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Micromagnetic simulations of 200-nm-diameter cobalt nanorings using a Reuleaux triangular geometry
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Journal of Magnetism and Magnetic Materials 305 (2006) 133–140 www.elsevier.com/locate/jmmm
Micromagnetic simulations of 200-nm-diameter cobalt nanorings using a Reuleaux triangular geometry J.J. Torres-Heredia, F. Lo´pez-Urı´ as, E. Mun˜oz-Sandoval Advanced Materials Department, IPICYT, Camino a la presa San Jose´ 2055, Col. Lomas 4a seccio´n 78216 San Luis Potosı´, S.L.P., Mexico Received 28 June 2005; received in revised form 27 September 2005 Available online 9 January 2006
Abstract Using micromagnetic simulations, we investigated the magnetic states and switching processes of Co nanorings with lateral dimensions of 200 nm. We propose a special geometry of nanorings that adopts different Reuleaux triangular shapes. Reuleaux’s triangles (RT) combine both the equilateral triangle and circular geometries. We studied the magnetic spin configurations of individual nanorings by varying the thickness and geometry of the nanomagnets. Our results demonstrated that in most nanomagnets exhibiting a thickness of less than 4 nm, there exists an onion-type state, which precedes either a twisted, double twisted, or cardioid state, when studying the magnetization reversal process. The hysteresis loops and magnetic states found in these RTs are compared with circular nanorings. r 2006 Elsevier B.V. All rights reserved. PACS: 75.60.Jk; 75.60.d; 75.75.+a keywords: Micromagnetic simulations; Magnetic nanorings; Hysteresis
1. Introduction Due to a large number of technological expectations in the magnetic recording industry, magnetic circular nanorings have been intensively studied [1–8]. Because of their outstanding magnetic properties, such nanomagnets have been proposed as building blocks of ultrahigh density vertical magnetic random access memory [9,10]. Particularly, a circular geometry eliminates problems caused by the sharp ends and assures reproducibility and stability of the magnetic switching, resulting in negligible magnetostatic interactions between the circular magnetization (magnetic vortex states) of the nanorings when they are embedded in nanoarrays. In addition, the hole of the ring permits lower exchange energy and an enhanced circular magnetization configuration. If the effects of magnetocrystalline anisotropy are negligible, the geometry is predicted to determine the microscopic spin structure of the magnetic states [11,12]. For example, Kla¨ui et al. [13] showed the influence of geometry on the spin structure of the head to Corresponding author. Tel.: +52 444 834 2000; fax: +52 444 834 2010.
E-mail address: [email protected] (E. Mun˜oz-Sandoval). 0304-8853/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2005.12.002
head wall in both Co and NiFe nanorings. In this account, different patterns and shapes have been constructed and their magnetic properties studied [14–22]. In particular, domain configurations and magnetization reversal have been studied in square rings. These exhibit an interesting mechanism that is able to control the domain wall motion within the nanomagnets [23]. However, the circular ring shape is still the most popular geometry for researchers that study the magnetic properties of nanomagnets and their possible applications. Recently, the effects of different geometrical constrictions, such as notches, on domain walls in circular nanorings have been investigated by Klau¨i et al. [24]. They concluded that the domain walls can be pinned and stabilized by the notches that modify the magnetic properties. Although the onion and vortex spin configurations in circular nanorings have been found in most reported results [25–28], other interesting magnetic metastable states have been discovered. Recently, Castan˜o et al. [29,30] found a vortex containing a 3601 twisted state in thin Co nanorings. This magnetic configuration, which survives over a range of fields around remanence, could be useful for electronic, magnetotransport, and magnetic random access memory applications [10]. In this context,
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micromagnetic simulation has been a successful theoretical tool to verify the magnetic properties of real nanomagnets of different sizes, morphologies, and magnetic materials [31–35]. This success is because micromagnetic theory includes the calculation of the exchange, anisotropy, and both Zeeman and magnetostatic energies, which are the main factors used to determine the magnetic properties of materials. In particular, micromagnetic simulations that use two-dimensional oriented micromagnetic framework (OOMMF) software [36] have been extremely useful in predicting and verifying the experimental results of different nanomagnets [37,38]. In order to study the magnetic properties of Co nanomagnets with a particular geometry, and to observe the effects of distorting the shape of the nanorings, we investigated the magnetic properties of Co nanorings with an outer diameter (OD) of 200 nm, a width of 40 nm, exhibiting the RT morphology [39] with different roundness in its vertices. In some nanorings, we avoided the presence of sharp corners, which significantly affected the magnetization reversal process, the coercive field, and remanence. The construction of the RTs used in this paper was performed as follows: (1) For the conventional RT without roundness in its vertices (RT-0), we drew arcs from each vertex of an equilateral triangle between the other two vertices, and the resulting geometry can be observed in Fig. 1(a); (2) For the RT with half roundness (RT-1/2), we
drew, in each vertex, a circumference with a ratio equal to half of the side of the equilateral triangle, and subsequently drew arcs from each vertex that join the opposed circumferences (see the resulting geometry in Fig. 1(b)); (3) For the RT with one-third roundness (RT-1/3), we drew, in each vertex, a circle with a ratio equal to one-third of the side of the equilateral triangle (see Fig. 1(c)). Different RTs with different roundness in their vertices are obtained in this way. Finally, the central parts of the RTs are removed in order to obtain the nanorings. We systematically calculated the hysteresis loops of these rings with 200 nm of outer diameter by varying the thickness from 2 to 20 nm and a fixing the width of 40 nm. In addition, we calculated the hysteresis loops of circular nanorings with similar dimensions and thickness. The calculations were computed by oriented micromagnetic framework (OOMMF) software [36]. The polycrystalline cobalt parameters were used as follows [27,40,41]: the exchange coupling Ax ¼ 1:4 1011 J=m, the saturated magnetization M s ¼ 1400 103 A=m, and uniaxial anisotropy constant K ¼ 0. The RTs and circular nanorings were divided into squared cells with 2-nm lengths allowing the spins to be free and to rotate in three dimensions, which is less than the exchange length of the Co material. In all cases the magnetic field was applied in the plane. We used a large value for the damping parameter (0.5) to calculate quasistatic domain patterns [42,43]. 2. Results
Fig. 1. Three different geometries for Reuleaux’s triangles (RTs): (a) RT-0, in which the vertices have not been rounded; (b) RT-1/3, in which the vertices have been rounded by a circle of ratio equal to one-third of the side of an equilateral triangle; (c) RT-1/2, in which the vertices were rounded by a circle with a diameter equal to the side of an equilateral triangle.
Fig. 2 shows the calculated hysteresis loops of Co nanorings as a function of the geometry and the thickness (2, 4 and 10 nm) of the nanorings with either the RT morphology or circular shapes. It is clearly observed in this figure that there are four types of hysteresis loops, showing one (Fig. 2(g)), two (Figs. 2(f and i)–l), three (Fig. 2(c, e, and h)), or four (Figs. 2(a, b, and d)) switching processes. In general, when the thickness is less than or equal to 10 nm, the remanent state corresponded to an onion-type state for all RT morphologies. In all these nanorings, this onion-type spin configuration preceded the first switching at a positively applied magnetic field, in which the two transverse walls [11] of this onion-type state are formed in the lower vertices of the RTs (the first one in the right-hand side and the second one in the left-hand side). For circular nanoring, the onion state was formed with the usual halves of the ring representing symmetrical single magnetic domains with the net magnetization oriented in the direction of the applied magnetic field. Notice that for RT-1/3 with a 2-nm thickness (Fig. 2(g)), only one switching process was observed, corresponding to a direct transition from an onion-type to an inverse onion-type state with no intermediate spin configuration. Hysteresis loops of RT-0 with either 2-nm or 4-nm thickness (Fig. 2(j and k), respectively) present two switching processes, the first related to an onion-type-totwisted-state transition, which remains during a small
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Fig. 2. Hysteresis loops for both cobalt circular and Reuleaux’s triangle (RT) nanorings with dimensions of 200 nm. The columns indicate the results for three different thicknesses, t ¼ 2; 4, and 10 nm. The first row corresponds to circular nanorings. From the second to fourth row, the geometry of the RTs is different. The magnetic field was applied along the x direction. The results were obtained using micromagnetic simulation (OOMMF) software. A detailed description of (h) and (a) are given in Figs. 4 and 5, respectively.
interval when applying magnetic field. The second switching process is due to the change of this twisted spin configuration to a reverse, onion-type magnetic state. This twisted state was formed at the left inferior vertex of the RT-0, with a net magnetization of almost zero. The ferromagnetic RT-1/3 nanoring, with 2-nm thickness, exhibited only one abrupt magnetic transition (Fig. 2(g)); however, there was an intermediate metastable twisted spin configuration with a small range of stability that corresponded to 301 of the arc angle between the domain walls (aperture) [29,30]. A second switching process occurred when this twisted state was destroyed and a reverse onion-type spin configuration appeared. Three switching processes were observed in the circular ring with 10-nm thickness, and in both RT-1/2 and RT-1/3, each exhibiting a 4-nm thickness (Fig. 2(c, e, and h), respectively). In these cases, the first switching corresponded to the onion-to-twisted-state transition, which is almost imperceptible in the circular ring. However, a large increase in exchange energy and a small reduction of the magnetostatic energy occurred in the RT cases, contrary to the circle where this transition involved a significant drop in magnetostatic energy together with a small increase in the exchange energy. The second switching process in the circular 10-nm ring, caused the disappearance of the
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twisted state and the emergence of a vortex that reduced both the magnetostatic and exchange energies. In the RTs, the second switching process eliminated the typical twisted state (vortex state containing a 3601 wall) and a modifiedtwisted state emerged (reverse onion state containing a 3601 wall). This process increased the magnetostatic energy and decreased the exchange energy. Finally, in the third switching process, the vortex state and the modified-twisted state were eliminated, in both circular and RTs, respectively, in order to reach the onion-type state. It is worth pointing out that in all RTs of 10-nm thickness, the intermediate twisted magnetic state was not formed. For these RTs, the magnetization reversal process took place by nucleation and annihilation of the magnetic vortex state and no other intermediate magnetic states were witnessed (see Fig. 2(f, i, and l)). The structures of the corresponding hysteresis loops of these 10-nm RTs nanomagnets had similar features, but the switching fields shifted to the right as the RT shape became sharply curved. A detailed description of the magnetization reversal mechanism for the 4-nm RT-0 and RT-1/3, and the 2-nm RT-1/2 and circular nanorings are depicted in Figs. 3–6. Fig. 3 shows the simulated hysteresis loop for RT-0 nanorings, with an outer diameter of 200 nm and a thickness of 4 nm. The development of magnetic states during magnetization reversal is also represented. As the magnitude of the externally applied field was reduced from negative saturation (point a), RT-0 gradually changed its spin configuration (point b) to reach the onion-type state in which the walls are at the lower extremes of the nanoring (point c). Additional reduction of the applied field caused these walls to start moving to the center of the lower part of RT-0 (point d). However, this movement was not energetically favorable due to the sharpness of the vertices that caused a great increase in exchange energy. Therefore, the last onion-type spin configuration allowed is shown as point e. In order to reduce the exchange energy, the left wall was anchored to the vertex of RT-0, and the right wall approached it to form a twisted state (281 of aperture) at point f. The existence of this twisted state was of very short duration and was destroyed when reaching point h, immediately after point g, in which the twisted state was reduced in size (241 of aperture). At point h, the onion-type spin configuration developed to a saturated state, which then transitioned to a reverse-onion state (point i). Fig. 4 presents the hysteresis loop and different magnetic spin configurations for 4 nm-thickness RT-1/3. After saturation (point a) and the formation of an onion-type spin configuration (point b), the domain walls of the oniontype state moved to a lower part of RT-1/3 and reached point c where a second switching process occurred and the nanomagnet suffered an abrupt change in magnetization (25% of the total inversion). Additionally, the formation of the twisted state (point d) of 43-nm circumferential extension (351 of aperture), along with a vortex-type state, was centered in the middle of RT-1/3. If the magnetic field was increased, the center of this vortex moved to the top
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Fig. 3. Hysteresis loop for the cobalt Reuleaux’s triangle (RT) nanoring, without roundness (RT-0) and thickness of t ¼ 4 nm. The images representing the different magnetic states are shown in the upward part of the figure. The labels (a–i) in the images represent the spin configurations in the different regions of the hysteresis curve. The label (a) corresponds to the saturated state; (b) represents the spin configuration previous to the formation of the onion state (label (c)); labels (d and e) show the evolution of the onion state; (f and g) correspond to the initial and last twisted states in the left vertex, respectively; and label (i) is the reverse onion state. The details of the different states during the magnetization reversal process are explained in the text.
part of RT-1/3, and the dimensions of the twisted state were reduced at point e (301 of aperture) where the magnetization suffered a drastic change due to the disappearance of the vortex state and formation of an onion-type configuration (point f). From point f to point g, the magnetization is slightly increased, keeping the spin configuration fairly constant. Notice that the twisted state persisted for a long interval of the magnetic field, with smaller dimensions (251 of aperture), coexisting with an
Fig. 4. Hysteresis loop for the cobalt Reuleaux’s triangle (RT) nanoring, with 1/3 of roundness (RT-1/3) and thickness of t ¼ 4 nm. The images representing the magnetic states are shown in the upward part of the figure. The labels (a–h) represent the spin configurations in the different regions of the hysteresis curve. The labels (a and h) correspond to the saturated and almost-reverse saturated state, respectively. Labels (b and c) correspond to onion states, in which the position of the domain walls is different. Labels (d and e) correspond to twisted states in the inferior part, with the position of the center of the intrinsic vortex slightly upwards in the case of the spin configuration represented by (e). Labels (f and g) represent a spin configuration in which there is a combination of a twisted state and an onion state, as seen in the schematic (f and g) magnetic representation images. The details of the different states during the magnetization reversal process are explained in the text.
onion state. The effect of increasing the applied field was the reduction of the size of the twisted state (201 of aperture) and the onion state almost disappeared. Because of the increase and competition of both magnetostatic and exchange energies, this configuration was not stable within the field and a third switching process occurred, reducing the total energy (point g–h) as consequence of the destruction of the twisted and reverse onion-type state and the formation of the reverse saturated state. When the sharpness of the vertices was reduced in RTs, we found interesting spin configurations. Fig. 5 shows the sequence of the spin configurations for RT-1/2 with 2-nm thick together with the calculated hysteresis loops. Similar to RT-1/3, the evolution of the saturated state to the twisted state was observed in this case (point a–d). However, the twisted state formed at point d transformed
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Fig. 5. Similar to the previous figure, the hysteresis loop for the RT-1/2 nanoring, with a thickness of t ¼ 2 nm is represented. The images above represent the corresponding spin configurations, in which the arrows depict the magnetic moments. Labels (a and l) correspond to the saturated and inversed saturated state, respectively; (b and c) are the onion states before transition to the twisted state (label (d)); (e) depicts a twisted state with a smaller size, and (f) is the twisted state with the center of the vortex moved upwards prior to the formation of magnetic state (g), in which we easily observed the cardioid state; (h and i) show the evolution of this configuration with their reduced size of both the twisted and cardioid states, prior to the destruction of the twisted state in (j); (k) represents the isolated cardioid state that coexists with the nanoring in a nearly saturated state; and (l) is the saturated state.
to the spin configuration shown in point f, where it is possible to observe an upward movement of the center of the vortex bounded to the twisted state (in point e, a size reduction of the twisted state was observed). This configuration led to a large increase of both the exchange and magnetostatic energies, which allowed the nanoring to reach the spin configuration shown in point g. This complex spin configuration consisted of three different walls: (1) a 3601 wall in the lower part of RT-1/2; (2) a 1801 wall state, which we call the ‘‘cardioid’’ state [44], in the left part; and (3) head to head wall the right part. This configuration caused the exchange and magnetostatic
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Fig. 6. Hysteresis loop for a circular nanoring with a thickness of t ¼ 2 nm. The images above represent the respective spin configurations, with arrows depicting magnetic moments. Label (a) is the onion state; (b and c) show the movement of the walls to form the twisted state in (d); (e) is the vortex state and is distorted with its center vortex slightly moved upwards, and the twisted state is reduced its size; (f) shows a complicated spin configuration in which the formation of two twisted states, shared with a reverse onion state, is clearly seen; (g) represents the two twisted states; (h) represents the twisted state alone with the remaining nanorings in a saturated state; and (i) is the corresponding saturated state.
energies to consistently increase. This 1801 wall transverse-type spin configuration has been found previously in strips [10], but not in rings. The magnetization reversal process, with increasing the applied field, occurred via the reduction of the size of both the twisted and the cardioid states (points g and i). In the switching process, represented for the small increase in magnetization from point i to j, the twisted state was destroyed considerably reducing both exchange and magnetostatic energies. The cardioid state
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was destroyed when the remaining nanoring spins were aligned in the direction of the applied field and almost the entire ring was saturated. This transition reduced considerably the exchange energy with almost no changes in the magnetostatic energy. Notice that the cardioid-type spin configuration was very stable and survived at 400 mT where minute changes in magnetization were observed (point g–k). The twisted state disappeared at point j, but the cardioid state remained for an additional 170 mT (point j–k). In order to compare the magnetic configurations of RTs in Fig. 6, the hysteresis loop of circular nanorings with 2nm thickness is shown. The spin configuration was transformed from a saturated state to a typical onion state (point a), then the walls started to move to a lower part of the nanoring (point b). At an applied magnetic field of zero, the magnetization jumped from point b to c and a very unstable twisted-type state was formed (point c). This twisted state had a large size (651 of aperture). This size was gradually reduced when applying the magnetic field, which simultaneously increased the magnetostatic and exchange energies during a very short interval of applied magnetic field. The stability of this state finished with a jump in magnetization from point c to d. At this point, the twisted state was reduced to 351 of aperture. From point d to point e, the magnetization increased linearly from a negative value (point f) to a positive value (point h), which caused a slower increase in energy. During this interval, the spin configuration was a vortex, coexisting with the 3601 wall (twisted state), with the magnetization between point d and point e of zero, and the size of the twisted state of 251 of aperture. The movement of the center of the vortex upwards within the nanoring produced a sharp increase in the magnetization and is represented by the jump from point e to f. This process increased abruptly both the magnetostatic and exchange energies, resulting in an intense competition between them. The spin configuration, at point f, corresponded to two twisted states coexisting with a reverse-onion state (see the spin configuration f in Fig. 6). The stability of this magnetic configuration was large (more than 200 mT) and was destroyed when the superior twisted state was eliminated, leaving the inferior twisted state (configuration h of Fig. 6). Notice that the remaining spins of the ring pointed in the direction of the applied field. Finally, the saturated state was achieved at point i. In order to summarize the results in Fig. 7, we show the dependence of the switching fields as a function of the thickness for the different nanorings studied here. Fig. 7a depicts the case of circular nanorings. In this figure, we find that for nanorings with thickness larger than 10 nm, two types of switching field are present. The vortex to reverse onion switching field increases with increasing the thickness (as observed by Klau¨i et al. in circular nanorings with larger OD [45]). In these nanorings, the behavior of vortex to reverse onion switching is not affected as the thickness is increased. If the thickness is reduced (4 nmoto12 nm), an
intermediate metastable twisted state appears. Here, the switching fields of the transition between the twisted and vortex states increase with decreasing nanoring thickness. These twisted states have been experimentally observed in nanorings with diameters smaller than 500 nm [29,30] and in strips [46]. This twisted state has been modeled in GMR multilayer films [47] and in pseudo spin valves [48]. For thinner circular nanorings (to6 nm), a quadruple switching occurs: (1) onion to twisted state; (2) twisted state to double twisted coexisting with a reverse-onion state; (3) this double twisted to twisted state; (4) twisted state to saturated state. In the case of RT-1/2 (Fig. 7b), the switching behavior for thick RT-1/2 nanorings (t46 nm) is similar that the circular case, except that vortex to reverse onion transition switching field begins to be constant at 18 nm of thickness. For thinner RT-1/2 nanorings (to8 nm), triple and quadruple switching are obtained. In this geometry, the diminishing of thickness produces the movement of domain walls and complex states (cardioid, twisted plus onion, etc.) are formed. As it can see in Fig. 7b, the switching fields of that transitions increase with decreasing the thickness. Similar switching behavior appears for RT-1/3 (Fig. 7c), except that the nanoring with 2 nm of thick; corresponding to the onion to twisted and the twisted to reverse onion switching fields. In the case of RT-0 (Fig. 7d) as well as the previous cases, the vortex to reverse onion switching field increases with increasing thickness. However, the onion to vortex switching field is not constant anymore but it increases with increasing thickness. For thinner RT-0 nanorings (to6 nm) only a double switching is exhibited (onion to twisted and twisted to reverse onion). All our nanorings had an OD of 200 nm and we studied their magnetic behaviors as a function of the thickness and geometry. We observed that the geometrical shape and thickness of the nanoring produced different magnetization reversal processes. Hysteresis loops with one, two, three, or four switching processes could be found, depending on thickness and morphology. In general, the reversal of magnetization started with the formation of an onion-type state. The formation of twisted, double twisted, or cardioid metastable states generally increased both the magnetostatic and exchange energies, which produced an intense competition between them. This increase in energies was large and avoided the formation of isolated vortex states in thin nanorings. In almost all cases, there were walls that transitioned to form a twisted state, except in sharply curved nanorings (RT-0) in which a wall is pinned. We observed that the magnetization reversal process was governed by the magnestostatic energy, which is several times larger than the exchange energy; however, the changes in the exchange energy determined the changes in magnetization and spin configuration. The reduction of thickness caused twisted, double twisted, and cardioid states to emerge, and sometimes, more complicated spin configurations. These spin configurations became more stable as the thickness was diminished. Thin,
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Fig. 7. Switching fields of polycrystalline Co nanorings with 200 nm outer diameter as a function of thickness for different geometries; (a) circle; (b) RT-1/2; (c) RT-1/3; (d) RT-0. The meaning of notation is explained in the text.
isolated RT-1/2 nanorings presented cardioid-type magnetic states with relatively large stabilities. In thick nanorings, the onion and vortex states considerably reduced the magnetostatic and exchange energies as their spin configurations were observed. In summary, the two-dimensional micromagnetic code, OOMMF, was used to study the mechanism by which the magnetization within RT, reversed under the action of an externally applied magnetic field. The magnetic behavior of both RT and circular nanorings were presented as function of the thickness and geometrical effects. In particular, we observed the presence of both twisted and cardioid states with large stabilities. We identified that in most thin RT nanorings (thickness less than 10 nm), a twisted state was present and an onion-type state comprised the nucleation of that state. For thin circular nanorings, we found two twisted states that coexisted with an onion state. Isolated vortex states were not observed for either thin RTs or circular nanorings. It is possible to use RT, by introducing asymmetry into the ring by distorting the shape of the
circular ring, in order to pin the walls in certain places to control magnetization reversal and apply magnetic fields where switching processes should occur [49]. This opens the possibility to control different magnetic states by changing the geometry of RTs by rounding their vertices. Our preliminary results, obtained by studying arrays of two RT nanorings, show a wide spectrum of magnetic spin configurations, particularly the double twisted and cardioid states. There exists the possibility of using the effects of the interaction between thin RT nanorings to control the switching processes in order to fabricate build magnetoelectronic devices. Acknowledgements Authors gratefully acknowledge the financial support of CONACyT (Mexico) through Grants J36909, J41452-F, 39577-F, and 39643-F, 41464 (Inter American Materials Collaboration). The authors wish to thank Prof. M. Terrones for comments and fruitful discussions.
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Journal of Magnetism and Magnetic Materials 305 (2006) 133–140 www.elsevier.com/locate/jmmm
Micromagnetic simulations of 200-nm-diameter cobalt nanorings using a Reuleaux triangular geometry J.J. Torres-Heredia, F. Lo´pez-Urı´ as, E. Mun˜oz-Sandoval Advanced Materials Department, IPICYT, Camino a la presa San Jose´ 2055, Col. Lomas 4a seccio´n 78216 San Luis Potosı´, S.L.P., Mexico Received 28 June 2005; received in revised form 27 September 2005 Available online 9 January 2006
Abstract Using micromagnetic simulations, we investigated the magnetic states and switching processes of Co nanorings with lateral dimensions of 200 nm. We propose a special geometry of nanorings that adopts different Reuleaux triangular shapes. Reuleaux’s triangles (RT) combine both the equilateral triangle and circular geometries. We studied the magnetic spin configurations of individual nanorings by varying the thickness and geometry of the nanomagnets. Our results demonstrated that in most nanomagnets exhibiting a thickness of less than 4 nm, there exists an onion-type state, which precedes either a twisted, double twisted, or cardioid state, when studying the magnetization reversal process. The hysteresis loops and magnetic states found in these RTs are compared with circular nanorings. r 2006 Elsevier B.V. All rights reserved. PACS: 75.60.Jk; 75.60.d; 75.75.+a keywords: Micromagnetic simulations; Magnetic nanorings; Hysteresis
1. Introduction Due to a large number of technological expectations in the magnetic recording industry, magnetic circular nanorings have been intensively studied [1–8]. Because of their outstanding magnetic properties, such nanomagnets have been proposed as building blocks of ultrahigh density vertical magnetic random access memory [9,10]. Particularly, a circular geometry eliminates problems caused by the sharp ends and assures reproducibility and stability of the magnetic switching, resulting in negligible magnetostatic interactions between the circular magnetization (magnetic vortex states) of the nanorings when they are embedded in nanoarrays. In addition, the hole of the ring permits lower exchange energy and an enhanced circular magnetization configuration. If the effects of magnetocrystalline anisotropy are negligible, the geometry is predicted to determine the microscopic spin structure of the magnetic states [11,12]. For example, Kla¨ui et al. [13] showed the influence of geometry on the spin structure of the head to Corresponding author. Tel.: +52 444 834 2000; fax: +52 444 834 2010.
E-mail address: [email protected] (E. Mun˜oz-Sandoval). 0304-8853/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2005.12.002
head wall in both Co and NiFe nanorings. In this account, different patterns and shapes have been constructed and their magnetic properties studied [14–22]. In particular, domain configurations and magnetization reversal have been studied in square rings. These exhibit an interesting mechanism that is able to control the domain wall motion within the nanomagnets [23]. However, the circular ring shape is still the most popular geometry for researchers that study the magnetic properties of nanomagnets and their possible applications. Recently, the effects of different geometrical constrictions, such as notches, on domain walls in circular nanorings have been investigated by Klau¨i et al. [24]. They concluded that the domain walls can be pinned and stabilized by the notches that modify the magnetic properties. Although the onion and vortex spin configurations in circular nanorings have been found in most reported results [25–28], other interesting magnetic metastable states have been discovered. Recently, Castan˜o et al. [29,30] found a vortex containing a 3601 twisted state in thin Co nanorings. This magnetic configuration, which survives over a range of fields around remanence, could be useful for electronic, magnetotransport, and magnetic random access memory applications [10]. In this context,
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micromagnetic simulation has been a successful theoretical tool to verify the magnetic properties of real nanomagnets of different sizes, morphologies, and magnetic materials [31–35]. This success is because micromagnetic theory includes the calculation of the exchange, anisotropy, and both Zeeman and magnetostatic energies, which are the main factors used to determine the magnetic properties of materials. In particular, micromagnetic simulations that use two-dimensional oriented micromagnetic framework (OOMMF) software [36] have been extremely useful in predicting and verifying the experimental results of different nanomagnets [37,38]. In order to study the magnetic properties of Co nanomagnets with a particular geometry, and to observe the effects of distorting the shape of the nanorings, we investigated the magnetic properties of Co nanorings with an outer diameter (OD) of 200 nm, a width of 40 nm, exhibiting the RT morphology [39] with different roundness in its vertices. In some nanorings, we avoided the presence of sharp corners, which significantly affected the magnetization reversal process, the coercive field, and remanence. The construction of the RTs used in this paper was performed as follows: (1) For the conventional RT without roundness in its vertices (RT-0), we drew arcs from each vertex of an equilateral triangle between the other two vertices, and the resulting geometry can be observed in Fig. 1(a); (2) For the RT with half roundness (RT-1/2), we
drew, in each vertex, a circumference with a ratio equal to half of the side of the equilateral triangle, and subsequently drew arcs from each vertex that join the opposed circumferences (see the resulting geometry in Fig. 1(b)); (3) For the RT with one-third roundness (RT-1/3), we drew, in each vertex, a circle with a ratio equal to one-third of the side of the equilateral triangle (see Fig. 1(c)). Different RTs with different roundness in their vertices are obtained in this way. Finally, the central parts of the RTs are removed in order to obtain the nanorings. We systematically calculated the hysteresis loops of these rings with 200 nm of outer diameter by varying the thickness from 2 to 20 nm and a fixing the width of 40 nm. In addition, we calculated the hysteresis loops of circular nanorings with similar dimensions and thickness. The calculations were computed by oriented micromagnetic framework (OOMMF) software [36]. The polycrystalline cobalt parameters were used as follows [27,40,41]: the exchange coupling Ax ¼ 1:4 1011 J=m, the saturated magnetization M s ¼ 1400 103 A=m, and uniaxial anisotropy constant K ¼ 0. The RTs and circular nanorings were divided into squared cells with 2-nm lengths allowing the spins to be free and to rotate in three dimensions, which is less than the exchange length of the Co material. In all cases the magnetic field was applied in the plane. We used a large value for the damping parameter (0.5) to calculate quasistatic domain patterns [42,43]. 2. Results
Fig. 1. Three different geometries for Reuleaux’s triangles (RTs): (a) RT-0, in which the vertices have not been rounded; (b) RT-1/3, in which the vertices have been rounded by a circle of ratio equal to one-third of the side of an equilateral triangle; (c) RT-1/2, in which the vertices were rounded by a circle with a diameter equal to the side of an equilateral triangle.
Fig. 2 shows the calculated hysteresis loops of Co nanorings as a function of the geometry and the thickness (2, 4 and 10 nm) of the nanorings with either the RT morphology or circular shapes. It is clearly observed in this figure that there are four types of hysteresis loops, showing one (Fig. 2(g)), two (Figs. 2(f and i)–l), three (Fig. 2(c, e, and h)), or four (Figs. 2(a, b, and d)) switching processes. In general, when the thickness is less than or equal to 10 nm, the remanent state corresponded to an onion-type state for all RT morphologies. In all these nanorings, this onion-type spin configuration preceded the first switching at a positively applied magnetic field, in which the two transverse walls [11] of this onion-type state are formed in the lower vertices of the RTs (the first one in the right-hand side and the second one in the left-hand side). For circular nanoring, the onion state was formed with the usual halves of the ring representing symmetrical single magnetic domains with the net magnetization oriented in the direction of the applied magnetic field. Notice that for RT-1/3 with a 2-nm thickness (Fig. 2(g)), only one switching process was observed, corresponding to a direct transition from an onion-type to an inverse onion-type state with no intermediate spin configuration. Hysteresis loops of RT-0 with either 2-nm or 4-nm thickness (Fig. 2(j and k), respectively) present two switching processes, the first related to an onion-type-totwisted-state transition, which remains during a small
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Fig. 2. Hysteresis loops for both cobalt circular and Reuleaux’s triangle (RT) nanorings with dimensions of 200 nm. The columns indicate the results for three different thicknesses, t ¼ 2; 4, and 10 nm. The first row corresponds to circular nanorings. From the second to fourth row, the geometry of the RTs is different. The magnetic field was applied along the x direction. The results were obtained using micromagnetic simulation (OOMMF) software. A detailed description of (h) and (a) are given in Figs. 4 and 5, respectively.
interval when applying magnetic field. The second switching process is due to the change of this twisted spin configuration to a reverse, onion-type magnetic state. This twisted state was formed at the left inferior vertex of the RT-0, with a net magnetization of almost zero. The ferromagnetic RT-1/3 nanoring, with 2-nm thickness, exhibited only one abrupt magnetic transition (Fig. 2(g)); however, there was an intermediate metastable twisted spin configuration with a small range of stability that corresponded to 301 of the arc angle between the domain walls (aperture) [29,30]. A second switching process occurred when this twisted state was destroyed and a reverse onion-type spin configuration appeared. Three switching processes were observed in the circular ring with 10-nm thickness, and in both RT-1/2 and RT-1/3, each exhibiting a 4-nm thickness (Fig. 2(c, e, and h), respectively). In these cases, the first switching corresponded to the onion-to-twisted-state transition, which is almost imperceptible in the circular ring. However, a large increase in exchange energy and a small reduction of the magnetostatic energy occurred in the RT cases, contrary to the circle where this transition involved a significant drop in magnetostatic energy together with a small increase in the exchange energy. The second switching process in the circular 10-nm ring, caused the disappearance of the
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twisted state and the emergence of a vortex that reduced both the magnetostatic and exchange energies. In the RTs, the second switching process eliminated the typical twisted state (vortex state containing a 3601 wall) and a modifiedtwisted state emerged (reverse onion state containing a 3601 wall). This process increased the magnetostatic energy and decreased the exchange energy. Finally, in the third switching process, the vortex state and the modified-twisted state were eliminated, in both circular and RTs, respectively, in order to reach the onion-type state. It is worth pointing out that in all RTs of 10-nm thickness, the intermediate twisted magnetic state was not formed. For these RTs, the magnetization reversal process took place by nucleation and annihilation of the magnetic vortex state and no other intermediate magnetic states were witnessed (see Fig. 2(f, i, and l)). The structures of the corresponding hysteresis loops of these 10-nm RTs nanomagnets had similar features, but the switching fields shifted to the right as the RT shape became sharply curved. A detailed description of the magnetization reversal mechanism for the 4-nm RT-0 and RT-1/3, and the 2-nm RT-1/2 and circular nanorings are depicted in Figs. 3–6. Fig. 3 shows the simulated hysteresis loop for RT-0 nanorings, with an outer diameter of 200 nm and a thickness of 4 nm. The development of magnetic states during magnetization reversal is also represented. As the magnitude of the externally applied field was reduced from negative saturation (point a), RT-0 gradually changed its spin configuration (point b) to reach the onion-type state in which the walls are at the lower extremes of the nanoring (point c). Additional reduction of the applied field caused these walls to start moving to the center of the lower part of RT-0 (point d). However, this movement was not energetically favorable due to the sharpness of the vertices that caused a great increase in exchange energy. Therefore, the last onion-type spin configuration allowed is shown as point e. In order to reduce the exchange energy, the left wall was anchored to the vertex of RT-0, and the right wall approached it to form a twisted state (281 of aperture) at point f. The existence of this twisted state was of very short duration and was destroyed when reaching point h, immediately after point g, in which the twisted state was reduced in size (241 of aperture). At point h, the onion-type spin configuration developed to a saturated state, which then transitioned to a reverse-onion state (point i). Fig. 4 presents the hysteresis loop and different magnetic spin configurations for 4 nm-thickness RT-1/3. After saturation (point a) and the formation of an onion-type spin configuration (point b), the domain walls of the oniontype state moved to a lower part of RT-1/3 and reached point c where a second switching process occurred and the nanomagnet suffered an abrupt change in magnetization (25% of the total inversion). Additionally, the formation of the twisted state (point d) of 43-nm circumferential extension (351 of aperture), along with a vortex-type state, was centered in the middle of RT-1/3. If the magnetic field was increased, the center of this vortex moved to the top
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Fig. 3. Hysteresis loop for the cobalt Reuleaux’s triangle (RT) nanoring, without roundness (RT-0) and thickness of t ¼ 4 nm. The images representing the different magnetic states are shown in the upward part of the figure. The labels (a–i) in the images represent the spin configurations in the different regions of the hysteresis curve. The label (a) corresponds to the saturated state; (b) represents the spin configuration previous to the formation of the onion state (label (c)); labels (d and e) show the evolution of the onion state; (f and g) correspond to the initial and last twisted states in the left vertex, respectively; and label (i) is the reverse onion state. The details of the different states during the magnetization reversal process are explained in the text.
part of RT-1/3, and the dimensions of the twisted state were reduced at point e (301 of aperture) where the magnetization suffered a drastic change due to the disappearance of the vortex state and formation of an onion-type configuration (point f). From point f to point g, the magnetization is slightly increased, keeping the spin configuration fairly constant. Notice that the twisted state persisted for a long interval of the magnetic field, with smaller dimensions (251 of aperture), coexisting with an
Fig. 4. Hysteresis loop for the cobalt Reuleaux’s triangle (RT) nanoring, with 1/3 of roundness (RT-1/3) and thickness of t ¼ 4 nm. The images representing the magnetic states are shown in the upward part of the figure. The labels (a–h) represent the spin configurations in the different regions of the hysteresis curve. The labels (a and h) correspond to the saturated and almost-reverse saturated state, respectively. Labels (b and c) correspond to onion states, in which the position of the domain walls is different. Labels (d and e) correspond to twisted states in the inferior part, with the position of the center of the intrinsic vortex slightly upwards in the case of the spin configuration represented by (e). Labels (f and g) represent a spin configuration in which there is a combination of a twisted state and an onion state, as seen in the schematic (f and g) magnetic representation images. The details of the different states during the magnetization reversal process are explained in the text.
onion state. The effect of increasing the applied field was the reduction of the size of the twisted state (201 of aperture) and the onion state almost disappeared. Because of the increase and competition of both magnetostatic and exchange energies, this configuration was not stable within the field and a third switching process occurred, reducing the total energy (point g–h) as consequence of the destruction of the twisted and reverse onion-type state and the formation of the reverse saturated state. When the sharpness of the vertices was reduced in RTs, we found interesting spin configurations. Fig. 5 shows the sequence of the spin configurations for RT-1/2 with 2-nm thick together with the calculated hysteresis loops. Similar to RT-1/3, the evolution of the saturated state to the twisted state was observed in this case (point a–d). However, the twisted state formed at point d transformed
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Fig. 5. Similar to the previous figure, the hysteresis loop for the RT-1/2 nanoring, with a thickness of t ¼ 2 nm is represented. The images above represent the corresponding spin configurations, in which the arrows depict the magnetic moments. Labels (a and l) correspond to the saturated and inversed saturated state, respectively; (b and c) are the onion states before transition to the twisted state (label (d)); (e) depicts a twisted state with a smaller size, and (f) is the twisted state with the center of the vortex moved upwards prior to the formation of magnetic state (g), in which we easily observed the cardioid state; (h and i) show the evolution of this configuration with their reduced size of both the twisted and cardioid states, prior to the destruction of the twisted state in (j); (k) represents the isolated cardioid state that coexists with the nanoring in a nearly saturated state; and (l) is the saturated state.
to the spin configuration shown in point f, where it is possible to observe an upward movement of the center of the vortex bounded to the twisted state (in point e, a size reduction of the twisted state was observed). This configuration led to a large increase of both the exchange and magnetostatic energies, which allowed the nanoring to reach the spin configuration shown in point g. This complex spin configuration consisted of three different walls: (1) a 3601 wall in the lower part of RT-1/2; (2) a 1801 wall state, which we call the ‘‘cardioid’’ state [44], in the left part; and (3) head to head wall the right part. This configuration caused the exchange and magnetostatic
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Fig. 6. Hysteresis loop for a circular nanoring with a thickness of t ¼ 2 nm. The images above represent the respective spin configurations, with arrows depicting magnetic moments. Label (a) is the onion state; (b and c) show the movement of the walls to form the twisted state in (d); (e) is the vortex state and is distorted with its center vortex slightly moved upwards, and the twisted state is reduced its size; (f) shows a complicated spin configuration in which the formation of two twisted states, shared with a reverse onion state, is clearly seen; (g) represents the two twisted states; (h) represents the twisted state alone with the remaining nanorings in a saturated state; and (i) is the corresponding saturated state.
energies to consistently increase. This 1801 wall transverse-type spin configuration has been found previously in strips [10], but not in rings. The magnetization reversal process, with increasing the applied field, occurred via the reduction of the size of both the twisted and the cardioid states (points g and i). In the switching process, represented for the small increase in magnetization from point i to j, the twisted state was destroyed considerably reducing both exchange and magnetostatic energies. The cardioid state
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was destroyed when the remaining nanoring spins were aligned in the direction of the applied field and almost the entire ring was saturated. This transition reduced considerably the exchange energy with almost no changes in the magnetostatic energy. Notice that the cardioid-type spin configuration was very stable and survived at 400 mT where minute changes in magnetization were observed (point g–k). The twisted state disappeared at point j, but the cardioid state remained for an additional 170 mT (point j–k). In order to compare the magnetic configurations of RTs in Fig. 6, the hysteresis loop of circular nanorings with 2nm thickness is shown. The spin configuration was transformed from a saturated state to a typical onion state (point a), then the walls started to move to a lower part of the nanoring (point b). At an applied magnetic field of zero, the magnetization jumped from point b to c and a very unstable twisted-type state was formed (point c). This twisted state had a large size (651 of aperture). This size was gradually reduced when applying the magnetic field, which simultaneously increased the magnetostatic and exchange energies during a very short interval of applied magnetic field. The stability of this state finished with a jump in magnetization from point c to d. At this point, the twisted state was reduced to 351 of aperture. From point d to point e, the magnetization increased linearly from a negative value (point f) to a positive value (point h), which caused a slower increase in energy. During this interval, the spin configuration was a vortex, coexisting with the 3601 wall (twisted state), with the magnetization between point d and point e of zero, and the size of the twisted state of 251 of aperture. The movement of the center of the vortex upwards within the nanoring produced a sharp increase in the magnetization and is represented by the jump from point e to f. This process increased abruptly both the magnetostatic and exchange energies, resulting in an intense competition between them. The spin configuration, at point f, corresponded to two twisted states coexisting with a reverse-onion state (see the spin configuration f in Fig. 6). The stability of this magnetic configuration was large (more than 200 mT) and was destroyed when the superior twisted state was eliminated, leaving the inferior twisted state (configuration h of Fig. 6). Notice that the remaining spins of the ring pointed in the direction of the applied field. Finally, the saturated state was achieved at point i. In order to summarize the results in Fig. 7, we show the dependence of the switching fields as a function of the thickness for the different nanorings studied here. Fig. 7a depicts the case of circular nanorings. In this figure, we find that for nanorings with thickness larger than 10 nm, two types of switching field are present. The vortex to reverse onion switching field increases with increasing the thickness (as observed by Klau¨i et al. in circular nanorings with larger OD [45]). In these nanorings, the behavior of vortex to reverse onion switching is not affected as the thickness is increased. If the thickness is reduced (4 nmoto12 nm), an
intermediate metastable twisted state appears. Here, the switching fields of the transition between the twisted and vortex states increase with decreasing nanoring thickness. These twisted states have been experimentally observed in nanorings with diameters smaller than 500 nm [29,30] and in strips [46]. This twisted state has been modeled in GMR multilayer films [47] and in pseudo spin valves [48]. For thinner circular nanorings (to6 nm), a quadruple switching occurs: (1) onion to twisted state; (2) twisted state to double twisted coexisting with a reverse-onion state; (3) this double twisted to twisted state; (4) twisted state to saturated state. In the case of RT-1/2 (Fig. 7b), the switching behavior for thick RT-1/2 nanorings (t46 nm) is similar that the circular case, except that vortex to reverse onion transition switching field begins to be constant at 18 nm of thickness. For thinner RT-1/2 nanorings (to8 nm), triple and quadruple switching are obtained. In this geometry, the diminishing of thickness produces the movement of domain walls and complex states (cardioid, twisted plus onion, etc.) are formed. As it can see in Fig. 7b, the switching fields of that transitions increase with decreasing the thickness. Similar switching behavior appears for RT-1/3 (Fig. 7c), except that the nanoring with 2 nm of thick; corresponding to the onion to twisted and the twisted to reverse onion switching fields. In the case of RT-0 (Fig. 7d) as well as the previous cases, the vortex to reverse onion switching field increases with increasing thickness. However, the onion to vortex switching field is not constant anymore but it increases with increasing thickness. For thinner RT-0 nanorings (to6 nm) only a double switching is exhibited (onion to twisted and twisted to reverse onion). All our nanorings had an OD of 200 nm and we studied their magnetic behaviors as a function of the thickness and geometry. We observed that the geometrical shape and thickness of the nanoring produced different magnetization reversal processes. Hysteresis loops with one, two, three, or four switching processes could be found, depending on thickness and morphology. In general, the reversal of magnetization started with the formation of an onion-type state. The formation of twisted, double twisted, or cardioid metastable states generally increased both the magnetostatic and exchange energies, which produced an intense competition between them. This increase in energies was large and avoided the formation of isolated vortex states in thin nanorings. In almost all cases, there were walls that transitioned to form a twisted state, except in sharply curved nanorings (RT-0) in which a wall is pinned. We observed that the magnetization reversal process was governed by the magnestostatic energy, which is several times larger than the exchange energy; however, the changes in the exchange energy determined the changes in magnetization and spin configuration. The reduction of thickness caused twisted, double twisted, and cardioid states to emerge, and sometimes, more complicated spin configurations. These spin configurations became more stable as the thickness was diminished. Thin,
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Fig. 7. Switching fields of polycrystalline Co nanorings with 200 nm outer diameter as a function of thickness for different geometries; (a) circle; (b) RT-1/2; (c) RT-1/3; (d) RT-0. The meaning of notation is explained in the text.
isolated RT-1/2 nanorings presented cardioid-type magnetic states with relatively large stabilities. In thick nanorings, the onion and vortex states considerably reduced the magnetostatic and exchange energies as their spin configurations were observed. In summary, the two-dimensional micromagnetic code, OOMMF, was used to study the mechanism by which the magnetization within RT, reversed under the action of an externally applied magnetic field. The magnetic behavior of both RT and circular nanorings were presented as function of the thickness and geometrical effects. In particular, we observed the presence of both twisted and cardioid states with large stabilities. We identified that in most thin RT nanorings (thickness less than 10 nm), a twisted state was present and an onion-type state comprised the nucleation of that state. For thin circular nanorings, we found two twisted states that coexisted with an onion state. Isolated vortex states were not observed for either thin RTs or circular nanorings. It is possible to use RT, by introducing asymmetry into the ring by distorting the shape of the
circular ring, in order to pin the walls in certain places to control magnetization reversal and apply magnetic fields where switching processes should occur [49]. This opens the possibility to control different magnetic states by changing the geometry of RTs by rounding their vertices. Our preliminary results, obtained by studying arrays of two RT nanorings, show a wide spectrum of magnetic spin configurations, particularly the double twisted and cardioid states. There exists the possibility of using the effects of the interaction between thin RT nanorings to control the switching processes in order to fabricate build magnetoelectronic devices. Acknowledgements Authors gratefully acknowledge the financial support of CONACyT (Mexico) through Grants J36909, J41452-F, 39577-F, and 39643-F, 41464 (Inter American Materials Collaboration). The authors wish to thank Prof. M. Terrones for comments and fruitful discussions.
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