic phase of polymers in experiments. It is because the polymers with the same composition may possess different magnetic properties, such as ferromagnetic, antiferromagnetic or paramagnetic phase. Furthermore, the polymers easily undergo phase transition (such as spin-flop transition) under different external conditions, which makes identification more difficult. Here we present a method based on the measurement of the behaviors of χ (Tm ) and Tm to investigate the magnetic properties of coordination polymers. The effect of magnetic field h on the longitudinal susceptibility in a spin S = 1/2 exchange anisotropic three-dimensional
370
Y. Xiang et al. / Solid State Communications 148 (2008) 369–373
Heisenberg antiferromagnet, is studied by the double-time Green’s function method within Tyablikov approximation. The calculation results indicated that the height χ (Tm ) and position Tm of the maximum of the longitudinal susceptibility display different behaviors related to the magnetic fields and exchange anisotropic parameters. These behaviors are very different from that in the exchange anisotropic Heisenberg ferromagnet in the magnetic field. The results are: (1) When the field h is weak, in a antiferromagnet, the height χ (Tm ) is a constant χ0 which is independent of field and exchange anisotropy, but the position Tm is only a function of the exchange anisotropy. While in a ferromagnet, both χ (Tm ) and Tm are a function of field and the exchange anisotropy. (2) When the field h is strong, in a antiferromagnet, χ (Tm ) becomes dependent of field and the exchange anisotropy, and χ (Tm ) and Tm are fitted satisfactory to power laws: χ (Tm ) − χ0 ∝ hd and TN − Tm ∝ hc , respectively. Here TN is the Neel temperature. On the contrary, in a ferromagnet, 0 χ(Tm ) and Tm are fitted to power laws: χ (Tm ) ∝ h−d and Tm − Tc ∝ c0 h , where Tc is the Curie temperature. In short, the investigation of χ(Tm ) and Tm in antiferromagnets are both meaningful in academic field and applications. The paper is organized as follows. In Section 2, the formalisms of Green’s function theory are described, and the basic self-consistent equations for the S = 1/2 exchange anisotropic antiferromagnet are obtained. In Section 3, the numerical results of the χ (Tm ) and Tm under weak h are presented. In Section 4, the numerical results of the χ(Tm ) and Tm under strong h are presented. In Section 5, a brief conclusion is given.
The equation of motion for G+− aa (t ) is given by i
+ − d
+ − Sai (t ); Sai Sai (t ), H ; Sai 0 (0) = 2ma · δ(t )δii0 + 0 (0) dt
+ = 2ma · δ(t )δii0 + h Sai (t ); Sai−0 (0) + J · η · ma ·
ρ
− J · mb ·
+ Sbj (t ); Sai−0 (0)
X
+ Sai (t ); Sai−0 (0) .
ρ
(3)
z Here the Tyablikov decoupling [15] was applied as Sai = ma
z
P
and Sbj = mb , and ρ denote a sum over the nearest-neighbor sites. After Fourier transforming the Eq. (3) with respect to the space and time variables, we obtain a algebraic equation that are +− easily for
+ solved the transformed Green’s function Gaa (k, ω) of − Sai (t ); Sai0 (0) , which can be evaluated as: +− +− ωG+− aa (k, ω) = 2ma + hGaa (k, ω) + J · η · ma · γk · Gba (k, ω)
− J · mb · γ0 · G+− aa (k, ω)
(4) * * i k ·ρ
where γ0 = Z is the coordination number and γk = . ρe Hereafter we consider the case of simple cubic lattice, with rk = 2(cos kx + cos ky + cos kz ) and r0 = 6. Similar to the G+− aa (t ), the equation of motion for G+− ( t ) is given by: ba
P
i 2. Method and discussion
+ d
+ − − Sbj (t ); Sai 0 (0) = h Sbj (t ); Sai0 (0)
dt
+ J · η · mb ·
We consider the S = 1/2 three-dimensional Heisenberg antiferromagnet (AFM) model with anisotropic spin exchange. Its Hamiltonian is given by [11]
X
+ Sai (t ); Sai−0 (0)
ρ
− J · ma ·
X η H = J [Saiz · Sbjz + (Sai+ · Sbj− + Sai− · Sbj+ )]
X
ρ
+ (t ); Sai−0 (0) . Sbj
(5)
+ And the transformed Green’s function G+− ba (k, ω) of hhSbj (t );
2
hiji
X
Sai0 (0)ii, which can be evaluated as: −
−h
X i
z Sai −h
X
z Sbj .
(1)
j
Here η denotes the exchange anisotropic parameters with 0 < η < 1, and hiji denotes a sum over the nearest-neighbor sites. J is the exchange coupling constant between neighboring spins and for AFM, J > 0. In order to keep the translational invariance, the AFM lattices are divided into two sublattices A and B, where ai refers to the lattice site i in the sublattice A, so does the bj. It is assumed that the spontaneous magnetization ma of sublattice A is in the +z direction, but mb of sublattice B is in the – z direction. h is the x applied longitudinal magnetic field, which is along the z axis. Sai , y z Sai and Sai represent the three components of the spin S = 1/2 ± operator for a spin at site i in sublattice A. Sai = Saix ± iSaiy are the spin raising and lowing operators, which satisfy the commutation + − relation [Sai , Sbj ] = 2Saiz δij δab and [Sai± , Sbjz ] = ∓Sai± δij δab . In order to calculate the magnetic properties of this model,
we introduce retarded Green’s function Aai (t ); Bbj (0) =
−iθ(t ) Aai (t ), Bbj (0) , where θ (t ) is the step function. The equation of motion for Green’s function follows in a straight forward fashion, which is given by [12–14] i
d
Aai (t ); Bbj (0) = δ(t )δij δab [Aai , Bbj ] dt
+ [Aai (t ), H ]; Bbj (0) .
− J · ma · γ0 · G+− ba (k, ω).
(6)
From Eqs. (4), (6), Gaa (k, ω) can be written as: +−
G+− aa (k, ω) =
2ma
ω1 − ω2
·
ω1 − Ea ω2 − Ea − ω − ω1 ω − ω2
,
(7)
where Ea = h − J · γ0 · ma , Eb = h − J · γ0 · mb ,
ω1,2 =
(Ea + Eb ) ±
p (Ea − Eb )2 + 4ma · mb · (J · η · γk )2 2
.
The correlation function of G+− aa (t ) is found to be
1 1 − + Sai − Saz = − ma . 0 (0)Sai (t ) i0 =i,t =0 = 2 2
With the help of the spectral theorem, we obtain the following expression for the magnetization of sublattice A: 1
1
3 Z
π
Z
π
Z
π
1
= 2π ω1 − ω2 −π −π −π ω1 × (ω1 − Ea ) · coth
2ma (2)
Now, we choose different operators A and B to form Green functions as follow:
+ − + − G+− G+− aa (t ) = Sai (t ); Sai0 (0) , ba (t ) = Sbj (t ); Sai0 (0) , DD EE DD EE + + G+− Sbj (t ); Sbj−0 (0) and G+− Sai (t ); Sbj−0 (0) . bb (t ) = ab (t ) =
+− +− ωG+− ba (k, ω) = hGba (k, ω) + J · η · mb · γk · Gaa (k, ω)
2kB T
− (ω2 − Ea ) · coth
ω2 2kB T
*
dk.
(8)
Following the same way, we obtain the following expression for the magnetization of sublattice B:
Y. Xiang et al. / Solid State Communications 148 (2008) 369–373
371
Fig. 1. Longitudinal susceptibilities χ (T , h) of AFM and FM vs. magnetic field h and temperature T with the same exchange anisotropy η = 0.5.
Fig. 3. The magnetic field h dependence on Tm of AFM and FM with different exchange anisotropy η.
Fig. 2. Longitudinal susceptibilities χ(T , h) of AFM and FM vs. exchange anisotropy η and temperature T under the same magnetic field h = 0.05.
Fig. 4. The magnetic field h dependence of m(Tm ) of AFM and FM with the different exchange anisotropy η.
3 Z π Z π Z π 1 1 = 2mb 2π ω1 − ω2 −π −π −π ω1 × (ω1 − Eb ) · coth
For the field h fixed, in the AFM, χ (Tm ) is found to be independent of the exchange anisotropy η. But its Tm shifts and gets smaller with the increasing η. However, in the FM, as η increases, χ (Tm ) increases, and Tm shifts and decreases, as shown in Figs. 2 and 3. In the end, when h → 0 and η → 1, the Tm of AFM will approach the Neel temperature TN ≈ 0.99, and the Tm of FM will approach the Curie temperature Tc ≈ 1.98. The magnetic field h dependence of the magnetization m(Tm ) of AFM and FM are plotted in Fig. 4 with different exchange anisotropy η. In AFM, the magnetization m(Tm ) at temperature Tm is linear with h, and is independence of η. The slope of the line is found to be χ0 . But in FM, m(Tm ) is nonlinear with h and η. Let us consider the Eqs. (8) and (9), when under the condition h → 0 and T → TN , where TN is the Neel temperature. By using ω1 → 0, ω2T2 → 0 and coth x|x→0 ≈ 1x , the Eq. (8) can be written 2T as:
1
2kB T
− (ω2 − Eb ) · coth
ω2 2kB T
*
dk.
(9)
From Eqs. (8) and (9), we can obtain the magnetization of AFM m = 12 (ma + mb ) and the longitudinal susceptibility χ = ∂∂mh . In the following, for the convenience we let J = kB = 1. When under different conditions h and η, the longitudinal susceptibility χ possess maximum values χ (Tm ) at the temperature location Tm , and the effects of h and η on the χ (Tm ) and Tm are described as follow. 3. Under weak magnetic field conditions When the field h is weak, in antiferromagnets, the height χ (Tm ) is a constant χ0 which is independent of field and exchange anisotropy, but the position Tm is only a function of the exchange anisotropy. While in ferromagnets, both χ (Tm ) and Tm are a function of field and the exchange anisotropy. These different behaviors between antiferromagnet (AFM) and ferromagnet (FM) are shown in Figs. 1–4. If the exchange anisotropy is given, Fig. 1 shows that in AFM, the height χ (Tm ) and position Tm of the longitudinal susceptibilities χ are independent of magnetic field h. Whereas in FM, as h increases, χ(Tm ) always decreases, but Tm increases.
h ma
=
1
3 Z
2π
π −π
Z
π −π
Z
π
4 · h · T · Ea
ω1 · ω2
−π
*
dk.
Then the following equations can be obtained: 4 · T · χa · (1 − γ0 · χa ) 1
= 1 2π
3 R π R π R π −π
−π
−π
1 1−γ0 ·(χa +χb )+χa ·χb ·(γ02 −η2 ·γk2 )
(10)
*
dk
4 · T · χb · (1 − γ0 · χb )
=
1
Rπ Rπ Rπ
1 3 2π
−π
−π
1
−π 1−γ0 ·(χa +χb )+χa ·χb ·(γ 2 −η2 ·γ 2 ) d k 0
*
k
.
(11)
372
Y. Xiang et al. / Solid State Communications 148 (2008) 369–373
Fig. 6. log–log plot of the magnetic field h dependence on Tm .
Fig. 5. Longitudinal susceptibility χ(h, T ) of AFM vs. the magnetic field h and temperature T .
From the Eqs. (10) and (11), we can have:
γ0 · (χa + χb ) = 1. Therefore, the longitudinal susceptibility maximum of AFM can be written as:
χa + χb
χ(Tm ) = χ0 =
2
=
1 2γ0
≈ 0.083.
(12)
The analytic result of Eq. (12) coincides well with the numerical calculation, and shows that χ (Tm ) is a constant which is independence of h and η. If we introduce the Eq. (12) into (10), we can obtain the position temperature is given by TN = Tm =
γ0 4 · C (η)
1
,
3 Z
Fig. 7. log–log plot of the magnetic field h dependence on χ(Tm ).
π
Z
π
Z
π
1
−π
1 − (η · γk /γ0 )
*
The analytic result of Eq. (13) shows that Tm decreases as the η increases.
Furthermore, our calculation results for the longitudinal susceptibility maximum χ (Tm ) and temperature position Tm of the AFM as a function of h are plotted logarithmically in Figs. 6 and 7, respectively. It is shown that χ (Tm ) and Tm is fitted satisfactory to power laws:
4. Under strong magnetic field conditions
χ (Tm ) − χ0 ∝ hd
When under strong magnetic field h, χ (Tm ) and Tm of AFM are the function of h and η, and their behaviors are far different from that in small h. Fig. 5 shows that in AFM, as the field h increases, χ (Tm ) gets higher, and Tm gets smaller. However, in FM, χ (Tm ) gets lower, and Tm gets bigger. These kinds of behaviors of χ (Tm ) and Tm can be used to investigate the magnetic property of coordination polymers. There exist two problems in this area. The first one is the synthesis of coordination polymers. Under different synthetic conditions, coordination polymers which possess the same chemical composition will display different magnetic properties which can’t be predicted before synthesis. Therefore, a method is needed to identify the different magnetic phases of polymers, such as ferromagnetic, antiferromagnetic and metamagnetic phase. The second problem is the phase transition of polymers. Under different conditions (h, T and η), the phases of polymers are easily changed to other phases, such as changing from the antiferromagnetic phase to the spinflop phase under magnetic field h, or changing from the antiferromagnetic phase to the paramagnetic phase under temperature T . Therefore, the way is needed to find this phase transition. Hence, the investigation of χ (Tm ) and Tm is an easy and reliable solution to overcome these two problems.
and
with C (η) =
2π
−π
−π
2
dk.
(13)
T N − T m ∝ hc ,
(14)
(15)
respectively. As mentioned in the introduction, the power laws in FM have obtained much concern in theory as well as experiment, and the corresponding power laws are expressed as follow: 0
χ ( T m ) ∝ h−d
(16)
and 0
T m − T c ∝ hc ,
(17)
where Tc is the Curie temperature. Comparing the Eqs. (14) and (15) with Eqs. (16) and (17), we can conclude that the power laws in AFM are different from those in FM. However, there is little study on the power laws in AFM. The discussion in this paper is a useful supplement in this field. As for the Tm , Tm > Tc exists in FM, while Tm < TN exists in AFM; As for the χ (Tm ), in FM, the power exponent −d0 < 0 (where d0 > 0), while in AFM, the power exponent d > 0, and term χ (Tm ) − χ0 replaces term χ (Tm ), which is present in FM.
Y. Xiang et al. / Solid State Communications 148 (2008) 369–373
In addition, the power exponents are found to be dependent on the exchange anisotropy. If the values of the exchange anisotropy are η = 0.25, 0.50, 0.75, the power exponents c = 2.036, 2.014, 2.006, and d = 2.013, 2.034, 2.076, respectively. From the above data, the exponents c and d possess opposite tendencies with varying η. Let us consider the analytical results of Eqs. (8) and (9). According to the calculation, when χ (T ) → χ (Tm ), there will be ma ≈ mb , and Ea = Eb = E0 . Applying this condition and the approximation coth x ≈ 1x + 3x , the Eqs. (8) and (9) can be simplified as: 1 m
= 4 · T · E0 · E0
+
3T
1
3 Z
2π
π −π
Z
π −π
Z
*
π
dk
E02 − (η · m · γk )2
−π
.
(18)
373
5. Conclusion The effect of magnetic field h on the longitudinal susceptibility in a spin S = 1/2 exchange anisotropic three-dimensional Heisenberg AFM, is studied by the double-time Green’s function method within Tyablikov approximation. The calculation results indicated that the height χ (Tm ) and position Tm of the maximum of longitudinal susceptibility display different behaviors related to the magnetic fields and exchange anisotropic parameters. These behaviors are very different from that in the exchange anisotropic Heisenberg FM in the magnetic field. Considering that there is currently little research on the power laws of AFM, this paper is a useful supplement in this field. Furthermore, the above theory results can be developed as an experimental method to investigate the magnetic property of coordination polymers, where the χ (Tm ) and Tm can be measured easily and accurately.
Let us introduce the following approximations into Eq. (18)
χ(Tm ) = χ0 + 1χ ,
Tm = TN − 1T ,
where 1χ χ0 = 2γ1 , 1T TN = 0 From Eq. (18) we can have 1≈
TN − 1T TN
+
E0 · m 3TN
Acknowledgements
m(Tm ) = m0 + 1m, γ0
4C (η)
, and 1m m0 = 2γh . 0
.
(19)
In addition, applying the derivative ∂∂h of Eq. (18), we can have, 1
γ
2 0
·
1 m
−
h·χ m2
· C 0 (η) =
m + χ · (h − 2γ0 m)
Rπ Rπ Rπ
6T 2
(20)
*
where C 0 (η) = ( 21π )3 −π −π −π d k [ (1−η·γ1 /γ )2 + (1+η·γ1 /γ )2 ]. k 0 k 0 After straightforward and approximately deduction, we have the analytical results
1T ≈
h2
(21)
12γ0
and
1χ ≈
h2 48 · γ0 · C 0 (η) · TN2
(22)
which yield the power laws TN − Tm ∝ hc , χ (Tm ) − χ0 ∝ hd , with the corresponding exponent c = 2.0, d = 2.0.
This work was supported by the National Natural Science Foundation of China (10674033) and the Science and Technology Program of Guangdong Province of China (2007B010600060). References [1] V. Markovich, E. Rozenberg, G. Gorodetsky, B. Revsin, J. Pelleg, I. Felner, Phys. Rev. B 62 (2000) 14186. [2] I. Junger, D. Ihle, J. Richter, A. KlÄumper, Phys. Rev. B 70 (2004) 104419. [3] J. Sznajd, Phys. Rev. B 64 (2001) 052401. [4] Ai-Yuan Hu, Yuan Chen, Li-Jun Peng, Physica B 393 (2007) 368. [5] Ai-Yuan Hu, Yuan Chen, Li-Jun Peng, J. Magn. Magn. Mater. 313 (2007) 366. [6] L. Siurakshina, D. Ihle, Phys. Rev. B 61 (2000) 14601. [7] Michelle L. Taliaferro, Fernando Palaciob, Joel S. Miller, J. Mater. Chem. 16 (2006) 2677. [8] Hong-Peng Jia, Wei Li, Zhan-Feng Ju, Jie Zhang, Dalton Trans. (2007) 3699. [9] Souhei Kaneko, Yoshihide Tsunobuchi, Shunsuke Sakurai, Shin-ichi Ohkoshi, Chem. Phys. Lett. 446 (2007) 292. [10] Long Jiang, Xiao-Long Feng, Tong-Bu Lu, Song Gao, Inorg. Chem. 45 (2006) 5018. [11] Huai-Yu Wang, Phys. Rev. B 69 (2004) 174431. [12] Jian-Hua Cai, et al., The Green’s Function Theory of Quantum statistics, Science Press, Beijing, 1982. [13] Norberto Majlis, The Quantum Theory of Magnetism, World Scientific Publishing Co. Pte. Td, Singapore, 2000. [14] Huai-Yu Wang, The Green’s Function Theory in Condensed Matter Physics, Science Press, Beijing, 2008. [15] S.V. Tyablikov, Method in Quantum Theory of Magnetism, Plenum, New York, 1967.