Discontinuity lines in superconducting films with predefined edge defects

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Discontinuity lines in superconducting

films with predefined edge defects

Jérémy Brisbois

Experimental Physics of Nanostructured Materials University of Liège, Belgium

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Collaborators

W. A. Ortiz M. Motta A. Silhanek O. Adami J. Avila Osses B. Vanderheyden P. Vanderbemden N. D. Nguyen Liège São Carlos Grenoble R. G. B. Kramer

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Superconducting currents

Screening currents

𝐵 = 0

𝐻

Bean model for a bulk superconductor

J

_c

H

₁

H

₂

𝛻 × 𝐻 = 𝐽𝑐

𝛻 × 𝐻 = 0

Where flux penetrates:

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Influence of defects

Defects are present in all materials, so it is critical to understand their influence on the superconducting properties.

What happens to the current streamlines in the presence of a defect?

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The role of defects

M. Campbell, J.E. Evetts, Critical Currents in Superconductors, Taylor & Francis, London, (1972)

𝑥

𝑦

𝑦 = 𝑎𝑥

2

+ 𝑐

𝑎 =

1 2𝑅

𝑐 = −𝑅

Bean model (in a bulk

superconductor) shows the appearance of

discontinuity lines,

where the current is sharply bending.

cylindrical cavity with radius 𝑅, where 𝐽_𝑐 = 0.

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Vortices don’t cross the d-lines

Vortices never cross the d-lines.

Th. Schuster et al., Phys. Rev. B 49, 3443 (1994)

𝐸 = 𝐸𝑐

𝐽

_𝑐

𝑛

𝐽

_𝑐

𝐽 < 𝐽

_𝑐

Close to the d-lines, 𝐽

𝐽_𝑐

< 1

:

𝐸

0

n → ∞

Therefore, the velocity of vortices at the d-lines is

v =

𝐸

𝐵

→ 0

.

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Gurevich and Friesen, PRB 62, 4004 (2000); PRB 63, 64521 (2001)

How far does the perturbation

propagate?

𝐿

_⊥

~ 𝑅𝑛

𝐿

_∥

~ 𝑅 𝑛

𝑛 ≈

𝐿

⊥

𝐿

_∥ 2

The perturbation propagates further for higher 𝑛

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Does the shape of the defect matter?

Rectangular defect

𝑦 = 𝑎 𝑥 −

𝑏

2

+ 𝑐

𝑎 =

1 2𝑅

𝑐 =

𝑅

2

There are two distinct branches separated by a distance

𝑏.

𝑅

𝑏/2

𝑥

𝑦

parabola 45° line

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Flux avalanches in superconductors

Smooth

flux penetration Flux avalanches

observed by magneto-optical imaging (MOI)

vortex motion

H increased dissipation in normal core

efficient heat removal T raises locally

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Indentations should act as nucleation

points for flux avalanches

Th. Schuster et al., Phys. Rev. B 54, 3514 (1996)

All the flux inside the d-lines must enter via channel 1, making it a distinguished place for the nucleation of a flux avalanche.

𝑄 = 𝐽. 𝐸 𝑑𝑥 𝑑𝑦

Heat:

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What happens in thin films?

J. I. Vestgarden et al., Phys. Rev. B 76, 174509 (2007) Defect size ~ 80 µm

 In thin films, ∆ can be larger than the indentation radius r0

 Larger indentations produce a larger ∆  For smaller values of n, smaller ∆

Locally enhanced Joule heating is predicted to facilitate nucleation of a thermal instability at the indentation, so avalanches are

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Motivations

1. So far, most of the investigations deal with indentations far larger than ξ and λ.

2. Previous investigations neither control nor study the shape of the indentations.

3. What parameters can be extracted from the shape of the d-lines emerging from the indentation ?

4. How does the distance between indentations affect the penetration ?

5. Do indentations trigger flux avalanches, as systematically predicted in the literature ?

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 Shape 13

Samples layout

800 µm 400 µm 𝑝 T10p10 𝑏 S 𝑅 𝑅 C T10 𝑏 𝑅 P R 𝑅 = 10 µm 𝑅 = 𝑏 = 0.5, 2, 5, 8 µm 𝑝 = 0, 10, 50, 100 µm T10p0 T10p10 T10p50 T10p100 T10 T0.5 T2 T5 T8

100 nm-thick Nb films grown on the same substrate

𝑅 = 10 µm, 𝑏 = 20 µm

T

𝑏 𝑅

J. Brisbois et al., Phys. Rev. B 93, 54521 (2016)

 Size

 Periodicity

 Roughness

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Determination of d-lines

0 50 100 150 200 250 300 350 0.0 0.2 0.4 0.6 0.8 1.0 Intensity (a.u.) x (µm) x y

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Influence of defect size

The defect sizes R deduced from the Bean model are far larger

than the predicted values.

-80 -60 -40 -20 0 20 40 60 80 0 20 40 60 80 100 120 140 T/T_c=0.68 T0.5 T8 y (µm) x (µm) 0 2 4 6 8 10 12 0 2 4 6 8 10 12 14 16 18 20 22 T/T_c=0.68 1/ 2a ( µm) R (µm) R

𝑦 = 𝑎𝑥

2

+ 𝑐

What are the possible sources of disagreement between experiments and the Bean model for longitudinal geometry?

(i) current crowding?

(ii) unrealistically high creep exponent n (iii) nonlocal nature of thin films?

(iv) field-dependent critical current density j_c?

𝑎 =

1 2𝑅

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16 0 10 20 30 40 50 60 70 80 0 20 40 60 80 100 120 140 T/T_c  0.67 T10 S C y (µm) x (µm)

(i) Influence of the defect shape

Current crowding plays a minor role in the parabola shape.

Accumulation of current lines (current crowding)

eases vortex penetration

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(ii) Influence of the creep exponent n

parabola widening as T increases (as n approaches 1).

-80 -60 -40 -20 0 20 40 60 80 0 20 40 60 80 100 120 140 T/T_c=0.65 T/T_c=0.89 y (µm) x (µm) T10 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.00 0.01 0.02 0.03 0.04 T10 S C a ( µm -1 ) T/T_c

𝑛 =

𝑈

0

𝑘

_𝐵

𝑇

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(iii-iv) Numerical simulations

The parabola should shrink as T increases (n decreases), but this is not verified in the experiments.

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Excess flux penetration

0 1 2 3 4 5 6 7 8 0 10 20 30 40 0 1 2 3 4 5 6 7 8 10 20 30 40 0 1 2 3 4 5 6 7 8 10 20 30 40 0 50 100 150 2000 1 2 3 4 5  (µm) ₀H (mT)  (µm) ₀H (mT) (H_m,_m) C and T0.5 T10 ₀H (mT)  (µm) S ₀H (mT) P S C T10 T0.5 D ( µ m) ∆ 𝐷 x y

The maximum excess flux penetration ∆_𝑚 increases as current crowding is more pronounced (S, T10).

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T dependence of ∆

_𝒎

and 𝑯

_𝒎

∆_𝑚 increases with 𝑇 (n

decreases), contrarily to what was predicted by simulations.

4.5 5.0 5.5 6.0 6.5 7.0 7.5 40 50 60 70 80 90 T10 S  m (µm) T (K) 4.5 5.0 5.5 6.0 6.5 7.0 7.5 0.0 0.5 1.0 1.5 2.0 2.5 T10 S H m (mT) T (K)

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Do indentations trigger flux avalanches?

4 smooth sides 2 smooth (short sides) + 2 rough (long sides) 2 smooth (short sides) + 2 rough (long sides)

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Conclusions

We demonstrate that the d lines encode information about  the demagnetization effects

 the size and shape of the defect  the creep exponent n

 the field dependence of the critical current density.

Against the common wisdom, indentations do not seem to be preferred places for triggering flux avalanches.

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