- Email: [email protected]
Surface effect of magnetization of nanosized magnetic clusters
12 August 2002
Physics Letters A 300 (2002) 641–647 www.elsevier.com/locate/pla
Surface effect of magnetization of nanosized magnetic clusters Huang Zhigao a,b,∗ , Du Youwei a a National Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China b Department of Physics, Fujian Teachers University, Fuzhou 350007, China
Received 15 April 2002; accepted 26 June 2002 Communicated by J. Flouquet
Abstract Based on the Monte Carlo method, magnetizations and spin-wave excitations at the surface, in the core for the clusters with different surface interaction, surface thickness and atom number are studied. It is found that, for the pure cluster, the Bloch T 3/2 law is well satisfied at low temperature (T < 0.5TC ), Bs = 3Bbulk and B increases drastically with the reducing atom number N; the magnetization and spin-wave excitations at the surfaces coated different exchange interaction are quite different from ones in the core. The simulated results are consistent with experimental facts, and are well discussed by the Bloch exponent law in the approximate crystalline approximation. 2002 Elsevier Science B.V. All rights reserved. PACS: 75.50.-m; 75.40.Mg; 75.70.-i Keywords: Magnetization; Cluster; Spin-wave
1. Introduction Magnetic properties of small clusters, granular films and polycrystalline containing magnetic clusters and nanoparticles are of great current interests due to the finding of new phenomena and the potential applications in high density memory devices. Recently, a large tunneling magnetoresistance (MR) ratio has been observed in tunnel-type nanostructures, such as, in Co/AI2 O3 /CoFe tunnel junctions [1], in polycrystalline Zn0.41 Fe2.59 O4 [2], in half-metallic Fe3 O4 [3]. It is thought that such large MR results from the interplay of the relative orientation of magnetic moments * Corresponding author.
E-mail address: [email protected] (H. Zhigao).
between clusters, the intrinsic characters of clusters [4–8]. Especially, the grain boundary and surface can strongly influence the magnetic properties of these materials. Surface spin disorder and interface magnetic anisotropy has been measured and considered [9–12]. Magnetization of surface and nanoparticle has become interested topics in theory and experiment for a long time [13–17]. However, up to now, the difference of the Bloch constant and Bloch exponent in M(T ) = M(0)(1 − BT b ) obtained from different theory and experiment still exists. The magnetic properties of small particles have been studied using the Monte Carlo method [18–20], but the influence of complex surface on magnetic properties do not still considered. In this Letter, based on above opening questions, magnetizations and spin-wave excitations at the surface, in the core for the clusters with different
0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 2 ) 0 0 8 8 7 - 3
642
H. Zhigao, D. Youwei / Physics Letters A 300 (2002) 641–647
surface interaction, surface thickness and atom number are studied using the Monte Carlo method.
2. Model and simulation technique The total spin Hamiltonian Hs of the cluster is described as 2 Hs = − Si · u i Jij Si · Sj − Ks ij
−H
Ri Rc
Siz ,
(1)
Fig. 1. The outermost incomplete filled shells and the coordination number for FCC spherical cluster containing 2123 magnetic atoms.
i
where ij is performed over the spin pairs at nearest-neighbor (NN) sites i and j with exchange interaction Jij , Ks surface anisotropy (here let Ks = 0), ui the unit vector of easy axis at the site i. Surface and bulk anisotropy also have a influence on the configurations of the cluster, which will be studied elsewhere. At first, the pictures of the shell, surface and core of the cluster are introduced. For instance, we consider a FCC spherical cluster containing 2123 magnetic atoms (N = 2123) with 48 shells. Fig. 1 shows the outermost incomplete filled shells and the coordination number. The number under every square means one of atom with the coordination number Z. For example, in 48th shell, there are both the coordination structures. First one, there are 6 atoms with Z = 4; second one, there are 24 atoms with Z = 6. From Fig. 1, it is seen that the coordination structure is complex. Now, the cluster is divided into the surface and core. The boundary between the surface and the core is a shell or a radius. For example, as seen in Fig. 1, 39th shell is defined as the boundary, and then atom layer from 39th to 49th shell is named the surface, while one from first to 39th is defined as the core. The exchange interaction Jij in Eq. (1) with the site i and j or one of them locating in the core is taken as J . Here, J is used as unit of temperature and energy, and let J = 1. While Jij with both i and j sites belong to the surface is taken as J N . Positive and negative J N means the ferromagnetic and antiferromagnetic coupling in the surface, respectively. Free boundary conditions are applied in all directions. In our study, three series are considered. First one, N = 2123, the outermost 9 shells with different magnitude of J N (J N = −1, −0.5, −0.1, 1) are treated as the surface. Second one, N = 2123,
the different thickness of surface layer corresponding to different ratio p with the same value of J N (J N = −0.1) is considered, where p is defined as the ratio of atom number at the surface to total one; third one, different atom number of the pure cluster where the interaction in the core and surface is same (J N = J = 1) is studied. Now, let us consider to simulate spin configurations of FCC magnetic clusters using standard Monte Carlo–Metropolis (MC) [21]. The principle of this algorithm is that: for an N spin system, every spin in the lattice has an orientation, which forms a spin configuration. According to Eq. (1), the magnetic energy of the system corresponding to some configuration Ei is computed. Updating the spin configuration in visiting atomic site randomly, a new configuration energy Ej is produced. Thus, the difference of energy between both old and new configurations is ∆E = Ej −Ei . If ∆E < 0, the new configuration is accepted. If ∆E > 0, the new configuration is accepted with the probability of a successful spin flip exp(−∆E/KB T ). The first 104 MC steps per spin were discarded for equilibrium and thermal averages were made with next 104 MC steps. In thermal average process, we started to store the simulated parameter values separated by 20 MC steps to break correlation between successive configurations.
3. Spin-wave excitation It is well known that magnetization M(T ) at low temperature decrease as Bloch’s T 3/2 [9,22–25]: M(T ) = M(0) 1 − B T 3/2 . (2)
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Ferromagnetic magnons have a quadratic dispersion relation at small wave vector q, h¯ ω(q) = Dq 2 , where D is the spin-wave stiffness constant. The coefficients B is related to D through the expression B = 2.612v0(kB /4πD)3/2 ; here g is the Lande factor, µB is the Bohr magneton, kB is the Boltzmann constant. The spin-wave stiffness constant is, in the approximate crystalline approximation, expressed as follows [24]: D = D(0) 1 − 2Z(∆a/a)δ , (3) where D(0) = ZSJ a 2 /3, a is lattice constant, J exchange constant, S atom spin, v0 atom volume, ∆a2 is the mean-square deviation of NN pair distance and δ is the structure fluctuation or exchange interaction fluctuation. From Eqs. (2), (3) we can derive
kB T 3/2 M(T ) = M(0) 1 − B J = M(0) 1 − BT 3/2 (4) and 3/2 −3/2 Z 1 − 2Z(∆a/a)δ B = 2.612v0 3/4πSa 2 −3/2 ∝ [Z]−3/2 1 − 2Z(∆a/a)δ , (5) where T = kB T /J , which is the reduced temperature corresponding to our Monte Carlo simulation. Eq. (4) has been widely used to study the spin-wave excitation of amorphous alloys [24,25]. In Eq. (5), the first term means the influence of the coordination number on B, second one does the role of the structure and exchange fluctuations on B. It is thought that Eq. (4) is also applicable to the cluster. The disordered spins [9,10], decreasing of the coordination and the deviation of NN pair distance at the surface of cluster, will make the magnitude of B change. For the cluster, another Bloch exponent law M(T ) = M(0)(1 − BT b ) is also often used to discuss the dependence of magnetization. Here b is the Bloch exponent.
4. Numerical results and discussion 4.1. The influence of exchange interaction at the surface on magnetization Now, we consider first series where N = 2123, and the outermost 9 shells (from 39th to 48th shell) with
Fig. 2. MC simulated and fitting magnetization as a function of temperature with J N = −1, −0.5, −0.1, 1: (a) total cluster, M; (b) at the surface, Ms; (c) in the core, Mb.
different magnitude of J N (J N = −1, −0.5, −0.1, 1) are treated as the surface. Fig. 2 shows the temperature dependencies of total, surface and core magnetizations (M, Ms, Mb) with MC simulation and fitting curves. From Fig. 2, it is found that the Bloch T 3/2 law is well satisfied at low temperature (T < 0.5TC ) for M and Mb except for M with J N = −0.1. A good linear relation is found in M vs. T curve with J N = −0.1, which responds to b = 1. However, for high temperature region (0.3TC < T < 0.9TC ), a better T 2 relation is fit in M(T ) with J N = −1, −0.5, 1.
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Now, we focus on the temperature dependence of Ms in the surface. From Fig. 2(b) and fitting data, we can get the fitting Ms(T ) expressions as follows: Ms(T ) = 0.98678 − 0.0468T 3/2 for J N = 1; Ms(T ) = −0.2597 + 1.112 exp(−T /4.471) for J N = −0.1; Ms(T ) = 0.4323 − 0.07691T for J N = −0.5; Ms(T ) = 0.2220 − 0.0353T for J N = −1. Different magnetization character in coated surface layers leads to the changes of magnetization behaviors of the clusters and produces different b value, which is consistent with the fact that the Bloch constant b decreases from 1.52 to 0.37 with decreasing size of iron crystalline coated Mg–F2 obtained by Zhang et al. [14]. Based on spin-wave excitation, Mills et al. predicted that the surface and bulk magnetization decrease with temperature according to the same T 3/2 law, but the Bloch constant of the surface is twice that of the bulk, Bs = 2Bbulk [26–28]. For J = J N = 1 (pure cluster), we get Bs = 0.0468, Bb = 0.0245, Bbulk = 0.0147. It is obtained that Bs = 2Bb , Bs = 3Bbulk , b = 3/2, which is accordant with one obtained experimentally by Pierce et al. for the magnetization of macroscopic surface [13,15], and the prediction of Mills et al. After coating the surface layer with antiferromagnetic coupling, the magnetic states at the surface become complex due to interplay of antiferromagnetic coupling and exchange interaction from the core. As J N = −0.1, the temperature dependence of Ms(T ) obeys exponent law. But for J N = −0.5, −1, a linear decrease of Ms(T ) exists, which is accordant with the quasilinear temperature dependence of the surface magnetization predicted by Rado [17]. Different coating layer influences obviously the magnetic state in the core as seen in Fig. 2(c). Although their M vs. T curves in the core all satisfy T 3/2 law, the Bloch constant decreases linearly with increasing of J N as shown in Fig. 3. This surface effect has significant for magnetoresistance effect in nanostructured materials [6]. 4.2. The influence of exchange interaction in the surface on magnetization The number of atom of cluster and the exchange interaction J N in the surface remain 2123 and −0.1, respectively. But, the ratio p of atom number at the surface to total is changed. Fig. 4 shows six typical sectional spin configurations at T = 0.01 and 3.31 with p = 0, 0.5, 0.8. Figs. 5 and 6 show magnetiza-
Fig. 3. The Bloch constant in the core as a function of J N with N = 2123.
Fig. 4. Six typical sectional spin configurations at T = 0.01 and 3.31 with p = 0, 0.5, 0.8.
tion as a function of P at T = 0.01 and T = 3.31 with H = 0, 2.0, respectively. It is clearly seen that: (1) the disorder degree in the surface increases with increasing p, which responds to the drastic decrease of magnetization; (2) at low temperature spins in the core retain well parallel array along initial direction (z axis), which responds to magnetization to be 1 constantly; (3) at higher temperature (T < TC ), spin configuration with p = 0 also remains basely order parallel array, but the spin direction has deviated from z axis. At the same time, there is still an interested phenomenon that under the external field the magnetization in the core decreases contrary. The above both phenomena may mean the existence of shape anisotropy and surface anisotropy originating from the breaking of symmetry at the surface of cluster [11,29,30], which will be studied further.
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Fig. 5. Magnetization as a function of P at T = 0.01 with H = 0, 2.0. M and M–H —total one; Ms and Ms–H —one at the surface; Mb and Mb–H —one in the core.
Fig. 7. Magnetization as a function of the temperature with N = 55, 683, 2123.
Fig. 6. Magnetization as a function of P at T = 3.31 with H = 0, 2.0. M and M–H —total one; Ms and Ms–H —one at the surface; Mb and Mb–H —one in the core.
Fig. 8. The N dependence of B. The inset presents B as a function of Z −3/2 .
4.3. Magnetization as a function of the size of cluster We study effect of atom number of the pure cluster (J N = J = 1) on magnetization. Fig. 7 shows M vs. T curves with N = 55, 683, 2123. It is found that at low temperature, Bloch T 3/2 law is obeyed well for the clusters with atom number from 55 to 2123. It is evident that below the Curie temperature magnetization increases with the size of cluster, which is similar to results by Merikoski et al. [19]. Using different MC steps to simulate the magnetizations of small clusters, we find that, for no enough MC steps, it is true that the decay with temperature of the magnetization is slowest for the smallest cluster at temperature well above the Curie temperature because of increasing fluctuations. However, for enough MC steps, the fluctuation can be reduced dramatically. Therefore, the temperature dependence of magnetization at tempera-
ture above Curie temperature is different from results by Merikoski et al. In our simulation, 40000 steps are used for small cluster, while only 3000 steps were visited in theirs. Fig. 8 shows the N dependence of B, and the inset presents B as a function of Z −3/2 . From the figure, both important results are obtained: first one, B increases drastically with reducing N , which is consistent with B as a function of iron crystalline size by Zhang [14,31]; second one, B is approximately proportional to Z −3/2 in both different slops. Now, we use Eq. (5) obtained in the approximate crystalline approximation to discuss the influence of the surface exchange interaction and the coordination number Z on B. For the FCC cluster, Z reduces drastically with decreasing N for N < 1000 as seen in Fig. 9. Fig. 9 and Eq. (5) make us understand easily the N and Z −3/2 dependencies of B found in Fig. 8. The change of slop in the inset of Fig. 8 should be explained as
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constant result from variant coordination number, the structure and exchange interaction fluctuation.
Acknowledgements This work was supported by Grant No. G1999064508 for National Key Project for Basic Research of China, and Foundation for University Key Teacher by the Ministry of Education. Fig. 9. The coordination number Z as a function of N for FCC structure spherical cluster.
References follows: the structure and exchange interaction fluctuation (∆a/a)δ increases with decreasing of Z. It is assumed that (∆a/a)δ ∝ 1/(c + Z); here c is constant. Then, we can obtain that the slop of B to Z −3/2 decreases with decreasing of Z. In addition, an obvious exchange fluctuation can be produced in the clusters coated with antiferromagnetic coupling surface layers. Therefore, the thicker the coated layer, the larger the fluctuation, the bigger the magnitude of B, which explain well the result in Fig. 3.
5. Conclusion In conclusion, magnetizations and spin-wave excitations at the surface, in the core for the clusters with different surface interaction, surface thickness and atom number N have been studied. It is found that, for the pure cluster, the Bloch T 3/2 law is well satisfied at low temperature (T < 0.5TC ), but Bloch constant B increases drastically with the reducing atom number N , and Bs = 3Bbulk ; the magnetization and spinwave excitations at the surfaces coated different exchange interaction have complex temperature dependence, which influences strongly the core and cluster magnetic states; the increasing of the thickness of the surface layer with antiferromagnetic interaction make the surface and cluster magnetization reduce fast, but spins in the core remains basely parallel array at T < TC . Compared the simulated results with experimental facts, both are consistent. According to the Bloch exponent law in the approximate crystalline approximation, it is evident that the changes of the Bloch
[1] J.S. Moodera, L.R. Kinder, T.M. Wong, Meservey, Phys. Rev. Lett. 74 (1995) 3273. [2] P. Chen, D.Y. Xing, Y.W. Du, J.M. Zhu, D. Feng, Phys. Rev. Lett. 87 (2001) 107202-1. [3] J.J. Versluijs, M.A. Bari, J.M.D. Coey, Phys. Rev. Lett. 87 (2001) 026601. [4] M. Holdenried, B. Hackenbroich, H. Micklitz, J. Magn. Magn. Mater. 231 (2001) L13. [5] N. Zhang, W. Ding, W. Zhong, D. Xing, Y. Du, Phys. Rev. B 56 (1997) 8138. [6] H. Zhigao, D. Youwei, unpublished. [7] C. Xu, Z.-Y. Li, I.E. Dikshtein, V.G. Shavrov, P.M. Hui, Phys. Lett. A 291 (2001) 325. [8] M. Ziese, Phys. Rev. B 60 (1999) R738. [9] R.H. Kodama, J. Magn. Magn. Mater. 200 (1999) 359. [10] R.H. Kodama, A.E. Berkowitz, Phys. Rev. Lett. 77 (1996) 394; R.H. Kodama, A.E. Berkowitz, J. Appl. Phys. 81 (1997) 5552; R.H. Kodama, A.E. Berkowitz, Phys. Rev. B 59 (1999) 6321. [11] M.M. Deshmukh, S. Kleff, S. Gueron, E. Bonet, A.N. Pasupathy, J. von Delft, D.C. Ralph, Phys. Rev. Lett. 87 (2001) 226801-1. [12] P. Vargas, D. Altbir, J. d’Albuquerque e Castro, J. Magn. Magn. Mater. 226–230 (2001) 603. [13] U. Gradmann, J. Magn. Magn. Mater. 100 (1991) 481. [14] D.J. Zhang, K.J. Klabunde, C.M. Sorensen, G.C. Hadjipanayis, Phys. Rev. B 58 (1998) 14167. [15] D.T. Pierce, R.J. Celotta, J. Unguris, H.C. Siegmann, Phys. Rev. B 26 (1982) 2566. [16] P.V. Hendriksen, S. Linderoth, P.A. Lindgard, Phys. Rev. B 48 (1993) 7259. [17] G.T. Rado, Phys. Rev. B 40 (1989) 407. [18] V. Wildpaner, Z. Phys. 270 (1974) 215. [19] J. Merikoski, J. Timonen, M. Manninen, Phys. Rev. Lett. 66 (1991) 938. [20] F. Aguilera-Granja, J.M. Montejano-Carrizales, J.L. MoranLopez, Phys. Lett. A 242 (1998) 255. [21] K. Binder, D.W. Heermann, Monte Carlo Simulation in Statistical Physics, Springer, Berlin, 1992. [22] C. Kittel, Introduction to Solid State Physics, 6th ed., Wiley, New York, 1986.
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[23] T. Kaneyoshi, Amorphous Magnetism, CRC Press, Boca Raton, 1984. [24] S.N. Kaul, T.V.S.M. Mohan Babu, J. Phys. Condens. Matter 1 (1989) 8509. [25] L. Heng, H. Zhigao, Intermetallics 8 (2000) 655. [26] D.L. Mills, A.A. Maradudin, J. Phys. Chem. Solids 28 (1967) 1855. [27] D.L. Mills, Solid State Phys. 4 (1971) 28.
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Physics Letters A 300 (2002) 641–647 www.elsevier.com/locate/pla
Surface effect of magnetization of nanosized magnetic clusters Huang Zhigao a,b,∗ , Du Youwei a a National Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China b Department of Physics, Fujian Teachers University, Fuzhou 350007, China
Received 15 April 2002; accepted 26 June 2002 Communicated by J. Flouquet
Abstract Based on the Monte Carlo method, magnetizations and spin-wave excitations at the surface, in the core for the clusters with different surface interaction, surface thickness and atom number are studied. It is found that, for the pure cluster, the Bloch T 3/2 law is well satisfied at low temperature (T < 0.5TC ), Bs = 3Bbulk and B increases drastically with the reducing atom number N; the magnetization and spin-wave excitations at the surfaces coated different exchange interaction are quite different from ones in the core. The simulated results are consistent with experimental facts, and are well discussed by the Bloch exponent law in the approximate crystalline approximation. 2002 Elsevier Science B.V. All rights reserved. PACS: 75.50.-m; 75.40.Mg; 75.70.-i Keywords: Magnetization; Cluster; Spin-wave
1. Introduction Magnetic properties of small clusters, granular films and polycrystalline containing magnetic clusters and nanoparticles are of great current interests due to the finding of new phenomena and the potential applications in high density memory devices. Recently, a large tunneling magnetoresistance (MR) ratio has been observed in tunnel-type nanostructures, such as, in Co/AI2 O3 /CoFe tunnel junctions [1], in polycrystalline Zn0.41 Fe2.59 O4 [2], in half-metallic Fe3 O4 [3]. It is thought that such large MR results from the interplay of the relative orientation of magnetic moments * Corresponding author.
E-mail address: [email protected] (H. Zhigao).
between clusters, the intrinsic characters of clusters [4–8]. Especially, the grain boundary and surface can strongly influence the magnetic properties of these materials. Surface spin disorder and interface magnetic anisotropy has been measured and considered [9–12]. Magnetization of surface and nanoparticle has become interested topics in theory and experiment for a long time [13–17]. However, up to now, the difference of the Bloch constant and Bloch exponent in M(T ) = M(0)(1 − BT b ) obtained from different theory and experiment still exists. The magnetic properties of small particles have been studied using the Monte Carlo method [18–20], but the influence of complex surface on magnetic properties do not still considered. In this Letter, based on above opening questions, magnetizations and spin-wave excitations at the surface, in the core for the clusters with different
0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 2 ) 0 0 8 8 7 - 3
642
H. Zhigao, D. Youwei / Physics Letters A 300 (2002) 641–647
surface interaction, surface thickness and atom number are studied using the Monte Carlo method.
2. Model and simulation technique The total spin Hamiltonian Hs of the cluster is described as 2 Hs = − Si · u i Jij Si · Sj − Ks ij
−H
Ri Rc
Siz ,
(1)
Fig. 1. The outermost incomplete filled shells and the coordination number for FCC spherical cluster containing 2123 magnetic atoms.
i
where ij is performed over the spin pairs at nearest-neighbor (NN) sites i and j with exchange interaction Jij , Ks surface anisotropy (here let Ks = 0), ui the unit vector of easy axis at the site i. Surface and bulk anisotropy also have a influence on the configurations of the cluster, which will be studied elsewhere. At first, the pictures of the shell, surface and core of the cluster are introduced. For instance, we consider a FCC spherical cluster containing 2123 magnetic atoms (N = 2123) with 48 shells. Fig. 1 shows the outermost incomplete filled shells and the coordination number. The number under every square means one of atom with the coordination number Z. For example, in 48th shell, there are both the coordination structures. First one, there are 6 atoms with Z = 4; second one, there are 24 atoms with Z = 6. From Fig. 1, it is seen that the coordination structure is complex. Now, the cluster is divided into the surface and core. The boundary between the surface and the core is a shell or a radius. For example, as seen in Fig. 1, 39th shell is defined as the boundary, and then atom layer from 39th to 49th shell is named the surface, while one from first to 39th is defined as the core. The exchange interaction Jij in Eq. (1) with the site i and j or one of them locating in the core is taken as J . Here, J is used as unit of temperature and energy, and let J = 1. While Jij with both i and j sites belong to the surface is taken as J N . Positive and negative J N means the ferromagnetic and antiferromagnetic coupling in the surface, respectively. Free boundary conditions are applied in all directions. In our study, three series are considered. First one, N = 2123, the outermost 9 shells with different magnitude of J N (J N = −1, −0.5, −0.1, 1) are treated as the surface. Second one, N = 2123,
the different thickness of surface layer corresponding to different ratio p with the same value of J N (J N = −0.1) is considered, where p is defined as the ratio of atom number at the surface to total one; third one, different atom number of the pure cluster where the interaction in the core and surface is same (J N = J = 1) is studied. Now, let us consider to simulate spin configurations of FCC magnetic clusters using standard Monte Carlo–Metropolis (MC) [21]. The principle of this algorithm is that: for an N spin system, every spin in the lattice has an orientation, which forms a spin configuration. According to Eq. (1), the magnetic energy of the system corresponding to some configuration Ei is computed. Updating the spin configuration in visiting atomic site randomly, a new configuration energy Ej is produced. Thus, the difference of energy between both old and new configurations is ∆E = Ej −Ei . If ∆E < 0, the new configuration is accepted. If ∆E > 0, the new configuration is accepted with the probability of a successful spin flip exp(−∆E/KB T ). The first 104 MC steps per spin were discarded for equilibrium and thermal averages were made with next 104 MC steps. In thermal average process, we started to store the simulated parameter values separated by 20 MC steps to break correlation between successive configurations.
3. Spin-wave excitation It is well known that magnetization M(T ) at low temperature decrease as Bloch’s T 3/2 [9,22–25]: M(T ) = M(0) 1 − B T 3/2 . (2)
H. Zhigao, D. Youwei / Physics Letters A 300 (2002) 641–647
643
Ferromagnetic magnons have a quadratic dispersion relation at small wave vector q, h¯ ω(q) = Dq 2 , where D is the spin-wave stiffness constant. The coefficients B is related to D through the expression B = 2.612v0(kB /4πD)3/2 ; here g is the Lande factor, µB is the Bohr magneton, kB is the Boltzmann constant. The spin-wave stiffness constant is, in the approximate crystalline approximation, expressed as follows [24]: D = D(0) 1 − 2Z(∆a/a)δ , (3) where D(0) = ZSJ a 2 /3, a is lattice constant, J exchange constant, S atom spin, v0 atom volume, ∆a2 is the mean-square deviation of NN pair distance and δ is the structure fluctuation or exchange interaction fluctuation. From Eqs. (2), (3) we can derive
kB T 3/2 M(T ) = M(0) 1 − B J = M(0) 1 − BT 3/2 (4) and 3/2 −3/2 Z 1 − 2Z(∆a/a)δ B = 2.612v0 3/4πSa 2 −3/2 ∝ [Z]−3/2 1 − 2Z(∆a/a)δ , (5) where T = kB T /J , which is the reduced temperature corresponding to our Monte Carlo simulation. Eq. (4) has been widely used to study the spin-wave excitation of amorphous alloys [24,25]. In Eq. (5), the first term means the influence of the coordination number on B, second one does the role of the structure and exchange fluctuations on B. It is thought that Eq. (4) is also applicable to the cluster. The disordered spins [9,10], decreasing of the coordination and the deviation of NN pair distance at the surface of cluster, will make the magnitude of B change. For the cluster, another Bloch exponent law M(T ) = M(0)(1 − BT b ) is also often used to discuss the dependence of magnetization. Here b is the Bloch exponent.
4. Numerical results and discussion 4.1. The influence of exchange interaction at the surface on magnetization Now, we consider first series where N = 2123, and the outermost 9 shells (from 39th to 48th shell) with
Fig. 2. MC simulated and fitting magnetization as a function of temperature with J N = −1, −0.5, −0.1, 1: (a) total cluster, M; (b) at the surface, Ms; (c) in the core, Mb.
different magnitude of J N (J N = −1, −0.5, −0.1, 1) are treated as the surface. Fig. 2 shows the temperature dependencies of total, surface and core magnetizations (M, Ms, Mb) with MC simulation and fitting curves. From Fig. 2, it is found that the Bloch T 3/2 law is well satisfied at low temperature (T < 0.5TC ) for M and Mb except for M with J N = −0.1. A good linear relation is found in M vs. T curve with J N = −0.1, which responds to b = 1. However, for high temperature region (0.3TC < T < 0.9TC ), a better T 2 relation is fit in M(T ) with J N = −1, −0.5, 1.
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Now, we focus on the temperature dependence of Ms in the surface. From Fig. 2(b) and fitting data, we can get the fitting Ms(T ) expressions as follows: Ms(T ) = 0.98678 − 0.0468T 3/2 for J N = 1; Ms(T ) = −0.2597 + 1.112 exp(−T /4.471) for J N = −0.1; Ms(T ) = 0.4323 − 0.07691T for J N = −0.5; Ms(T ) = 0.2220 − 0.0353T for J N = −1. Different magnetization character in coated surface layers leads to the changes of magnetization behaviors of the clusters and produces different b value, which is consistent with the fact that the Bloch constant b decreases from 1.52 to 0.37 with decreasing size of iron crystalline coated Mg–F2 obtained by Zhang et al. [14]. Based on spin-wave excitation, Mills et al. predicted that the surface and bulk magnetization decrease with temperature according to the same T 3/2 law, but the Bloch constant of the surface is twice that of the bulk, Bs = 2Bbulk [26–28]. For J = J N = 1 (pure cluster), we get Bs = 0.0468, Bb = 0.0245, Bbulk = 0.0147. It is obtained that Bs = 2Bb , Bs = 3Bbulk , b = 3/2, which is accordant with one obtained experimentally by Pierce et al. for the magnetization of macroscopic surface [13,15], and the prediction of Mills et al. After coating the surface layer with antiferromagnetic coupling, the magnetic states at the surface become complex due to interplay of antiferromagnetic coupling and exchange interaction from the core. As J N = −0.1, the temperature dependence of Ms(T ) obeys exponent law. But for J N = −0.5, −1, a linear decrease of Ms(T ) exists, which is accordant with the quasilinear temperature dependence of the surface magnetization predicted by Rado [17]. Different coating layer influences obviously the magnetic state in the core as seen in Fig. 2(c). Although their M vs. T curves in the core all satisfy T 3/2 law, the Bloch constant decreases linearly with increasing of J N as shown in Fig. 3. This surface effect has significant for magnetoresistance effect in nanostructured materials [6]. 4.2. The influence of exchange interaction in the surface on magnetization The number of atom of cluster and the exchange interaction J N in the surface remain 2123 and −0.1, respectively. But, the ratio p of atom number at the surface to total is changed. Fig. 4 shows six typical sectional spin configurations at T = 0.01 and 3.31 with p = 0, 0.5, 0.8. Figs. 5 and 6 show magnetiza-
Fig. 3. The Bloch constant in the core as a function of J N with N = 2123.
Fig. 4. Six typical sectional spin configurations at T = 0.01 and 3.31 with p = 0, 0.5, 0.8.
tion as a function of P at T = 0.01 and T = 3.31 with H = 0, 2.0, respectively. It is clearly seen that: (1) the disorder degree in the surface increases with increasing p, which responds to the drastic decrease of magnetization; (2) at low temperature spins in the core retain well parallel array along initial direction (z axis), which responds to magnetization to be 1 constantly; (3) at higher temperature (T < TC ), spin configuration with p = 0 also remains basely order parallel array, but the spin direction has deviated from z axis. At the same time, there is still an interested phenomenon that under the external field the magnetization in the core decreases contrary. The above both phenomena may mean the existence of shape anisotropy and surface anisotropy originating from the breaking of symmetry at the surface of cluster [11,29,30], which will be studied further.
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Fig. 5. Magnetization as a function of P at T = 0.01 with H = 0, 2.0. M and M–H —total one; Ms and Ms–H —one at the surface; Mb and Mb–H —one in the core.
Fig. 7. Magnetization as a function of the temperature with N = 55, 683, 2123.
Fig. 6. Magnetization as a function of P at T = 3.31 with H = 0, 2.0. M and M–H —total one; Ms and Ms–H —one at the surface; Mb and Mb–H —one in the core.
Fig. 8. The N dependence of B. The inset presents B as a function of Z −3/2 .
4.3. Magnetization as a function of the size of cluster We study effect of atom number of the pure cluster (J N = J = 1) on magnetization. Fig. 7 shows M vs. T curves with N = 55, 683, 2123. It is found that at low temperature, Bloch T 3/2 law is obeyed well for the clusters with atom number from 55 to 2123. It is evident that below the Curie temperature magnetization increases with the size of cluster, which is similar to results by Merikoski et al. [19]. Using different MC steps to simulate the magnetizations of small clusters, we find that, for no enough MC steps, it is true that the decay with temperature of the magnetization is slowest for the smallest cluster at temperature well above the Curie temperature because of increasing fluctuations. However, for enough MC steps, the fluctuation can be reduced dramatically. Therefore, the temperature dependence of magnetization at tempera-
ture above Curie temperature is different from results by Merikoski et al. In our simulation, 40000 steps are used for small cluster, while only 3000 steps were visited in theirs. Fig. 8 shows the N dependence of B, and the inset presents B as a function of Z −3/2 . From the figure, both important results are obtained: first one, B increases drastically with reducing N , which is consistent with B as a function of iron crystalline size by Zhang [14,31]; second one, B is approximately proportional to Z −3/2 in both different slops. Now, we use Eq. (5) obtained in the approximate crystalline approximation to discuss the influence of the surface exchange interaction and the coordination number Z on B. For the FCC cluster, Z reduces drastically with decreasing N for N < 1000 as seen in Fig. 9. Fig. 9 and Eq. (5) make us understand easily the N and Z −3/2 dependencies of B found in Fig. 8. The change of slop in the inset of Fig. 8 should be explained as
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constant result from variant coordination number, the structure and exchange interaction fluctuation.
Acknowledgements This work was supported by Grant No. G1999064508 for National Key Project for Basic Research of China, and Foundation for University Key Teacher by the Ministry of Education. Fig. 9. The coordination number Z as a function of N for FCC structure spherical cluster.
References follows: the structure and exchange interaction fluctuation (∆a/a)δ increases with decreasing of Z. It is assumed that (∆a/a)δ ∝ 1/(c + Z); here c is constant. Then, we can obtain that the slop of B to Z −3/2 decreases with decreasing of Z. In addition, an obvious exchange fluctuation can be produced in the clusters coated with antiferromagnetic coupling surface layers. Therefore, the thicker the coated layer, the larger the fluctuation, the bigger the magnitude of B, which explain well the result in Fig. 3.
5. Conclusion In conclusion, magnetizations and spin-wave excitations at the surface, in the core for the clusters with different surface interaction, surface thickness and atom number N have been studied. It is found that, for the pure cluster, the Bloch T 3/2 law is well satisfied at low temperature (T < 0.5TC ), but Bloch constant B increases drastically with the reducing atom number N , and Bs = 3Bbulk ; the magnetization and spinwave excitations at the surfaces coated different exchange interaction have complex temperature dependence, which influences strongly the core and cluster magnetic states; the increasing of the thickness of the surface layer with antiferromagnetic interaction make the surface and cluster magnetization reduce fast, but spins in the core remains basely parallel array at T < TC . Compared the simulated results with experimental facts, both are consistent. According to the Bloch exponent law in the approximate crystalline approximation, it is evident that the changes of the Bloch
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