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Ginzburg–Landau simulation for a vortex around a columnar defect in a superconducting film
ARTICLE IN PRESS Journal of Physics and Chemistry of Solids 69 (2008) 3301– 3303
Contents lists available at ScienceDirect
Journal of Physics and Chemistry of Solids journal homepage: www.elsevier.com/locate/jpcs
Ginzburg–Landau simulation for a vortex around a columnar defect in a superconducting film N. Nakai a,b,, N. Hayashi a,b, M. Machida a,b a b
CCSE, Japan Atomic Energy Agency, 6-9-3 Higashi-Ueno, Taito-ku, Tokyo 110-0015, Japan CREST(JST), 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan
a r t i c l e in fo
Keywords: A. Superconductors D. Superconductivity
abstract By means of numerical simulations based on Ginzburg–Landau theory, we study the vortex depinning from a columnar defect in a superconducting film. We evaluate the limiting thickness of the film, below which the depinning does not occur even under an application of the magnetic field perpendicular to the columnar defect. The limiting thickness is a measure of the pinning strength of the columnar defect. The dependence of this limiting thickness on the magnitude of the applied field is obtained for two types of columnar defects. & 2008 Elsevier Ltd. All rights reserved.
1. Introduction Moving vortices driven by an applied supercurrent lead to the ohmic heating (namely, energy loss) in superconductors. Therefore, it is necessary for an efficient application of superconductors under magnetic fields that the vortex motion is pinned by defects. Columnar defects are considered to provide strong vortex-pinning sources. It is important to investigate properties of a columnar defect as a vortex pining center. In this paper, we study the vortex pinning and depinning phenomena in a three-dimensional superconductor (i.e., a superconducting film with finite thickness) that contains a single columnar defect. Our analysis is based on the Ginzburg–Landau (GL) theory. Previously, the vortex pining and vortex dynamics were studied on the basis of the time-dependent (TD) GL theory [1–4]. Such studies were also extended to an anisotropic-gap (unconventional) superconductor [5]. Concerning a modeling for a columnar defect, a local suppression of the superconducting order parameter or critical temperature inside a line defect with a finite radius is, for example, considered [2,3]. This type of defect modeling assumes that the interior of the columnar defect is filled by the normal state, i.e., the metallic region. We call such a defect a metal defect in this paper. Another modeling imposes the following boundary condition at the defect interface. No current flows from a surrounding superconducting region to the interior of the defect [4]. (Refer to Ref. [6] for more details of this boundary condition.) It corresponds to a case that the columnar is filled by insulator. We call it Corresponding author at: CCSE, Japan Atomic Energy Agency, 6-9-3 HigashiUeno, Taito-ku, Tokyo 110-0015, Japan. E-mail address: [email protected] (N. Nakai).
0022-3697/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.jpcs.2008.06.075
an insulator defect. In the present study, we adopt these two modelings for the columnar defect and compare their potentiality as pinning center. In order to evaluate the strength of the columnar defect as vortex pinning center, first we prepare a vortex trapped along a columnar defect (see Fig. 1). We then apply an external magnetic field perpendicular to this columnar defect. For weak fields or small thickness Lz of the film, the vortex is kept pinned. However, the vortex depinning occurs for strong fields or large thickness. We evaluate threshold values of the film thickness as functions of the temperature and of the external magnetic field. We call it ‘‘limiting thickness’’.
2. GL equation We numerically solve the GL equation in order to evaluate the limiting thickness. In the present simulation, the superconducting order parameter D is fully calculated. However, we do not take account of the vector potential A in self-consistent manner for simplicity and for lack of CPU time. This simplification is justified in the extreme type-II limit. With the GL free energy F, the GL equation is written as " # 2 D 2 dðF=ðN0 D20 ÞÞ T D 1 q A D ¼ 1 þ ¼ 0, Tc i qr=x0 A0 D0 dðD =D0 Þ D0 D0 (1) where A0 ¼ f0 =ð2px0 Þ. D0 is the order parameter at zero temperature ðT ¼ 0Þ, T c is the critical temperature, x0 is the coherence length at T ¼ 0, and f0 is the flux quantum. The applied field H is written as H=H0 ¼ q=qðr=x0 Þ A=A0 , where 2 H0 ¼ f0 =ð2px0 Þ. The vector potential is given by using the
ARTICLE IN PRESS 3302
N. Nakai et al. / Journal of Physics and Chemistry of Solids 69 (2008) 3301–3303
One vortex in the columnar defect at the initial state Columnar defect
Superconductor
Lz Applied field
Ly
H // x Lx Fig. 1. Schematic configuration of a superconducting film and a columnar defect. The parallelepiped superconductor is surrounded by an insulator. The columnar defect along the z direction is situated at the center of the xy plane. The area of the xy plane is ðLx ; Ly Þ ¼ ð30x0 ; 30x0 Þ, and the columnar defect size in the xy plane is 1:0x0 1:0x0 . In numerical calculations, the system is discretized by employing cubic grids and the dimensions of the cubic unit are 0:5x0 0:5x0 0:5x0 . Before applying external magnetic fields (H ¼ 0), a vortex is trapped in the columnar defect. The external magnetic field Ha0 is applied along the x direction.
Top
Middle
Bottom
-10
-10 rx /ξ
0
0
Columnar Defect 0
0 10 -10
r y/ξ
0.5 Amplitude
10 rx / ξ0
0
1
0
0
ξ0 r y/
10 -10 0 Phase
Fig. 2. Spatial profiles of the order parameter D=D0 for T=T c ¼ 0:1, H ¼ ð0:1H0 ; 0; 0Þ, and Lz ¼ 9:0x0 . The amplitude (a), and the phase (b) of D=D0 . Here, D at the defect is set zero (i.e., the metal defect). From top to bottom layer, each plane is situated at rz =x0 ¼ 4:5, 0.0, and 4.5. The displayed region is centered at the columnar defect in the xy plane, and its range is 20x0 20x0 . The circular arrows schematically indicate the vortex, and the center of them corresponds to the vortex center. The small boxes indicate the location of the columnar defect.
symmetric gauge A ¼ 12 ðH rÞ. The boundary condition is written by 1 q A D ¼ 0, (2) i qr=x0 A0 D0 kn where the vector n is perpendicular to the interface between the superconductor and insulator. Then, no current flows along the direction of n at the interface. To obtain a converged solution for D, practically we solve the TDGL equation
qðD=D0 Þ dðF=ðN0 D20 ÞÞ ¼ . qðt=t 0 Þ dðD =D0 Þ
simulation to investigate the vortex depinning from the columnar defect. (I) At first, the TDGL equation is solved without external magnetic fields (H ¼ 0), and a converged solution for D with a vortex trapped inside the columnar defect is obtained. (II) Using this solution as an initial state, we perform the TDGL simulation by applying the field Hx ða0Þ. Finally, if the vortex core escapes from the columnar defect in the top and bottom of the xy planes as shown in Figs. 2(a) and (b), we regard that the vortex depinning occurs.
3. Result (3)
Here, time t is measured by t 0 [1]. When D does not change with time evolution anymore, we obtain a converged solution for D. This is equivalent with the numerical relaxation method. Our numerical simulation has been performed with the setup shown in Fig. 1. The following is the procedure of our numerical
Here, we show that the vortex depinning certainly occurs. The obtained profile of the order parameter D under a certain applied field is displayed in Fig. 2 for the metal defect case. In Fig. 2(a), the suppression of jDj at the vortex core can be found apart from the columnar defect. Indeed, in Fig. 2(b) there is a phase singularity at the same position where jDj is suppressed.
ARTICLE IN PRESS N. Nakai et al. / Journal of Physics and Chemistry of Solids 69 (2008) 3301–3303
15
Limiting thickness
Depinning
Pinning Insulator (Columnar) Metal (Columnar) Metal (Line) 0 0
0.2
3303
defect, while the vortex tilting is observed above the critical field or thickness. The critical thickness is ‘‘limiting thickness’’. In terms of the limiting thickness, we compare the metal and the insulator defect cases. Fig. 3(a) shows the field dependence of the limiting thickness for both cases. As the applied field H increases, the limiting thickness monotonically decreases. One also finds that the limiting thickness in the insulator defect case is larger than that of the metal defect one. That is, the vortex pinning energy in the insulator case is found to be larger than that of the metal one. Fig. 3(b) shows the temperature dependence of the limiting thickness. As the temperature T increases, the limiting thickness monotonically decreases. The decreasing rate is not so remarkable but observable. It is also found in the temperature dependence that the insulator defect is stronger at the pinning center than the metal one.
H/H0 4. Summary
15
Limiting thickness
Depinning
Pinning
Insulator (Columnar) Metal (Columnar) Metal (Line)
0 0
0.6 T/Tc
Fig. 3. The field dependence (a), and the temperature dependence (b) of the limiting thickness for the vortex pinning. T=T c ¼ 0:1 in (a), and H ¼ ð0:1H0 ; 0; 0Þ in (b). In each plot, data are for the insulator defect, the metal (columnar) defect, and the metal (line) defect from top to bottom. Here, the ‘‘(line)’’ means that the cross section of the defect is a point in the xy plane, while the ‘‘(columnar)’’ indicates that the defect has a certain range in the xy plane (refer to the caption of Fig. 1). The thickness is measured by x0 .
By applying a field perpendicular to the columnar defect, the vortex has an option to simply tilt or keep the line inside the columnar defect. The tilting is regarded as the vortex depinning, in which the phase singular point shifts toward the x direction from the columnar defect in the top plane, while it does toward the opposite direction in the bottom plane. Fig. 2 just corresponds to a critical situation between the depinning and the pinning. Below the critical field or thickness, the vortex still remains inside the
We calculated the limiting thickness for the depinning triggered by applying the magnetic field perpendicular to the columnar defect. We numerically obtained the following results. The limiting thickness, below which the vortex line is pinned inside the defect, is clearly dependent on the applied-field magnitude, while its temperature dependence is not so clear. In both the field and temperature dependences, it is found that the insulator columnar defect behaves as a stronger pinning center than the metal one, through the comparison of the limiting thickness. In conclusion, we focused on the simple problem, i.e., the depinning of the single vortex from the single columnar defect in order to understand how the vortex depinning is affected by the nature of the line defect. Consequently, we found that the insulator defect is stronger pinning center than the metal one for the tilting field application in all temperature range. The calculation is presently not complete due to ignorance of self-consistency for the vector potential. However, we believe that the result is qualitatively the same because those results completely coincide in the type-II limit. We will confirm the present result by the full self-consistent calculation and examine the electric-field generation induced by the depinning dynamics in the future. References [1] R. Kato, Y. Enomoto, S. Maekawa, Phys. Rev. B 44 (1991) 6916. [2] M. Machida, H. Kaburaki, Phys. Rev. Lett. 75 (1995) 3178. [3] G.W. Crabtree, D.O. Gunter, H.G. Kaper, A.E. Koshelev, G.K. Leaf, V.M. Vinokur, Phys. Rev. B 61 (2000) 1446. [4] A.R.de C. Romaguera, M.M. Doria, Eur. Phys. J. B 42 (2004) 3. [5] Y. Matsunaga, M. Ichioka, K. Machida, Phys. Rev. B 70 (2004) 100502(R). [6] P.G. de Gennes, Superconductivity of Metals and Alloys, W.A. Benjamin, New York, 1966 reprinted by Perseus Books (1999).
Contents lists available at ScienceDirect
Journal of Physics and Chemistry of Solids journal homepage: www.elsevier.com/locate/jpcs
Ginzburg–Landau simulation for a vortex around a columnar defect in a superconducting film N. Nakai a,b,, N. Hayashi a,b, M. Machida a,b a b
CCSE, Japan Atomic Energy Agency, 6-9-3 Higashi-Ueno, Taito-ku, Tokyo 110-0015, Japan CREST(JST), 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan
a r t i c l e in fo
Keywords: A. Superconductors D. Superconductivity
abstract By means of numerical simulations based on Ginzburg–Landau theory, we study the vortex depinning from a columnar defect in a superconducting film. We evaluate the limiting thickness of the film, below which the depinning does not occur even under an application of the magnetic field perpendicular to the columnar defect. The limiting thickness is a measure of the pinning strength of the columnar defect. The dependence of this limiting thickness on the magnitude of the applied field is obtained for two types of columnar defects. & 2008 Elsevier Ltd. All rights reserved.
1. Introduction Moving vortices driven by an applied supercurrent lead to the ohmic heating (namely, energy loss) in superconductors. Therefore, it is necessary for an efficient application of superconductors under magnetic fields that the vortex motion is pinned by defects. Columnar defects are considered to provide strong vortex-pinning sources. It is important to investigate properties of a columnar defect as a vortex pining center. In this paper, we study the vortex pinning and depinning phenomena in a three-dimensional superconductor (i.e., a superconducting film with finite thickness) that contains a single columnar defect. Our analysis is based on the Ginzburg–Landau (GL) theory. Previously, the vortex pining and vortex dynamics were studied on the basis of the time-dependent (TD) GL theory [1–4]. Such studies were also extended to an anisotropic-gap (unconventional) superconductor [5]. Concerning a modeling for a columnar defect, a local suppression of the superconducting order parameter or critical temperature inside a line defect with a finite radius is, for example, considered [2,3]. This type of defect modeling assumes that the interior of the columnar defect is filled by the normal state, i.e., the metallic region. We call such a defect a metal defect in this paper. Another modeling imposes the following boundary condition at the defect interface. No current flows from a surrounding superconducting region to the interior of the defect [4]. (Refer to Ref. [6] for more details of this boundary condition.) It corresponds to a case that the columnar is filled by insulator. We call it Corresponding author at: CCSE, Japan Atomic Energy Agency, 6-9-3 HigashiUeno, Taito-ku, Tokyo 110-0015, Japan. E-mail address: [email protected] (N. Nakai).
0022-3697/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.jpcs.2008.06.075
an insulator defect. In the present study, we adopt these two modelings for the columnar defect and compare their potentiality as pinning center. In order to evaluate the strength of the columnar defect as vortex pinning center, first we prepare a vortex trapped along a columnar defect (see Fig. 1). We then apply an external magnetic field perpendicular to this columnar defect. For weak fields or small thickness Lz of the film, the vortex is kept pinned. However, the vortex depinning occurs for strong fields or large thickness. We evaluate threshold values of the film thickness as functions of the temperature and of the external magnetic field. We call it ‘‘limiting thickness’’.
2. GL equation We numerically solve the GL equation in order to evaluate the limiting thickness. In the present simulation, the superconducting order parameter D is fully calculated. However, we do not take account of the vector potential A in self-consistent manner for simplicity and for lack of CPU time. This simplification is justified in the extreme type-II limit. With the GL free energy F, the GL equation is written as " # 2 D 2 dðF=ðN0 D20 ÞÞ T D 1 q A D ¼ 1 þ ¼ 0, Tc i qr=x0 A0 D0 dðD =D0 Þ D0 D0 (1) where A0 ¼ f0 =ð2px0 Þ. D0 is the order parameter at zero temperature ðT ¼ 0Þ, T c is the critical temperature, x0 is the coherence length at T ¼ 0, and f0 is the flux quantum. The applied field H is written as H=H0 ¼ q=qðr=x0 Þ A=A0 , where 2 H0 ¼ f0 =ð2px0 Þ. The vector potential is given by using the
ARTICLE IN PRESS 3302
N. Nakai et al. / Journal of Physics and Chemistry of Solids 69 (2008) 3301–3303
One vortex in the columnar defect at the initial state Columnar defect
Superconductor
Lz Applied field
Ly
H // x Lx Fig. 1. Schematic configuration of a superconducting film and a columnar defect. The parallelepiped superconductor is surrounded by an insulator. The columnar defect along the z direction is situated at the center of the xy plane. The area of the xy plane is ðLx ; Ly Þ ¼ ð30x0 ; 30x0 Þ, and the columnar defect size in the xy plane is 1:0x0 1:0x0 . In numerical calculations, the system is discretized by employing cubic grids and the dimensions of the cubic unit are 0:5x0 0:5x0 0:5x0 . Before applying external magnetic fields (H ¼ 0), a vortex is trapped in the columnar defect. The external magnetic field Ha0 is applied along the x direction.
Top
Middle
Bottom
-10
-10 rx /ξ
0
0
Columnar Defect 0
0 10 -10
r y/ξ
0.5 Amplitude
10 rx / ξ0
0
1
0
0
ξ0 r y/
10 -10 0 Phase
Fig. 2. Spatial profiles of the order parameter D=D0 for T=T c ¼ 0:1, H ¼ ð0:1H0 ; 0; 0Þ, and Lz ¼ 9:0x0 . The amplitude (a), and the phase (b) of D=D0 . Here, D at the defect is set zero (i.e., the metal defect). From top to bottom layer, each plane is situated at rz =x0 ¼ 4:5, 0.0, and 4.5. The displayed region is centered at the columnar defect in the xy plane, and its range is 20x0 20x0 . The circular arrows schematically indicate the vortex, and the center of them corresponds to the vortex center. The small boxes indicate the location of the columnar defect.
symmetric gauge A ¼ 12 ðH rÞ. The boundary condition is written by 1 q A D ¼ 0, (2) i qr=x0 A0 D0 kn where the vector n is perpendicular to the interface between the superconductor and insulator. Then, no current flows along the direction of n at the interface. To obtain a converged solution for D, practically we solve the TDGL equation
qðD=D0 Þ dðF=ðN0 D20 ÞÞ ¼ . qðt=t 0 Þ dðD =D0 Þ
simulation to investigate the vortex depinning from the columnar defect. (I) At first, the TDGL equation is solved without external magnetic fields (H ¼ 0), and a converged solution for D with a vortex trapped inside the columnar defect is obtained. (II) Using this solution as an initial state, we perform the TDGL simulation by applying the field Hx ða0Þ. Finally, if the vortex core escapes from the columnar defect in the top and bottom of the xy planes as shown in Figs. 2(a) and (b), we regard that the vortex depinning occurs.
3. Result (3)
Here, time t is measured by t 0 [1]. When D does not change with time evolution anymore, we obtain a converged solution for D. This is equivalent with the numerical relaxation method. Our numerical simulation has been performed with the setup shown in Fig. 1. The following is the procedure of our numerical
Here, we show that the vortex depinning certainly occurs. The obtained profile of the order parameter D under a certain applied field is displayed in Fig. 2 for the metal defect case. In Fig. 2(a), the suppression of jDj at the vortex core can be found apart from the columnar defect. Indeed, in Fig. 2(b) there is a phase singularity at the same position where jDj is suppressed.
ARTICLE IN PRESS N. Nakai et al. / Journal of Physics and Chemistry of Solids 69 (2008) 3301–3303
15
Limiting thickness
Depinning
Pinning Insulator (Columnar) Metal (Columnar) Metal (Line) 0 0
0.2
3303
defect, while the vortex tilting is observed above the critical field or thickness. The critical thickness is ‘‘limiting thickness’’. In terms of the limiting thickness, we compare the metal and the insulator defect cases. Fig. 3(a) shows the field dependence of the limiting thickness for both cases. As the applied field H increases, the limiting thickness monotonically decreases. One also finds that the limiting thickness in the insulator defect case is larger than that of the metal defect one. That is, the vortex pinning energy in the insulator case is found to be larger than that of the metal one. Fig. 3(b) shows the temperature dependence of the limiting thickness. As the temperature T increases, the limiting thickness monotonically decreases. The decreasing rate is not so remarkable but observable. It is also found in the temperature dependence that the insulator defect is stronger at the pinning center than the metal one.
H/H0 4. Summary
15
Limiting thickness
Depinning
Pinning
Insulator (Columnar) Metal (Columnar) Metal (Line)
0 0
0.6 T/Tc
Fig. 3. The field dependence (a), and the temperature dependence (b) of the limiting thickness for the vortex pinning. T=T c ¼ 0:1 in (a), and H ¼ ð0:1H0 ; 0; 0Þ in (b). In each plot, data are for the insulator defect, the metal (columnar) defect, and the metal (line) defect from top to bottom. Here, the ‘‘(line)’’ means that the cross section of the defect is a point in the xy plane, while the ‘‘(columnar)’’ indicates that the defect has a certain range in the xy plane (refer to the caption of Fig. 1). The thickness is measured by x0 .
By applying a field perpendicular to the columnar defect, the vortex has an option to simply tilt or keep the line inside the columnar defect. The tilting is regarded as the vortex depinning, in which the phase singular point shifts toward the x direction from the columnar defect in the top plane, while it does toward the opposite direction in the bottom plane. Fig. 2 just corresponds to a critical situation between the depinning and the pinning. Below the critical field or thickness, the vortex still remains inside the
We calculated the limiting thickness for the depinning triggered by applying the magnetic field perpendicular to the columnar defect. We numerically obtained the following results. The limiting thickness, below which the vortex line is pinned inside the defect, is clearly dependent on the applied-field magnitude, while its temperature dependence is not so clear. In both the field and temperature dependences, it is found that the insulator columnar defect behaves as a stronger pinning center than the metal one, through the comparison of the limiting thickness. In conclusion, we focused on the simple problem, i.e., the depinning of the single vortex from the single columnar defect in order to understand how the vortex depinning is affected by the nature of the line defect. Consequently, we found that the insulator defect is stronger pinning center than the metal one for the tilting field application in all temperature range. The calculation is presently not complete due to ignorance of self-consistency for the vector potential. However, we believe that the result is qualitatively the same because those results completely coincide in the type-II limit. We will confirm the present result by the full self-consistent calculation and examine the electric-field generation induced by the depinning dynamics in the future. References [1] R. Kato, Y. Enomoto, S. Maekawa, Phys. Rev. B 44 (1991) 6916. [2] M. Machida, H. Kaburaki, Phys. Rev. Lett. 75 (1995) 3178. [3] G.W. Crabtree, D.O. Gunter, H.G. Kaper, A.E. Koshelev, G.K. Leaf, V.M. Vinokur, Phys. Rev. B 61 (2000) 1446. [4] A.R.de C. Romaguera, M.M. Doria, Eur. Phys. J. B 42 (2004) 3. [5] Y. Matsunaga, M. Ichioka, K. Machida, Phys. Rev. B 70 (2004) 100502(R). [6] P.G. de Gennes, Superconductivity of Metals and Alloys, W.A. Benjamin, New York, 1966 reprinted by Perseus Books (1999).