- Email: [email protected]
Magnetization of small dot arrays with dipolar interacting nanoparticles
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 320 (2008) 285–291 www.elsevier.com/locate/jmmm
Magnetization of small dot arrays with dipolar interacting nanoparticles C. Zhua,b, L.F. Zhanga, C. Xua, a
Department of Physics, Jiangsu Key Laboratory of Thin Films, Suzhou University, Suzhou 215006, China b Department of Physics, College of Science, Southern Yangtze University, Wuxi 214122, China Received 16 October 2006; received in revised form 10 May 2007 Available online 12 June 2007
Abstract We study the magnetization of a patterned array of cylinders using the Landau–Lifshitz–Gilbert equation. The direction of the external magnetizing field may vary from in-plane orientation to out-of-plane orientation. The magnetization is found to depend on the direction of the external magnetic field and the strength of dipolar interaction. The hysteresis loops show strongly size-dependent effects, as a result of the competition among the dipolar interaction, perpendicular anisotropy and Zeeman energy. The remanence and the switching field are anisotropic with respect to the direction of the applied magnetic field. r 2007 Elsevier B.V. All rights reserved. PACS: 75.60.Ej; 75.60.Jk; 75.75.+a Keywords: Patterned array; Magnetization; Dipolar interaction
1. Introduction Patterned arrays of single-domain nanodots are the subject of substantial current research interest due to their potential applications in future magnetic data storage and magnetic sensing devices [1–7]. In patterned arrays, the dipolar interaction between particles can strongly affect the static magnetic ordering and magnetization processes [8–13]. Detailed studies on the magnetic properties in such systems are, therefore, essential to the design of novel devices. Kayali et al. [10] and Stamps et al. [14,15] investigated the magnetic ordering and hysteresis behavior of small arrays of cylinders with magnetic moments constrained to be in plane, i.e., parallel to the plane of the array. They found that magnetic ordering and hysteresis are strongly affected by the size of the array and the direction of the external applied field. In experiments, Ross et al. [4] studied patterned arrays of cylindrical nanoparticles of large aspect (height/diameter) ratio. These particles exhibit singleCorresponding author. Tel./fax: +86 512 65112597.
E-mail addresses: [email protected] (C. Zhu), [email protected] (C. Xu). 0304-8853/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2007.06.002
domain behavior and their easy axes are aligned perpendicular to the substrate surface, as a result of both shape anisotropy and magnetocrystalline anisotropy. For the weak dipolar interaction regime studied in their experiments, these particles have their moments aligned either ‘‘up’’ or ‘‘down’’, with respect to the direction perpendicular to the array. The strength of dipolar interaction is related to the separation between the magnetic particles and the relative orientations of the interacting magnetic moments. Therefore, a finite array of magnetic dots with magnetic moments not restricted to lie parallel to the plane of the array due to possibly large perpendicular anisotropy is expected to exhibit different properties from an array consisting only of in-plane magnetic moments [10,14,15]. In particular, the effects of dipolar interaction would be important in high-density patterned arrays. In previous work, we studied the magnetic properties of an array of small dots [8,9]. The hysteresis loops are found to be highly sensitive to the dipolar coupling strength, for an external magnetic field applied either parallel or perpendicular to the plane of the array. The behavior of the in-plane hysteresis loops is dominated by the strong dipolar interaction between particles, while that of the outof-plane hysteresis loops is dominated by the perpendicular
ARTICLE IN PRESS C. Zhu et al. / Journal of Magnetism and Magnetic Materials 320 (2008) 285–291
286
anisotropy. In the present paper, we consider finite square arrays of single-domain cylindrical particles, the magnetic moments of which can point at any direction in threedimensional space. We assume that the temperature is sufficiently low so that the superparamagnetic effect can be neglected. We study the dependence of the hysteresis loop on the direction of the external magnetic field. The remanence under different magnetization processes and the switching field as a function of the direction of the external magnetic field are also investigated. The behavior in small arrays of different sizes is also studied. The equilibrium configuration of magnetic moments is invoked to explain the dependence of the remanence and switching field on the direction of the applied field, system size, and the strength of dipolar interaction. 2. Model of calculation The hysteresis of a system of coupled magnetic moments can be studied by numerically solving the Landau–Lifshitz–Gilbert (LLG) equations. This approach can be used to study time-dependent relaxation to static order and the dynamics of the magnetic moments in the array. We consider a square array with identical magnetic particles located at the lattice sites. Each particle is treated as a single-domain cylindrical particle with an effective magnetic moment m. The ith particle is located at ri ¼ ^ where a is the lattice constant [16] and p and pax^ þ qay, q are integers (see Fig. 1). The motion of the magnetic moment m in the array can be described by the LLG equation dmi a dmi ¼ gmi Hi þ mi , (1) m dt dt where g is the gyromagnetic ratio and the parameter a controls the rate of dissipation. The effective magnetic field Hi at site i includes the external magnetic field he , the dipolar field hd;i and a single-particle effective anisotropy field hk;i . With these definitions, the effective field Hi can be written as Hi ¼ hk;i þ he þ hd;i
(2)
strength is represented by the parameter hd given by hd ¼
m VM s ¼ 3 , a3 a
(5)
where V is the volume of a particle. The orientation of mi can be specified by the angles ðyi ; fi Þ, where y is the polar angle and f is the azimuthal angle (Fig. 1). Writing t0 ¼ tM s gð1 þ a2 Þ1 , the orientation of the magnetic moments mi as a function of time t0 can be found by solving the following equations [17]: H iy dyi H ix H ix ¼ sin f þ cos f þ a cos yi cos fi i i dt0 Ms Ms Ms H iy H iz þ cos yi sin fi sin yi , ð6Þ Ms Ms dfi cos yi cos fi H ix cos yi sin fi H iy H iz ¼ þ dt0 sin yi Ms sin yi Ms Ms sin fi H ix cos fi H iy a , sin yi M s sin yi M s
ð7Þ
where H ix ; H iy ; H iz are the three components of Hi . We consider finite square arrays of size N N with N ¼ 3; . . . ; 8. The magnetizing process in the presence of an external magnetic field he is studied by solving Eqs. (6) and (7) numerically using the fourth-order Runge–Kutta method. To start the calculations, a strong field of strength H 0 is applied to the array in order to set the system into a saturated state. The field strength is then gradually decreased in increments of dh to H 0 , followed by an increase back to þH 0 . At each value of the applied field, the orientations of the moments are obtained after the system has settled into a stable state. We define M h to be the component of the magnetization along the direction of the applied field, and M 0 the saturation magnetization of the array under the influence of an applied field, i.e., M h ¼ M 0 cosðM; he Þ. In our calculations, we assume a large reduced anisotropy field hk =M s ¼ 2:0 [18,19] pointing perpendicular to the array. This serves to model singledomain particles in which both shape anisotropy and magnetocrystalline anisotropy are important. This rather large anisotropy field may arise in slim cylindrical particles.
with hk;i ¼ hk mz;i z^ ¼ ð2K=M s Þmz;i z^,
hd;i
" # X 3ðrij mj Þrij mj ¼ 3 r5ij rij jai X 3ð^rij m ^ j Þ^rij m ^j ¼ hd . 3 r~ij jai
z
(3)
θ
ð4Þ
Here, K is the anisotropy constant, M s is the saturation magnetization, mz;i ¼ M z =M s , r^ij is the unit vector pointing ^ j is the unit vector of from dipole moment mi to mj , m moment mj , and r~ij ¼ rij =a. The dipolar interaction
φ
he
y
x a Fig. 1. Geometry of the array. The polar angle y and azimuthal angle f specify the orientation of the applied magnetic field. Identical particles form a square array of lattice constant a.
ARTICLE IN PRESS C. Zhu et al. / Journal of Magnetism and Magnetic Materials 320 (2008) 285–291
In the presence of such a large anisotropy, the magnetic moments can be aligned perpendicular to the plane even for the strong dipolar interaction considered in the present paper when the applied magnetic field is absent. We take a large damping constant ða ¼ 1:0Þ to shorten the time it takes for the system to achieve equilibrium. The choice of a, while reducing the time to reach the steady state, does not affect the final stable configuration of the magnetic moments.
Fig. 2(b)and (e) represents a moderate dipolar coupling, and hd ¼ 0:7hk in Fig. 2(c)and (f) represents a strong dipolar coupling. In Fig. 2(a)–(c), the external magnetic field lies in the y–z plane and three typical polar angles y ¼ 0 ; 45 , and 90 of the field are considered. In Fig. 2(a), at weak dipolar coupling ðhd ¼ 0:05hk Þ, the hysteresis loop is quite wide (see dashed line ) for y ¼ 0 . This is due to the large single-particle anisotropy. When the external magnetic field is applied at y ¼ 45 , the anisotropy field of each particle becomes weaker along the external field and the hysteresis loop becomes narrower, when compared with the case of y ¼ 0 . When the external field lies in the plane of the array (y ¼ 90 ), the hysteresis loop closes. In this case, the perpendicular uniaxial anisotropy of the particles has no effect on the irreversible magnetizing process when we sweep the external field.
3. Results and discussion
1.0
1.0
0.5
0.5 Mh/M0
Mh/M0
Fig. 2 shows the hysteresis loops for a 3 3 array for several values of hd representing different dipolar interaction strengths. The strength hd ¼ 0:05hk in Fig. 2(a) and (d) represents a weak dipolar coupling, hd ¼ 0:3hk in
0.0
hd=0.05hk
0.0
-1.0
-1.0
1.0
1.0
0.5
0.5
0.0 hd=0.3hk
-0.5
-1.0 1.0
hd=0.05hk
-0.5
Mh/M0
Mh/M0
-0.5
0.0
-0.5
hd=0.3hk
-1.0 1.0
θ=0°
φ=0°
θ=45° 0.5 Mh/M0
Mh/M0
φ=45°
θ=90°
0.5
287
0.0 hd=0.7hk
-0.5
0.0 hd=0.7hk
-0.5
-1.0
-1.0 -3
-2
-1
0 he/Ms
1
2
3
-3
-2
-1
0
1
2
3
he/Ms
Fig. 2. The hysteresis loops for a 3 3 array for several values of hd representing different dipolar interaction strengths. In the left column, the dipolar interaction strengths are (a) hd ¼ 0:05hk , (b) hd ¼ 0:3hk , and (c) hd ¼ 0:7hk . Results for an external magnetic field applied in the y–z plane with y ¼ 0 , 45 , and 90 , respectively, are shown. In the right column, the dipolar interaction strengths are (d) hd ¼ 0:05hk , (e) hd ¼ 0:3hk , and (f) hd ¼ 0:7hk . Results for an external magnetic field applied in the x–y plane with f ¼ 0 and 45 , respectively, are shown.
ARTICLE IN PRESS C. Zhu et al. / Journal of Magnetism and Magnetic Materials 320 (2008) 285–291
288
However, as the dipolar interaction increases, the hysteresis loop at weak dipolar coupling splits into three loops. As an example, we show the results for the case of hd ¼ 0:3hk in Fig. 2(b). In this case, the dipolar interaction apparently affects the magnetizing process. It induces more
1.0
energy barriers and wells so that there are more irreversible magnetizing paths. Therefore, the multi-loops magnetization are usually observed in dipolar interacting arrays with perpendicular anisotropy. If the dipolar interaction increases further, e.g., the hd ¼ 0:7hk case in Fig. 2(c), the 1.0
θ=0°
θ=0°
θ=45°
θ=45° 0.5
θ=90°
N=4 Mh/M0
Mh/M0
0.5 N=4
0.0
0.0
-0.5
-0.5
-1.0
-1.0
1.0
1.0 N=5 N=5
0.5 Mh/M0
Mh/M0
0.5
0.0
0.0
-0.5
-0.5
-1.0
-1.0
1.0
1.0 N=6 N=6
0.5 Mh/M0
Mh/M0
0.5
0.0
0.0
-0.5
-0.5
-1.0
-1.0
1.0
1.0 N=7 N=7
0.5 Mh/M0
Mh/M0
0.5
0.0
0.0
-0.5
-0.5
-1.0
-1.0 -3
-2
-1
0 he/ms
1
2
3
-3
-2
-1
0 he/ms
1
2
3
Fig. 3. Hysteresis loops for N N arrays with N ¼ 4; . . . ; 7 for hd ¼ 0:7hk . In (a)–(d), the external magnetic field is applied in the y–z plane with y ¼ 0 ; 45 and 90 , respectively. In (e)–(h), the external magnetic field is applied in the x–y plane with f ¼ 0 and 45 , respectively.
ARTICLE IN PRESS C. Zhu et al. / Journal of Magnetism and Magnetic Materials 320 (2008) 285–291
1.0
0.8 (A)
0.6
Mr/M0
Mr/M0
0.12 θ=90°
0.06 0.00
0.4
0
15
30
45
φ (°) 0.2 (B) 0.0
1.5
hs/M0
magnetization curve becomes anhysteresis when the external field is applied at y ¼ 0 . The anisotropy energy barrier is now suppressed by the strong dipolar interaction. The orientations of the magnetic moments are not controlled by the single-particle anisotropy in this case, but instead depend on the interaction between neighboring moments. The moments are no longer pointing perpendicular to the plane of the array. Thus, the abrupt jumps in Fig. 2(a) and (b) at y ¼ 0 , which are associated with the switching of some moments in the array [9], disappear when the dipolar interaction is strong. Interestingly, for strong dipolar coupling, hysteresis loops appear when the external field is applied at y ¼ 90 . It is because, while the external field tends to align the moments so that they are all parallel to each other, the strong dipolar interaction tends to align the moments parallel to the plane in the structure of antiparallel ferromagnetic chains or local vortices [10,20]. It is this competition between the dipolar interaction and Zeeman energy that leads to the in-plane hysteresis of magnetization. In Fig. 2(d)–(f), the dependence of the magnetization on the in-plane external magnetic field is also studied. Here, we choose two typical azimuthal angles of f ¼ 0 and 45 . Again, at weak dipolar interaction, the in-plane magnetization curve is closed. In this case, the competition between the single-particle anisotropy and Zeeman energy plays the dominant role. Since the easy anisotropy axes of the particles are perpendicular to the external magnetic field, the components of the magnetic moments along the þz- or z-direction are identical. Hence, the magnetization shows an anhysteresis behavior and it is almost independent of the direction of in-plane external magnetic field. For strong dipolar interaction, however, the hysteresis loop shows anisotropic behavior that is strongly dependent on the direction of the external magnetic field. This is due to the anisotropic nature of the dipolar interaction. Fig. 3 shows the magnetization in arrays of sizes 4 4, 5 5, 6 6, and 7 7 at a strong dipolar interaction of hd ¼ 0.7hk . For a given direction of the external field, the hysteresis loops are sensitive to the size of these small arrays. As the array increases in size, the number of dipole moments also increases. This changes the collective behavior of the moments, and shows up in the magnetizing behavior. Such complexity in the M h –he relation is a typical character in small arrays, and the sensitivity to size becomes weaker as the size becomes larger. (We have performed calculations on larger arrays, but the results are not shown here.) The corresponding dependence of the remanence and the switching field are investigated at weak and strong dipolar coupling strengths (hd ¼ 0.05hk in Fig. 4 and hd ¼ 0.7hk in Fig. 5). The switching field refers to the field that makes the magnetization along the external magnetic field transit from positive magnetization to negative magnetization. The reduced remanence keeps constant when he is applied in the x–z plane except for y 90 . The results imply that the magnetic moments are ordered in such a way that they point perpendicular to the
289
1.0
N=3 N=4 0.5
N=5 N=6 N=7
0.0
0
30
60
90
θ (°) Fig. 4. The dependence of reduced remanence M r /M 0 and reduced switching field hs =M 0 on y for hd ¼ 0:05hk and N ¼ 3; . . . ; 7. The external magnetic field is applied in the x–z plane. The inset shows the dependence of M r =M 0 on f for hd ¼ 0:05hk . The configurations (A) and (B) show the orientations of magnetic moments in the corresponding arrays.
plane, due to the strong perpendicular anisotropy (see configuration (A) in Fig. 4(a)). When he is close to in-plane or in-plane, the evolution of the magnetic moments will be affected by the weak dipolar interaction. As an example, consider a field he along the x-direction (y ¼ 90 ). Starting from an initial saturated state with all moments pointing along þx-direction, the moments will evolve to an antiferromagnetic state with all the magnetic moments pointing perpendicular to the plane (see configuration (B) in Fig. 4(a)) when he is swept to zero. The configuration (B) implies a reduced remanence of 1=N for odd N and 0 when even N. This is the reason that the reduced remanence
ARTICLE IN PRESS C. Zhu et al. / Journal of Magnetism and Magnetic Materials 320 (2008) 285–291
290
0.4
0.4
φ=45°
Mr/M0
Mr/M0
φ=0°
0.2
(B)
(A)
(C)
0.0
(D)
0.0 N=3 N=4 N=5 N=6
1.0
1.0
N=7
hs/M0
hs/M0
0.2
0.5
0.0
0.5
0.0 0
30
60
90
θ (°)
0
30
60
90
θ (°)
Fig. 5. The dependence of reduced remanence M r =M 0 and reduced switching field hs =M 0 on y for hd ¼ 0:7hk and N ¼ 3; . . . ; 7. The insets show the configurations of the magnetic moments. The configurations (A)–(D) show the orientations of magnetic moments in the corresponding arrays.
takes on different constant values for different system sizes, as shown in the inset of Fig. 4(a) for in-plane he . The reduced switching field decreases as the direction of he changes from y ¼ 0 to 90 (see Fig. 4(b)). The larger is the component of he in the x-direction, the smaller is the switching field. The results show that for he close to inplane, the system size effect is small for weak dipolar interactions. For strong dipolar interactions (see Fig. 5), both the reduced remanence and reduced switching field exhibit responses that are sensitive to the direction of the applied magnetic field. In Fig. 5(a) and (b), the external magnetic field is applied at different polar angles y for a fixed azimuthal angle f ¼ 0 . For N ¼ 3 and 4, M r =M 0 takes on a constant over the whole range of y. For N ¼ 3 and he 0, the magnetic moments evolve to a structure of antiparallel ferromagnetic chains as shown in the configuration (A) in Fig. 5(a). This configuration gives M r =M 0 13. For N ¼ 4, however, when he approaches 0, the magnetic moments evolve to a vortex structure as shown in the configuration (C) in Fig. 5(c). This configuration gives M r =M 0 0. For N ¼ 5, 6, and 7, M r =M 0 show sharp changes at certain values of the angles y. The results indicate that the configuration of magnetic moments corresponding to the remanent state changes as the direction of the sweeping field varies. Fig. 5(b) shows the reduced switching field as a function of y for fixed f ¼ 0 . To explain the features, we again take
N ¼ 3 and 4 as examples. For N ¼ 3, the component of he along the x-direction is small when y is small. The field cannot change the in-plane magnetic-moment configuration (see (A) in Fig. 5(a)). The component of he along the zdirection will lead to a non-vanishing magnetization. If he is strong enough, the magnetization will have a positive component along the external magnetic field so that the magnetization is switched by the external magnetic field. As y increases, the component of he in the z-direction decreases, for fixed magnitude he . Therefore, we need to increase he in order to switch the magnetization. So the switching field increases with y. However, when y is increased further, the x-component of he becomes large enough to change the magnetic moments from the configuration (see (A) in Fig. 5(a)) of six moments pointing approximately along the þx-direction and three moments pointing approximately along the x-direction to the configuration of having three moments pointing along the þx-direction and six moments along the x-direction. When y is increased further, the x-component of he increases for fixed magnitude he . Thus, a smaller he is sufficient to switch the magnetization. Therefore, we see a decreasing behavior in hs =M 0 as y is further increased. For N ¼ 4, the magnetic moments show a local vortex structure (see (C) in Fig. 5(c)) in the remanent state. When an external magnetic field is applied, the in-plane component of the magnetization is zero and the z-component of the magnetization is non-zero as long as he a0, and hence
ARTICLE IN PRESS C. Zhu et al. / Journal of Magnetism and Magnetic Materials 320 (2008) 285–291
causing a switching of the magnetization along he . So the switching field hs approaches zero for N ¼ 4 (see Fig. 5(b)). The sharp changes of hs =M 0 for N ¼ 5; 6, and 7 are related to the changes in the configuration of the magnetic moments at the remanent state. When the external magnetic field is swept through different values of y at a fixed angle f ¼ 45 (see Fig. 5(c) and (d)), the reduced remanence shows sharp changes at small y for all the system sizes considered here. Comparing the cases of N ¼ 3 in Fig. 5(c) to (a), the magnetic moments can evolve to a vortex-like structure (see (B) in Fig. 5(a) at small y for f ¼ 45 ), and thus the remanence decreases. For N ¼ 4, the magnetic moments can evolve to a chain-like structure (see configuration (D) in Fig. 5(c)) at large y (f ¼ 45 ), and hence the system could have a non-zero remanence. The behavior of hs =M 0 versus y shown in Fig. 5(d) is similar to the case of N ¼ 3 in Fig. 5(b), except that there are sharp changes at small values of the angle y. This is also related to the change in the configuration of the magnetic moments at the remanent state. 4. Summary We examined the hysteresis in a planar array of coupled magnetic moments by solving the LLG equations numerically.. The magnetic moment of each particle can point at any direction in space. The hysteresis loops of small arrays of nanoparticles are sensitive to the direction of the external magnetic field. Due to dipolar interaction, the system size effect is significant in small arrays. The magnetization is determined by the competition among the dipolar interaction, perpendicular anisotropy, and Zeeman energy. At weak dipolar interaction, the hysteresis loop closes for an in-plane external magnetic field. For strong dipolar interaction, the magnetization becomes anhysteresis when the external magnetic field is applied perpendicular to the plane containing the array. Multiloops in the magnetization curve appear at moderate and strong dipolar interactions. The remanence and the switching field show strong dependence on the direction of the applied field and thus these quantities are anisotropic. Such
291
anisotropy becomes significant at strong dipolar couplings. These features may be useful in the applications of small arrays as devices, and different behavior would result by controlling the dipolar coupling strength between particles. Acknowledgments This work was supported by the Jiangsu provincial Education Foundation under Grant no. 06KJD140165. We thank P.M. Hui for suggestions on revising an earlier version of the paper. References [1] H.S. Nalwa (Ed.), Magnetic Nanostructures, American Scientific Publishers, 2002. [2] J.I. Martı´ n, J. Nogue´s, K. Liu, J.L. Vicent, I.K. Schuller, J. Magn. Magn. Mater. 256 (2003) 449. [3] J. Bai, H. Takahoshi, H. Ito, H. Satio, S. Ishio, J. Appl. Phys. 91 (2004) 6848. [4] C.A. Ross, et al., J. Appl. Phys. 91 (2002) 6848; M.C. Abraham, et al., J. Appl. Phys. 89 (2001) 5667. [5] X. Zhu, P. Gru¨tter, V. Metlushko, B. Ilic, Phys. Rev. B 66 (2002) 024423. [6] C.A. Ross, et al., Phys. Rev. B 65 (2002) 144417. [7] S. Sun, C.B. Murray, D. Weller, L. Folks, A. Moser, Science 287 (2000) 1989. [8] L.F. Zhang, C. Xu, P.M. Hui, Y.Q. Ma, J. Appl. Phys. 97 (2005) 103912. [9] C. Xu, P.M. Hui, L.F. Zhang, Y.Q. Ma, J.H. Zhou, Z.Y. Li, Eur. Phys. J. B 46 (2005) 475. [10] M. Amin Kayali, W.M. Saslow, Phys. Rev. B 70 (2004) 174404. [11] J. Garcı´ a-Otero, M. Porto, J. Rivas, A. Bunde, Phys. Rev. Lett. 84 (2000) 167. [12] D. Kechrakos, K.N. Trohidou, Phys. Rev. B 58 (1998) 12169. [13] M. El-Hilo, K. O’Grady, R.W. Chantrell, J. Appl. Phys. 76 (1994) 6811. [14] R.L. Stamps, R.E. Camley, Phys. Rev. B 60 (1999) 11694. [15] R.L. Stamps, R.E. Camley, J. Magn. Magn. Mater. 177–181 (1998) 813. [16] Paola R Arias, D. Altbir, M. Bahiana, J. Phys. Condens. Matter 17 (2005) 1625. [17] L.F. Zhang, C. Xu, Y.Q. Ma, Phys. Lett. A 338 (2005) 373. [18] C.A. Ross, et al., Phys. Rev. B 65 (2000) 144417. [19] Dieter Weller, et al., IEEE Trans. Magn. 36 (2000) 10. [20] K. De’Bell, A.B. MacIsaac, J.P. Whitehead, Rev. Mod. Phys. 72 (2000) 225.
Journal of Magnetism and Magnetic Materials 320 (2008) 285–291 www.elsevier.com/locate/jmmm
Magnetization of small dot arrays with dipolar interacting nanoparticles C. Zhua,b, L.F. Zhanga, C. Xua, a
Department of Physics, Jiangsu Key Laboratory of Thin Films, Suzhou University, Suzhou 215006, China b Department of Physics, College of Science, Southern Yangtze University, Wuxi 214122, China Received 16 October 2006; received in revised form 10 May 2007 Available online 12 June 2007
Abstract We study the magnetization of a patterned array of cylinders using the Landau–Lifshitz–Gilbert equation. The direction of the external magnetizing field may vary from in-plane orientation to out-of-plane orientation. The magnetization is found to depend on the direction of the external magnetic field and the strength of dipolar interaction. The hysteresis loops show strongly size-dependent effects, as a result of the competition among the dipolar interaction, perpendicular anisotropy and Zeeman energy. The remanence and the switching field are anisotropic with respect to the direction of the applied magnetic field. r 2007 Elsevier B.V. All rights reserved. PACS: 75.60.Ej; 75.60.Jk; 75.75.+a Keywords: Patterned array; Magnetization; Dipolar interaction
1. Introduction Patterned arrays of single-domain nanodots are the subject of substantial current research interest due to their potential applications in future magnetic data storage and magnetic sensing devices [1–7]. In patterned arrays, the dipolar interaction between particles can strongly affect the static magnetic ordering and magnetization processes [8–13]. Detailed studies on the magnetic properties in such systems are, therefore, essential to the design of novel devices. Kayali et al. [10] and Stamps et al. [14,15] investigated the magnetic ordering and hysteresis behavior of small arrays of cylinders with magnetic moments constrained to be in plane, i.e., parallel to the plane of the array. They found that magnetic ordering and hysteresis are strongly affected by the size of the array and the direction of the external applied field. In experiments, Ross et al. [4] studied patterned arrays of cylindrical nanoparticles of large aspect (height/diameter) ratio. These particles exhibit singleCorresponding author. Tel./fax: +86 512 65112597.
E-mail addresses: [email protected] (C. Zhu), [email protected] (C. Xu). 0304-8853/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2007.06.002
domain behavior and their easy axes are aligned perpendicular to the substrate surface, as a result of both shape anisotropy and magnetocrystalline anisotropy. For the weak dipolar interaction regime studied in their experiments, these particles have their moments aligned either ‘‘up’’ or ‘‘down’’, with respect to the direction perpendicular to the array. The strength of dipolar interaction is related to the separation between the magnetic particles and the relative orientations of the interacting magnetic moments. Therefore, a finite array of magnetic dots with magnetic moments not restricted to lie parallel to the plane of the array due to possibly large perpendicular anisotropy is expected to exhibit different properties from an array consisting only of in-plane magnetic moments [10,14,15]. In particular, the effects of dipolar interaction would be important in high-density patterned arrays. In previous work, we studied the magnetic properties of an array of small dots [8,9]. The hysteresis loops are found to be highly sensitive to the dipolar coupling strength, for an external magnetic field applied either parallel or perpendicular to the plane of the array. The behavior of the in-plane hysteresis loops is dominated by the strong dipolar interaction between particles, while that of the outof-plane hysteresis loops is dominated by the perpendicular
ARTICLE IN PRESS C. Zhu et al. / Journal of Magnetism and Magnetic Materials 320 (2008) 285–291
286
anisotropy. In the present paper, we consider finite square arrays of single-domain cylindrical particles, the magnetic moments of which can point at any direction in threedimensional space. We assume that the temperature is sufficiently low so that the superparamagnetic effect can be neglected. We study the dependence of the hysteresis loop on the direction of the external magnetic field. The remanence under different magnetization processes and the switching field as a function of the direction of the external magnetic field are also investigated. The behavior in small arrays of different sizes is also studied. The equilibrium configuration of magnetic moments is invoked to explain the dependence of the remanence and switching field on the direction of the applied field, system size, and the strength of dipolar interaction. 2. Model of calculation The hysteresis of a system of coupled magnetic moments can be studied by numerically solving the Landau–Lifshitz–Gilbert (LLG) equations. This approach can be used to study time-dependent relaxation to static order and the dynamics of the magnetic moments in the array. We consider a square array with identical magnetic particles located at the lattice sites. Each particle is treated as a single-domain cylindrical particle with an effective magnetic moment m. The ith particle is located at ri ¼ ^ where a is the lattice constant [16] and p and pax^ þ qay, q are integers (see Fig. 1). The motion of the magnetic moment m in the array can be described by the LLG equation dmi a dmi ¼ gmi Hi þ mi , (1) m dt dt where g is the gyromagnetic ratio and the parameter a controls the rate of dissipation. The effective magnetic field Hi at site i includes the external magnetic field he , the dipolar field hd;i and a single-particle effective anisotropy field hk;i . With these definitions, the effective field Hi can be written as Hi ¼ hk;i þ he þ hd;i
(2)
strength is represented by the parameter hd given by hd ¼
m VM s ¼ 3 , a3 a
(5)
where V is the volume of a particle. The orientation of mi can be specified by the angles ðyi ; fi Þ, where y is the polar angle and f is the azimuthal angle (Fig. 1). Writing t0 ¼ tM s gð1 þ a2 Þ1 , the orientation of the magnetic moments mi as a function of time t0 can be found by solving the following equations [17]: H iy dyi H ix H ix ¼ sin f þ cos f þ a cos yi cos fi i i dt0 Ms Ms Ms H iy H iz þ cos yi sin fi sin yi , ð6Þ Ms Ms dfi cos yi cos fi H ix cos yi sin fi H iy H iz ¼ þ dt0 sin yi Ms sin yi Ms Ms sin fi H ix cos fi H iy a , sin yi M s sin yi M s
ð7Þ
where H ix ; H iy ; H iz are the three components of Hi . We consider finite square arrays of size N N with N ¼ 3; . . . ; 8. The magnetizing process in the presence of an external magnetic field he is studied by solving Eqs. (6) and (7) numerically using the fourth-order Runge–Kutta method. To start the calculations, a strong field of strength H 0 is applied to the array in order to set the system into a saturated state. The field strength is then gradually decreased in increments of dh to H 0 , followed by an increase back to þH 0 . At each value of the applied field, the orientations of the moments are obtained after the system has settled into a stable state. We define M h to be the component of the magnetization along the direction of the applied field, and M 0 the saturation magnetization of the array under the influence of an applied field, i.e., M h ¼ M 0 cosðM; he Þ. In our calculations, we assume a large reduced anisotropy field hk =M s ¼ 2:0 [18,19] pointing perpendicular to the array. This serves to model singledomain particles in which both shape anisotropy and magnetocrystalline anisotropy are important. This rather large anisotropy field may arise in slim cylindrical particles.
with hk;i ¼ hk mz;i z^ ¼ ð2K=M s Þmz;i z^,
hd;i
" # X 3ðrij mj Þrij mj ¼ 3 r5ij rij jai X 3ð^rij m ^ j Þ^rij m ^j ¼ hd . 3 r~ij jai
z
(3)
θ
ð4Þ
Here, K is the anisotropy constant, M s is the saturation magnetization, mz;i ¼ M z =M s , r^ij is the unit vector pointing ^ j is the unit vector of from dipole moment mi to mj , m moment mj , and r~ij ¼ rij =a. The dipolar interaction
φ
he
y
x a Fig. 1. Geometry of the array. The polar angle y and azimuthal angle f specify the orientation of the applied magnetic field. Identical particles form a square array of lattice constant a.
ARTICLE IN PRESS C. Zhu et al. / Journal of Magnetism and Magnetic Materials 320 (2008) 285–291
In the presence of such a large anisotropy, the magnetic moments can be aligned perpendicular to the plane even for the strong dipolar interaction considered in the present paper when the applied magnetic field is absent. We take a large damping constant ða ¼ 1:0Þ to shorten the time it takes for the system to achieve equilibrium. The choice of a, while reducing the time to reach the steady state, does not affect the final stable configuration of the magnetic moments.
Fig. 2(b)and (e) represents a moderate dipolar coupling, and hd ¼ 0:7hk in Fig. 2(c)and (f) represents a strong dipolar coupling. In Fig. 2(a)–(c), the external magnetic field lies in the y–z plane and three typical polar angles y ¼ 0 ; 45 , and 90 of the field are considered. In Fig. 2(a), at weak dipolar coupling ðhd ¼ 0:05hk Þ, the hysteresis loop is quite wide (see dashed line ) for y ¼ 0 . This is due to the large single-particle anisotropy. When the external magnetic field is applied at y ¼ 45 , the anisotropy field of each particle becomes weaker along the external field and the hysteresis loop becomes narrower, when compared with the case of y ¼ 0 . When the external field lies in the plane of the array (y ¼ 90 ), the hysteresis loop closes. In this case, the perpendicular uniaxial anisotropy of the particles has no effect on the irreversible magnetizing process when we sweep the external field.
3. Results and discussion
1.0
1.0
0.5
0.5 Mh/M0
Mh/M0
Fig. 2 shows the hysteresis loops for a 3 3 array for several values of hd representing different dipolar interaction strengths. The strength hd ¼ 0:05hk in Fig. 2(a) and (d) represents a weak dipolar coupling, hd ¼ 0:3hk in
0.0
hd=0.05hk
0.0
-1.0
-1.0
1.0
1.0
0.5
0.5
0.0 hd=0.3hk
-0.5
-1.0 1.0
hd=0.05hk
-0.5
Mh/M0
Mh/M0
-0.5
0.0
-0.5
hd=0.3hk
-1.0 1.0
θ=0°
φ=0°
θ=45° 0.5 Mh/M0
Mh/M0
φ=45°
θ=90°
0.5
287
0.0 hd=0.7hk
-0.5
0.0 hd=0.7hk
-0.5
-1.0
-1.0 -3
-2
-1
0 he/Ms
1
2
3
-3
-2
-1
0
1
2
3
he/Ms
Fig. 2. The hysteresis loops for a 3 3 array for several values of hd representing different dipolar interaction strengths. In the left column, the dipolar interaction strengths are (a) hd ¼ 0:05hk , (b) hd ¼ 0:3hk , and (c) hd ¼ 0:7hk . Results for an external magnetic field applied in the y–z plane with y ¼ 0 , 45 , and 90 , respectively, are shown. In the right column, the dipolar interaction strengths are (d) hd ¼ 0:05hk , (e) hd ¼ 0:3hk , and (f) hd ¼ 0:7hk . Results for an external magnetic field applied in the x–y plane with f ¼ 0 and 45 , respectively, are shown.
ARTICLE IN PRESS C. Zhu et al. / Journal of Magnetism and Magnetic Materials 320 (2008) 285–291
288
However, as the dipolar interaction increases, the hysteresis loop at weak dipolar coupling splits into three loops. As an example, we show the results for the case of hd ¼ 0:3hk in Fig. 2(b). In this case, the dipolar interaction apparently affects the magnetizing process. It induces more
1.0
energy barriers and wells so that there are more irreversible magnetizing paths. Therefore, the multi-loops magnetization are usually observed in dipolar interacting arrays with perpendicular anisotropy. If the dipolar interaction increases further, e.g., the hd ¼ 0:7hk case in Fig. 2(c), the 1.0
θ=0°
θ=0°
θ=45°
θ=45° 0.5
θ=90°
N=4 Mh/M0
Mh/M0
0.5 N=4
0.0
0.0
-0.5
-0.5
-1.0
-1.0
1.0
1.0 N=5 N=5
0.5 Mh/M0
Mh/M0
0.5
0.0
0.0
-0.5
-0.5
-1.0
-1.0
1.0
1.0 N=6 N=6
0.5 Mh/M0
Mh/M0
0.5
0.0
0.0
-0.5
-0.5
-1.0
-1.0
1.0
1.0 N=7 N=7
0.5 Mh/M0
Mh/M0
0.5
0.0
0.0
-0.5
-0.5
-1.0
-1.0 -3
-2
-1
0 he/ms
1
2
3
-3
-2
-1
0 he/ms
1
2
3
Fig. 3. Hysteresis loops for N N arrays with N ¼ 4; . . . ; 7 for hd ¼ 0:7hk . In (a)–(d), the external magnetic field is applied in the y–z plane with y ¼ 0 ; 45 and 90 , respectively. In (e)–(h), the external magnetic field is applied in the x–y plane with f ¼ 0 and 45 , respectively.
ARTICLE IN PRESS C. Zhu et al. / Journal of Magnetism and Magnetic Materials 320 (2008) 285–291
1.0
0.8 (A)
0.6
Mr/M0
Mr/M0
0.12 θ=90°
0.06 0.00
0.4
0
15
30
45
φ (°) 0.2 (B) 0.0
1.5
hs/M0
magnetization curve becomes anhysteresis when the external field is applied at y ¼ 0 . The anisotropy energy barrier is now suppressed by the strong dipolar interaction. The orientations of the magnetic moments are not controlled by the single-particle anisotropy in this case, but instead depend on the interaction between neighboring moments. The moments are no longer pointing perpendicular to the plane of the array. Thus, the abrupt jumps in Fig. 2(a) and (b) at y ¼ 0 , which are associated with the switching of some moments in the array [9], disappear when the dipolar interaction is strong. Interestingly, for strong dipolar coupling, hysteresis loops appear when the external field is applied at y ¼ 90 . It is because, while the external field tends to align the moments so that they are all parallel to each other, the strong dipolar interaction tends to align the moments parallel to the plane in the structure of antiparallel ferromagnetic chains or local vortices [10,20]. It is this competition between the dipolar interaction and Zeeman energy that leads to the in-plane hysteresis of magnetization. In Fig. 2(d)–(f), the dependence of the magnetization on the in-plane external magnetic field is also studied. Here, we choose two typical azimuthal angles of f ¼ 0 and 45 . Again, at weak dipolar interaction, the in-plane magnetization curve is closed. In this case, the competition between the single-particle anisotropy and Zeeman energy plays the dominant role. Since the easy anisotropy axes of the particles are perpendicular to the external magnetic field, the components of the magnetic moments along the þz- or z-direction are identical. Hence, the magnetization shows an anhysteresis behavior and it is almost independent of the direction of in-plane external magnetic field. For strong dipolar interaction, however, the hysteresis loop shows anisotropic behavior that is strongly dependent on the direction of the external magnetic field. This is due to the anisotropic nature of the dipolar interaction. Fig. 3 shows the magnetization in arrays of sizes 4 4, 5 5, 6 6, and 7 7 at a strong dipolar interaction of hd ¼ 0.7hk . For a given direction of the external field, the hysteresis loops are sensitive to the size of these small arrays. As the array increases in size, the number of dipole moments also increases. This changes the collective behavior of the moments, and shows up in the magnetizing behavior. Such complexity in the M h –he relation is a typical character in small arrays, and the sensitivity to size becomes weaker as the size becomes larger. (We have performed calculations on larger arrays, but the results are not shown here.) The corresponding dependence of the remanence and the switching field are investigated at weak and strong dipolar coupling strengths (hd ¼ 0.05hk in Fig. 4 and hd ¼ 0.7hk in Fig. 5). The switching field refers to the field that makes the magnetization along the external magnetic field transit from positive magnetization to negative magnetization. The reduced remanence keeps constant when he is applied in the x–z plane except for y 90 . The results imply that the magnetic moments are ordered in such a way that they point perpendicular to the
289
1.0
N=3 N=4 0.5
N=5 N=6 N=7
0.0
0
30
60
90
θ (°) Fig. 4. The dependence of reduced remanence M r /M 0 and reduced switching field hs =M 0 on y for hd ¼ 0:05hk and N ¼ 3; . . . ; 7. The external magnetic field is applied in the x–z plane. The inset shows the dependence of M r =M 0 on f for hd ¼ 0:05hk . The configurations (A) and (B) show the orientations of magnetic moments in the corresponding arrays.
plane, due to the strong perpendicular anisotropy (see configuration (A) in Fig. 4(a)). When he is close to in-plane or in-plane, the evolution of the magnetic moments will be affected by the weak dipolar interaction. As an example, consider a field he along the x-direction (y ¼ 90 ). Starting from an initial saturated state with all moments pointing along þx-direction, the moments will evolve to an antiferromagnetic state with all the magnetic moments pointing perpendicular to the plane (see configuration (B) in Fig. 4(a)) when he is swept to zero. The configuration (B) implies a reduced remanence of 1=N for odd N and 0 when even N. This is the reason that the reduced remanence
ARTICLE IN PRESS C. Zhu et al. / Journal of Magnetism and Magnetic Materials 320 (2008) 285–291
290
0.4
0.4
φ=45°
Mr/M0
Mr/M0
φ=0°
0.2
(B)
(A)
(C)
0.0
(D)
0.0 N=3 N=4 N=5 N=6
1.0
1.0
N=7
hs/M0
hs/M0
0.2
0.5
0.0
0.5
0.0 0
30
60
90
θ (°)
0
30
60
90
θ (°)
Fig. 5. The dependence of reduced remanence M r =M 0 and reduced switching field hs =M 0 on y for hd ¼ 0:7hk and N ¼ 3; . . . ; 7. The insets show the configurations of the magnetic moments. The configurations (A)–(D) show the orientations of magnetic moments in the corresponding arrays.
takes on different constant values for different system sizes, as shown in the inset of Fig. 4(a) for in-plane he . The reduced switching field decreases as the direction of he changes from y ¼ 0 to 90 (see Fig. 4(b)). The larger is the component of he in the x-direction, the smaller is the switching field. The results show that for he close to inplane, the system size effect is small for weak dipolar interactions. For strong dipolar interactions (see Fig. 5), both the reduced remanence and reduced switching field exhibit responses that are sensitive to the direction of the applied magnetic field. In Fig. 5(a) and (b), the external magnetic field is applied at different polar angles y for a fixed azimuthal angle f ¼ 0 . For N ¼ 3 and 4, M r =M 0 takes on a constant over the whole range of y. For N ¼ 3 and he 0, the magnetic moments evolve to a structure of antiparallel ferromagnetic chains as shown in the configuration (A) in Fig. 5(a). This configuration gives M r =M 0 13. For N ¼ 4, however, when he approaches 0, the magnetic moments evolve to a vortex structure as shown in the configuration (C) in Fig. 5(c). This configuration gives M r =M 0 0. For N ¼ 5, 6, and 7, M r =M 0 show sharp changes at certain values of the angles y. The results indicate that the configuration of magnetic moments corresponding to the remanent state changes as the direction of the sweeping field varies. Fig. 5(b) shows the reduced switching field as a function of y for fixed f ¼ 0 . To explain the features, we again take
N ¼ 3 and 4 as examples. For N ¼ 3, the component of he along the x-direction is small when y is small. The field cannot change the in-plane magnetic-moment configuration (see (A) in Fig. 5(a)). The component of he along the zdirection will lead to a non-vanishing magnetization. If he is strong enough, the magnetization will have a positive component along the external magnetic field so that the magnetization is switched by the external magnetic field. As y increases, the component of he in the z-direction decreases, for fixed magnitude he . Therefore, we need to increase he in order to switch the magnetization. So the switching field increases with y. However, when y is increased further, the x-component of he becomes large enough to change the magnetic moments from the configuration (see (A) in Fig. 5(a)) of six moments pointing approximately along the þx-direction and three moments pointing approximately along the x-direction to the configuration of having three moments pointing along the þx-direction and six moments along the x-direction. When y is increased further, the x-component of he increases for fixed magnitude he . Thus, a smaller he is sufficient to switch the magnetization. Therefore, we see a decreasing behavior in hs =M 0 as y is further increased. For N ¼ 4, the magnetic moments show a local vortex structure (see (C) in Fig. 5(c)) in the remanent state. When an external magnetic field is applied, the in-plane component of the magnetization is zero and the z-component of the magnetization is non-zero as long as he a0, and hence
ARTICLE IN PRESS C. Zhu et al. / Journal of Magnetism and Magnetic Materials 320 (2008) 285–291
causing a switching of the magnetization along he . So the switching field hs approaches zero for N ¼ 4 (see Fig. 5(b)). The sharp changes of hs =M 0 for N ¼ 5; 6, and 7 are related to the changes in the configuration of the magnetic moments at the remanent state. When the external magnetic field is swept through different values of y at a fixed angle f ¼ 45 (see Fig. 5(c) and (d)), the reduced remanence shows sharp changes at small y for all the system sizes considered here. Comparing the cases of N ¼ 3 in Fig. 5(c) to (a), the magnetic moments can evolve to a vortex-like structure (see (B) in Fig. 5(a) at small y for f ¼ 45 ), and thus the remanence decreases. For N ¼ 4, the magnetic moments can evolve to a chain-like structure (see configuration (D) in Fig. 5(c)) at large y (f ¼ 45 ), and hence the system could have a non-zero remanence. The behavior of hs =M 0 versus y shown in Fig. 5(d) is similar to the case of N ¼ 3 in Fig. 5(b), except that there are sharp changes at small values of the angle y. This is also related to the change in the configuration of the magnetic moments at the remanent state. 4. Summary We examined the hysteresis in a planar array of coupled magnetic moments by solving the LLG equations numerically.. The magnetic moment of each particle can point at any direction in space. The hysteresis loops of small arrays of nanoparticles are sensitive to the direction of the external magnetic field. Due to dipolar interaction, the system size effect is significant in small arrays. The magnetization is determined by the competition among the dipolar interaction, perpendicular anisotropy, and Zeeman energy. At weak dipolar interaction, the hysteresis loop closes for an in-plane external magnetic field. For strong dipolar interaction, the magnetization becomes anhysteresis when the external magnetic field is applied perpendicular to the plane containing the array. Multiloops in the magnetization curve appear at moderate and strong dipolar interactions. The remanence and the switching field show strong dependence on the direction of the applied field and thus these quantities are anisotropic. Such
291
anisotropy becomes significant at strong dipolar couplings. These features may be useful in the applications of small arrays as devices, and different behavior would result by controlling the dipolar coupling strength between particles. Acknowledgments This work was supported by the Jiangsu provincial Education Foundation under Grant no. 06KJD140165. We thank P.M. Hui for suggestions on revising an earlier version of the paper. References [1] H.S. Nalwa (Ed.), Magnetic Nanostructures, American Scientific Publishers, 2002. [2] J.I. Martı´ n, J. Nogue´s, K. Liu, J.L. Vicent, I.K. Schuller, J. Magn. Magn. Mater. 256 (2003) 449. [3] J. Bai, H. Takahoshi, H. Ito, H. Satio, S. Ishio, J. Appl. Phys. 91 (2004) 6848. [4] C.A. Ross, et al., J. Appl. Phys. 91 (2002) 6848; M.C. Abraham, et al., J. Appl. Phys. 89 (2001) 5667. [5] X. Zhu, P. Gru¨tter, V. Metlushko, B. Ilic, Phys. Rev. B 66 (2002) 024423. [6] C.A. Ross, et al., Phys. Rev. B 65 (2002) 144417. [7] S. Sun, C.B. Murray, D. Weller, L. Folks, A. Moser, Science 287 (2000) 1989. [8] L.F. Zhang, C. Xu, P.M. Hui, Y.Q. Ma, J. Appl. Phys. 97 (2005) 103912. [9] C. Xu, P.M. Hui, L.F. Zhang, Y.Q. Ma, J.H. Zhou, Z.Y. Li, Eur. Phys. J. B 46 (2005) 475. [10] M. Amin Kayali, W.M. Saslow, Phys. Rev. B 70 (2004) 174404. [11] J. Garcı´ a-Otero, M. Porto, J. Rivas, A. Bunde, Phys. Rev. Lett. 84 (2000) 167. [12] D. Kechrakos, K.N. Trohidou, Phys. Rev. B 58 (1998) 12169. [13] M. El-Hilo, K. O’Grady, R.W. Chantrell, J. Appl. Phys. 76 (1994) 6811. [14] R.L. Stamps, R.E. Camley, Phys. Rev. B 60 (1999) 11694. [15] R.L. Stamps, R.E. Camley, J. Magn. Magn. Mater. 177–181 (1998) 813. [16] Paola R Arias, D. Altbir, M. Bahiana, J. Phys. Condens. Matter 17 (2005) 1625. [17] L.F. Zhang, C. Xu, Y.Q. Ma, Phys. Lett. A 338 (2005) 373. [18] C.A. Ross, et al., Phys. Rev. B 65 (2000) 144417. [19] Dieter Weller, et al., IEEE Trans. Magn. 36 (2000) 10. [20] K. De’Bell, A.B. MacIsaac, J.P. Whitehead, Rev. Mod. Phys. 72 (2000) 225.