Effect of pinning on the vortex motion in superconducting strip

Accepted Manuscript

Effect of pinning on the vortex motion in superconducting strip Zhidong Chen , Huadong Yong , Youhe Zhou PII: DOI: Reference:

S0921-4534(18)30085-6 10.1016/j.physc.2018.06.005 PHYSC 1253371

To appear in:

Physica C: Superconductivity and its applications

Received date: Revised date: Accepted date:

23 February 2018 9 May 2018 14 June 2018

Please cite this article as: Zhidong Chen , Huadong Yong , Youhe Zhou , Effect of pinning on the vortex motion in superconducting strip, Physica C: Superconductivity and its applications (2018), doi: 10.1016/j.physc.2018.06.005

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Effect of pinning on the vortex motion in superconducting strip Zhidong Chen, Huadong Yong*, Youhe Zhou* Key Laboratory of Mechanics on Disaster and Environment in Western China

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Ministry of Education of China, and Department of Mechanics and Engineering Sciences, College of Civil Engineering and Mechanics, Lanzhou University, Lanzhou, Gansu 730000, People’s Republic of China

Abstract

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We study the effect of different sizes and shapes of pinning centers on the vortex motion in superconducting strip in the presence of external current and magnetic field using the time-dependent Ginzburg-Landau theory. The size of pinning centers obviously affects the threshold current corresponding to the appearance of phase-slip

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line. The larger size results in a considerable decrease of the threshold current and a larger output voltage. For the same size, the different shapes of pinning centers lead to

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different behaviors of vortices. The dynamic behavior of vortices is dependent on the minimum distance 𝑑𝑚 between pinning center and the sample surface and the

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maximum length 𝑙𝑚 of the pinning center paralleling with the sample edge. However, the obvious changes are observed in the dynamic of vortex motion in superconductor

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with different pinning centers.

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1. INTRODUCTION The discovery of high-temperature superconducting material promotes the

practical application of superconductor in a variety of areas, such as LHC (Large Hadron Collider) [1], ITER (International Thermonuclear Experimental Reactor) [2], energy storage system [3], electric power application [4] etc. Due to their superior * E-mail address:

[email protected] (H.D. Yong)

* E-mail address:

[email protected] (Y.H. Zhou)

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performance, a number of researches on the behavior of type-II superconductor have been presented [5-8]. The superconducting strip has high-current-carrying capacity in high magnetic field [9]. However, the energy dissipation will occur during the motion of vortex in superconductors. Thus, it is necessary to consider the mechanism of the

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vortex motion and phase-slip centers (PSCs), which are two major factors induce the energy dissipation in the current-carrying superconducting thin films [10, 11]. In phase-slip state, the oscillation amplitudes of superconducting order parameter are not

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constant along the kinematic vortices motion in the presence of external current [12-14]. The velocity of these vortices [15] (𝑣𝐾 ≈ 105 𝑚/𝑠) is much larger than the speed of Abrikosov vortices [16] (𝑣𝐴 ≈ 103 𝑚/𝑠). Such kinematic vortexes are shown

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by experiment [17, 18] and simulations based on the time-dependent Ginzburg– Landau (TDGL) equation [13]. With the development of modern microfabrication

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techniques, the mesoscopic superconducting sample attracts much attention in which

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the size is comparable to the coherence length 𝜉. Because of the influence of the size and geometry, the PSCs are converted into phase-slip lines (PSLs) [19]. It is well

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known that phase-line state leads to the resistive state in the superconductor, where

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the local temperature increases and the superconductor may experience transition from the condensed state to the normal state [20]. This phenomenon was reported by numerical simulations with TDGL [21]. This work investigates the effect of pinning on the vortex motion in the superconductor strip with the external current. We are interested in the fact that the pinning strongly affects the response of the superconductor to external electric and

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magnetic fields [22, 23]. Larkin et al [23] demonstrated the influence of weak pinning with high densities of inclusions. Ovchinniko et al [24] investigated the critical-current value in the superconductor with strong pinning centers, i.e., low densities of strong impurities. The current distribution is very sensitive to the

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inhomogeneities in current-carrying superconductors [13]. Berdiyorov et al [25, 26] showed that the dynamics of current-driven phase-slip centers in superconducting strips with numerical simulations and found that pinning obviously reduces the

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threshold current of the transition from Meissner state to mixed state. He et al [27, 28] have presented the dynamics of the vortices in a mesoscopic thin superconducting strip with slits. The resistive state transition current becomes larger with increasing

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the titled angle of slit. And Xue et al [29, 30] investigated the dynamics of vortex in the mesoscopic superconducting ring under the current and magnetic field. In our

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simulation, the trend of vortex motion by introducing pinning centers with different

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shapes and sizes is presented. By increasing the size of pinning centers, the threshold current for the appearance of PSL in the type-II superconductors becomes smaller. At

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a low current only the Abrikosov vortex exits in the superconductor, and the vortex

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will move fast (kinematic state) at larger currents. With further increase of current, the PSLs appear and the whole system tends to transit into the normal state. The paper is organized as follows. A brief introduction to our theoretical

approach is presented in section 2. Section 3 presents the current-voltage characteristics of the sample with different pinning centers and the free energy characteristics of the sample. The conclusions and summarization are in section 4.

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2. THEORETICAL FORMALISM In this section, we consider a thin superconducting strip with length L, width W and thickness 𝑑(𝑑 ≪ 𝜉, 𝜆 ) in a uniform magnetic field 𝐻 (𝐻 = 0.05 ) along the

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z-direction. The transport current 𝐼 is along the x-direction through normal-metal contact. The pinning is introduced by inclusions with different shapes, and only the diamond pinning centers are presented in Fig.1. The inclusions are made of normal

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metal material or other superconducting material with lower critical temperature (𝑇𝑐1 < 𝑇𝑐 ) which are arranged with array structure (4 ∗ 7). The dynamical properties of vortices are investigated for four different cases: the square-pinning case, the

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diamond-pinning case, the ellipse-pinning case and the rectangular-pinning case. In order to study the effect of different pinning centers on type-II superconductor,

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we use the following form of the generalized TDGL equations [31, 32]:

2  2   2   i       iA    f (r )   2  2 t  1   2   t







  div Im  *    iA 

2



(1)

(2)

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u

where the distance is scaled by coherence length 𝜉(𝑇) and the unit of time is GL relaxation time 𝑡𝐺𝐿 = 𝜋ℏ/8𝑘𝐵 𝑇𝑢. We express the electrostatic potential 𝜑 in units of 𝜑0 = ℏ/2𝑒𝑡𝐺𝐿 and vector potential 𝐀 is scaled of 𝐻𝑐2 𝜉(𝐻𝑐2 = 𝛷0 /2𝑒𝜉 2 , and 𝛷0 is the quantum of magnetic flux) [31]. The current density is measured in

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𝑗0 = 𝑐𝛷0 /8𝜋 2 𝜆2 𝜉

.

The

parameter

𝑓(𝑟⃗) = 1 − 𝑇/𝑇𝑐

describes

the

𝑇𝑐 -nonhomogeneity in the model [32]: for 𝑓(𝑟⃗) < 1 (𝑇𝑐1 < 𝑇𝑐 , in this article we choose 𝑓(𝑟) = 0 for pinning) the superconductivity is weaker inside the pinning than the basic superconducting material, leading to enhanced attraction of flux

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vortices. We use the parameter 𝛾 = 2𝜏𝐸 Ψ0 /ℏ to characterize the chosen material with 𝜏𝐸 meaning the inelastic collision time and 𝜓0 is the value of order parameter at 𝑇 = 0 and 𝐻 = 0. The parameter u is the measure of the different relaxation time.

assign 𝐀 = (−𝐻𝑦/2, 𝐻𝑥/2, 0).

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Because the width of the strip is much smaller than the effective penetration depth, we

In this section, the length L and width W of the strip are chosen to be 32𝜉 and 16𝜉 . The parameter 𝑢 and 𝛾 are taken as 𝑢 = 5.79 and 𝛾 = 10 [31]. The

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environment temperature is close to 𝑇𝑐 , and we neglect the heating effects in our

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simulations. Superconductor-vacuum boundary conditions (∇ − i𝐀)𝜓|𝑛 = 0, ∇𝜑|𝑛 =

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0 are adopted along y-direction and normal-superconductor boundary conditions 𝜓|𝑦=0,𝑤 = 0, ∇𝜑|𝑛 = −𝑗𝑎 are used along x-direction. For our simulation, we solved

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the TDGL equations by COMSOL [33].

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And the total free energy of the superconductor is given by [34] 2 2  2 1 4 F         i  A    2   A  0 H  d   2  

(3)

Because 𝐀 and 𝐇 are assumed to be constants in our system, the last item 1

(magnetic energy and diamagnetic energy) is neglected. We use 𝑓 = |𝜓|2 + 2 |𝜓|4 + |(−𝑖∇ − 𝐀)𝜓|2 to simplify the calculation of the free energy.

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3. NUMERICAL RESULTS 3.1 Square-pinning and diamond-pinning cases Next, we consider effects of pinning centers with different shapes and sizes and choose square and diamond pinning centers. Two groups of pinning centers with same

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shape and different areas (𝑠1 = 1.00 and 𝑠2 = 1.57) are investigated. We first set the properties of square-pinning sample as a reference for our further analysis. The pinning considerably reduces the threshold current of the transition from Meissner

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state to mixed state [26]. The average voltage (I-V) characteristics is shown in Fig.2 (A). With the applied current increasing, there is no vortex in the sample up to a threshold current where the superconductor turns into mixed states (vortex state) from

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Meissner state. The resistance of sample is always nonzero due to the normal-superconductor boundary condition. The Abrikosov vortex appears at a lower

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currents range, i.e. 0.22𝑗0 < 𝑗 < 0.26𝑗0 (see red point 1 in Fig.2 (A)), resulting in a

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slight difference in slope of I-V curve (see the inset highlighted by canary yellow block in Fig.2 (A)) and the snapshot of |𝜓|2 is shown in panel 1(a). Since the

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external magnetic field is present, the vortex first appears at the middle of the right

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edge of sample, crosses the sample and disappears at the opposite edge. As the current is smaller than 𝑗𝑐1 , the system is in a nonequilibrium resistive state, i.e. nonequilibrium Abrikosov vortex state. Beyond the current 𝑗𝑐1, the velocity of vortex becomes larger and first PSL is induced in the sample through the middle row of pinning centers. The obvious deformation of vortices can be observed (see white ellipses in panel 1-4 in Fig.2) [13, 35]. With further increase of the applied current,

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more PSLs appear in the sample. They prefer to be arranged axially symmetrically, while the PSLs begin to merge and separate irregularly as the current is close to the point where the entire sample will transform into normal state. The different number of the PSLs correspond to the different resistance of the sample, which lead to

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different slopes in I-V curve. As the area of pinning center increases, the critical current 𝑗𝑐1 (𝑗𝑐1|𝑠=1.00 = 0.27𝑗0 , 𝑗𝑐1|𝑠=1.57 = 0.26𝑗0 ) becomes smaller and a PSL appears more easily in the

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sample since the penetration barrier for the vortices is lower and the attraction of vortex is increasing by introducing the larger pinning centers. Another reason may be that the larger pinning centers is closer to the sample edge that reduces the [36]. In the large current region, more and more PSLs appear in

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edge/surface barrier

the sample, leading to a jump in the I-V curves and an increase of the

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sample-resistance. Just like the situation of appearance of one PSL, the two PSLs

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appear with a lower current in superconductor with larger pinning which is illustrated at the points 2 and 3 in Fig.2 (A). It is interesting that the introducing of vortex just

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results in a slight increasing (near 𝑗 = 0.22𝑗0 ) in the I-V curve(see the insets in Fig.2).

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The reason is that the small speed of the vortex makes a little change of sample resistance from a superconducting state to the resistive state, while a larger vortex speed leads to PSLs and an obvious change of slopes in I-V curve. The mechanism is that the motion of vortex results in the magnetic flux change over time and leads to the change in the resistance of superconductor. The other interesting thing is the absolute value of Cooper-pair density distribution becomes approximately symmetric

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in a dynamic stable state. Then we discuss the diamond-pinning case. Compared with the square-pinning case, they exhibit similar behaviors in the both different areas of pinning center samples (𝑠1 = 1.00 and 𝑠2 = 1.57). For the different sizes, the impacts of pinning

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centers are similar and obvious. A larger area leads to a noticeable higher voltage with the same current and a smaller threshold current (including from superconducting state to resistive state, from Abrikosov vortex state to the kinematic vortex states).

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There are some differences between the two cases. For a larger transport current, the third PSL appears in a lower current in diamond-pinning case and the voltage of square-pinning case is a little smaller in the same state (such as the same number of

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PSLs) at the same current. The main factor can be the minimum distance 𝑑𝑚 between the pinning center and the sample surface along the x-axis and the maximum

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length 𝑙𝑚 of the pinning center parallel to the current flow direction along the y-axis,

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which are shown in Fig.3. To explore the differences, the ellipse-pinning case and

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rectangle-pining case will be investigated in next section.

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3.2 Ellipse-pinning and rectangle-pining cases To find out the reason of those differences, we choose the same area of pinning

center for two cases (𝑠2 = 1.57) and place the pinning center with two different angles (𝛼1 = 0°, 𝛼2 = 90°), which is defined as the 𝛼1 -case and the 𝛼2 -case. Fig.3 shows the schematic of the ellipse-pinning case and the rectangle-pinning case. The longer axis and shorter axis of ellipse-pinning centers are 𝑎 = 2.0, 𝑏 = 1.0. And the

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side lengths of rectangle-pinning center are 𝑙 = 2.507, 𝑤 = 0.626. The behavior of vortex motion of 𝛼2 -cases are similar to the 𝑠2 -square-pinning case. However, there are obvious differences between 𝛼1 -cases and other 𝑠2 - square-cases. In 𝛼1 -case, the two-vortices state replace the one-PSL state (see the panel 1(a) and panel 2(b) for red

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point 1(a) and blue point 2(b) in Fig.4). And after threshold current 𝑗𝑐1 = 0.27𝑗0 , the two vortices directly turn into two PSLs in the same symmetry position (panels 1(a) and 2(a) for the red points 1(a) and 2(a) in Fig.4(B)), which results in obvious change

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of slope and a jump in the I-V curve. The minimum distance 𝑑𝑚 is major factor for this phenomenon (see Fig.3). Meanwhile, the larger 𝑑𝑚 indicates the larger penetration potential barrier [36]. And due to the suppressed superconductivity, the

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closer location of defect to the edge of the sample leads to a smaller threshold currents [27]. For same reasons, another noticeable difference is the threshold current where

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the new PSL appears in the sample. The 𝛼2 -rectangle-sample turns one PSL state to

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two PSLs state in a larger current (𝑗𝑐2 = 0.28𝑗0 ). And in the 𝛼1 -rectangle-sample, the number of PSLs increases from two to three in a smaller current (𝑗𝑐3 = 0.32𝑗0). And

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by comparing 𝛼1 -cases and 𝛼2 -cases, we find that the voltage of samples in the

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similar state increases with the maximum length 𝑙𝑚 increasing. This is due to the reasons that the vortices prefer to move along the row of pinning centers and the pinning centers create an effective path for the motion. The larger 𝑙𝑚 makes a greater area for motion of vortexes [26]. That leads to a larger space for low-density of Cooper pairs, which means a larger higher-resistance zone. 3.3 Discussion

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According to the panel 1-4 in Fig.2 and panel 1-4 in Fig.4, we can find the position of phase-slip lines is determined by pinning centers and the current. At a small current, there are only the Abrikosov vortexes in samples. The vortex first appears from the middle of the surface and moves through the middle row of pinning

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centers along x-axis in a lower speed. Because the slow vortex cannot lead to a larger increase of the sample resistance, the differences of voltage in same current among these cases are negligible. After 𝑗𝑐1, the PSL appears in the middle of sample except

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the 𝛼1 -cases. When the sample finally is in a dynamic equilibrium status, the phase-slip lines are in symmetrical distribution to minimize their mutual repulsion and the free energy. Another reason of the symmetry is the symmetrical arrangement of

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pinning centers.

In Fig.5, we analyze the vortex behaviors by presenting the free energy and

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voltage of samples versus time for four cases (square, diamond, 𝛼2 -ellipse and

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𝛼1 -rectangle). The samples are all in Abrikosov vortex state at 𝑗 = 0.24𝑗0 . In this state, the motion of vortex, which results in energy dissipation and an obvious voltage characterizes

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oscillation,

the

𝐹𝐸(𝑡)

(free

energy-time)

curve

and

𝑉(𝑡)

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(voltage-time) curve. As noted above, all cases have the similar features. However, with increasing current, the trend becomes different and the increase of the free energy and voltage is obvious. We choose 𝑗 = 0.29𝑗0 and 𝑗 = 0.31𝑗0 since in these currents all cases have the same number of phase-slip lines. Because the PSLs are unstable at the initial stage of the mixed state formation, both of 𝐹𝐸(𝑡) curve and 𝑉(𝑡) curve show that the sample is in an unstable state when the sample turns into

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mixed states in Fig.5. The voltage and free energy reach the maximal values before the appearance of stable PSLs at larger current. The sample achieves dynamic balance after a long time. The free energy and the voltage tend to keep the stable values. Each curve has a certain average value with small oscillations when the stable phase-slip

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lines formed in the samples. While at 𝑗 = 0.31𝑗0, the current is close to the critical current where the sample turns the resistive state into normal state. Those figures demonstrate the conclusion, samples tend to show similar vortex motion when they

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have the comparable size and same arrangement of pinning centers. Moreover, there is a close relation between free energy and voltage. As mentioned above, both of the free energy and voltage are affected by of motion of vortex, so that the 𝐹𝐸(𝑡) curves

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4. CONCLUSIONS

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are similar to the 𝑉(𝑡) curves.

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In conclusions, we have studied the effect of size and shape of pinning center on the response of superconducting strips under an external current by using the time

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dependent Ginzburg-Landau theory. We found that the size and shape of pinning

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center play an important role in the performance of superconductor, which is related to the penetration barrier and the effective path for the superconducting vortex motion. The threshold current decreases with the increases of size of pinning center. Then we have presented detailed discussions on the shape factors. The smaller minimum distance 𝑑𝑚 between pinning center and the sample surface leads to the smaller surface barrier and threshold current. For a larger maximum length 𝑙𝑚 , it results in a

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larger output voltage due to the fact the pinning centers provide the effective path for vortex motion. Acknowledgments This work was supported by National Natural Science Foundation of China (Nos.

Universities (lzujbky-2017-k18) and 111 Project, B14044.

[1] A. Ballarino, Supercond. Sci. Technol. 27 (2014) 044024.

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11327802 and 11472120), the Fundamental Research Funds for the Central

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[33] COMSOL.

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Pinning(Tc1)

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[36] D.Y. Vodolazov, I. Maksimov, E. Brandt, Physica C. 384 (2003) 211-226.

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Y

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Fig.1. A thin superconducting strip (size 𝐿 × 𝑊 × 𝑑, thickness𝑑 ≪ 𝜉) containing

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arranged pinning centers (4 × 7 array) with critical temperature 𝑇𝑐1 < 𝑇𝑐 in the

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presence of an applied current 𝐈 and a perpendicular magnetic field 𝐻 = 0.05.

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1.2

1.2

s1=1.00 s2=1.57

A 1.0

s1=1.00 s2=1.57

B 1.0

4(b) 4(a)

V/V0

0.12

0.6

0.8

V/V0

0.18

0.8

0.18 0.12

0.6

0.4

0.20

0.22

0.24

0.26

0.28

3(a)

0.4

2(b)

2(a)

0.2

0.06 0.20

0.2

1(b) 1(b) 0.0 0.00

0.05

0.10

0.15

0.20

0.25

0.30

j/j0

0.0 0.00

0.05

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3(b)

0.06

0.22

0.24

0.10

0.26

0.15

0.28

0.20

0.25

0.30

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j/j0

Y

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X

Fig.2. I-V characteristics of the sample with size 16𝜉 × 32𝜉 and a square array of

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pinning center for different sizes 𝑎 = 1𝜉 (red curve and area 𝑠1 = 1.00𝜉 2 ) and

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𝑎 = 1.253𝜉 (blue curve and area 𝑠2 = 1.57𝜉 2 ). Panel A and B show the I-V curve for square and diamond pinning centers. The Insets in the main panel represent the

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lower part of the I-V curve. Panels 1–4 show snapshots of the Cooper-pair density |𝜓|2 for different resistive states corresponding to the I-V curves. And inset a and b show the 𝑠1 -case and 𝑠2 -case in panels 1-3. Panel 4 shows the difference between square-pinning case and diamond-pinning case. White ellipses in panel 1-4 highlight some of the vortices in the sample.

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Boundary

2.507

2.000

Superconductor

CR IP T

0.626

1.000

X

AN US

Y

AC

CE

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M

Fig.3. Schematic view of the different shape and size of pinning center in the sample.

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1.2

1.2

α1=0

A 1.0

α2=90o 0.18

0.8

B

o

1.0 0.8

α1=0o α2=90o

0.60

4

0.48

V/V0

0.36

0.12

0.6

0.6

3

0.24

0.06

2(a)

0.12

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0.20 0.22 0.24 0.26 0.28

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j/j0

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2(b)

0.28

1(b) 0.05

1(a)

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0.4

0.10

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0.20

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0.30

j/j0

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Fig.4. Panels A and B (A for the ellipse-case, B for the rectangle-case) show I-V

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characteristics of the sample with size 16𝜉 × 32𝜉 in the presence of an array of pinning center of different shapes and angles (red curve for 𝛼1 = 0o and blue curve

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for 𝛼2 = 90𝑜 ) for the inhomogeneity coefficient of 𝑓(𝑟) = 0. The pinning centers of

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all cases has the same area 𝑠 = 1.57𝜉 2 . The Insets in the main panel represent the lower part of the I-V curve. Panels 1–4 show snapshots of the Cooper-pair density

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|𝜓|2 and the resistive states for current values indicated on the I-V curves.

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B

A 1.2

0.24j0 0.29j0 0.31j0

1

-80 -120

1

1.0

0.24j0 0.29j0 0.31j0

0.8 0.6

-160

0.4 0.2

-200

0.0

2

2

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V/V0

FreeEnergy

0.6 -160 -200

3

-80

0.4 0.2 0.0

3

1.2 1.0

-120

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-160

0.4 0.2

-200

4

-80

AN US

0.0

-60

CR IP T

0.8

-120

4

1.0 0.8

-120

0.6

-160

0.4 0.2

-200

0.0

0

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2000

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t/t0

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0

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t/t0

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Fig.5. the curves of the free energy (A) and the voltage (B) versus time for four cases. Panel 1-4 shows the curves of square (1), diamond (2), 𝛼2 -ellipse (3) and

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𝛼1 -rectangle (4) respectively.