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Superconductor Bilayer Structure
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ScienceDirect Physics Procedia 65 (2015) 85 – 88
27th International Symposium on Superconductivity, ISS 2014
Vortex Structure in Chiral Helimagnet /Superconductor Bilayer Structure Saoto Fukuia*, Masaru Katob, Yoshihiko Togawac a Department of Materials Science, Osaka Prefecture University, 1-1 Gakuen-cho, Naka-ku, Sakai 599-8531, Japan Department of Mathematical Sciences, Osaka Prefecture University, 1-1 Gakuen-cho, Naka-ku, Sakai 599-8531, Japan c Department of Physics and Electronics, Osaka Prefecture University, 1-1 Gakuen-cho, Naka-ku, Sakai 599-8531, Japan b
Abstract We studied vortex structures in a chiral helimagnet/ superconductor bilayer structure by solving the Ginzburg-Landau equations. We found that vortex structures depend on the magnetic distribution in the chiral helimagnet. In addition, when the chiral helimagnet forms a chiral soliton lattice where the helical rotation of the magnetizations loosens periodically under an external magnetic field, vortices gather in the region where the directions of the magnetization are parallel to the external magnetic field. It is expected that vortices cannot move from one region to the next region, so that the pinning effect appears. © by by Elsevier B.V.B.V. This is an open access article under the CC BY-NC-ND license © 2015 2015The TheAuthors. Authors.Published Published Elsevier (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the ISS 2014 Program Committee. Peer-review under responsibility of the ISS 2014 Program Committee Keywords: superconductor / magnet bilayer; chiral helimagnet; Ginzburg-Landau equations; vortex; finite element method
1. Introduction Studying vortex structures is one of important elements for the application of a superconductor. One of topics in these studies is changes of external environments. For example, it is known theoretically and experimentally that in a ferromagnet/ superconductor bilayer structure, the magnetic structure in the ferromagnet affects the distribution of vortices and the change of the magnetization in the superconductor under an external field [1]. These vortex structures in the ferromagnet/ superconductor layer structure have been studied actively, but there are few studies about the vortex structure with other magnets, for example, paramagnets, anti-ferromagnets, or chiral helimagnets. We focus on the chiral helimagnet [2]. The chiral helimagnet is a magnet whose spins align like the helical rotation along one direction. The origin of this structure comes from the competition between the ferromagnetic exchange interaction and the Dzyaloshinsky-Moriya interaction, which tilts spins slightly. We expect that the chiral helimagnet/ superconductor layer structure has a unique vortex structure. In addition, when an external magnetic field is applied, the helical rotation of the spins in the chiral helimagnet loosens periodically, which is called a chiral soliton lattice. In addition, we expect the change of the vortex structure due to the change of the period of the soliton lattice.
* Corresponding author. Tel.: +81-72-254-9368; fax: +81-72-254-9916. E-mail address: [email protected]
1875-3892 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the ISS 2014 Program Committee doi:10.1016/j.phpro.2015.05.133
86
Saoto Fukui et al. / Physics Procedia 65 (2015) 85 – 88
We study the vortex structure in the chiral helimagnet/ superconductor bilayer structure using the Ginzburg-Landau equations. In this report, we show the distributions of vortices at large magnetic fields where the period of the soliton lattice becomes longer. 2. Model We consider a bilayer system that consists of an s-wave superconductor and a chiral helimagnet. For simplicity, we assume the superconductor is a two dimensional system and the chiral helimagnet affects the superconductor through a magnetic field of the chiral helimagnets. We solve the Ginzburg-Landau equations. 2
D | \ |2 E | \ |2 \
4S J c
1 § 2e · ¨ A¸ \ 4m © i c ¹
0,
(1)
4S c
e ½ 2e2 * \ *\ \\ * \ \ A¾ , (2) ® 2 mi mc ¯ ¿ where Ƚ ൌ ߙ ሺܶ െ ܶ ሻ, ܶ is a temperature, ܶ is a critical temperature, ߙ and ߚሺ Ͳሻ are coefficients, ߰ is the order parameter in the superconductor, ݉ is the electron mass, ݁ is the electron charge, A is a vector potential, J is a supercurrent density, and the magnetic field ࡴୣ୶୲ comes from the chiral helimagnet and an applied external field. The vector potential A is obtained from H = curl A. In order to obtain the magnetic field, we use the Hamiltonian for chiral helimagnets [1]; curl H Hext
H
J ¦ Sn Sn 1 D ¦ Sn u Sn 1 2PB H z ¦ Snz , n
(3)
n
n
where J is exchange coefficient, S is the spin in the chiral helimagnet, D is the vector value of the DzyaloshinskyMoriya interaction, ɊB is the Bohr magneton, and Happl is the applied magnetic field. By representing ܁ in terms of the polar coordinates Sn S (sin Tn cos I ,sin Tn sin I ,cosTn ) and performing the variation with respect to ߠ , we obtain the polar angle T x , § § H* ·· (4) T x 2sin 1 ¨ sn ¨ x | k ¸¸ S, ¸¸ ¨ ¨ k ¹¹ © © where the continuum limit is taken, and ሺݔȁ݇ሻ is the Jacobi’s elliptic function and k is the modulus (Ͳ ൏ ݇ ൏ ͳ), which is determined by
SM 4 H
*
E (k ) , k
(5)
where E(k) is the second kind complete elliptic integral, ߮ ൌ ିଵ ܦȀ ܬand כ ܪൌ ʹߤ Happl Ȁሺඥܬଶ ܦଶ ܵሻ. The period of the chiral soliton lattice L is determined the following equation:
L
2k K(k )
(6) , H* where K(k) is the first kind complete elliptic integral. The period of the chiral soliton lattice depends on the external field in Eq. (6). The relation between the magnetic field H and the period of the distribution of the magnetization in the chiral helimagnet in Eq. (6) is shown in Fig. 1. In Fig. 1, the period increases rapidly at the large magnetic field. When we solve the Ginzburg-Landau equations with the magnetic field from the chiral helimagnet, we use the finite element method [4]. In the finite element method, we divide the system into triangular elements and calculate the order parameter and the vector potential in each element.
Fig. 1. The relation between the period of the helix and the magnetic field H as a function of Eq. (6).
Saoto Fukui et al. / Physics Procedia 65 (2015) 85 – 88
3. Result We study two dimensional ʹߦܮ ൈ ͶͲߦ superconductors, where L is the period of the chiral soliton lattice and [ 0 is the coherence length at ൌ Ͳ , and we consider only the perpendicular magnetic field from magnetization of the chiral helimagnet. Also we consider long period cases where the external field is rather large. Magnetic field distributions are shown in Figs. 2(a)-(f). Under these field distributions, we solve the Ginzburg-Landau equations Eqs. (1) and (2) and we obtain \ ( x) and A( x) . These distributions of the order parameter are shown in Figs. 2(g), (h), and (i), for the field distributions of Figs. 2(a), (b), and (c), respectively. From these figures, we can see that the configurations of vortices depend on the magnetization distribution of the chiral helimagnet. In Figs. 2(g) and (h), there are no vortices in the small field region. From these results, we expect that vortices cannot move through the small field region, and that the magnetization of the chiral helimagnet restricts the movement of vortices. Therefore, the vortex pinning is expected. In Figs. 2(b) and (h), when the period of the chiral helimagnet becomes longer, a triangular lattice of the vortices is formed. In Fig. 2(i), however, when the external field increases, vortices appear even in the small field region and the distribution of the vortices is like that on the ferromagnet. Therefore, the distribution of the vortex structure can be controlled by the external field, which affects the period of the chiral soliton lattice.
Fig. 2. The distributions of magnetic field in the chiral helimagnet are shown for k=0.88795 (a), 0.951047 (b) and 0.97989 (c), respectively. The perpendicular magnetic field ሺࡴୣ୶୲ ሻ௭ in the chiral helimagnet are shown at the same values for k = 0.88795 (d), 0.951047 (e), and 0.97989 (f). The distributions of the order parameter in the superconductor are shown at the same values for k = 0.88795 (g), 0.951047 (h), and 0.97989 (i), where the dark spots show vortices.
4. Summary We studied the chiral helimagnet/ superconductor bilayer structure at the large magnetic field where the period of the soliton lattice becomes longer. We used Ginzburg-Landau equations and finite elements method, and found that the distribution of the vortices depends on the magnetic distribution of the chiral helimagnet. Therefore we expect that vortex structures can be controlled by the external field, which affects the period of the chiral soliton lattice. In the soliton lattice, vortices form triangle lattices in the large field and the longer region, and there are no vortices in the small field region. We expect that vortices cannot move to the next region, so the pinning effect is expected. In this report, we studied the vortex structure at large magnetic fields. In the future, we will study the vortex structure at small magnetic fields, where the antiparallel field regions exist. Moreover, we considered only the effect of the perpendicular magnetic field from magnetization of chiral helimagnet, so we will take into account the others component of the magnetic field. Finally, we neglected the effect of the superconductivity, such as the perfect diamagnetism, on the chiral helimagnet. Therefore, we will study the vortex structure in the case that the superconductor and the chiral helimagnet are affected with each other.
87
88
Saoto Fukui et al. / Physics Procedia 65 (2015) 85 – 88
Acknowledgements This work was supported by JSPS KAKENHI Grant Number 26400367. We thank to Y. Higashi, M. Kashiwagi, N. Fujita, M. Umeda, R. Onishi, N. Nakai, N. Hayashi, S. Mori, and Y. Ishii for useful discussion.
References [1] I. F. Lyuksyutov, V. L. Pokrovsky, Adv. Phys. 54 (2005) 1. [2] Y. Togawa, T. Koyama, K. Takayanagi, S. Mori, Y. Kousaka, J. Akimitsu, S. Nishihara, K. Inoue, A. S. Ovchinnikov, J. Kishine, Phys. Rev. Lett. 108 (2012) 108 [3] J. Kishine, K. Inoue, Y. Yoshida, Prog. Theor. Phys. Suppl. 159 (2005) 82 [4] M. Kato, T. Ishida, T. Koyama, M. Machida, A. Gabovich (Eds), Superconductors-Materials, Properties and Applications, Intech, 2013, pp 319342
ScienceDirect Physics Procedia 65 (2015) 85 – 88
27th International Symposium on Superconductivity, ISS 2014
Vortex Structure in Chiral Helimagnet /Superconductor Bilayer Structure Saoto Fukuia*, Masaru Katob, Yoshihiko Togawac a Department of Materials Science, Osaka Prefecture University, 1-1 Gakuen-cho, Naka-ku, Sakai 599-8531, Japan Department of Mathematical Sciences, Osaka Prefecture University, 1-1 Gakuen-cho, Naka-ku, Sakai 599-8531, Japan c Department of Physics and Electronics, Osaka Prefecture University, 1-1 Gakuen-cho, Naka-ku, Sakai 599-8531, Japan b
Abstract We studied vortex structures in a chiral helimagnet/ superconductor bilayer structure by solving the Ginzburg-Landau equations. We found that vortex structures depend on the magnetic distribution in the chiral helimagnet. In addition, when the chiral helimagnet forms a chiral soliton lattice where the helical rotation of the magnetizations loosens periodically under an external magnetic field, vortices gather in the region where the directions of the magnetization are parallel to the external magnetic field. It is expected that vortices cannot move from one region to the next region, so that the pinning effect appears. © by by Elsevier B.V.B.V. This is an open access article under the CC BY-NC-ND license © 2015 2015The TheAuthors. Authors.Published Published Elsevier (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the ISS 2014 Program Committee. Peer-review under responsibility of the ISS 2014 Program Committee Keywords: superconductor / magnet bilayer; chiral helimagnet; Ginzburg-Landau equations; vortex; finite element method
1. Introduction Studying vortex structures is one of important elements for the application of a superconductor. One of topics in these studies is changes of external environments. For example, it is known theoretically and experimentally that in a ferromagnet/ superconductor bilayer structure, the magnetic structure in the ferromagnet affects the distribution of vortices and the change of the magnetization in the superconductor under an external field [1]. These vortex structures in the ferromagnet/ superconductor layer structure have been studied actively, but there are few studies about the vortex structure with other magnets, for example, paramagnets, anti-ferromagnets, or chiral helimagnets. We focus on the chiral helimagnet [2]. The chiral helimagnet is a magnet whose spins align like the helical rotation along one direction. The origin of this structure comes from the competition between the ferromagnetic exchange interaction and the Dzyaloshinsky-Moriya interaction, which tilts spins slightly. We expect that the chiral helimagnet/ superconductor layer structure has a unique vortex structure. In addition, when an external magnetic field is applied, the helical rotation of the spins in the chiral helimagnet loosens periodically, which is called a chiral soliton lattice. In addition, we expect the change of the vortex structure due to the change of the period of the soliton lattice.
* Corresponding author. Tel.: +81-72-254-9368; fax: +81-72-254-9916. E-mail address: [email protected]
1875-3892 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the ISS 2014 Program Committee doi:10.1016/j.phpro.2015.05.133
86
Saoto Fukui et al. / Physics Procedia 65 (2015) 85 – 88
We study the vortex structure in the chiral helimagnet/ superconductor bilayer structure using the Ginzburg-Landau equations. In this report, we show the distributions of vortices at large magnetic fields where the period of the soliton lattice becomes longer. 2. Model We consider a bilayer system that consists of an s-wave superconductor and a chiral helimagnet. For simplicity, we assume the superconductor is a two dimensional system and the chiral helimagnet affects the superconductor through a magnetic field of the chiral helimagnets. We solve the Ginzburg-Landau equations. 2
D | \ |2 E | \ |2 \
4S J c
1 § 2e · ¨ A¸ \ 4m © i c ¹
0,
(1)
4S c
e ½ 2e2 * \ *\ \\ * \ \ A¾ , (2) ® 2 mi mc ¯ ¿ where Ƚ ൌ ߙ ሺܶ െ ܶ ሻ, ܶ is a temperature, ܶ is a critical temperature, ߙ and ߚሺ Ͳሻ are coefficients, ߰ is the order parameter in the superconductor, ݉ is the electron mass, ݁ is the electron charge, A is a vector potential, J is a supercurrent density, and the magnetic field ࡴୣ୶୲ comes from the chiral helimagnet and an applied external field. The vector potential A is obtained from H = curl A. In order to obtain the magnetic field, we use the Hamiltonian for chiral helimagnets [1]; curl H Hext
H
J ¦ Sn Sn 1 D ¦ Sn u Sn 1 2PB H z ¦ Snz , n
(3)
n
n
where J is exchange coefficient, S is the spin in the chiral helimagnet, D is the vector value of the DzyaloshinskyMoriya interaction, ɊB is the Bohr magneton, and Happl is the applied magnetic field. By representing ܁ in terms of the polar coordinates Sn S (sin Tn cos I ,sin Tn sin I ,cosTn ) and performing the variation with respect to ߠ , we obtain the polar angle T x , § § H* ·· (4) T x 2sin 1 ¨ sn ¨ x | k ¸¸ S, ¸¸ ¨ ¨ k ¹¹ © © where the continuum limit is taken, and ሺݔȁ݇ሻ is the Jacobi’s elliptic function and k is the modulus (Ͳ ൏ ݇ ൏ ͳ), which is determined by
SM 4 H
*
E (k ) , k
(5)
where E(k) is the second kind complete elliptic integral, ߮ ൌ ିଵ ܦȀ ܬand כ ܪൌ ʹߤ Happl Ȁሺඥܬଶ ܦଶ ܵሻ. The period of the chiral soliton lattice L is determined the following equation:
L
2k K(k )
(6) , H* where K(k) is the first kind complete elliptic integral. The period of the chiral soliton lattice depends on the external field in Eq. (6). The relation between the magnetic field H and the period of the distribution of the magnetization in the chiral helimagnet in Eq. (6) is shown in Fig. 1. In Fig. 1, the period increases rapidly at the large magnetic field. When we solve the Ginzburg-Landau equations with the magnetic field from the chiral helimagnet, we use the finite element method [4]. In the finite element method, we divide the system into triangular elements and calculate the order parameter and the vector potential in each element.
Fig. 1. The relation between the period of the helix and the magnetic field H as a function of Eq. (6).
Saoto Fukui et al. / Physics Procedia 65 (2015) 85 – 88
3. Result We study two dimensional ʹߦܮ ൈ ͶͲߦ superconductors, where L is the period of the chiral soliton lattice and [ 0 is the coherence length at ൌ Ͳ , and we consider only the perpendicular magnetic field from magnetization of the chiral helimagnet. Also we consider long period cases where the external field is rather large. Magnetic field distributions are shown in Figs. 2(a)-(f). Under these field distributions, we solve the Ginzburg-Landau equations Eqs. (1) and (2) and we obtain \ ( x) and A( x) . These distributions of the order parameter are shown in Figs. 2(g), (h), and (i), for the field distributions of Figs. 2(a), (b), and (c), respectively. From these figures, we can see that the configurations of vortices depend on the magnetization distribution of the chiral helimagnet. In Figs. 2(g) and (h), there are no vortices in the small field region. From these results, we expect that vortices cannot move through the small field region, and that the magnetization of the chiral helimagnet restricts the movement of vortices. Therefore, the vortex pinning is expected. In Figs. 2(b) and (h), when the period of the chiral helimagnet becomes longer, a triangular lattice of the vortices is formed. In Fig. 2(i), however, when the external field increases, vortices appear even in the small field region and the distribution of the vortices is like that on the ferromagnet. Therefore, the distribution of the vortex structure can be controlled by the external field, which affects the period of the chiral soliton lattice.
Fig. 2. The distributions of magnetic field in the chiral helimagnet are shown for k=0.88795 (a), 0.951047 (b) and 0.97989 (c), respectively. The perpendicular magnetic field ሺࡴୣ୶୲ ሻ௭ in the chiral helimagnet are shown at the same values for k = 0.88795 (d), 0.951047 (e), and 0.97989 (f). The distributions of the order parameter in the superconductor are shown at the same values for k = 0.88795 (g), 0.951047 (h), and 0.97989 (i), where the dark spots show vortices.
4. Summary We studied the chiral helimagnet/ superconductor bilayer structure at the large magnetic field where the period of the soliton lattice becomes longer. We used Ginzburg-Landau equations and finite elements method, and found that the distribution of the vortices depends on the magnetic distribution of the chiral helimagnet. Therefore we expect that vortex structures can be controlled by the external field, which affects the period of the chiral soliton lattice. In the soliton lattice, vortices form triangle lattices in the large field and the longer region, and there are no vortices in the small field region. We expect that vortices cannot move to the next region, so the pinning effect is expected. In this report, we studied the vortex structure at large magnetic fields. In the future, we will study the vortex structure at small magnetic fields, where the antiparallel field regions exist. Moreover, we considered only the effect of the perpendicular magnetic field from magnetization of chiral helimagnet, so we will take into account the others component of the magnetic field. Finally, we neglected the effect of the superconductivity, such as the perfect diamagnetism, on the chiral helimagnet. Therefore, we will study the vortex structure in the case that the superconductor and the chiral helimagnet are affected with each other.
87
88
Saoto Fukui et al. / Physics Procedia 65 (2015) 85 – 88
Acknowledgements This work was supported by JSPS KAKENHI Grant Number 26400367. We thank to Y. Higashi, M. Kashiwagi, N. Fujita, M. Umeda, R. Onishi, N. Nakai, N. Hayashi, S. Mori, and Y. Ishii for useful discussion.
References [1] I. F. Lyuksyutov, V. L. Pokrovsky, Adv. Phys. 54 (2005) 1. [2] Y. Togawa, T. Koyama, K. Takayanagi, S. Mori, Y. Kousaka, J. Akimitsu, S. Nishihara, K. Inoue, A. S. Ovchinnikov, J. Kishine, Phys. Rev. Lett. 108 (2012) 108 [3] J. Kishine, K. Inoue, Y. Yoshida, Prog. Theor. Phys. Suppl. 159 (2005) 82 [4] M. Kato, T. Ishida, T. Koyama, M. Machida, A. Gabovich (Eds), Superconductors-Materials, Properties and Applications, Intech, 2013, pp 319342