Numerical Ginzburg-Landau studies of Jc in 2D and 3D polycrystalline superconductors

G.J.Carty and D P HampshireSuperconductivity Group, Department of Physics, University of Durham, South Road,

Durham DH1 3LE, UKCalculated critical current density values are presented for simulated 2D and 3D polycrystalline superconductors. In high applied magnetic fields Jc in 2D is similar in functional form and magnitude to that predicted by the Kramer model, while a second mechanism operates in low fields, consistent with the experimental data originally review by Kramer. In 3D superconductors Jc follows the Kramer-like b–½(1 – b)2 reduced-field dependence throughout the entire field range, as is observed in state-of-the-art materials.

We acknowledge the support of Duncan Rand, Lydia Heck, Thomas Winiecki and the Durham University Scholarship Fund

1.Calculation MethodA semi-implicit numerical TDGL algorithm is used, with the normal metal components simulated with a Usadel-derived model[1].

The 3D polycrystalline superconductor is simulated as a packing of truncated octahedra, thus giving identical grains separated by constant-width boundaries, but without any continuous planar grain boundaries running through the material.

The edges of the superconductor are diffused over a width of 10ξ to lower the overall surface barrier.

The grain boundaries consist of a non-superconducting region that is wider than the inner region where ρ is increased by electron scattering, consistent with the strong strain dependence of Tc [2] and avoiding Hc3 effects.

[1] G. Carty, M. Machida, and D. P. Hampshire, Phys. Rev. B 71, 144507 (2005). [2] J. W. Ekin, Cryogenics 20, 611 (1980)[3] E. K. Kramer, J. Elec. Mat. 4, 839 (1975)

5. Conclusions

( )( )p

fmk-= ´ -

2.51 224 2

200

23.6 10 1c

p

B TF b b

3. 3D resultsThe simulated 3D superconductor is made up of truncated octahedral grains, as shown in Fig. 4 (the diffused edges are as shown in Fig. 1)

Simple ρN = ρS grain boundaries are used, due to 3D computational limitations.

Figure 5 shows that in both high and low E-fields, Jc in 3D is about 20% of its value for the equivalent 2D system.

In Fig. 6 we see that fluxons within the grains of the superconductor are almost completely straight – all significant bending is on entering and leaving grain boundaries.

As in the 2D system, dissipation in the 3D system results from flux flow along grain boundaries.

-4 0 0 4 0

-1 0 0

-5 0

0

5 0

1 0 0

-4 0 0 4 0

-100

-50

0

50

100-5 0 0 5 0 1 0 0

-1 0

1 0

-5 0 0 5 0 1 0 0

-1 0

1 0

-1 0 0 -5 0 0 5 0 1 0 0

-1 0

1 0

-1 0 0 -5 0 0 5 0 1 0 0

-1 0

1 0

0 0 .2 0 .4 0 .6 0 .8 1 .0

Figure 6: |ψ|2 in slices of simulated 3Dsuperconductor at H = 0.151Hc2

0.0 0.2 0.4 0.6 0.8 1.00.00

0.01

0.02

0.03

0.04

0.05

= 10, Grain Size 30, N =

S

J c½B¼

(0¼

Hc2

¾/

)

Local field (Bc2)

2D high-E2D low-E3D high-E3D low-E

Figure 5: Kramer plot comparing Jc in 2D and3D superconductors at high and low E-fields

Figure 4: 3D packing of truncated octahedra

2. 2D results Normal

Fullysuperconducting

Main material(constant ρ)

Grain boundary(increased ρ)Edges matched byperiodic boundaryconditions

Figure 1: Diagram of the simulated2D superconductor system

2D calculations use a slice through the 3D structure, as shown in Fig. 1.

When plotted on a Kramer plot, Jc for the 2D system has a ‘knee’ at H ~ 0.8Hc2, suggesting that Jc results from two different mechanisms.

The inner grain boundary resistivity does not affect Jc – this can be seen in Fig. 2.

Changing kappa, or the grain size may move the position of the knee, but not the high-field reduced-field dependence.

Fluxons within the grain boundaries, visible in the enlarged inset of Fig. 3, move for J > Jc these grain boundary fluxons thus determine the critical state.

Figure 2: Kramer plot for 2D κ = 10 superconductorwith 30ξ grains, demonstrating two mechanisms

and ρN/ρS-independence of Jc

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.000

0.005

0.010

0.015

0.020

0.025

0.030

Mechanism 2

Mechanism

1

= 10, grain size 30

J c½B¼

(0¼

Hc2

¾/

)

Mean Local Field (Bc2)

N = 2

S

N = 5

S

N = 10

S

N = 20

S

Figure 3: Plots of a) |ψ|2 and b) magnitude of normalcurrent at H = 0.43Hc2

- 5 0 0 5 0

- 1 0 0

- 5 0

0

5 0

1 0 0

1 5 0

2 0 3 0 4 0

- 4 0

- 3 0

- 2 0

a )

- 5 0 0 5 0

- 1 0 0

- 5 0

0

5 0

1 0 0

1 5 0

b )

0

0 . 0 1

0 . 1

1

1 0 - 7

1 0 - 6

1 0 - 5

1 0 - 4

1 0 - 3

Magnetization Jc measurements in both 2D and 3D polycrystalline superconductors have been simulated using time-dependent Ginzburg-Landau theory. In both cases, Jc is determined by flux flow along grain boundaries.

In 2D, there are two mechanisms determining Jc, related to the state of ordering of fluxons within the grains. The low-field mechanism is dependent on grain size, while the high-field mechanism is not.

In 3D, the Jc for high B-fields is about 20% of the value for the equivalent 2D system. Jc and the pinning force Fp have the correct experimental dependences - low E-field Fp is given by:

4. Comparison with Experimental DataTwo distinct mechanisms determining Jc, crossing over at a specific reduced field b, as seen in our 2D computational results, are also observed experimentally, such as in the Nb3Sn data in Fig. 7.

Consistent with our 2D computational results, only the low-field mechanism is microstructure-dependent.

0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

8

10

12

Nb3Sn

J c1/2 B1/

4 (104 A

1/2 T

1/4 m

1)

b

Unalloyed 0.1% CO2

3.7% Ta + 0.15% CO2

Lower substrate T 0.01% Si + C

2H

6

0.002% Bi + 0.15% CO2

Figure 7: Kramer plot for Nb3Sn samples withvarious metallic and gaseous dopants[3]. An average

Bc2 value = 20.5 T is assumed