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Ginzburg-Landau Calculations of Circular Mo80Ge20 Plates with Sector Defect
Available online at www.sciencedirect.com
ScienceDirect Physics Procedia 81 (2016) 93 – 96
28th International Symposium on Superconductivity, ISS 2015, November 16-18, 2015, Tokyo, Japan
Ginzburg-Landau calculations of circular Mo80Ge20 plates with sector defect Vu The Danga, *, Ho Thanh Huya,b, Hitoshi Matsumotoa, Hiroki Miyoshia, Shigeyuki Miyajimaa,c, Hiroaki Shishidoa,c, Masaru Katoc,d, Takekazu Ishidaa,c b
a Department of Physics and Electronics, Osaka Prefecture University, Sakai, Osaka 599-8531, Japan Department of Physics and Electronics, University of Sciences, Vietnam National University HCMC, Vietnam c Institute for Nanofabrication Research, Osaka Prefecture University, Sakai, Osaka 599-8531, Japan d Department of Mathematical Science, Osaka Prefecture University, Sakai, Osaka 599-8531, Japan
Abstract We present the theoretical calculations on vortex structures in a nanosized superconducting circle with a deficit sector. The numerical calculations of Ginzburg–Landau equation has been carried out with the aid of the finite element method, which is convenient to treat an arbitrarily shaped superconductor. We found that the vortices form an arc structure or a partial shell structure in a deficient circle plate, and mirror symmetry can be seen with respect to the sector deficit. Due to the vortex-vortex interaction and the boundary confinement effect, we also found the evolution of double (outer and inner) shell structure as a function of vorticity. Our theoretical studies will be compared to the experimental studies. © Published by Elsevier B.V. B.V. This is an open access article under the CC BY-NC-ND license ©2016 2016The TheAuthors. Authors. Published by Elsevier (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the ISS 2015 Program Committee. Peer-review under responsibility of the ISS 2015 Program Committee Keywords: Vortex states; Ginzburg-Landau calculations; SQUID microscope
1.
Introduction
The distribution of vortices in mesoscopic superconducting disks, of which the size is comparable to the coherence length ξ and the magnetic penetration depth λ, has been explored to discover exotic vortex states. Such features cannot be observed for a bulk superconductor, but it could be caused in the mesoscopic system driven by the vortex–vortex interaction and the boundary–vortex interaction. The formation of different vortex configuration was dependent on geometry of pattern, the external magnetic field, and vorticity [1, 2, 3]. The symmetric vortex structure has been investigated by using the regular polygons with mirror symmetric lines. Several recent studies on mesoscopic disks have clarified the question of how vortices are arranged by the sample geometry. For a small circular disk, the formation of a ring-like structure was arranged under the influence of the sample boundary plays a
* Corresponding author Tel.: +81-72-254-9260 fax: +81-72-254-9498 E-mail address: [email protected]
1875-3892 © 2016 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 2015 Program Committee doi:10.1016/j.phpro.2016.04.036
94
Vu The Dang et al. / Physics Procedia 81 (2016) 93 – 96
crucial role on vortex distribution [2, 4]. In the case of the noncircular geometry such as an equilateral triangle, [5, 6] or a square [7, 8], the symmetry (C3, C4) of the boundary condition tends to impose the particular vortex arrangement in those system. For instance, with the same vorticity L, the vortex configuration is different in square and triangle plates [9]. Beside the standard pentagon, several studies introduced an artificial element such as a pinning center to make the vortex configuration asymmetric [1] which was used to observed the vortex interaction, or changing sides of pentagon to become star to make complex structure [10] which observed a distinction configuration of vortex in concave decagon. Both theoretical and experimental approaches have used to explore the distribution of vortices, i.e., the nonlinear Ginzburg–Landau theory and the London approximation were used [11, 12, 13] while scanning SQUID microscopy was applied to small superconducting Mo80Ge20 plates [3, 5]. In this paper, we present theoretical prediction about the distribution of vortex in a circle disk with sector defect (Pakman [14]). In this pattern, there are several features. (1) The super current is inflected when it follows along the sample edge of Pakman shapes. At the deep corner of the deficit sector, the magnetic field is weakened, and hence the vortices tend to penetrate into the interior of the plate, in which we consider that the sector-shaped defect acts as a gate way for allowing ͵ሺݔଷ ǡ ݕଷ ሻ the intentional entrance of vortices into the circle οଷ ʹሺݔଶ ǡ ݕଶ ሻ Ȝଷ disk. It is our interest to see not only the symmetry in οଶ Ȝଶ vortex configuration but also the influence of the ͳሺݔଵ ǡ ݕଵ ሻ open angle of the deficit sector upon the shell (b) οଵ structure of vortices. We consider that the number of Ȝଵ vortices to form shell structure was affected by the (a) angle of each sector defect because of the limited Fig.1.(a) Division of a Pakman superconductor into a number of triangular length of the partial shell (arc) length. The vortexelements; (b) a triangular element. At each node, values of an order vortex interaction and the supercurrent confinement parameter and a magnetic vector potential are defined. would lead to the formation of the two-shell structure. We found that the number of vortices at an outer shell was dependent on the number of inner-shell vortices when we increased the magnetic field systematically. We investigated the vortex profiles in the Pakman-shaped plate both theoretically and experimentally. Although we found a reasonable agreement between the theoretical predictions and the experimental findings by using a scanning SQUID microscope on the small Pakman plate, we would like to explain the theoretical results on vortex configuration in the present paper. 2. Theoretical formalism We used the Ginzburg-Landau (GL) calculation in our preceding studies [1, 12]. We rebuilt order parameter and structure of circular plates with sector defect by means of finite element method. We consider a Pakman-shaped plate consisting of twodimensional nanosized superconducting film, to which the external magnetic field H was applied perpendicular. In order to obtain stable vortex structures, the GL free energy is written in terms of a complex order parameter as: ଵ
ఈ
ଶ
ඥఉ
࣠ሺȟǡ ܣሻ ൌ ݀ ȳ ቈ ൬ඥߚȁȟȁଶ
ଶ
൰
ଵ ସ
ଶ
ቚቀ
ଶ
ܣቁ οቚ
ȁൈିுȁమ ଼గ
ଵ ଼గ
ሺ݀݅ܣݒሻଶ
(1)
where Δ(r) is an order parameter of a superconductor, A is the magnetic vector potential and H is the external magnetic field. The parameter β and α are a positive constant, α depends on temperature as α = α(0)(1- T/Tc). In this study, we consider that the temperature of the system is well below the transition temperature Tc, and hence is assumed to be positive. οሺݔǡ ݕሻ ൌ ܰଵ ሺݔǡ ݕሻοଵ ܰଶ ሺݔǡ ݕሻοଶ ܰଷ ሺݔǡ ݕሻοଷ
(2)
ሺݔǡ ݕሻ ൌ ܰଵ ሺݔǡ ݕሻଵ ܰଶ ሺݔǡ ݕሻଶ ܰଷ ሺݔǡ ݕሻଷ
(3)
where ܰ ሺݔǡ ݕሻ ൌ ሺܽ ܾ ݔ ܿ ݕሻȀʹܵ ሺ݅ ൌ ͳǡʹǡ͵ሻare area coordinates, and ȟ and ۯ are the values of the order parameter and the magnetic vector potential at i-th node, respectively (see Fig. 1b). ൌ ݔ ݕ , ൌ ݕ െ ݕ and ܥ ൌ ݔ െ ݕ are defined using coordinates of nodes ሺݔ ǡ ݕ ݖ ሻ of the triangular element and ܵ is an area of the element. Using this expansion for minimizing the free energy, we obtain following equations for the order parameter, ଶ σൣܲ ሺሼܣሽሻ ܲଶோ ሺሼοሽሻ൧ܴ݁ο ൣܳ ሺሼܣሽሻ ܳ ሺሼοሽሻ൧݉ܫο ൌ ܸோ ሺሼοሽሻ ଶ ଶூ σൣെܳ ሺሼܣሽሻ ܳ ሺሼοሽሻ൧ܴ݁ο ൣܲ ሺሼܣሽሻ ܲ ሺሼοሽሻ൧݉ܫο ൌ ܸூ ሺሼοሽሻ
(4) (5)
where the coefficients are defined as ܲ ሺሼܣሽሻ ൌ σఈୀ௫ǡ௬ ܭఈఈ σఈୀ௫ǡ௬ σభమ ܫభభ ܣభఈ ܣమఈ െ ோ ଶቀ ቁ ܲ ூ ሺሼܣሽሻ
ೕ కమ
σభమ ܫభమ ቆቀ͵ቁ ܴ݁ȟభ ܴ݁ȟమ ቀͳቁ ݉ܫȟభ ݉ܫȟమ ቇ ͳ ͵ ଶ ఈ ఈ ఈ ൯ܣ െ ܬ ܳ ሺሼܣሽሻ ൌ మ σఈୀ௫ǡ௬ σభ൫ܬ భ భ భ ൌ క
ଵ
కమ
(6) (7) (8)
95
Vu The Dang et al. / Physics Procedia 81 (2016) 93 – 96 ଶ ሺሼܣሽሻ ൌ ܳ
ܸ
ோ ଶቀ ቁ ூ ሺሼȟሽሻ
ଵ కమ
σభమ ܫభమ ݉ܫȟభ ܴ݁ȟమ
(9)
ܴ݁ȟయ ଶ ൰൱ ൌ మ σభమయ ܫభమయ ൭ȟభ ȟכమ ൬ క ݉ܫȟయ
(10)
Here, ߦ ൌ ߦ Ȁඥͳ െ ܶȀܶ is the coherence length at temperature T, where ߦ is the coherence length at zero temperature. Integrals ఈ …, are defined in Ref. [15]. For the magnetic vector potential of products of area coordinates over the element, such as ܫ , ܭఈఈ , ܬ A, we can get similar equation using integrals of area coordinates. The Maxwell equation is solved self-consistently with a randomly distributed initial order parameter structure and we obtain a stable state at fixed temperature and external field by solving Eqs. (4) and (5). Because of the nonlinearity of the GL equation, the free energies (Eq. (1)) of the locally stable states are determined as the most stable state by comparing several locally stables states obtained by the different initial states. Our numerical method is of the
(a)
(b)
(c)
(d)
(e)
(f)
Fig 2. Vortex distribution in superconducting circular disk with sector defect with a circumradius of R = 20ξ0 and a GL parameter of κ = 10 in terms of spatial distribution of the amplitude of the order parameter Δ under various different external fields. (a) h = 0.02; (b) h=0.025; (c) h = 0.03; (d) h=0.035; (e) h=0.04; (f) h=0.045
advantage because the boundary condition of the order parameter is automatically satisfied. Therefore, we decided to employ the finite element method in conducting the systematic calculations. 3. Results and discussion We calculated the distribution of vortex in a Pakman-shaped plate with a radius of R = 20ξ0 and a GL parameter of κ = 10. The magnetic field is applied in perpendicular to the Pakman plate at a temperature of T/Tc = 0.5, because Tc of our Mo80Ge20 sample is 7.3 K and our experimental measurements are performed at T = 4 K (T/Tc = 0.55). Figure 2 shows the GL calculationss in a single structrure with the order parameter Δ under various external fields. Applied magnetic field H was represented by a normalized magnetic field of h = H/Hc2. The mangetic field H ranges up to 10.5% of the upper critical field Hc2. Figure 2 shows the vortex configuration in a single shell, the green line is the mirror symmetry line of vortex configuration. As the normalized magnetic field h increases we find that the vorticity L grows in a step-wise manner upon increasing applied field as follows: (1) First, the Meissner state appears (figure 2(a)); (2) then, one vortex penetrates at tip of indent and stays on the mirror symmetry line of the Pakman plate (figure 2(b)); (3) for vorticity L = 2, the symmetry line divided two vortices into left and right (figure 2(c)); (4) third, the four and five vortices were formed in an arc-shaped manner. In the case of odd vorticity, one vortex stays on the symmetry line while the others were divided into left and right of the mirror symmetry line. They are the same in case of even vorticity while no vortices stay on the symmetry line (figures 2(d)–(f)). In Figs. 2(b)-(f), we found that the symmetry line of vortex configuration is in good agreement with the Pakman symmetry line. As pointed out by Huy et al. [4] about the circle disk, the rotational freedom of the circle is infinite while the Pakman plate has only one symmetry line at the position of sector defect. The maximum of vorticity at single shell structure and starting shell structure was depended on the angle of sector defect. Since the area of vortex penetration increases as the opening angle of the sector defect decreases. If the magnetic field increases to h=0.061, the vortex configuration becomes a two-shell structure. In Fig. 3, we show the evolution of vortex images in shell structure. It evolves with applied magnetic field as follows: (1) L = (1,5); L = (1,6); L = (1,7) one vortex sits near tip indent and others made outer shell in an arc-shaped manner (figures 3(a)–(c)), respectively. In this case, we can clearly see that a distance between two vortices in an out shell becomes larger when a vortex in an inner shell sits between the two vortices in the outer shell. A difference in space decreases when an outer shell vorticity is seven. This can be explained that a vortex in the inner shell is forced by the two vortices located in the outer shell. This makes the distance larger when the vorticity of the outer shell is large enough to accommodate more vortices. We also found the relationship of vorticity between the outer shell and inner shell. (2) According to Alstrøm et al. [17], the energy barrier at the boundary is the lowest at the triangular defect and therefore all vortices enter into superconducting disk through this defect. As the magnetic field increases from h=0.061 of Fig. 3(a), a new vortex is introduced through the tip of indent. One vortex was forced toward the outer shell by joining of new vortex. Then, the vorticity of the outer shell becomes six. This would be the same until the vorticity of the outer shell becomes seven in Fig. 3. Further increase of the applied magnetic field does not lead to the increase of the vorticity of the outer shell, but the vorticity of the inner shell increases up to two. They can be explained that the vorticity of the outer shell is large enough to change the force balance for one vortex of the inner shell. Therefore, a new vortex would stay in the inner shell until the vorticity in the inner shell increases up to two L=(2,7) of Fig. 3(d). This repeats the same situation when the shell structure grows up from L= (2,7) to L=(2,8) and (4,8). A step size of the vorticity growth in the outer shell decreases while that of the inner-shell increases. They can be explained by the confinement of supercurrents and the vortex-vortex interaction. The number of vortices in the inner shell and that
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Vu The Dang et al. / Physics Procedia 81 (2016) 93 – 96
of the outer shell must be balanced to minimize the free energy coming from the vortex distribution and the boundary-vortex and vortex-vortex interactions. As pointed out by Kokubo et al. [2, 16] for the cases of the vorticity more than 6, the vortex configuration in formation of concentric shell rings of vortices, the vorticity at inner shell will increase when the outer shell vorticity is large enough. As pointed out by Huy et al. [1], the shell structure of pentagon starts a formation of the shell structure at L= (1,5). The growth in vorticity from L = 6 to 8 results in the formation of a shell configuration in which a single vortex nearly stays at the center while the others vortices form the outer shell. The two-shell structure remains for the state L = (1, 7), where one vortex is newly added to the outer shell. The number of vortices in the inner shell starts to grow at L = 9, resulting in the subsequent (2, 7) and (2, 8) configurations with L = 10. Further experimental investigations are necessary to reveal the nature of novel vortex states of the circular Mo80Ge20 disk with sector defect.
2
4
(f) 0
0
(e) |Δ|/ Δ
0
(d) |Δ|/ Δ
0
|Δ|/ Δ
0
8
2
(c)
(b) |Δ|/ Δ
0
(a) |Δ|/ Δ
1
1
1
8
7
|Δ|/ Δ
7
6
5
Fig 3. Vortex distribution in superconducting d t duct circular Mo80Ge20 plates with sector defect f wi fec with i a circumradius diius oof R = 20ξ0 and a GL L parameter par of κ = 10 in terms of spatial distribution of the amplitude of the order parameter Δ under various different external fields. (a) h= 0.061; (b) h=0.065; (c) h=0.07; (d) h=0.078; (e) h=0.08; (f) h=0.09.
4.
Conclusion We have performed numerical simulations of vortex structures in a Pakman superconducting plate with R=20ξ0 under an external field. The Ginzburg–Landau equation can be conducted by employing the finite element method of an arbitrarily formed shape to obtain the distribution of vortices inside. We found the symmetric vortex structure was affected by the geometry of the Pakman and the boundary interaction. We noticed that the rule of shell filling in the double shell structure was influenced as a function of applied magnetic field into the Pakman disk. Beginning of the two-shell structure occurs at the vorticity L = (1,5) and the vorticity in the inner shell increases up to two when the outer-shell evolves large enough up to seven vortices. When the innershell becomes four, the outer shell grows up eight vortices. In the present studies, the experiment on the circular Mo80Ge20 plate with sector defect is conducted systematically by using the Ginzburg-Landau calculations to explain a tendency of vortex evolution. Detailed comparison between theoretical predictions and experimental findings is in progress. Acknowledgements This work was partly supported by Grant-in-Aid for challenging Exploratory Research (No. 25600018, No. 15K13979) from JSPS, Grant-in-Aid for Young Scientists (B) (No. 26820130 and No. 26800192) from JSPS, and Grant-in-Aid for Scientific Research (S) (No. 23226019) from JSPS. References H T Huy et al., “Vortex states in de facto mesoscopic Mo80Ge20 pentagon plates,” Supercond. Sci. Technol, 26 (2013) 065001. N. Kokubo et al., Phys. Rev. B, 82 (2010) 014501. N. Kokubo et al., J. Phys. Soc. Jpn, 83 (2014) 083704. H. T. Huy et al., Physica C , 484 (2013) 86–90. L. R. E. Cabral and Albino Aguiar J, Phys. Rev. B, 80 (2009) 214533. L. R. E. Cabral, J. Barba-Ortega, C. C. de Souza Silva and Albino Aguiar, J., Physica C, 470 (2011) 786. L. F. Chibotaru, A. Ceulemans, V. Bruyndoncx and V. V. Moshchalkov, Nature, 408 (2000) 14. P. J. Pereira, L. F. Chibotaru and V. V. Moshchalkov, Phys. Rev. B, 84 (2011) 144504. B. J. Baelus, and F. M. Peeters, Phys. Rev. B, 65 (2002) 104515. T. Ishida, “Vortices in small concave decagon Mo,” in COST MP1201 Workshop, Arcachon-France, 2015. B. J. Baelus, L. R. E. Cabral, F. M. Peeters, Phys. Rev. B, 69 (2004) 064506. M. Kato et al., Physica C, 494 (2013) 124. V. R. Misko, B. Xu, F. M. Peeters, Phys. Rev. B, 76 (2007) 024516. Pakman is the logo originally created by Namco Inc. M. Kato, T. Ishida, T. Koyama, M. Machida, in: Alexander Gabovich (Ed.), Superconductors – Materials, Properties and Applications, InTech, Rijeka, no. 319, 2012. [16] N. Kokubo et al., J. Phys. Soc. Jpn, 83 (2014) 083704. [17] T.S. Alstrøm et al., Acta Appl. Math., 115 (2010) 63-74. [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
ScienceDirect Physics Procedia 81 (2016) 93 – 96
28th International Symposium on Superconductivity, ISS 2015, November 16-18, 2015, Tokyo, Japan
Ginzburg-Landau calculations of circular Mo80Ge20 plates with sector defect Vu The Danga, *, Ho Thanh Huya,b, Hitoshi Matsumotoa, Hiroki Miyoshia, Shigeyuki Miyajimaa,c, Hiroaki Shishidoa,c, Masaru Katoc,d, Takekazu Ishidaa,c b
a Department of Physics and Electronics, Osaka Prefecture University, Sakai, Osaka 599-8531, Japan Department of Physics and Electronics, University of Sciences, Vietnam National University HCMC, Vietnam c Institute for Nanofabrication Research, Osaka Prefecture University, Sakai, Osaka 599-8531, Japan d Department of Mathematical Science, Osaka Prefecture University, Sakai, Osaka 599-8531, Japan
Abstract We present the theoretical calculations on vortex structures in a nanosized superconducting circle with a deficit sector. The numerical calculations of Ginzburg–Landau equation has been carried out with the aid of the finite element method, which is convenient to treat an arbitrarily shaped superconductor. We found that the vortices form an arc structure or a partial shell structure in a deficient circle plate, and mirror symmetry can be seen with respect to the sector deficit. Due to the vortex-vortex interaction and the boundary confinement effect, we also found the evolution of double (outer and inner) shell structure as a function of vorticity. Our theoretical studies will be compared to the experimental studies. © Published by Elsevier B.V. B.V. This is an open access article under the CC BY-NC-ND license ©2016 2016The TheAuthors. Authors. Published by Elsevier (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the ISS 2015 Program Committee. Peer-review under responsibility of the ISS 2015 Program Committee Keywords: Vortex states; Ginzburg-Landau calculations; SQUID microscope
1.
Introduction
The distribution of vortices in mesoscopic superconducting disks, of which the size is comparable to the coherence length ξ and the magnetic penetration depth λ, has been explored to discover exotic vortex states. Such features cannot be observed for a bulk superconductor, but it could be caused in the mesoscopic system driven by the vortex–vortex interaction and the boundary–vortex interaction. The formation of different vortex configuration was dependent on geometry of pattern, the external magnetic field, and vorticity [1, 2, 3]. The symmetric vortex structure has been investigated by using the regular polygons with mirror symmetric lines. Several recent studies on mesoscopic disks have clarified the question of how vortices are arranged by the sample geometry. For a small circular disk, the formation of a ring-like structure was arranged under the influence of the sample boundary plays a
* Corresponding author Tel.: +81-72-254-9260 fax: +81-72-254-9498 E-mail address: [email protected]
1875-3892 © 2016 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 2015 Program Committee doi:10.1016/j.phpro.2016.04.036
94
Vu The Dang et al. / Physics Procedia 81 (2016) 93 – 96
crucial role on vortex distribution [2, 4]. In the case of the noncircular geometry such as an equilateral triangle, [5, 6] or a square [7, 8], the symmetry (C3, C4) of the boundary condition tends to impose the particular vortex arrangement in those system. For instance, with the same vorticity L, the vortex configuration is different in square and triangle plates [9]. Beside the standard pentagon, several studies introduced an artificial element such as a pinning center to make the vortex configuration asymmetric [1] which was used to observed the vortex interaction, or changing sides of pentagon to become star to make complex structure [10] which observed a distinction configuration of vortex in concave decagon. Both theoretical and experimental approaches have used to explore the distribution of vortices, i.e., the nonlinear Ginzburg–Landau theory and the London approximation were used [11, 12, 13] while scanning SQUID microscopy was applied to small superconducting Mo80Ge20 plates [3, 5]. In this paper, we present theoretical prediction about the distribution of vortex in a circle disk with sector defect (Pakman [14]). In this pattern, there are several features. (1) The super current is inflected when it follows along the sample edge of Pakman shapes. At the deep corner of the deficit sector, the magnetic field is weakened, and hence the vortices tend to penetrate into the interior of the plate, in which we consider that the sector-shaped defect acts as a gate way for allowing ͵ሺݔଷ ǡ ݕଷ ሻ the intentional entrance of vortices into the circle οଷ ʹሺݔଶ ǡ ݕଶ ሻ Ȝଷ disk. It is our interest to see not only the symmetry in οଶ Ȝଶ vortex configuration but also the influence of the ͳሺݔଵ ǡ ݕଵ ሻ open angle of the deficit sector upon the shell (b) οଵ structure of vortices. We consider that the number of Ȝଵ vortices to form shell structure was affected by the (a) angle of each sector defect because of the limited Fig.1.(a) Division of a Pakman superconductor into a number of triangular length of the partial shell (arc) length. The vortexelements; (b) a triangular element. At each node, values of an order vortex interaction and the supercurrent confinement parameter and a magnetic vector potential are defined. would lead to the formation of the two-shell structure. We found that the number of vortices at an outer shell was dependent on the number of inner-shell vortices when we increased the magnetic field systematically. We investigated the vortex profiles in the Pakman-shaped plate both theoretically and experimentally. Although we found a reasonable agreement between the theoretical predictions and the experimental findings by using a scanning SQUID microscope on the small Pakman plate, we would like to explain the theoretical results on vortex configuration in the present paper. 2. Theoretical formalism We used the Ginzburg-Landau (GL) calculation in our preceding studies [1, 12]. We rebuilt order parameter and structure of circular plates with sector defect by means of finite element method. We consider a Pakman-shaped plate consisting of twodimensional nanosized superconducting film, to which the external magnetic field H was applied perpendicular. In order to obtain stable vortex structures, the GL free energy is written in terms of a complex order parameter as: ଵ
ఈ
ଶ
ඥఉ
࣠ሺȟǡ ܣሻ ൌ ݀ ȳ ቈ ൬ඥߚȁȟȁଶ
ଶ
൰
ଵ ସ
ଶ
ቚቀ
ଶ
ܣቁ οቚ
ȁൈିுȁమ ଼గ
ଵ ଼గ
ሺ݀݅ܣݒሻଶ
(1)
where Δ(r) is an order parameter of a superconductor, A is the magnetic vector potential and H is the external magnetic field. The parameter β and α are a positive constant, α depends on temperature as α = α(0)(1- T/Tc). In this study, we consider that the temperature of the system is well below the transition temperature Tc, and hence is assumed to be positive. οሺݔǡ ݕሻ ൌ ܰଵ ሺݔǡ ݕሻοଵ ܰଶ ሺݔǡ ݕሻοଶ ܰଷ ሺݔǡ ݕሻοଷ
(2)
ሺݔǡ ݕሻ ൌ ܰଵ ሺݔǡ ݕሻଵ ܰଶ ሺݔǡ ݕሻଶ ܰଷ ሺݔǡ ݕሻଷ
(3)
where ܰ ሺݔǡ ݕሻ ൌ ሺܽ ܾ ݔ ܿ ݕሻȀʹܵ ሺ݅ ൌ ͳǡʹǡ͵ሻare area coordinates, and ȟ and ۯ are the values of the order parameter and the magnetic vector potential at i-th node, respectively (see Fig. 1b). ൌ ݔ ݕ , ൌ ݕ െ ݕ and ܥ ൌ ݔ െ ݕ are defined using coordinates of nodes ሺݔ ǡ ݕ ݖ ሻ of the triangular element and ܵ is an area of the element. Using this expansion for minimizing the free energy, we obtain following equations for the order parameter, ଶ σൣܲ ሺሼܣሽሻ ܲଶோ ሺሼοሽሻ൧ܴ݁ο ൣܳ ሺሼܣሽሻ ܳ ሺሼοሽሻ൧݉ܫο ൌ ܸோ ሺሼοሽሻ ଶ ଶூ σൣെܳ ሺሼܣሽሻ ܳ ሺሼοሽሻ൧ܴ݁ο ൣܲ ሺሼܣሽሻ ܲ ሺሼοሽሻ൧݉ܫο ൌ ܸூ ሺሼοሽሻ
(4) (5)
where the coefficients are defined as ܲ ሺሼܣሽሻ ൌ σఈୀ௫ǡ௬ ܭఈఈ σఈୀ௫ǡ௬ σభమ ܫభభ ܣభఈ ܣమఈ െ ோ ଶቀ ቁ ܲ ூ ሺሼܣሽሻ
ೕ కమ
σభమ ܫభమ ቆቀ͵ቁ ܴ݁ȟభ ܴ݁ȟమ ቀͳቁ ݉ܫȟభ ݉ܫȟమ ቇ ͳ ͵ ଶ ఈ ఈ ఈ ൯ܣ െ ܬ ܳ ሺሼܣሽሻ ൌ మ σఈୀ௫ǡ௬ σభ൫ܬ భ భ భ ൌ క
ଵ
కమ
(6) (7) (8)
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Vu The Dang et al. / Physics Procedia 81 (2016) 93 – 96 ଶ ሺሼܣሽሻ ൌ ܳ
ܸ
ோ ଶቀ ቁ ூ ሺሼȟሽሻ
ଵ కమ
σభమ ܫభమ ݉ܫȟభ ܴ݁ȟమ
(9)
ܴ݁ȟయ ଶ ൰൱ ൌ మ σభమయ ܫభమయ ൭ȟభ ȟכమ ൬ క ݉ܫȟయ
(10)
Here, ߦ ൌ ߦ Ȁඥͳ െ ܶȀܶ is the coherence length at temperature T, where ߦ is the coherence length at zero temperature. Integrals ఈ …, are defined in Ref. [15]. For the magnetic vector potential of products of area coordinates over the element, such as ܫ , ܭఈఈ , ܬ A, we can get similar equation using integrals of area coordinates. The Maxwell equation is solved self-consistently with a randomly distributed initial order parameter structure and we obtain a stable state at fixed temperature and external field by solving Eqs. (4) and (5). Because of the nonlinearity of the GL equation, the free energies (Eq. (1)) of the locally stable states are determined as the most stable state by comparing several locally stables states obtained by the different initial states. Our numerical method is of the
(a)
(b)
(c)
(d)
(e)
(f)
Fig 2. Vortex distribution in superconducting circular disk with sector defect with a circumradius of R = 20ξ0 and a GL parameter of κ = 10 in terms of spatial distribution of the amplitude of the order parameter Δ under various different external fields. (a) h = 0.02; (b) h=0.025; (c) h = 0.03; (d) h=0.035; (e) h=0.04; (f) h=0.045
advantage because the boundary condition of the order parameter is automatically satisfied. Therefore, we decided to employ the finite element method in conducting the systematic calculations. 3. Results and discussion We calculated the distribution of vortex in a Pakman-shaped plate with a radius of R = 20ξ0 and a GL parameter of κ = 10. The magnetic field is applied in perpendicular to the Pakman plate at a temperature of T/Tc = 0.5, because Tc of our Mo80Ge20 sample is 7.3 K and our experimental measurements are performed at T = 4 K (T/Tc = 0.55). Figure 2 shows the GL calculationss in a single structrure with the order parameter Δ under various external fields. Applied magnetic field H was represented by a normalized magnetic field of h = H/Hc2. The mangetic field H ranges up to 10.5% of the upper critical field Hc2. Figure 2 shows the vortex configuration in a single shell, the green line is the mirror symmetry line of vortex configuration. As the normalized magnetic field h increases we find that the vorticity L grows in a step-wise manner upon increasing applied field as follows: (1) First, the Meissner state appears (figure 2(a)); (2) then, one vortex penetrates at tip of indent and stays on the mirror symmetry line of the Pakman plate (figure 2(b)); (3) for vorticity L = 2, the symmetry line divided two vortices into left and right (figure 2(c)); (4) third, the four and five vortices were formed in an arc-shaped manner. In the case of odd vorticity, one vortex stays on the symmetry line while the others were divided into left and right of the mirror symmetry line. They are the same in case of even vorticity while no vortices stay on the symmetry line (figures 2(d)–(f)). In Figs. 2(b)-(f), we found that the symmetry line of vortex configuration is in good agreement with the Pakman symmetry line. As pointed out by Huy et al. [4] about the circle disk, the rotational freedom of the circle is infinite while the Pakman plate has only one symmetry line at the position of sector defect. The maximum of vorticity at single shell structure and starting shell structure was depended on the angle of sector defect. Since the area of vortex penetration increases as the opening angle of the sector defect decreases. If the magnetic field increases to h=0.061, the vortex configuration becomes a two-shell structure. In Fig. 3, we show the evolution of vortex images in shell structure. It evolves with applied magnetic field as follows: (1) L = (1,5); L = (1,6); L = (1,7) one vortex sits near tip indent and others made outer shell in an arc-shaped manner (figures 3(a)–(c)), respectively. In this case, we can clearly see that a distance between two vortices in an out shell becomes larger when a vortex in an inner shell sits between the two vortices in the outer shell. A difference in space decreases when an outer shell vorticity is seven. This can be explained that a vortex in the inner shell is forced by the two vortices located in the outer shell. This makes the distance larger when the vorticity of the outer shell is large enough to accommodate more vortices. We also found the relationship of vorticity between the outer shell and inner shell. (2) According to Alstrøm et al. [17], the energy barrier at the boundary is the lowest at the triangular defect and therefore all vortices enter into superconducting disk through this defect. As the magnetic field increases from h=0.061 of Fig. 3(a), a new vortex is introduced through the tip of indent. One vortex was forced toward the outer shell by joining of new vortex. Then, the vorticity of the outer shell becomes six. This would be the same until the vorticity of the outer shell becomes seven in Fig. 3. Further increase of the applied magnetic field does not lead to the increase of the vorticity of the outer shell, but the vorticity of the inner shell increases up to two. They can be explained that the vorticity of the outer shell is large enough to change the force balance for one vortex of the inner shell. Therefore, a new vortex would stay in the inner shell until the vorticity in the inner shell increases up to two L=(2,7) of Fig. 3(d). This repeats the same situation when the shell structure grows up from L= (2,7) to L=(2,8) and (4,8). A step size of the vorticity growth in the outer shell decreases while that of the inner-shell increases. They can be explained by the confinement of supercurrents and the vortex-vortex interaction. The number of vortices in the inner shell and that
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of the outer shell must be balanced to minimize the free energy coming from the vortex distribution and the boundary-vortex and vortex-vortex interactions. As pointed out by Kokubo et al. [2, 16] for the cases of the vorticity more than 6, the vortex configuration in formation of concentric shell rings of vortices, the vorticity at inner shell will increase when the outer shell vorticity is large enough. As pointed out by Huy et al. [1], the shell structure of pentagon starts a formation of the shell structure at L= (1,5). The growth in vorticity from L = 6 to 8 results in the formation of a shell configuration in which a single vortex nearly stays at the center while the others vortices form the outer shell. The two-shell structure remains for the state L = (1, 7), where one vortex is newly added to the outer shell. The number of vortices in the inner shell starts to grow at L = 9, resulting in the subsequent (2, 7) and (2, 8) configurations with L = 10. Further experimental investigations are necessary to reveal the nature of novel vortex states of the circular Mo80Ge20 disk with sector defect.
2
4
(f) 0
0
(e) |Δ|/ Δ
0
(d) |Δ|/ Δ
0
|Δ|/ Δ
0
8
2
(c)
(b) |Δ|/ Δ
0
(a) |Δ|/ Δ
1
1
1
8
7
|Δ|/ Δ
7
6
5
Fig 3. Vortex distribution in superconducting d t duct circular Mo80Ge20 plates with sector defect f wi fec with i a circumradius diius oof R = 20ξ0 and a GL L parameter par of κ = 10 in terms of spatial distribution of the amplitude of the order parameter Δ under various different external fields. (a) h= 0.061; (b) h=0.065; (c) h=0.07; (d) h=0.078; (e) h=0.08; (f) h=0.09.
4.
Conclusion We have performed numerical simulations of vortex structures in a Pakman superconducting plate with R=20ξ0 under an external field. The Ginzburg–Landau equation can be conducted by employing the finite element method of an arbitrarily formed shape to obtain the distribution of vortices inside. We found the symmetric vortex structure was affected by the geometry of the Pakman and the boundary interaction. We noticed that the rule of shell filling in the double shell structure was influenced as a function of applied magnetic field into the Pakman disk. Beginning of the two-shell structure occurs at the vorticity L = (1,5) and the vorticity in the inner shell increases up to two when the outer-shell evolves large enough up to seven vortices. When the innershell becomes four, the outer shell grows up eight vortices. In the present studies, the experiment on the circular Mo80Ge20 plate with sector defect is conducted systematically by using the Ginzburg-Landau calculations to explain a tendency of vortex evolution. Detailed comparison between theoretical predictions and experimental findings is in progress. Acknowledgements This work was partly supported by Grant-in-Aid for challenging Exploratory Research (No. 25600018, No. 15K13979) from JSPS, Grant-in-Aid for Young Scientists (B) (No. 26820130 and No. 26800192) from JSPS, and Grant-in-Aid for Scientific Research (S) (No. 23226019) from JSPS. References H T Huy et al., “Vortex states in de facto mesoscopic Mo80Ge20 pentagon plates,” Supercond. Sci. Technol, 26 (2013) 065001. N. Kokubo et al., Phys. Rev. B, 82 (2010) 014501. N. Kokubo et al., J. Phys. Soc. Jpn, 83 (2014) 083704. H. T. Huy et al., Physica C , 484 (2013) 86–90. L. R. E. Cabral and Albino Aguiar J, Phys. Rev. B, 80 (2009) 214533. L. R. E. Cabral, J. Barba-Ortega, C. C. de Souza Silva and Albino Aguiar, J., Physica C, 470 (2011) 786. L. F. Chibotaru, A. Ceulemans, V. Bruyndoncx and V. V. Moshchalkov, Nature, 408 (2000) 14. P. J. Pereira, L. F. Chibotaru and V. V. Moshchalkov, Phys. Rev. B, 84 (2011) 144504. B. J. Baelus, and F. M. Peeters, Phys. Rev. B, 65 (2002) 104515. T. Ishida, “Vortices in small concave decagon Mo,” in COST MP1201 Workshop, Arcachon-France, 2015. B. J. Baelus, L. R. E. Cabral, F. M. Peeters, Phys. Rev. B, 69 (2004) 064506. M. Kato et al., Physica C, 494 (2013) 124. V. R. Misko, B. Xu, F. M. Peeters, Phys. Rev. B, 76 (2007) 024516. Pakman is the logo originally created by Namco Inc. M. Kato, T. Ishida, T. Koyama, M. Machida, in: Alexander Gabovich (Ed.), Superconductors – Materials, Properties and Applications, InTech, Rijeka, no. 319, 2012. [16] N. Kokubo et al., J. Phys. Soc. Jpn, 83 (2014) 083704. [17] T.S. Alstrøm et al., Acta Appl. Math., 115 (2010) 63-74. [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]