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Step phenomenon of vortices flow in superconductor
Physica C 470 (2010) S836–S837
Contents lists available at ScienceDirect
Physica C journal homepage: www.elsevier.com/locate/physc
Step phenomenon of vortices flow in superconductor D.Q. Shi a,*, X.B. Xu b, X.B. Zhu a,c, L. Wang a, Q. Li a, S.X. Dou a a
Institute for Superconducting and Electronic Materials, University of Wollongong, Wollongong 2522, Australia Department of Physics, Nanjing University, Nanjing 210093, PR China c Key Laboratory of Materials Physics, Institute of Solid State Physics, Chinese Academy of Sciences, Heifei 230031, PR China b
a r t i c l e
i n f o
Article history: Accepted 9 November 2009 Available online 12 November 2009 Keywords: Type-II superconductor Vortex Molecular dynamics simulation
a b s t r a c t We investigate numerically the transport of vortices in superconductors under a driving force by using Langevin dynamics. The velocity-driving force characteristic (Vx–FL curve) of superconductors includes a region with series of stages. The impact of the initial states was investigated and there is a dramatic difference between the initial state produced by the current annealing process (CAP) and the well pinned state (WPS). Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction The step phenomenon in type-II superconductor is that at the onset of particular voltages, there are abrupt steps in the I–V curves which are associated with sharp peaks in the differential resistance. Experiments on different superconductors, both low critical temperature ðT c Þ and high T c types, have shown this features [1]. Many simulations have been conducted to simulate and explain this phenomenon [2]. Theoretical understanding of plastic depinning and plastic flow is far from complete, particularly in high temperature and low field conditions. In this report, we numerically study the step phenomenon via a molecular dynamics simulation (MDS) based on a random pinning arrangement. To connect our simulations to the experiments, we adopt high temperature and low field values found to be appropriate for such a sample through dc transport measurement. 2. Model The overdamped Langevin equation of motion for a vortex in position ri is [3]:
Fi ¼
Np Nv X X dr i F vv ðr i r j Þ þ F v p ðr i r pk Þ þ F L þ F Ti ¼ g dt j–i k
where F i is the total force acting on vortex i, Fvv and F v p are the forces due to vortex–vortex and vortex–pin interactions, respectively, F L is the driving force due to the current J ðF L / U0 J ZÞ, * Corresponding author. Tel.: +61 2 42215727; fax: +61 2 42215731. E-mail address: [email protected] (D.Q. Shi). 0921-4534/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2009.11.080
where U0 is the flux quantum, FT is the thermal stochastic force, g is the Bardeen–Stephen friction coefficient: g / U0 Bc2 =qn; N v the number of vortices, Np the number of pinning centers, and r pk the position of the kth pinning center. The total number of vortices Nv ¼ 484 is used. Similar results are obtained for larger systems. We employ penetration length k0 ¼ 690 Å´, the length of the vortex s = 12 Å´, while the pinning strength at zero temperature is fpv 0 ¼ 6f 0 , and g0 ¼ 1:4 10—17 kg=s and the range of pinning force rp ¼ 0:2k. The details of methodology have been reported elsewhere [3]. 3. Results and discussion At first, we use a current annealing process (CAP) to obtain the initial state for molecular dynamics simulation. Here, we choose the temperature of 0:9T c and a small field in order to simulate the transport I–V measurements which are usually at high temperature and self-field. The sample is under a driving force scan of L dF =dt of 0:01— 0:001f 0 =t 0 . Based on a typical statistic calculation from the simulation, about 97% of the vortices have been pinned by the pinning sites in the CAP state. Fig. 1 shows typical plots of the average velocity of the vortex in the x direction against the driving force F L under different fields. Except for the field, other parameters are the same and expressed as: N v ¼ N p ; T ¼ 0 : 9T c ; fpv 0 ¼ 6f 0 and r ¼ 0:5rp0 . The F L scan speed L is dF =dt ¼ 0:01f 0 =t0 . For the plot with B = 30 G, which means vortices are diluted, there are four rectangular steps, and the jumps are very sharp. This curve exactly simulated the reported I–V measurement curve of NbSe2 [4]. With the increase of the field, corresponding to the increase of the vortex density, the number of steps is reduced, and the jumps are not sharp.
S837
D.Q. Shi et al. / Physica C 470 (2010) S836–S837
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Fig. 1. V x —F curves from the numerical simulations under different fields. The inset (a) shows an enlargement of the indicated region. The inset (b) contains the L differential plot of dV=dF of the V x —F L curve for a 30 G field on the semi-log scale.
(b)
CAP WPS
0.8
Fig. 3. (a) Typical V x —F L loop for the WPS state with Np:Nv = 0.6. Inset is a typical V x —F L loop for the CAP state with the same Np:Nv ratio. The solid symbols indicate the ascending plot, and open symbols are for the descending plot. (b) Corresponding N pv ¼ N v F L , the number of pinning vortices normalised by N v , vs. F L for both the WPA (open symbol) and CAP (solid symbol) loops in (a).
Np = 0.2 Nv 10
Np = 0.4 Nv
8
the vortices have been pinned in the initial state of CAP, which means that there are only 7% interstitial vortices. By contrast, there are 40% interstitial vortices in the WPS state. The bump in the ascending plot results from the motions of numerous interstitial vortices. Fig. 3b also indicates that following the increasing driving force the moving interstitial vortices of WPS are relocated into pinning sites and form many multiquanta vortices.
Np = 0.6 Nv Np = 0.8 Nv Np = 1.0 Nv
6 4 2
4. Conclusions
0 0.0
0.5
1.0
1.5
2.0
FL Fig. 2. Effect of pinning site density on the V x —F L curves for the WPS state.
We also have conducted simulations starting from a well pinning state (WPS). In the WPS, 100% of the vortices were set into the pinning sites. Other parameters are the same as with the current annealing process. Fig. 2 shows the plots of V x —F L curves for the samples with different N p : N v ratios. At the ratio of 1.0 ðN v ¼ N p Þ, there is no step at all in the ascending curve, which means that the J c has been a significantly increased. Fig. 3a compared the two V x —F L loops with the N v : N p ¼ 0:6 from the initial states of the WPS and CAP processes. For the WPS case, the ascending plot has a broad bump from the beginning, which makes the descending plot crossing over the ascending plot. Fig. 3b shows the number of pinned vortices N pv normalized by N v as a function of driving force. It can be seen that about 93% of
Based on Langevin dynamics simulation, we study the transport behaviors of vortices in plastic flowing states under a driving force. We find that the repinned effect causes step phenomenon in the velocity–force characteristics. Besides, the simulation results indicate that the initial state of vortices has a significant impact on this normal transport behaviour of vortex. Acknowledgement This work was supported by Australian Research Council under Project Nos. DP 0666771 and LX377331014. References [1] W. Henderson, E.Y. Andrei, et al., Phys. Rev. Lett. 77 (2007) 2077; G.D. Anna et al., Phys. Rev. Lett. 75 (1995) 3521; M. Bar-Sadan et al., J. Supercond. 17 (2004) 497. [2] Niels Gronbech-Jensen et al., Phys. Rev. Lett. 76 (1996) 2985; C. Reichhardt, G.T. Zimanyi, N. Gronbech-Jensen, Phys. Rev. B 64 (2001) 014501. [3] X.B. Xu et al., Phys. Rev. Lett. 101 (2008) 147002. [4] W. Henderson et al., Phys. Rev. Lett. 77 (1996) 2077; A. Gupta, R. Jagannathan, E.I. Cooper, et al., Appl. Phys. Lett. 52 (1988) 2077.
Contents lists available at ScienceDirect
Physica C journal homepage: www.elsevier.com/locate/physc
Step phenomenon of vortices flow in superconductor D.Q. Shi a,*, X.B. Xu b, X.B. Zhu a,c, L. Wang a, Q. Li a, S.X. Dou a a
Institute for Superconducting and Electronic Materials, University of Wollongong, Wollongong 2522, Australia Department of Physics, Nanjing University, Nanjing 210093, PR China c Key Laboratory of Materials Physics, Institute of Solid State Physics, Chinese Academy of Sciences, Heifei 230031, PR China b
a r t i c l e
i n f o
Article history: Accepted 9 November 2009 Available online 12 November 2009 Keywords: Type-II superconductor Vortex Molecular dynamics simulation
a b s t r a c t We investigate numerically the transport of vortices in superconductors under a driving force by using Langevin dynamics. The velocity-driving force characteristic (Vx–FL curve) of superconductors includes a region with series of stages. The impact of the initial states was investigated and there is a dramatic difference between the initial state produced by the current annealing process (CAP) and the well pinned state (WPS). Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction The step phenomenon in type-II superconductor is that at the onset of particular voltages, there are abrupt steps in the I–V curves which are associated with sharp peaks in the differential resistance. Experiments on different superconductors, both low critical temperature ðT c Þ and high T c types, have shown this features [1]. Many simulations have been conducted to simulate and explain this phenomenon [2]. Theoretical understanding of plastic depinning and plastic flow is far from complete, particularly in high temperature and low field conditions. In this report, we numerically study the step phenomenon via a molecular dynamics simulation (MDS) based on a random pinning arrangement. To connect our simulations to the experiments, we adopt high temperature and low field values found to be appropriate for such a sample through dc transport measurement. 2. Model The overdamped Langevin equation of motion for a vortex in position ri is [3]:
Fi ¼
Np Nv X X dr i F vv ðr i r j Þ þ F v p ðr i r pk Þ þ F L þ F Ti ¼ g dt j–i k
where F i is the total force acting on vortex i, Fvv and F v p are the forces due to vortex–vortex and vortex–pin interactions, respectively, F L is the driving force due to the current J ðF L / U0 J ZÞ, * Corresponding author. Tel.: +61 2 42215727; fax: +61 2 42215731. E-mail address: [email protected] (D.Q. Shi). 0921-4534/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2009.11.080
where U0 is the flux quantum, FT is the thermal stochastic force, g is the Bardeen–Stephen friction coefficient: g / U0 Bc2 =qn; N v the number of vortices, Np the number of pinning centers, and r pk the position of the kth pinning center. The total number of vortices Nv ¼ 484 is used. Similar results are obtained for larger systems. We employ penetration length k0 ¼ 690 Å´, the length of the vortex s = 12 Å´, while the pinning strength at zero temperature is fpv 0 ¼ 6f 0 , and g0 ¼ 1:4 10—17 kg=s and the range of pinning force rp ¼ 0:2k. The details of methodology have been reported elsewhere [3]. 3. Results and discussion At first, we use a current annealing process (CAP) to obtain the initial state for molecular dynamics simulation. Here, we choose the temperature of 0:9T c and a small field in order to simulate the transport I–V measurements which are usually at high temperature and self-field. The sample is under a driving force scan of L dF =dt of 0:01— 0:001f 0 =t 0 . Based on a typical statistic calculation from the simulation, about 97% of the vortices have been pinned by the pinning sites in the CAP state. Fig. 1 shows typical plots of the average velocity of the vortex in the x direction against the driving force F L under different fields. Except for the field, other parameters are the same and expressed as: N v ¼ N p ; T ¼ 0 : 9T c ; fpv 0 ¼ 6f 0 and r ¼ 0:5rp0 . The F L scan speed L is dF =dt ¼ 0:01f 0 =t0 . For the plot with B = 30 G, which means vortices are diluted, there are four rectangular steps, and the jumps are very sharp. This curve exactly simulated the reported I–V measurement curve of NbSe2 [4]. With the increase of the field, corresponding to the increase of the vortex density, the number of steps is reduced, and the jumps are not sharp.
S837
D.Q. Shi et al. / Physica C 470 (2010) S836–S837
0.9
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Fig. 1. V x —F curves from the numerical simulations under different fields. The inset (a) shows an enlargement of the indicated region. The inset (b) contains the L differential plot of dV=dF of the V x —F L curve for a 30 G field on the semi-log scale.
(b)
CAP WPS
0.8
Fig. 3. (a) Typical V x —F L loop for the WPS state with Np:Nv = 0.6. Inset is a typical V x —F L loop for the CAP state with the same Np:Nv ratio. The solid symbols indicate the ascending plot, and open symbols are for the descending plot. (b) Corresponding N pv ¼ N v F L , the number of pinning vortices normalised by N v , vs. F L for both the WPA (open symbol) and CAP (solid symbol) loops in (a).
Np = 0.2 Nv 10
Np = 0.4 Nv
8
the vortices have been pinned in the initial state of CAP, which means that there are only 7% interstitial vortices. By contrast, there are 40% interstitial vortices in the WPS state. The bump in the ascending plot results from the motions of numerous interstitial vortices. Fig. 3b also indicates that following the increasing driving force the moving interstitial vortices of WPS are relocated into pinning sites and form many multiquanta vortices.
Np = 0.6 Nv Np = 0.8 Nv Np = 1.0 Nv
6 4 2
4. Conclusions
0 0.0
0.5
1.0
1.5
2.0
FL Fig. 2. Effect of pinning site density on the V x —F L curves for the WPS state.
We also have conducted simulations starting from a well pinning state (WPS). In the WPS, 100% of the vortices were set into the pinning sites. Other parameters are the same as with the current annealing process. Fig. 2 shows the plots of V x —F L curves for the samples with different N p : N v ratios. At the ratio of 1.0 ðN v ¼ N p Þ, there is no step at all in the ascending curve, which means that the J c has been a significantly increased. Fig. 3a compared the two V x —F L loops with the N v : N p ¼ 0:6 from the initial states of the WPS and CAP processes. For the WPS case, the ascending plot has a broad bump from the beginning, which makes the descending plot crossing over the ascending plot. Fig. 3b shows the number of pinned vortices N pv normalized by N v as a function of driving force. It can be seen that about 93% of
Based on Langevin dynamics simulation, we study the transport behaviors of vortices in plastic flowing states under a driving force. We find that the repinned effect causes step phenomenon in the velocity–force characteristics. Besides, the simulation results indicate that the initial state of vortices has a significant impact on this normal transport behaviour of vortex. Acknowledgement This work was supported by Australian Research Council under Project Nos. DP 0666771 and LX377331014. References [1] W. Henderson, E.Y. Andrei, et al., Phys. Rev. Lett. 77 (2007) 2077; G.D. Anna et al., Phys. Rev. Lett. 75 (1995) 3521; M. Bar-Sadan et al., J. Supercond. 17 (2004) 497. [2] Niels Gronbech-Jensen et al., Phys. Rev. Lett. 76 (1996) 2985; C. Reichhardt, G.T. Zimanyi, N. Gronbech-Jensen, Phys. Rev. B 64 (2001) 014501. [3] X.B. Xu et al., Phys. Rev. Lett. 101 (2008) 147002. [4] W. Henderson et al., Phys. Rev. Lett. 77 (1996) 2077; A. Gupta, R. Jagannathan, E.I. Cooper, et al., Appl. Phys. Lett. 52 (1988) 2077.