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Critical Current Trends in Type II Superconductors withPinning

Jacob GordonLyman Briggs College/Department of Physics, Michigan State University

Abstract

When Type II superconductors are in a magnetic field above a characteristic field

H1 (as they are in most superconductor applications) field lines penetrate the

superconducting area, thus creating vortices. By adding impurities to the superconductor,

those vortices can be pinned allowing more current to flow through the superconductor

without moving the vortices and ruining the very low resistance the superconductor had.

The effects of varying the sizes, numbers, and orientations of these normal inclusions are

simulated using the Ginzburg-Landau equation and finite elements method.

Introduction

Type I superconductors are traditionally thought of when someone mentions

superconductors. Superconductors are identified by two traits: perfect conductivity and

perfect diamagnetism. Perfect conductivity means, quite literally, there is no resistance

within a superconductor. Perfect diamagnetism means that the superconducting material

repels all external magnetic fields from itself. However, if a magnetic field is strong

enough, the superconductor can no longer expel it, and thus its superconductivity breaks

down and it returns to a normal state1. This applies to Type I superconductors, however

that is not what is studied here.

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There are two characteristic lengths for superconductors: the London penetration

depth λ and the coherence length ξ . The penetration depth is the depth a magnetic field

can penetrate into a superconductor. The coherence length is the distance over which the

density of electron pairs can vary without increasing the energy. The ratio between them

is κ , which is equal to ξλ . In Type I superconductors, 21κ then the superconductor behaves completely differently. These are known as

Type II superconductors1.

In Type II superconductors, instead of breaking down under excessive magnetic

fields they allow the magnetic field lines to penetrate only in specific areas, where the

field lines get pinched together. These areas in Type II superconductors are called

vortices. They are normal (non-superconducting) regions with super currents rotating

around them. Due to this ability of Type II superconductors, they are able to reach mixed

states, where they have both superconducting and normal attributes. The field above

which vortices start appearing is known as Hc1. If you continue to increase the field, the

number of vortices will increase and eventually the material will be completely normal.

The field above which the material is no longer superconducting is known as Hc21. Figure

1 shows 13 by 13ξ Type II superconducting plains with no current running through them

at different magnetic fields. The red area is superconducting while the blue areas are the

normal regions associated with vortices. By the time you reach a field of 5.0 T,

superconductivity is lost.

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Figure 1: Steady state vortex configurations at different magnetic fields.

Though Type II superconductors still have perfect conductivity in areas of the

material, if a strong enough current is run through the material the current causes the

vortices to move. This is a problem because as the vortices move they create resistance

through Lorenz forces1; thus ruining the perfect conductivity. The highest current at

which the vortices are still able to reach a steady state is known as the Critical Current.

There are, however, various ways to pin the vortices. They can be held in place by

adding small areas of normal material (material that is not superconducting in the same

H = 2.0 T H = 3.0 T

H = 5.0 T

H = 2.5 T

H = 4.5 TH = 4.0 TH = 3.5 T

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conditions, or conditions anywhere near the conditions that the superconductor is at) to

the superconductor. This is called ‘pinning’ the vortices. They are naturally attracted to

these normal inclusions. This additional force on the vortices, holding them in place,

allows for a higher current to run through without moving the vortices. This increases the

critical current.

Simulations of Type II Superconductors

Simulations were run with different numbers of normal inclusion, different

configurations, and different size normal inclusions using a code based on a time-

dependent Ginzburg-Landau equation. The code outputs the value of the order parameter

ψ for points in the sample at every time step. ψ varies from zero to one and measures

the density of cooper pairs in the superconductor1. If ψ is one, then the material is

completely in a superconduting state at that location. If it is zero, it is completely in a

normal state. The number of locations within the square sampled (at each time-step) can

be changed by changing a parameter within the code called “n_grid”. Increasing this

makes the grid over the sample finer. At each point on the grid ψ is measured for every

time-step. This allows contour maps to be created for each time-step, and when the time-

steps are combined videos can be generated showing the movement of vortices through

the square sample. Thus, the critical current can be found by finding the highest current

at which the vortices still reach a steady state.

When running the simulations, first parameters within the code need to be set for

the specific trial. Results come from simulations of a 13 by 13ξ sample in a 2.5 T

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magnetic field. n_grid is 18, which creates a 35 by 35 grid, yielding 1225 data points for

each time step. Other parameters within the code that need to be changed depending on

the trial’s conditions are the number of normal inclusions and the radius of normal

inclusions. The locations of the inclusions and the current running through the sample

are set within a text file in the code’s directory. The currents these simulations were run

at are all multiples of the critical current of the 1.3 by 1.3ξ case with no normal

inclusions in a magnetic field of 2.5 T, called Jc. This current was taken from Yanzhi

Zhang’s results2. The output data was viewed and analyzed using tecplot. The pictures

from Figure 1 and the others like Figure 1’s were generated using tecplot.

Results

First simulations were run without normal inclusions at different currents. Figure

2 displays the steady state configurations of vortices at sample currents approaching the

critical current and at the critical current. Notice how the vortices slowly get pushed

further to the right by the stronger currents coming in from the left; this trend continues

above the critical current as well. However, above the critical current vortices are pushed

off the right side of the square, and a new vortex spawns from the edge of the left side.

This continues, creating a stream of vortices across the super conductor. Generally, the

vortices reach a state where the new vortices follow the same path as the ones leaving.

The moving vortices move faster at higher currents. Also note, often many of the

vortices are not part of the stream, and if they move at all, it is just small movements as

other vortices pass by.

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Figure 2: Steady state positions of vortices at different currents.

Simulations were also run to determine the effects of adding more vortices to the

superconductor. Figure 3 shows the steady states reached in those simulations. Trials

were run at currents of 0, 1.0Jc, 1.3Jc, 1.6Jc and 1.8Jc and with five, seven, nine and

eleven normal inclusions. The normal inclusions were arranged to have symmetry over

horizontal and vertical bisectors. The results show a clear increase in critical current up

to nine normal inclusions. At 1.8Jc, both the cases with eleven and nine normal

inclusions were unstable. However the “stream” of vortices in the case with nine was

moving slower than the “stream” in the case with eleven. This may imply that the case

with nine actually has a higher critical current, but this cannot be determined without

more tests. Either way, adding that many normal inclusions is impractical because it does

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not leave sufficient area unaffected by the normal regains. Note that the majority of the

sample’s area in the cases of Figure 3 are no longer superconducting, leaving only the

edges. This might be avoided by using more normal inclusions with a smaller radius.

Figure 3: Steady state positions of vortices at different currents for differentnumbers of normal inclusions

Increasing the size (starting from a radius of zero) of the normal inclusions causes

a parabolic-like trend overall where it increases, then decreases. Figure 4a and 4b show

this parabolic-like behavior. The boxes below the graphs show the orientation that the

normal inclusions are in. The fit lines are only rough estimates of what is actually

happening. Notice the orientation with four normal inclusions reaches a much higher

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peak critical current than the two normal inclusion orientation. This is consistent with

Figure 3’s results.

Figure 4a Figure 4b

Figure 4c is a graph of critical current against normal inclusion size, and has one normal

inclusion centered in the sample. It shows a similar peak and drop-off as Figure 4a and

Figure 4b do, but then at a much higher normal inclusion radius, the critical current

increases again. Testing the other two configurations at a radius of 3ξ was not done

because at that radius their normal inclusions would have been overlapping. This raises

the question of how the critical current behaves between the radiuses tested. Are there

then many more hidden minimums?

Critical Current vs. Size of Two Virtical Line Oriented Normal Inclutions

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5

Size of normal inclution (in xi)

Crit

ical

Cur

rent

(in

J c)

Critical Current vs. Size of Square Oriented Normal Inclutions

0

0.5

1

1.5

2

0 0.5 1 1.5 2 2.5

Size of normal inclutions (in xi)

Crit

ical

Cur

rent

(in

J c)

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Figure 4c

The next variable tested was the orientation of the normal inclusions. It was

found that orientation matters; it matters a lot. Figure 5 shows data (presented in the

same way as Figure 4a-c) for three different ways to position four normal inclusions. All

three have very different critical currents above 1ξ and different peak locations. It

should be noted however, that Series 3’s line of fit is clearly flawed. The critical current

cannot become negative, and, based on how the system was behaving with normal

inclusions of 1.5 and 2ξ , the critical current probably will not start to increase again. In

fact, the reason for the sudden drop-off in the critical current is that-for this orientation-

when the radius of the normal inclusions gets to 1.5ξ , they become very close to the

edge of the system. Looking at the time evolution of the system at that radius, it appears

Critical Current vs. Size of Single Centered Normal Inclution

00.20.40.60.8

11.21.4

0 0.5 1 1.5 2 2.5 3 3.5

Size of normal inclution (in xi)

Crit

ical

Cur

rent

(in

Jc)

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that the normal inclusion is actually helping pull the spawning vortices off of the wall.

This newly pulled vortex then exerts a repulsive force on the vortices around it, and,

backed by the push of the current, it is able to push the vortex near the left edge of the

system off (which was made easier by the normal inclusion very close to the left wall of

the system). Thus the critical current for these cases drops well below Jc.

Figure 5

On the other hand, if two vortexes grow too close together, it also reduces the

critical current. Figure 6 shows that even with the same amount of normal material area,

having more, separate normal inclusion offers a higher critical current. The bottom

middle and bottom right cases have a critical current in the same range, which may be

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because the vortices in the bottom middle configuration are already so close together that

it is acting much like the bottom right configuration. Normal inclusions nearing each

other reduce the critical current for two reasons. When two vortices are very close

together, they repel each other. Thus, when two normal inclusions have vortices in them,

those vortices are getting pushed away from each other. The normal inclusions also

create a pull on near by vortices. So if a vortex is in a normal inclusion, there could be

another normal inclusion nearby pulling the vortex away from the one it is already in.

Both of these reduce the normal inclusion’s ability to hold onto vortices, and thus lower

the critical current.

Figure 6: This shows the highest current steady state vortex configuration for differentnormal inclusion orientations with the same amount of area of normal material (in ξ 2) dis the distance between the edges of normal inclusions

Conclusion

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These results imply that the critical current can be maximized by optimizing the

normal inclusion orientation in two ways. First, by balancing the ratio between the

distance between the edges of the system and the edges of the normal inclusions with the

distance between the edges of the normal inclusions and other inclusions. It appears from

the simulations that the highest critical current will appear when the distance between the

edges of the square and the edges of the normal inclusions should be the larger one, but

with more simulations that could be found out for certain. Note, the ratio does not

depend on the radius of the normal inclusions, but the actual distances do depend because

we are measuring from the edges of the normal inclusions and not from their centers.

Second, by choosing an orientation that allows more space between the edges of the

system and the edges of the vortices. For example, a square orientation is better than a

diamond orientation. Looking at the time evolution of the case without normal inclusions,

it will be seen that the vortices stop in one orientation, and then, eventually move again.

This final steady state orientation is preferred to the systems prior orientation because

there is more space between the vortices.

The results do not show any evidence that there is a trend for different normal

inclusion radii that is consistent across different numbers of normal inclusions, or across

different orientations. However, they do show that some radii are preferred to others. So

once an orientation has been chosen, simulations could be run to find what the best

normal inclusion radius is for that orientation. Remember that the locations of the normal

inclusions will have to be adjusted depending on the radius, assuming that future research

is consistent with the findings here.

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If what has been suggested here can be done, then Type II superconductors can be

created with much higher critical currents, increasing their efficiency and usefulness.

Hopefully, further research into the trends touched on here will make that possible.

Acknowledgment

The author would like to thank Professor Max Gunzburger and Yanzhi Zhang for

guidance and support on this project and for the program used, Florida State University’s

Department of Scientific Computing for use of its facilities and resources, and The

National Science Foundation for its funding.

References

1Tinkham, Michael, “Introduction to Superconductivity” Second Edition, Dover

Publications Inc., Mineola, New York, 2004, pg 2, 3, 5-7, 9, 11-13

2Yanzhi Zhang, “Result”, Florida State University Department of Scientific Computing

(Unfinished Manuscript)