- Email: [email protected]
Simulation study for the orientation of the driven vortex lattice in an amorphous superconductor
Physica C 469 (2009) 1106–1109
Contents lists available at ScienceDirect
Physica C journal homepage: www.elsevier.com/locate/physc
Simulation study for the orientation of the driven vortex lattice in an amorphous superconductor N. Nakai *, N. Hayashi, M. Machida 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
i n f o
Article history: Available online 31 May 2009 PACS: 74.25.Qt 74.20.De
a b s t r a c t We investigate the orientation of the vortex lattice driven by an applied current by means of numerical simulations based on the time-dependent Ginzburg–Landau (TDGL) theory. A lattice order is restored by a current driving of vortices under the influence of random vortex pinnings. The orientation of the moving vortex lattice is different between the presence and the absence of vortex pinnings. We show results of TDGL simulations for these phenomena. Ó 2009 Elsevier B.V. All rights reserved.
Keywords: Vortex lattice Flux pinning Ginzburg–Landau theory
1. Introduction Much attention has been focused on vortices in type-II superconductors under magnetic fields. The vortices are driven into motion by applying currents. The moving vortices induce the electric field, and therefore the response of vortices to the applied current is detected by measuring the electric resistivity. The vortex pinnings by defects in materials affect the vortex motion and consequently influence the resistivity. In this way, the vortex and vortex pinning play an important role for electromagnetic properties of superconductors. The vortices tend to form a lattice because of the repulsive interaction between them. It is known that the vortex pinnings disturb the lattice order in a static state. On the other hand, it has been discussed that a lattice order is restored by a current driving of vortices [1–4]. The dynamical lattice formation and deformation can be studied in a controlled way by applying currents, which stimulates broad interests in a wide range of science. Recently, an ordered motion of vortices with small velocity was observed in a clean superconductor by the scanning tunneling microscopy [5,6]. A vortex lattice flow was investigated in an amorphous superconductor by the mode-locking resonance technique [7,8]. These experiments revealed that the orientation of
* Corresponding author. Address: Center for Computational Science and e-Systems, Japan Atomic Energy Agency, 6-9-3 Higashi-Ueno, Taito-ku, Tokyo 110-0015, Japan. Tel.: +81 3 5246 2511; fax: +81 3 5246 2537. E-mail address: [email protected] (N. Nakai). 0921-4534/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2009.05.208
the moving vortex lattice depends on the temperature and applied magnetic field. The lattice vector is perpendicular (parallel) to the direction of vortex motion in an intermediate-field region (in lowand high-field regions) in an amorphous superconductor [8]. Here, the lattice vector indicates the direction of the nearest neighbor vortex. As discussed later, it is expected that the lattice vector tends to be parallel to the direction of vortex motion in the absence of vortex pinnings. Therefore, the above experimental observation in an amorphous superconductor implies that the vortex pinning clearly influences the orientation of the moving vortex lattice. By a molecular-dynamics (MD) simulation and an analytic study, it was indeed predicted that the lattice vector is perpendicular to the direction of vortex motion in the presence of vortex pinnings [9,10]. However, the magnetic-field dependence of the lattice orientation has not been understood yet. The experiments show that the vortices seem to move along the lattice vector near the upper critical field. Since the problem of the field dependence has still remained unsolved, further theoretical studies are required. Thus, simulation studies based on the time-dependent Ginzburg–Landau (TDGL) theory are expected to give a conclusive answer to the problem. As a first step, we have performed the TDGL simulations at an intermediate magnetic field. In this paper, we report that the lattice vector certainly points to the direction perpendicular to the vortex motion under the influence of a random distribution of vortex pinnings, which is consistent with the experimental observation. This is the first TDGL simulation aiming to address the above issue.
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N. Nakai et al. / Physica C 469 (2009) 1106–1109
2. Ginzburg–Landau and Maxwell equations
0.9 To obtain the time development of the superconducting order parameter D and vector potential A, we numerically solve the TDGL equation coupled with the Maxwell one. Those equations are written in a dimensionless form as
"( ) 2 # D 2 D 1 @ A D 1 T ; þ D D0 Tc i @r=n0 A0 D0 0
@ðA=A0 Þ D 1 @ A D D 1 @ A D þ ¼ D0 @ðt=t 0 Þ i @r=n0 A0 D0 D0 i @r=n0 A0 D0 @ H 2 : 2j @r=n0 H0
50
(a) ð1Þ
ð2Þ
Eq. (2) means jn ¼ js jt with the normal current jn , supercurrent js and total current jt , where jn ¼ rE ¼ r@A=@t. The scalar potential is set equal to zero. In Eqs. (1) and (2) T c is the transition temperature and j is the Ginzburg–Landau (GL) parameter. We shall introduce the local suppression of T c ðrÞ, which acts as vortex pinning. The order parameter D, time t, vector potential A, and magnetic field H are normalized by D0 ; t 0 ¼ 8pj2 n20 r=c2 ; A0 ¼ /0 =ð2pn0 Þ, and H0 ¼ /0 =ð2pn20 Þ, respectively. We have defined the order parameter D0 and coherence length n0 at zero temperature, the normal-state longitudinal conductance r, light velocity c, and flux quantum /0 . We consider a two dimensional system in the xy plane. The magnetic field Ha is applied perpendicular to the plane. We discretize the system into a grid and use the link variable Rr U ijl ¼ exp½i rij ðAl =A0 Þdl=n0 , when solving the TDGL and Maxwell equations [11–13]. Here, l stands for x or y. The gauge-invariant differential terms are then replaced as
ij Al D 1 @ 1 U l Dj = D0 Di = D0 ! ; i @rl =n0 A0 D0 i al 2 Al 1 @ D i @rl =n0 A0 D0 !
1.0
U ijl Di =D0 þ U jk l Dk =D0 2Dj =D0 a2l
with the step size al ðl ¼ x; yÞ and the sequential positions i; j; k along the l coordinate on the grid. The dimensions of a unit cell of the grid are ax ay . The magnetic field H is obtained by the R R Stokes’ theorem, exp½i S ðH=H0 Þ ndS=n20 ¼ exp½i c ðA=A0 Þ dl=n0 , which can be calculated by using the product of link variables. In our simulations, the size of the system is Lx Ly ¼ 50n0 200n0 with the grid unit n0 n0 . The external current is applied in the x direction, and a periodic boundary condition is imposed in this direction. The system edges perpendicular to the y direction are considered as interfaces between a superconductor and a normal metal, near which T c is set to be suppressed as shown in Fig. 1a. This T c suppression along the interface assists the vortex entrance into the system. Thus, the interface does not significantly affect the dynamical lattice formation. The bulk transition temperature is denoted by T c0 . When investigating the influence of vortex pinnings, we introduce vortex-pinning sites, each of which is a point defect where T c is locally suppressed within the range 0:9 6 T c =T c0 6 1. The positions of pinning sites and the degree of T c suppression are distributed randomly as shown in Fig. 1b. The number of pinning sites is 138 in the system 50n0 200n0 . The temperature T, applied magnetic field Ha , applied current density jx , and GL parameter j are set T=T c0 ¼ 0:5; Ha =H0 ¼ 0:2; jx =j0 ¼ 3 104 with j0 ¼ /0 =ð2pn30 Þ, and j ¼ 3, respectively. The applied current in the x direction generates a magnetic-field gradient in the y direction. Therefore, at the two boundary edges per-
1.5
Tc/Tc0
@ðD=D0 Þ 1 ¼ @ðt=t0 Þ 12
(b)
Tc/Tc0
0
-100
100
-25
-50 25
Fig. 1. (a) Spatical profile of the transition temperature T c ðrÞ normalized by the bulk transition temperature T c0 . (b) Distribution of random vortex pinnings used in the present simulations. The region of 50n0 100n0 is displayed, which is a part of the system 50n0 200n0 .
pendicular to the y axis, we set the external fields at the values consistent with the applied current and the applied magnetic field. D is set zero at these edges. The minimal time interval is set 0:01t 0 when solving the TDGL and Maxwell equations. We continue to solve the time development of these equations until an ordered lattice structure is stabilized after a steady vortex motion is attained. 3. Results We first consider a system without random vortex pinnings. A snapshot of moving vortices at a certain moment is shown in Fig. 2a, where the circles represent the positions of vortices. The vortices move from the top to the bottom in the negative y-direction. Another snapshot after the time interval t=t0 ¼ 1000 is presented in Fig. 2b. Then, the lattice structure of moving vortices is found to be already stabilized. We confirm that this lattice structure is unchanged permanently. The simulation result also shows that the lattice vector is parallel to the direction of vortex motion as denoted schematically in the lower panel of Fig. 2. Moreover, we find that this orientation of the vortex lattice is not determined by an influence of the interface perpendicular to the y axis. When the applied current is weak, it is expected that an interface effect appears and the vortices tend to align parallel to the interface along which T c is suppressed as shown in Fig. 1a. However, the present result indicates that the vortices align perpendicular to the interface. Therefore, we conclude that, in the absence of random vortex pinnings, the moving vortex lattice is always oriented so that the lattice vector is parallel to the direction of motion. On the other hand, when the applied current becomes substantially large, the moving lattice is destroyed by spontaneous nucleation of dislocations. A large field gradient due to a large applied current results in a significant spatial gradient of the density of vortices, by which dislocations of the vortex lattice are generated. Our result concerning the dislocations nucleation is consistent with the previous simulation study [4]. Next, we consider a vortex lattice flow under the influence of random vortex pinnings. The collective flow is simulated in this paper, while the plastic flow has been studied by another TDGL
1108
N. Nakai et al. / Physica C 469 (2009) 1106–1109
(a) t/t0=0
-25
Vortex
25
(b) t/t0=1000
-25
25
50
-50
(a) t/t0=0
-25
Vortex
25
Applied Current
(b) t/t0=1000
-25
25
50
-50
Pinning site
Fig. 2. (a) Snapshot of the moving vortex lattice in the absence of random vortex pinnings. The circular lines represent the contour of the order parameter at jD=D0 j ¼ 0:5, and each circle indicates the vortex position. (b) Snapshot after the time interval 1000t 0 . The lattice vector is parallel to the direction of motion as denoted schematically in the lower panel.
simulation [14]. The moving vortices form the steady lattice structure as shown in Fig. 3a and b, even in the presence of random vortex pinnings. When a current is not applied, vortices do not exhibit such a lattice order. The lattice order is restored by the current driving of vortices. The lattice vector is perpendicular to the direction of vortex motion under the influence of vortex pinnings as denoted schematically in the lower panel of Fig. 3. This result is interpreted as follows. Each vortex pinning induces a displacement of a vortex lattice. In the vortex flow region, the displacement is expected to frequently occur along the direction of vortex motion. On the other hand, on a vortex lattice a displacement along the next nearest neighbor (NNN) direction is more favorable than along the nearest neighbor (NN) direction because the repulsive interaction between vortices is related with their distance. That is, the vortex distance along the NNN direction is larger than the NN direction, and therefore, the NNN-direction displacement gives rise to a less elastic energy loss. Hence, the NNN direction of the lattice tends to coincide with the direction of vortex motion. This mechanism has been analyzed also by a MD simulation [9]. The same orientation has been experimentally observed in an amorphous superconductor [8], which is consistent with the above analysis. It has also been observed in a clean superconductor that the lattice vector is parallel to the direction of vortex motion at rather small vortex velocity [5]. When the vortex velocity is very small, an analysis based on the elastic energy of the displacement would become worse. Further analyses based on careful numerical simulations will be necessary in the case of the small vortex velocity.
Direction of motion
Direction of motion
Applied Current
Fig. 3. (a) Snapshot of the moving lattice in the presence of random vortex pinnings. The circular lines represent the contour of the order parameter at jD=D0 j ¼ 0:5, and each circle indicates the vortex position. (b) Snapshot after the time interval 1000t0 . The distribution of vortex pinnings shown in Fig. 1b is superimposed. The lattice vector is perpendicular to the direction of motion as denoted schematically in the lower panel.
4. Summary We have performed the numerical simulations to study the effect of random vortex pinnings on the vortex-lattice dynamics, on the basis of the time-dependent Ginzburg–Landau theory. In the absence of vortex pinnings, the lattice vector is parallel to the direction of vortex motion. On the other hand, in the presence of vortex pinnings and in the collective vortex flow phase, we found that the lattice vector is perpendicular to the direction of vortex motion, which is consistent with the experimental observation under intermediate magnetic fields in an amorphous superconductor [8]. The present study demonstrates that the TDGL simulations are reliable for studying the orientation problem of the moving vortex lattice under the influence of random vortex pinnings. Further TDGL analyses for elucidating the origin of the magnetic-field dependence observed experimentally are left for future studies.
Acknowledgements We thank N. Kokubo, S. Okuma and N. Nishida for helpful discussion. The work was partially supported by the Priority Area ‘‘Physics of new quantum phases in superclean materials” (Grant No. 18043022) from the Ministry of Education, Culture, Sports, Science and Technology of Japan. One of us (M.M.) acknowledges support by JSPS Core-to-Core Program-Strategic Research Networks, ‘‘Nanoscience and Engineering in Superconductivity (NES)”.
N. Nakai et al. / Physica C 469 (2009) 1106–1109
References [1] A.T. Fiory, Phys. Rev. Lett. 27 (1971) 501. [2] G. Li, E.Y. Andrei, Z.L. Xiao, P. Shuk, M. Greenblatt, Phys. Rev. Lett. 96 (2006) 017009. [3] A.E. Koshelev, V.M. Vinokur, Phys. Rev. Lett. 73 (1994) 3580. [4] D.W. Braun, G.W. Crabtree, H.G. Kaper, A.E. Koshelev, G.K. Leaf, D.M. Levine, V.M. Vinokur, Phys. Rev. Lett. 76 (1996) 831. [5] A.M. Troyanovski, J. Aarts, P.H. Kes, Nature 399 (1999) 665. [6] N. Nishida, unpublished.
[7] [8] [9] [10] [11] [12] [13]
1109
S. Okuma, J. Inoue, N. Kokubo, Phys. Rev. B 76 (2007) 172503. N. Kokubo, B. Shinozaki, P.H. Kes, Physica C 468 (2008) 581. S.O. Valenzuela, Phys. Rev. Lett. 88 (2002) 247003. A. Schmid, W. Hauger, J. Low Temp. Phys. 11 (1973) 667. R. Kato, Y. Enomoto, S. Maekawa, Phys. Rev. B 47 (1993) 8016. M. Machida, H. Kaburaki, Phys. Rev. Lett. 75 (1995) 3178. G.W. Crabtree, D.O. Gunter, H.G. Kaper, A.E. Koshelev, G.K. Leaf, V.M. Vinokur, Phys. Rev. B 61 (2000) 1446. [14] N. Nakai, N. Hayashi, M. Machida, Physica C 468 (2008) 1270.
Contents lists available at ScienceDirect
Physica C journal homepage: www.elsevier.com/locate/physc
Simulation study for the orientation of the driven vortex lattice in an amorphous superconductor N. Nakai *, N. Hayashi, M. Machida 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
i n f o
Article history: Available online 31 May 2009 PACS: 74.25.Qt 74.20.De
a b s t r a c t We investigate the orientation of the vortex lattice driven by an applied current by means of numerical simulations based on the time-dependent Ginzburg–Landau (TDGL) theory. A lattice order is restored by a current driving of vortices under the influence of random vortex pinnings. The orientation of the moving vortex lattice is different between the presence and the absence of vortex pinnings. We show results of TDGL simulations for these phenomena. Ó 2009 Elsevier B.V. All rights reserved.
Keywords: Vortex lattice Flux pinning Ginzburg–Landau theory
1. Introduction Much attention has been focused on vortices in type-II superconductors under magnetic fields. The vortices are driven into motion by applying currents. The moving vortices induce the electric field, and therefore the response of vortices to the applied current is detected by measuring the electric resistivity. The vortex pinnings by defects in materials affect the vortex motion and consequently influence the resistivity. In this way, the vortex and vortex pinning play an important role for electromagnetic properties of superconductors. The vortices tend to form a lattice because of the repulsive interaction between them. It is known that the vortex pinnings disturb the lattice order in a static state. On the other hand, it has been discussed that a lattice order is restored by a current driving of vortices [1–4]. The dynamical lattice formation and deformation can be studied in a controlled way by applying currents, which stimulates broad interests in a wide range of science. Recently, an ordered motion of vortices with small velocity was observed in a clean superconductor by the scanning tunneling microscopy [5,6]. A vortex lattice flow was investigated in an amorphous superconductor by the mode-locking resonance technique [7,8]. These experiments revealed that the orientation of
* Corresponding author. Address: Center for Computational Science and e-Systems, Japan Atomic Energy Agency, 6-9-3 Higashi-Ueno, Taito-ku, Tokyo 110-0015, Japan. Tel.: +81 3 5246 2511; fax: +81 3 5246 2537. E-mail address: [email protected] (N. Nakai). 0921-4534/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2009.05.208
the moving vortex lattice depends on the temperature and applied magnetic field. The lattice vector is perpendicular (parallel) to the direction of vortex motion in an intermediate-field region (in lowand high-field regions) in an amorphous superconductor [8]. Here, the lattice vector indicates the direction of the nearest neighbor vortex. As discussed later, it is expected that the lattice vector tends to be parallel to the direction of vortex motion in the absence of vortex pinnings. Therefore, the above experimental observation in an amorphous superconductor implies that the vortex pinning clearly influences the orientation of the moving vortex lattice. By a molecular-dynamics (MD) simulation and an analytic study, it was indeed predicted that the lattice vector is perpendicular to the direction of vortex motion in the presence of vortex pinnings [9,10]. However, the magnetic-field dependence of the lattice orientation has not been understood yet. The experiments show that the vortices seem to move along the lattice vector near the upper critical field. Since the problem of the field dependence has still remained unsolved, further theoretical studies are required. Thus, simulation studies based on the time-dependent Ginzburg–Landau (TDGL) theory are expected to give a conclusive answer to the problem. As a first step, we have performed the TDGL simulations at an intermediate magnetic field. In this paper, we report that the lattice vector certainly points to the direction perpendicular to the vortex motion under the influence of a random distribution of vortex pinnings, which is consistent with the experimental observation. This is the first TDGL simulation aiming to address the above issue.
1107
N. Nakai et al. / Physica C 469 (2009) 1106–1109
2. Ginzburg–Landau and Maxwell equations
0.9 To obtain the time development of the superconducting order parameter D and vector potential A, we numerically solve the TDGL equation coupled with the Maxwell one. Those equations are written in a dimensionless form as
"( ) 2 # D 2 D 1 @ A D 1 T ; þ D D0 Tc i @r=n0 A0 D0 0
@ðA=A0 Þ D 1 @ A D D 1 @ A D þ ¼ D0 @ðt=t 0 Þ i @r=n0 A0 D0 D0 i @r=n0 A0 D0 @ H 2 : 2j @r=n0 H0
50
(a) ð1Þ
ð2Þ
Eq. (2) means jn ¼ js jt with the normal current jn , supercurrent js and total current jt , where jn ¼ rE ¼ r@A=@t. The scalar potential is set equal to zero. In Eqs. (1) and (2) T c is the transition temperature and j is the Ginzburg–Landau (GL) parameter. We shall introduce the local suppression of T c ðrÞ, which acts as vortex pinning. The order parameter D, time t, vector potential A, and magnetic field H are normalized by D0 ; t 0 ¼ 8pj2 n20 r=c2 ; A0 ¼ /0 =ð2pn0 Þ, and H0 ¼ /0 =ð2pn20 Þ, respectively. We have defined the order parameter D0 and coherence length n0 at zero temperature, the normal-state longitudinal conductance r, light velocity c, and flux quantum /0 . We consider a two dimensional system in the xy plane. The magnetic field Ha is applied perpendicular to the plane. We discretize the system into a grid and use the link variable Rr U ijl ¼ exp½i rij ðAl =A0 Þdl=n0 , when solving the TDGL and Maxwell equations [11–13]. Here, l stands for x or y. The gauge-invariant differential terms are then replaced as
ij Al D 1 @ 1 U l Dj = D0 Di = D0 ! ; i @rl =n0 A0 D0 i al 2 Al 1 @ D i @rl =n0 A0 D0 !
1.0
U ijl Di =D0 þ U jk l Dk =D0 2Dj =D0 a2l
with the step size al ðl ¼ x; yÞ and the sequential positions i; j; k along the l coordinate on the grid. The dimensions of a unit cell of the grid are ax ay . The magnetic field H is obtained by the R R Stokes’ theorem, exp½i S ðH=H0 Þ ndS=n20 ¼ exp½i c ðA=A0 Þ dl=n0 , which can be calculated by using the product of link variables. In our simulations, the size of the system is Lx Ly ¼ 50n0 200n0 with the grid unit n0 n0 . The external current is applied in the x direction, and a periodic boundary condition is imposed in this direction. The system edges perpendicular to the y direction are considered as interfaces between a superconductor and a normal metal, near which T c is set to be suppressed as shown in Fig. 1a. This T c suppression along the interface assists the vortex entrance into the system. Thus, the interface does not significantly affect the dynamical lattice formation. The bulk transition temperature is denoted by T c0 . When investigating the influence of vortex pinnings, we introduce vortex-pinning sites, each of which is a point defect where T c is locally suppressed within the range 0:9 6 T c =T c0 6 1. The positions of pinning sites and the degree of T c suppression are distributed randomly as shown in Fig. 1b. The number of pinning sites is 138 in the system 50n0 200n0 . The temperature T, applied magnetic field Ha , applied current density jx , and GL parameter j are set T=T c0 ¼ 0:5; Ha =H0 ¼ 0:2; jx =j0 ¼ 3 104 with j0 ¼ /0 =ð2pn30 Þ, and j ¼ 3, respectively. The applied current in the x direction generates a magnetic-field gradient in the y direction. Therefore, at the two boundary edges per-
1.5
Tc/Tc0
@ðD=D0 Þ 1 ¼ @ðt=t0 Þ 12
(b)
Tc/Tc0
0
-100
100
-25
-50 25
Fig. 1. (a) Spatical profile of the transition temperature T c ðrÞ normalized by the bulk transition temperature T c0 . (b) Distribution of random vortex pinnings used in the present simulations. The region of 50n0 100n0 is displayed, which is a part of the system 50n0 200n0 .
pendicular to the y axis, we set the external fields at the values consistent with the applied current and the applied magnetic field. D is set zero at these edges. The minimal time interval is set 0:01t 0 when solving the TDGL and Maxwell equations. We continue to solve the time development of these equations until an ordered lattice structure is stabilized after a steady vortex motion is attained. 3. Results We first consider a system without random vortex pinnings. A snapshot of moving vortices at a certain moment is shown in Fig. 2a, where the circles represent the positions of vortices. The vortices move from the top to the bottom in the negative y-direction. Another snapshot after the time interval t=t0 ¼ 1000 is presented in Fig. 2b. Then, the lattice structure of moving vortices is found to be already stabilized. We confirm that this lattice structure is unchanged permanently. The simulation result also shows that the lattice vector is parallel to the direction of vortex motion as denoted schematically in the lower panel of Fig. 2. Moreover, we find that this orientation of the vortex lattice is not determined by an influence of the interface perpendicular to the y axis. When the applied current is weak, it is expected that an interface effect appears and the vortices tend to align parallel to the interface along which T c is suppressed as shown in Fig. 1a. However, the present result indicates that the vortices align perpendicular to the interface. Therefore, we conclude that, in the absence of random vortex pinnings, the moving vortex lattice is always oriented so that the lattice vector is parallel to the direction of motion. On the other hand, when the applied current becomes substantially large, the moving lattice is destroyed by spontaneous nucleation of dislocations. A large field gradient due to a large applied current results in a significant spatial gradient of the density of vortices, by which dislocations of the vortex lattice are generated. Our result concerning the dislocations nucleation is consistent with the previous simulation study [4]. Next, we consider a vortex lattice flow under the influence of random vortex pinnings. The collective flow is simulated in this paper, while the plastic flow has been studied by another TDGL
1108
N. Nakai et al. / Physica C 469 (2009) 1106–1109
(a) t/t0=0
-25
Vortex
25
(b) t/t0=1000
-25
25
50
-50
(a) t/t0=0
-25
Vortex
25
Applied Current
(b) t/t0=1000
-25
25
50
-50
Pinning site
Fig. 2. (a) Snapshot of the moving vortex lattice in the absence of random vortex pinnings. The circular lines represent the contour of the order parameter at jD=D0 j ¼ 0:5, and each circle indicates the vortex position. (b) Snapshot after the time interval 1000t 0 . The lattice vector is parallel to the direction of motion as denoted schematically in the lower panel.
simulation [14]. The moving vortices form the steady lattice structure as shown in Fig. 3a and b, even in the presence of random vortex pinnings. When a current is not applied, vortices do not exhibit such a lattice order. The lattice order is restored by the current driving of vortices. The lattice vector is perpendicular to the direction of vortex motion under the influence of vortex pinnings as denoted schematically in the lower panel of Fig. 3. This result is interpreted as follows. Each vortex pinning induces a displacement of a vortex lattice. In the vortex flow region, the displacement is expected to frequently occur along the direction of vortex motion. On the other hand, on a vortex lattice a displacement along the next nearest neighbor (NNN) direction is more favorable than along the nearest neighbor (NN) direction because the repulsive interaction between vortices is related with their distance. That is, the vortex distance along the NNN direction is larger than the NN direction, and therefore, the NNN-direction displacement gives rise to a less elastic energy loss. Hence, the NNN direction of the lattice tends to coincide with the direction of vortex motion. This mechanism has been analyzed also by a MD simulation [9]. The same orientation has been experimentally observed in an amorphous superconductor [8], which is consistent with the above analysis. It has also been observed in a clean superconductor that the lattice vector is parallel to the direction of vortex motion at rather small vortex velocity [5]. When the vortex velocity is very small, an analysis based on the elastic energy of the displacement would become worse. Further analyses based on careful numerical simulations will be necessary in the case of the small vortex velocity.
Direction of motion
Direction of motion
Applied Current
Fig. 3. (a) Snapshot of the moving lattice in the presence of random vortex pinnings. The circular lines represent the contour of the order parameter at jD=D0 j ¼ 0:5, and each circle indicates the vortex position. (b) Snapshot after the time interval 1000t0 . The distribution of vortex pinnings shown in Fig. 1b is superimposed. The lattice vector is perpendicular to the direction of motion as denoted schematically in the lower panel.
4. Summary We have performed the numerical simulations to study the effect of random vortex pinnings on the vortex-lattice dynamics, on the basis of the time-dependent Ginzburg–Landau theory. In the absence of vortex pinnings, the lattice vector is parallel to the direction of vortex motion. On the other hand, in the presence of vortex pinnings and in the collective vortex flow phase, we found that the lattice vector is perpendicular to the direction of vortex motion, which is consistent with the experimental observation under intermediate magnetic fields in an amorphous superconductor [8]. The present study demonstrates that the TDGL simulations are reliable for studying the orientation problem of the moving vortex lattice under the influence of random vortex pinnings. Further TDGL analyses for elucidating the origin of the magnetic-field dependence observed experimentally are left for future studies.
Acknowledgements We thank N. Kokubo, S. Okuma and N. Nishida for helpful discussion. The work was partially supported by the Priority Area ‘‘Physics of new quantum phases in superclean materials” (Grant No. 18043022) from the Ministry of Education, Culture, Sports, Science and Technology of Japan. One of us (M.M.) acknowledges support by JSPS Core-to-Core Program-Strategic Research Networks, ‘‘Nanoscience and Engineering in Superconductivity (NES)”.
N. Nakai et al. / Physica C 469 (2009) 1106–1109
References [1] A.T. Fiory, Phys. Rev. Lett. 27 (1971) 501. [2] G. Li, E.Y. Andrei, Z.L. Xiao, P. Shuk, M. Greenblatt, Phys. Rev. Lett. 96 (2006) 017009. [3] A.E. Koshelev, V.M. Vinokur, Phys. Rev. Lett. 73 (1994) 3580. [4] D.W. Braun, G.W. Crabtree, H.G. Kaper, A.E. Koshelev, G.K. Leaf, D.M. Levine, V.M. Vinokur, Phys. Rev. Lett. 76 (1996) 831. [5] A.M. Troyanovski, J. Aarts, P.H. Kes, Nature 399 (1999) 665. [6] N. Nishida, unpublished.
[7] [8] [9] [10] [11] [12] [13]
1109
S. Okuma, J. Inoue, N. Kokubo, Phys. Rev. B 76 (2007) 172503. N. Kokubo, B. Shinozaki, P.H. Kes, Physica C 468 (2008) 581. S.O. Valenzuela, Phys. Rev. Lett. 88 (2002) 247003. A. Schmid, W. Hauger, J. Low Temp. Phys. 11 (1973) 667. R. Kato, Y. Enomoto, S. Maekawa, Phys. Rev. B 47 (1993) 8016. M. Machida, H. Kaburaki, Phys. Rev. Lett. 75 (1995) 3178. G.W. Crabtree, D.O. Gunter, H.G. Kaper, A.E. Koshelev, G.K. Leaf, V.M. Vinokur, Phys. Rev. B 61 (2000) 1446. [14] N. Nakai, N. Hayashi, M. Machida, Physica C 468 (2008) 1270.