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Quantum tunneling of a large spin via exchange interaction in single-molecule magnets: a perturbative approach
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 272–276 (2004) 1104–1105
Quantum tunneling of a large spin via exchange interaction in single-molecule magnets: a perturbative approach Gwang-Hee Kim* Department of Physics, Sejong University, Seoul 143-747, Republic of Korea
Abstract The level splittings in two coupled single-molecule magnets with anisotropic exchange interaction are computed from the perturbative results. It is found that the anisotropic exchange interaction plays an important role in the level splittings and shifts of the resonant field. The results are discussed in comparison with a recent experiment. r 2003 Elsevier B.V. All rights reserved. PACS: 75.45.+j; 75.50.Tt Keywords: Macroscopic quantum tunneling; Molecular magnet
Single-molecule magnets (SMMs) provide exciting opportunities for observing quantum phenomena in nanosize magnets [1]. Many efforts have been made to understand their mechanism by considering crystals of SMMs as consisting of giant spins interacting with environmental degree of freedom. Until now, most of the study has neglected intermolecular exchange interaction. Recently, however, it has been reported [2,3] that in many SMMs exchange interactions between SMMs lead to a significant influence on the quantum properties of SMMs. It is therefore very interesting to understand the effect of the exchange interaction on magnetization tunneling. In this work, we deal with anisotropic exchange interaction in the giant-spin Hamiltonian and calculate the tunnel splitting of the level crossing based on the perturbation method. Let us consider the spin Hamiltonian describing two coupled SMMs H¼
2 X i¼1
Hi þ
X
Ja S# 1a S# 2a ;
ð1Þ
a¼x;y;z
where Hi ¼ DS# 2iz þ Htrans Hz S#iz ; D is the anisotroi py constant and Htrans includes the transverse anisoi tropy or field. The two SMMs are coupled by *Corresponding author: Tel.: +82-2-3408-3211; fax: +82-2499-5620. E-mail address: [email protected] (G.-H. Kim).
anisotropic exchange interaction Jþ ¼ ðJx þ Jy Þ=4; Jþþ ¼ ðJx Jy Þ=4: The system has ð2S1 þ 1Þð2S2 þ 1Þ degenerate energy levels which in the absence of the transverse terms of Eq. (1) are labeled by the spin projection M1 and M2 on the z-axis and given by ð0Þ EM ¼ DðM12 þ M22 Þ þ JM1 M2 : It can be easily 1 ;M2 ð0Þ checked P that for the longitudinal field Hzð0Þ ¼ ðEM 0 ;M 0 1 2 ð0Þ 2 0 EM1 ;M2 Þ= i¼1 ðMi Mi Þ; the energy levels are degenerate as EM10 ;M20 ¼ EM1 ;M2 : The corresponding energy levels are obtained for the resonant fields (Hzð0Þ ) in Fig. 1. Tunneling among the ð2S1 þ 1Þð2S2 þ 1Þ energy states is allowed by the transverse terms containing S#xi and S# yi : In the case of small transverse terms which is relevant for the dimer, the level splittings can be calculated in a more direct way using the high-order perturbation method. We have selected 13 level crossings (Fig. 1 and Table 1) which are divided into different sources of tunnel splittings. The level splittings in transitions 8 and 9 are induced by the transverse anisotropy in the xy-plane [BðS# 2xi S#2yi Þ]. The level splittings in transitions 1 and 12 are induced by both the transverse anisotropy term and the transverse field ½Hx ðS# x1 þ S# x2 Þ: These transitions are similar to those of the single-spin model with the difference that the exchange interaction leads to shifts of the resonant fields [4]. However, in order to observe the tunnel transitions in other cases, we need to include the anisotropic exchange term in the Hamiltonian (1). As an
0304-8853/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2003.12.755
ARTICLE IN PRESS G.-H. Kim / Journal of Magnetism and Magnetic Materials 272–276 (2004) 1104–1105
Table 1 The level splittings ðDEÞ of 13 tunnel transitions in Fig. 1 ðB ¼ 0:033 K [5] and Hx BJþþ BJþ B0:01 K for illustration), the resonant field ðHzð0Þ Þ; and the physical origins which induce level splittings
-16
11 4
2
-18
3
5
-20
Energy (K)
1 2 3 4 5 6 7 8 9 10 11 12 13
9
6
-22
7
10
-24
-26
-28
1
-30 0.0
12
8
0.2
0.4
0.6
0.8
1.0
1105
Hzð0Þ ðTÞ
DE ðKÞ
Sources
0:0336 0.0999 0.180 0.252 0.343 0.378 0.457 0.504 0.571 0.682 0.982 1.04 1.19
1:37 108 2:79 106 1:63 104 1:01 106 9:13 107 1:15 106 8:54 107 2:09 105 2:09 105 8:27 103 6:88 107 2:45 106 1:43 106
B; Hx Jþþ ; B; Jþ ; B; Jþ ; B Jþþ ; B Jþ ; B Jþþ ; B B B Jþ ; B Jþþ ; B; B; Hx Jþ ; B;
Hx Hx
Hx Hx
13
1.2
1.4
H z(T)
Fig. 1. Energy level of two coupled spins S ¼ 92 at Hzð0Þ with D ¼ 0:72 K; and Jz ¼ 0:01 K in low lying energy states of Mn4 : The numbers, labeled from 1 to 13, indicate the transitions claimed as the observed tunnel resonances in Ref. [3], where 1: ð9=2; 9=2Þ3ð9=2; 9=2Þ; 2: ð9=2; 5=2Þ3ð7=2; 7=2Þ; 3: ð5=2; 9=2Þ3ð7=2; 7=2Þ; 4: ð9=2; 5=2Þ3ð9=2; 3=2Þ; 5: ð9=2; 7=2Þ3ð9=2; 5=2Þ; 6: ð9=2; 7=2Þ3ð9=2; 5=2Þ; 7: ð9=2; 9=2Þ3ð7=2; 9=2Þ; 8: ð9=2; 9=2Þ3ð7=2; 9=2Þ; 9: ð9=2; 9=2Þ3ð9=2; 7=2Þ; 10: ð7=2; 9=2Þ3ð7=2; 7=2Þ; 11: ð9=2; 9=2Þ3ð7=2; 7=2Þ; 12: ð9=2; 9=2Þ3ð5=2; 9=2Þ; 13: ð9=2; 9=2Þ3ð7=2; 7=2Þ: For clarity, degenerate states such as ðM1 ; M2 Þ and ðM2 ; M1 Þ are not both listed.
example, the level splitting of the degenerate level pair ð92; 92Þ and ð72; 92Þ denoted by a number 7 in Table 1 is estimated from the perturbative result given by ðM2 M20 1Þ=2 M 2 1 X B 0 DEM1 ;M2 ;M10 ;M20 ¼ 2Jþþ g1 4D k¼M20 G½ðk þ aÞ=2G½ðk þ b þ 1Þ=2 f ðkÞ ; G½ðM20 þ aÞ=2G½ðM2 þ bÞ=2 ð2Þ
where g1 ¼ ½½ðS þ M2 Þ!ðS M20 Þ!=ðS M2 Þ!ðS þ M20 Þ! ðS þ M10 þ 1ÞðS M10 Þ1=2 ; f ðkÞ ¼ 1=½ðM2 k 1Þ!!ðk M20 Þ!!; a ¼ M20P þ ðHzð0Þ Jz M10 Þ=D; b ¼ M2 þ ðHzð0Þ 0 Jz M1 Þ=D; and increases in steps of 2. Other level splittings illustrated in Table 1 have been found by using the perturbation method and will be discussed elsewhere. This work was supported by Korea Research Foundation Grant (KRF-2002-041-C00104).
References [1] J.R. Friedman, et al., Phys. Rev. Lett. 76 (1996) 3830; L. Thomas, et al., Nature 383 (1996) 145; C. Sangregorio, et al., Phys. Rev. Lett. 78 (1997) 4645; R. Caciuffo, et al., Phys. Rev. Lett. 81 (1998) 4744; S.M. Aubin, et al., J. Am. Chem. Soc. 120 (1998) 4991. [2] W. Wernsdorfer, et al., Nature 416 (2002) 406. [3] W. Wernsdorfer, et al., Phys. Rev. Lett. 89 (2002) 197201. [4] G.-H. Kim, Phys. Rev. B 67 (2003) 024421. [5] W. Wernsdorfer, et al., Phys. Rev. B 65 (2002) 180403.
Journal of Magnetism and Magnetic Materials 272–276 (2004) 1104–1105
Quantum tunneling of a large spin via exchange interaction in single-molecule magnets: a perturbative approach Gwang-Hee Kim* Department of Physics, Sejong University, Seoul 143-747, Republic of Korea
Abstract The level splittings in two coupled single-molecule magnets with anisotropic exchange interaction are computed from the perturbative results. It is found that the anisotropic exchange interaction plays an important role in the level splittings and shifts of the resonant field. The results are discussed in comparison with a recent experiment. r 2003 Elsevier B.V. All rights reserved. PACS: 75.45.+j; 75.50.Tt Keywords: Macroscopic quantum tunneling; Molecular magnet
Single-molecule magnets (SMMs) provide exciting opportunities for observing quantum phenomena in nanosize magnets [1]. Many efforts have been made to understand their mechanism by considering crystals of SMMs as consisting of giant spins interacting with environmental degree of freedom. Until now, most of the study has neglected intermolecular exchange interaction. Recently, however, it has been reported [2,3] that in many SMMs exchange interactions between SMMs lead to a significant influence on the quantum properties of SMMs. It is therefore very interesting to understand the effect of the exchange interaction on magnetization tunneling. In this work, we deal with anisotropic exchange interaction in the giant-spin Hamiltonian and calculate the tunnel splitting of the level crossing based on the perturbation method. Let us consider the spin Hamiltonian describing two coupled SMMs H¼
2 X i¼1
Hi þ
X
Ja S# 1a S# 2a ;
ð1Þ
a¼x;y;z
where Hi ¼ DS# 2iz þ Htrans Hz S#iz ; D is the anisotroi py constant and Htrans includes the transverse anisoi tropy or field. The two SMMs are coupled by *Corresponding author: Tel.: +82-2-3408-3211; fax: +82-2499-5620. E-mail address: [email protected] (G.-H. Kim).
anisotropic exchange interaction Jþ ¼ ðJx þ Jy Þ=4; Jþþ ¼ ðJx Jy Þ=4: The system has ð2S1 þ 1Þð2S2 þ 1Þ degenerate energy levels which in the absence of the transverse terms of Eq. (1) are labeled by the spin projection M1 and M2 on the z-axis and given by ð0Þ EM ¼ DðM12 þ M22 Þ þ JM1 M2 : It can be easily 1 ;M2 ð0Þ checked P that for the longitudinal field Hzð0Þ ¼ ðEM 0 ;M 0 1 2 ð0Þ 2 0 EM1 ;M2 Þ= i¼1 ðMi Mi Þ; the energy levels are degenerate as EM10 ;M20 ¼ EM1 ;M2 : The corresponding energy levels are obtained for the resonant fields (Hzð0Þ ) in Fig. 1. Tunneling among the ð2S1 þ 1Þð2S2 þ 1Þ energy states is allowed by the transverse terms containing S#xi and S# yi : In the case of small transverse terms which is relevant for the dimer, the level splittings can be calculated in a more direct way using the high-order perturbation method. We have selected 13 level crossings (Fig. 1 and Table 1) which are divided into different sources of tunnel splittings. The level splittings in transitions 8 and 9 are induced by the transverse anisotropy in the xy-plane [BðS# 2xi S#2yi Þ]. The level splittings in transitions 1 and 12 are induced by both the transverse anisotropy term and the transverse field ½Hx ðS# x1 þ S# x2 Þ: These transitions are similar to those of the single-spin model with the difference that the exchange interaction leads to shifts of the resonant fields [4]. However, in order to observe the tunnel transitions in other cases, we need to include the anisotropic exchange term in the Hamiltonian (1). As an
0304-8853/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2003.12.755
ARTICLE IN PRESS G.-H. Kim / Journal of Magnetism and Magnetic Materials 272–276 (2004) 1104–1105
Table 1 The level splittings ðDEÞ of 13 tunnel transitions in Fig. 1 ðB ¼ 0:033 K [5] and Hx BJþþ BJþ B0:01 K for illustration), the resonant field ðHzð0Þ Þ; and the physical origins which induce level splittings
-16
11 4
2
-18
3
5
-20
Energy (K)
1 2 3 4 5 6 7 8 9 10 11 12 13
9
6
-22
7
10
-24
-26
-28
1
-30 0.0
12
8
0.2
0.4
0.6
0.8
1.0
1105
Hzð0Þ ðTÞ
DE ðKÞ
Sources
0:0336 0.0999 0.180 0.252 0.343 0.378 0.457 0.504 0.571 0.682 0.982 1.04 1.19
1:37 108 2:79 106 1:63 104 1:01 106 9:13 107 1:15 106 8:54 107 2:09 105 2:09 105 8:27 103 6:88 107 2:45 106 1:43 106
B; Hx Jþþ ; B; Jþ ; B; Jþ ; B Jþþ ; B Jþ ; B Jþþ ; B B B Jþ ; B Jþþ ; B; B; Hx Jþ ; B;
Hx Hx
Hx Hx
13
1.2
1.4
H z(T)
Fig. 1. Energy level of two coupled spins S ¼ 92 at Hzð0Þ with D ¼ 0:72 K; and Jz ¼ 0:01 K in low lying energy states of Mn4 : The numbers, labeled from 1 to 13, indicate the transitions claimed as the observed tunnel resonances in Ref. [3], where 1: ð9=2; 9=2Þ3ð9=2; 9=2Þ; 2: ð9=2; 5=2Þ3ð7=2; 7=2Þ; 3: ð5=2; 9=2Þ3ð7=2; 7=2Þ; 4: ð9=2; 5=2Þ3ð9=2; 3=2Þ; 5: ð9=2; 7=2Þ3ð9=2; 5=2Þ; 6: ð9=2; 7=2Þ3ð9=2; 5=2Þ; 7: ð9=2; 9=2Þ3ð7=2; 9=2Þ; 8: ð9=2; 9=2Þ3ð7=2; 9=2Þ; 9: ð9=2; 9=2Þ3ð9=2; 7=2Þ; 10: ð7=2; 9=2Þ3ð7=2; 7=2Þ; 11: ð9=2; 9=2Þ3ð7=2; 7=2Þ; 12: ð9=2; 9=2Þ3ð5=2; 9=2Þ; 13: ð9=2; 9=2Þ3ð7=2; 7=2Þ: For clarity, degenerate states such as ðM1 ; M2 Þ and ðM2 ; M1 Þ are not both listed.
example, the level splitting of the degenerate level pair ð92; 92Þ and ð72; 92Þ denoted by a number 7 in Table 1 is estimated from the perturbative result given by ðM2 M20 1Þ=2 M 2 1 X B 0 DEM1 ;M2 ;M10 ;M20 ¼ 2Jþþ g1 4D k¼M20 G½ðk þ aÞ=2G½ðk þ b þ 1Þ=2 f ðkÞ ; G½ðM20 þ aÞ=2G½ðM2 þ bÞ=2 ð2Þ
where g1 ¼ ½½ðS þ M2 Þ!ðS M20 Þ!=ðS M2 Þ!ðS þ M20 Þ! ðS þ M10 þ 1ÞðS M10 Þ1=2 ; f ðkÞ ¼ 1=½ðM2 k 1Þ!!ðk M20 Þ!!; a ¼ M20P þ ðHzð0Þ Jz M10 Þ=D; b ¼ M2 þ ðHzð0Þ 0 Jz M1 Þ=D; and increases in steps of 2. Other level splittings illustrated in Table 1 have been found by using the perturbation method and will be discussed elsewhere. This work was supported by Korea Research Foundation Grant (KRF-2002-041-C00104).
References [1] J.R. Friedman, et al., Phys. Rev. Lett. 76 (1996) 3830; L. Thomas, et al., Nature 383 (1996) 145; C. Sangregorio, et al., Phys. Rev. Lett. 78 (1997) 4645; R. Caciuffo, et al., Phys. Rev. Lett. 81 (1998) 4744; S.M. Aubin, et al., J. Am. Chem. Soc. 120 (1998) 4991. [2] W. Wernsdorfer, et al., Nature 416 (2002) 406. [3] W. Wernsdorfer, et al., Phys. Rev. Lett. 89 (2002) 197201. [4] G.-H. Kim, Phys. Rev. B 67 (2003) 024421. [5] W. Wernsdorfer, et al., Phys. Rev. B 65 (2002) 180403.