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Magnetization patterns simulations of Fe, Ni, Co, and permalloy individual nanomagnets
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 294 (2005) e7–e12 www.elsevier.com/locate/jmmm
Magnetization patterns simulations of Fe, Ni, Co, and permalloy individual nanomagnets F. Lo´pez-Urı´ as, J.J. Torres-Heredia, E. Mun˜oz-Sandoval Advanced Materials Department, IPICYT, Camino a la presa San Jose´ 2055, Col. Lomas 4a Seccio´n, 78216 San Luis Potosı´, SLP, Mexico Available online 18 April 2005
Abstract Hysteresis loop behaviours are studied in circular, triangular and Reuleaux’s triangle (RT) of Fe, Co, Ni, and permalloy nanomagnets using micromagnetic simulations. The size and morphology of the nanomagnets are analyzed for three different thickness (10, 20, and 40 nm). For the triangle and RT shapes, our results reveal that for all magnetic material considered and in the low thickness (10 nm) the hysteresis prefer to be open, showing important coercive fields and remanence. However, when the thickness is increased (40 nm) almost all hysteresis loops are closed. Finally, the different mechanism of the magnetization reversal are investigated by monitoring the spin configuration as a function of the applied magnetic field. r 2005 Elsevier B.V. All rights reserved. PACS: 75.60.Ej; 75.60.jk; 75.75.+a Keywords: Nanostructures; Micromagnetic simulations; Nanomagnetism
1. Introduction Since two-dimensional magnetic nanoparticles were suggested for nanorecording or magnetic random access memories [1–3], the efforts to fabricate and study novel submicron magnetic particles have grown exponentially [4–7]. This search for understanding the underlying physics Corresponding author. Tel.: +52 444 8342000;
fax: +52 444 8342040. E-mail address: fl[email protected] (F. Lo´pez-Urı´ as).
in the equilibrium magnetic states and magnetization reversal has lead the community to test out different lithography and advanced fabrication techniques and modern experimental tools for magnetic characterization. In addition, recently several micromagnetic models have been developed to explain the magnetic configurations presented in these small systems [8,9]. In particular, the effects of the shape, size or material of the elements on the magnetization reversal and have intensively been studied using micromagnetic models using different typical geometrical structures
0304-8853/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2005.03.044
ARTICLE IN PRESS e8
F. Lo´pez-Urı´as et al. / Journal of Magnetism and Magnetic Materials 294 (2005) e7–e12
like squares, rectangular, circular, elliptical, diamond-shape [10–15]. In particular, the thickness dependence have been systematically studied in ferromagnetic squares [10] and in Permalloy rectangles [11]. Micromagnetic simulations using two-dimensional Object Oriented MicroMagnetic Framework (OOMMF) software [16] have been useful to predict and verify experimental results of some of these nanomagnets [17,18]. In order to investigate the influence of the shape of nanoelements and the magnetic material employed on the magnetization reversal process, here using OOMMF code we present the micromagnetic simulation of a new geometrical structure: the Reuleaux’s triangles (RT) [19] varying thickness for a fixed both ‘‘base’’ and ‘‘height’’ of 200 nm and compare with circular and triangular shapes of similar dimensions. The geometrical properties of this figure produce interesting spin configuration depending on thickness. The exchange couplings used in the calculation are AX ¼ 1:4, 1.3, 0.9, 2.1 1011 J/m, and the saturated magnetizations M S ¼ 1400, 860, 490 and 1700 103 A/m to Co, Py, Ni and Fe, respectively. Cobalt and Permalloy are selected in polycrystalline form and for Ni and Fe a cubic magnetic anisotropy constant of K 1 ¼ 5:7 and 48 103 J/m3, were respectively used. The RT is divided into squared cell with 2 nm of length where the spins are free to rotate in three dimensions. The damping parameter selected was 0.5 for all cases. The magnetic field is applied in the plane of the nanomagnets and it is varied from 0.03 to 0.03 T and reverse.
2. Results Fig. 1 shows the entire set of calculated hysteresis loops for Fe, Co, Ni, and permalloy nanomagnets with Reuleaux’s triangle, circle, and triangle geometry, and 10 nm of thickness. For the circle, the evolution of magnetic configuration as function of the applied magnetic field is the previously found in many research papers [5,7,20,21]. As the magnetic field is reduced from saturated state, two types of magnetization behaviour are found depending on the magnetic
material of nanoelements: the saturated state remains stable until a certain magnetic applied field where the magnetization is abruptly changed to zero due to vortex formation (Fe and Co) and where the magnetization reversal occurs by coherent rotation [5] (Ni) or buckling [21] (Py) without magnetic vortex formation. For the triangle, Fig. 1 shows only one type of hysteresis curve. However, two different reversal mechanisms are observed: one where a C state is the intermediate state before switching (Ni an Py case) and the other where a complex buckling state is the intermediate state before switching (Co and Fe nanomagnets). We have named this O state [see Fig. 2(a)]. As the magnetization pattern resembles an O spin arrangement. For RT, there are three types of hysteresis loops: (1) Fe and Co nanomagnets, where for Fe the C, O [see Fig. 2(b)], distorted O [Fig. 2(c)] and vortex states gradually appear in the magnetization reversal process, with the biggest coercive fields, and for Co, which is similar to the Fe case, except that the distorted O state is absent in the hysteresis loop and the coercive field is smaller; (2) Ni and Py nanomagnets, where there are no vortex nucleations and the switching with high remanence and a small coercive field is a characteristic of single domain magnetic response. In order to investigate the effect of the thickness on the magnetic properties of the nanomagnets, we perform micromagnetic simulations for 20 and 40 nm thickness. In Fig. 3, we show the results for 40 nm thickness. The magnetic spin configurations in these nanomagnets are typical (C, and vortex states). An important change is observed with respect to 10 nm nanomagnets for Ni and Py where the vortex is present now. With the increase of the thickness, dramatic changes are observed for the Co and Fe triangle case, where after an O state, a vortex configuration is nucleated and a abrupt change in magnetization is observed with zero remanence. Apparently there are no changes in the hysteresis loop for the Ni triangle case with respect to 10 nm thickness. However, the state before switching is not a C state but a distorted Y state [22]. Hysteresis loops for Fe and Co RT-shaped nanomagnets, show
ARTICLE IN PRESS F. Lo´pez-Urı´as et al. / Journal of Magnetism and Magnetic Materials 294 (2005) e7–e12
REULEAUX
CIRCLE
e9
TRIANGLE
Fe
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
Normalized magnetization
Co
Ni
Py
-0.03
0
0.03
-0.03 0 0.03 Magnetic field (T)
-0.03
0
0.03
Fig. 1. Hysteresis loops for the Reuleaux triangle (RT), circle, and the triangle for the Fe, Co, Ni, and permalloy nanomagnets with 10 nm of thickness.
Fig. 2. Illustration of representative spin arrangements for nanomagnets with a 10 nm of thickness: (a) Fe triangle, (b) Co Reuleaux’s triangle, and (c) Fe Reuleaux’s triangle.
ARTICLE IN PRESS F. Lo´pez-Urı´as et al. / Journal of Magnetism and Magnetic Materials 294 (2005) e7–e12
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REULEAUX
CIRCLE
TRIANGLE
Fe
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
Normalized magnetization
Co
Ni
Py
-0.03
0
0.03
-0.03 0 0.03 Magnetic field (T)
-0.03
0
0.03
Fig. 3. Hysteresis loops for the Reuleaux triangle(RT), circle, and the triangle for the Fe, Co, Ni, and permalloy nanomagnets with 40 nm of thickness.
important differences in the curve with respect to 10 nm thickness, but the reversal mechanism are the same. However, the effects on the coercive fields are important as can be seen in Fig. 3(a)–(c). The combination of geometric characteristic of triangles and circles in RT-shape nanomagnets produces an apparently new metastable configuration, the O state. The effect of the material element employed is the most important in Fe case where a multi-step switching on thin nanomagnets is
observed (see Fig. 4), and the H2M hysteresis loop obtained by the simulation and the magnetization configurations for Fe nanomagnets with RT-shape and 10 nm thickness is shown.
3. Conclusions In summary, the two-dimensional micromagnetic code OOMMF was used to study the
ARTICLE IN PRESS F. Lo´pez-Urı´as et al. / Journal of Magnetism and Magnetic Materials 294 (2005) e7–e12
e11
particular, on the nucleation of annihilation of the vortex state.
Acknowledgements Authors gratefully acknowledge the financial support of CONACyT (Mexico) through grants J36909-E, J41452-F, 39577-F, 39643-F, and G-25851-E. References
Fig. 4. Hysteresis loop for the Reuleaux’s triangle (RT) nanomagnet of Fe with a thickness of t ¼ 10 nm. The different labels (a–f) represent the spin configurations in the different regions of the hysteresis curve. The labels (a) and (f) correspond to the saturated or almost saturated states, (b) corresponds to ‘‘C’’ state, label (c) corresponds to the O state, labels (d) and (e) are vortex state configurations.
mechanism by which the magnetization within each nanomagnet formed fromFe, Co, Ni and Py with 10, 20 and 40 nm thickness, reverses under action of an external applied field. We identified that in most of the cases of triangles and RT a vortex state is presented and an O state is the scenario for the nucleation of that state. As a natural consequence of this work, we are study the RT-shaped nanorings with round edges, a modified version of RT, isolated and arrays, to investigate the effect of removing the central part on the magnetic properties of the nanomagnets, in
[1] C. Stamm, F. Marty, A. Vaterlaus, V. Weich, S. Egger, U. Maier, U. Ramsperger, H. Fuhrmann, D. Pescia, Science 282 (1998) 449. [2] J.G. Zhu, Y. Zheng, G.A. Prinz, J. Appl. Phys. 87 (2000) 6668. [3] X.C. Zhu, J.G. Zhu, IEEE Trans. Magn. 39 (2003) 2854. [4] R.P. Cowburn, A.O. Adeyeye, M.E. Welland, Phys. Rev. Lett. 81 (1998) 5414. [5] R.P. Cowburn, D.K. Koltsov, A.O. Adeyeye, M.E. Welland, D.M. Tricker, Phys. Rev. Lett. 83 (1999) 1042. [6] M. Hehn, K. Ounadjela, J.-P. Bucher, F. Rousseaux, D. Decanini, B. Bartenlian, C. Chappert, Science 272 (1996) 1782. [7] T. Shinjo, T. Okuno, R. Hassdorf, K. Shigeto, T. Ono, Science 289 (2000) 930. [8] O. Fruchart, B. Kevorkian, J.C. Toussaint, Phys. Rev. B 63 (2001) 174418. [9] W. Scholz, K. Yu. Guslienko, V. Novosad, D. Suess, T. Schefl, R.W. Chantrell, J. Fidler, J. Magn. Magn. Mater. 266 (2003) 155. [10] D. Goll, G. Schutz, H. Kronmuller, Phys. Rev. B 67 (2003) 094414. [11] R. Hertel, Z. Metall. 93 (2002) 957. [12] R.P. Cowburn, M.E. Welland, Phys. Rev. B 58 (1998) 9217. [13] P.H.W. Ridley, G.W. Roberts, R.W. Chantrell, J. Appl. Phys. 87 (2000) 5523. [14] R. Hertel, H. Kronmu¨ller, J. Appl. Phys. 85 (1999) 6190. [15] A. Lebib, S.P. Li, M. Natali, Y. Chen, J. Appl. Phys. 89 (2001) 3892. [16] M.J. Donahue, D.G. Porter, OOMMF User’s Guide, Version 1.0, Interagency Report NISTIR 6376, Gaithersburg, MD, 1999. [17] N. Dao, S.L. Whittenburg, R.P. Cowburn, J. Appl. Phys. 90 (2001) 5235. [18] An extense list of publications that have used OOMMF can found in http://math.nist.gov/oommf/bibliography. html [19] A Reuleaux’s triangle (RT) is a figure with the least area of any curve of given constant width [23]. It is easily constructed by drawing arcs from each vertex of an
ARTICLE IN PRESS e12
F. Lo´pez-Urı´as et al. / Journal of Magnetism and Magnetic Materials 294 (2005) e7–e12
equilateral triangle between the other two vertices. The resultant figure has a constant width equal to the length of the interior triangle side. [20] G. Gubbiotti, G. Carlotti, F. Nizzoli, R. Zivieri, T. Okuno, T. Shinjo, IEEE Trans. Magn. 38 (2002) 2532.
[21] J.K. Ha, R. Hertel, J. Kirschner, Phys. Rev. B 67 (2003) 224432. [22] D.K. Koltsov, R.P. Cowburn, M.E. Welland, J. Appl. Phys. 88 (2000) 5315. [23] W. Blaschke, Math. Ann. 76 (1915) 504.
Journal of Magnetism and Magnetic Materials 294 (2005) e7–e12 www.elsevier.com/locate/jmmm
Magnetization patterns simulations of Fe, Ni, Co, and permalloy individual nanomagnets F. Lo´pez-Urı´ as, J.J. Torres-Heredia, E. Mun˜oz-Sandoval Advanced Materials Department, IPICYT, Camino a la presa San Jose´ 2055, Col. Lomas 4a Seccio´n, 78216 San Luis Potosı´, SLP, Mexico Available online 18 April 2005
Abstract Hysteresis loop behaviours are studied in circular, triangular and Reuleaux’s triangle (RT) of Fe, Co, Ni, and permalloy nanomagnets using micromagnetic simulations. The size and morphology of the nanomagnets are analyzed for three different thickness (10, 20, and 40 nm). For the triangle and RT shapes, our results reveal that for all magnetic material considered and in the low thickness (10 nm) the hysteresis prefer to be open, showing important coercive fields and remanence. However, when the thickness is increased (40 nm) almost all hysteresis loops are closed. Finally, the different mechanism of the magnetization reversal are investigated by monitoring the spin configuration as a function of the applied magnetic field. r 2005 Elsevier B.V. All rights reserved. PACS: 75.60.Ej; 75.60.jk; 75.75.+a Keywords: Nanostructures; Micromagnetic simulations; Nanomagnetism
1. Introduction Since two-dimensional magnetic nanoparticles were suggested for nanorecording or magnetic random access memories [1–3], the efforts to fabricate and study novel submicron magnetic particles have grown exponentially [4–7]. This search for understanding the underlying physics Corresponding author. Tel.: +52 444 8342000;
fax: +52 444 8342040. E-mail address: fl[email protected] (F. Lo´pez-Urı´ as).
in the equilibrium magnetic states and magnetization reversal has lead the community to test out different lithography and advanced fabrication techniques and modern experimental tools for magnetic characterization. In addition, recently several micromagnetic models have been developed to explain the magnetic configurations presented in these small systems [8,9]. In particular, the effects of the shape, size or material of the elements on the magnetization reversal and have intensively been studied using micromagnetic models using different typical geometrical structures
0304-8853/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2005.03.044
ARTICLE IN PRESS e8
F. Lo´pez-Urı´as et al. / Journal of Magnetism and Magnetic Materials 294 (2005) e7–e12
like squares, rectangular, circular, elliptical, diamond-shape [10–15]. In particular, the thickness dependence have been systematically studied in ferromagnetic squares [10] and in Permalloy rectangles [11]. Micromagnetic simulations using two-dimensional Object Oriented MicroMagnetic Framework (OOMMF) software [16] have been useful to predict and verify experimental results of some of these nanomagnets [17,18]. In order to investigate the influence of the shape of nanoelements and the magnetic material employed on the magnetization reversal process, here using OOMMF code we present the micromagnetic simulation of a new geometrical structure: the Reuleaux’s triangles (RT) [19] varying thickness for a fixed both ‘‘base’’ and ‘‘height’’ of 200 nm and compare with circular and triangular shapes of similar dimensions. The geometrical properties of this figure produce interesting spin configuration depending on thickness. The exchange couplings used in the calculation are AX ¼ 1:4, 1.3, 0.9, 2.1 1011 J/m, and the saturated magnetizations M S ¼ 1400, 860, 490 and 1700 103 A/m to Co, Py, Ni and Fe, respectively. Cobalt and Permalloy are selected in polycrystalline form and for Ni and Fe a cubic magnetic anisotropy constant of K 1 ¼ 5:7 and 48 103 J/m3, were respectively used. The RT is divided into squared cell with 2 nm of length where the spins are free to rotate in three dimensions. The damping parameter selected was 0.5 for all cases. The magnetic field is applied in the plane of the nanomagnets and it is varied from 0.03 to 0.03 T and reverse.
2. Results Fig. 1 shows the entire set of calculated hysteresis loops for Fe, Co, Ni, and permalloy nanomagnets with Reuleaux’s triangle, circle, and triangle geometry, and 10 nm of thickness. For the circle, the evolution of magnetic configuration as function of the applied magnetic field is the previously found in many research papers [5,7,20,21]. As the magnetic field is reduced from saturated state, two types of magnetization behaviour are found depending on the magnetic
material of nanoelements: the saturated state remains stable until a certain magnetic applied field where the magnetization is abruptly changed to zero due to vortex formation (Fe and Co) and where the magnetization reversal occurs by coherent rotation [5] (Ni) or buckling [21] (Py) without magnetic vortex formation. For the triangle, Fig. 1 shows only one type of hysteresis curve. However, two different reversal mechanisms are observed: one where a C state is the intermediate state before switching (Ni an Py case) and the other where a complex buckling state is the intermediate state before switching (Co and Fe nanomagnets). We have named this O state [see Fig. 2(a)]. As the magnetization pattern resembles an O spin arrangement. For RT, there are three types of hysteresis loops: (1) Fe and Co nanomagnets, where for Fe the C, O [see Fig. 2(b)], distorted O [Fig. 2(c)] and vortex states gradually appear in the magnetization reversal process, with the biggest coercive fields, and for Co, which is similar to the Fe case, except that the distorted O state is absent in the hysteresis loop and the coercive field is smaller; (2) Ni and Py nanomagnets, where there are no vortex nucleations and the switching with high remanence and a small coercive field is a characteristic of single domain magnetic response. In order to investigate the effect of the thickness on the magnetic properties of the nanomagnets, we perform micromagnetic simulations for 20 and 40 nm thickness. In Fig. 3, we show the results for 40 nm thickness. The magnetic spin configurations in these nanomagnets are typical (C, and vortex states). An important change is observed with respect to 10 nm nanomagnets for Ni and Py where the vortex is present now. With the increase of the thickness, dramatic changes are observed for the Co and Fe triangle case, where after an O state, a vortex configuration is nucleated and a abrupt change in magnetization is observed with zero remanence. Apparently there are no changes in the hysteresis loop for the Ni triangle case with respect to 10 nm thickness. However, the state before switching is not a C state but a distorted Y state [22]. Hysteresis loops for Fe and Co RT-shaped nanomagnets, show
ARTICLE IN PRESS F. Lo´pez-Urı´as et al. / Journal of Magnetism and Magnetic Materials 294 (2005) e7–e12
REULEAUX
CIRCLE
e9
TRIANGLE
Fe
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
Normalized magnetization
Co
Ni
Py
-0.03
0
0.03
-0.03 0 0.03 Magnetic field (T)
-0.03
0
0.03
Fig. 1. Hysteresis loops for the Reuleaux triangle (RT), circle, and the triangle for the Fe, Co, Ni, and permalloy nanomagnets with 10 nm of thickness.
Fig. 2. Illustration of representative spin arrangements for nanomagnets with a 10 nm of thickness: (a) Fe triangle, (b) Co Reuleaux’s triangle, and (c) Fe Reuleaux’s triangle.
ARTICLE IN PRESS F. Lo´pez-Urı´as et al. / Journal of Magnetism and Magnetic Materials 294 (2005) e7–e12
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REULEAUX
CIRCLE
TRIANGLE
Fe
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
Normalized magnetization
Co
Ni
Py
-0.03
0
0.03
-0.03 0 0.03 Magnetic field (T)
-0.03
0
0.03
Fig. 3. Hysteresis loops for the Reuleaux triangle(RT), circle, and the triangle for the Fe, Co, Ni, and permalloy nanomagnets with 40 nm of thickness.
important differences in the curve with respect to 10 nm thickness, but the reversal mechanism are the same. However, the effects on the coercive fields are important as can be seen in Fig. 3(a)–(c). The combination of geometric characteristic of triangles and circles in RT-shape nanomagnets produces an apparently new metastable configuration, the O state. The effect of the material element employed is the most important in Fe case where a multi-step switching on thin nanomagnets is
observed (see Fig. 4), and the H2M hysteresis loop obtained by the simulation and the magnetization configurations for Fe nanomagnets with RT-shape and 10 nm thickness is shown.
3. Conclusions In summary, the two-dimensional micromagnetic code OOMMF was used to study the
ARTICLE IN PRESS F. Lo´pez-Urı´as et al. / Journal of Magnetism and Magnetic Materials 294 (2005) e7–e12
e11
particular, on the nucleation of annihilation of the vortex state.
Acknowledgements Authors gratefully acknowledge the financial support of CONACyT (Mexico) through grants J36909-E, J41452-F, 39577-F, 39643-F, and G-25851-E. References
Fig. 4. Hysteresis loop for the Reuleaux’s triangle (RT) nanomagnet of Fe with a thickness of t ¼ 10 nm. The different labels (a–f) represent the spin configurations in the different regions of the hysteresis curve. The labels (a) and (f) correspond to the saturated or almost saturated states, (b) corresponds to ‘‘C’’ state, label (c) corresponds to the O state, labels (d) and (e) are vortex state configurations.
mechanism by which the magnetization within each nanomagnet formed fromFe, Co, Ni and Py with 10, 20 and 40 nm thickness, reverses under action of an external applied field. We identified that in most of the cases of triangles and RT a vortex state is presented and an O state is the scenario for the nucleation of that state. As a natural consequence of this work, we are study the RT-shaped nanorings with round edges, a modified version of RT, isolated and arrays, to investigate the effect of removing the central part on the magnetic properties of the nanomagnets, in
[1] C. Stamm, F. Marty, A. Vaterlaus, V. Weich, S. Egger, U. Maier, U. Ramsperger, H. Fuhrmann, D. Pescia, Science 282 (1998) 449. [2] J.G. Zhu, Y. Zheng, G.A. Prinz, J. Appl. Phys. 87 (2000) 6668. [3] X.C. Zhu, J.G. Zhu, IEEE Trans. Magn. 39 (2003) 2854. [4] R.P. Cowburn, A.O. Adeyeye, M.E. Welland, Phys. Rev. Lett. 81 (1998) 5414. [5] R.P. Cowburn, D.K. Koltsov, A.O. Adeyeye, M.E. Welland, D.M. Tricker, Phys. Rev. Lett. 83 (1999) 1042. [6] M. Hehn, K. Ounadjela, J.-P. Bucher, F. Rousseaux, D. Decanini, B. Bartenlian, C. Chappert, Science 272 (1996) 1782. [7] T. Shinjo, T. Okuno, R. Hassdorf, K. Shigeto, T. Ono, Science 289 (2000) 930. [8] O. Fruchart, B. Kevorkian, J.C. Toussaint, Phys. Rev. B 63 (2001) 174418. [9] W. Scholz, K. Yu. Guslienko, V. Novosad, D. Suess, T. Schefl, R.W. Chantrell, J. Fidler, J. Magn. Magn. Mater. 266 (2003) 155. [10] D. Goll, G. Schutz, H. Kronmuller, Phys. Rev. B 67 (2003) 094414. [11] R. Hertel, Z. Metall. 93 (2002) 957. [12] R.P. Cowburn, M.E. Welland, Phys. Rev. B 58 (1998) 9217. [13] P.H.W. Ridley, G.W. Roberts, R.W. Chantrell, J. Appl. Phys. 87 (2000) 5523. [14] R. Hertel, H. Kronmu¨ller, J. Appl. Phys. 85 (1999) 6190. [15] A. Lebib, S.P. Li, M. Natali, Y. Chen, J. Appl. Phys. 89 (2001) 3892. [16] M.J. Donahue, D.G. Porter, OOMMF User’s Guide, Version 1.0, Interagency Report NISTIR 6376, Gaithersburg, MD, 1999. [17] N. Dao, S.L. Whittenburg, R.P. Cowburn, J. Appl. Phys. 90 (2001) 5235. [18] An extense list of publications that have used OOMMF can found in http://math.nist.gov/oommf/bibliography. html [19] A Reuleaux’s triangle (RT) is a figure with the least area of any curve of given constant width [23]. It is easily constructed by drawing arcs from each vertex of an
ARTICLE IN PRESS e12
F. Lo´pez-Urı´as et al. / Journal of Magnetism and Magnetic Materials 294 (2005) e7–e12
equilateral triangle between the other two vertices. The resultant figure has a constant width equal to the length of the interior triangle side. [20] G. Gubbiotti, G. Carlotti, F. Nizzoli, R. Zivieri, T. Okuno, T. Shinjo, IEEE Trans. Magn. 38 (2002) 2532.
[21] J.K. Ha, R. Hertel, J. Kirschner, Phys. Rev. B 67 (2003) 224432. [22] D.K. Koltsov, R.P. Cowburn, M.E. Welland, J. Appl. Phys. 88 (2000) 5315. [23] W. Blaschke, Math. Ann. 76 (1915) 504.