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Collective weak pinning model of vortex dissipation in SRF cavities Danilo Liarte, James Sethna, Daniel Hall, Matthias Liepe TTC Topical Cornell University Workshop - RF Superconductivity: Pushing Cavity Performance Limits 5.5 a)

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SLIDE 1

Collective weak pinning model of vortex dissipation in SRF cavities

Danilo Liarte, James Sethna, Daniel Hall, Matthias Liepe Cornell University

TTC Topical Workshop - RF Superconductivity: Pushing Cavity Performance Limits

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SLIDE 2

Outline

Superconductor interface Pinning centers Vortex line HRF

a) Introduction b) Collective weak pinning c) Results d) Final remarks

a) b) c)

Mean free path (nm)

100 101 102 103

R0,B/Btrapped (nΩ/mG)

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 Experimental data Gurevich theory, ℓp = 75ℓ

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SLIDE 3

Outline

Superconductor interface Pinning centers Vortex line HRF

a) Introduction b) Collective weak pinning c) Results d) Final remarks

a) b) c)

Mean free path (nm)

100 101 102 103

R0,B/Btrapped (nΩ/mG)

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 Experimental data Gurevich theory, ℓp = 75ℓ

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SLIDE 4

Gaining insight on trapped flux

Sam POSEN Introduction and flux expulsion measurements on fine grain cavities 08:30 - 08:55 IARC Auditorium, Fermilab

  • Dr. Pashupati DHAKAL

Flux expulsion measurements in large grain cavities 08:55 - 09:20 IARC Auditorium, Fermilab Shreyas BALACHANDRAN Variations in bulk flux trapping by MO imaging in Nb 09:20 - 09:45 IARC Auditorium, Fermilab Ivan SADOVSKI Theoretical insights on pinning 09:45 - 10:05 IARC Auditorium, Fermilab Zuhawn SUNG Point pinning versus grain boundary pinning – physics and techniques 10:05 - 10:25 IARC Auditorium, Fermilab Coffee Break 10:25 - 10:45 IARC Auditorium, Fermilab

  • Dr. Mattia CHECCHIN

Theoretical models of flux expulsion and dissipation 10:45 - 11:10 IARC Auditorium, Fermilab

  • Dr. Danilo LIARTE

Flux losses due to weak collective pinning 11:10 - 11:35 IARC Auditorium, Fermilab

  • Dr. Akira MIYAZAKI

Vortex dissipation in Nb/Cu films 11:35 - 12:00 IARC Auditorium, Fermilab Ryan PORTER Flux dissipation in Nb3Sn Films 12:00 - 12:25 IARC Auditorium, Fermilab

2

  • H(t)

u(z,t)

Gurevich & Ciovati PRB 2013

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SLIDE 5

Sensitivity of 𝑆" to trapped flux

Mean free path (nm)

100 101 102 103

R0,B/Btrapped (nΩ/mG)

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

Experimental data Gurevich theory, ℓp = 75ℓ

Gonnella et al. J. Appl. Phys. 2016

Dependence on MFP (Gonnella) Dependence on RF field (Hall)

Hall et al. IPAC 2017

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SLIDE 6

Outline

Superconductor interface Pinning centers Vortex line HRF

a) Introduction b) Collective weak pinning c) Results d) Final remarks

a) b) c)

Mean free path (nm)

100 101 102 103

R0,B/Btrapped (nΩ/mG)

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 Experimental data Gurevich theory, ℓp = 75ℓ

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SLIDE 7

𝑂

  • fluctuations and collective weak pinning

𝐺

&'( ≅

𝑔

&'( + 𝑜 𝜊+ 𝑀

  • ℇ&'( 𝑀1 = ℇ34567'1 𝑀1

For 𝑀 > 𝑀1, a vortex can bend to find a favorable position in the pinning potential, cutting off the square-root growth of 𝐺

&'(. accumulated pinning force

Pinning f orces add up randomly; only fluctuations can pin the line.

𝑁𝑣̈ = 𝑔

< + 𝑔 > + 𝑔 & + 𝑔 3 + 𝑔 ?

viscous Lorentz pinning elastic Magnus inertial

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SLIDE 8

𝑂

  • fluctuations and collective weak pinning

ℇ&'( 𝑀1 = ℇ34567'1 𝑀1

𝑁𝑣̈ = 𝑔

< + 𝑔 > + 𝑔 & + 𝑔 3 + 𝑔 ?

viscous Lorentz pinning elastic Magnus inertial

𝐺

&'( 𝑀1 = 𝐺 >@A3(7B(𝑘E, 𝑀1)

Pinning force & depinning current… 𝐺

&'(

𝑀1 = 𝐼1

+𝜊 𝑘E

𝑘@

(cgs units)

A vortex breaks up into segments of size 𝑀1; each will compete with the Lorentz force.

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SLIDE 9

Outline

Superconductor interface Pinning centers Vortex line HRF

a) Introduction b) Collective weak pinning c) Results d) Final remarks

a) b) c)

Mean free path (nm)

100 101 102 103

R0,B/Btrapped (nΩ/mG)

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 Experimental data Gurevich theory, ℓp = 75ℓ

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SLIDE 10

Collective weak pinning at low frequency

Derivation and analytical solution

  • 1

2 3 4 5 6

  • 4
  • 2

2 4 z [μm] y [μm]

0 = 𝑔

> + 𝑔 & ?J + 𝑔 3

Lorentz MF pinning elastic

𝑧LL = 𝛽 − 𝛾 sin 𝑢 𝜀(𝑨)

Solution for Nb3Sn at 20mT RF field and 1mA/µm2 depinning current

𝑧V 𝑨 = 𝑏 𝑢 − 𝛾 sin 𝑢 𝑨 − |𝛽| 2 𝑨+ 𝑧 < 𝑧∗ 𝑧\ 𝑨 = |𝛽| 2 𝑨 − 𝛾 |𝛽| ]

+

𝑧 > 𝑧∗ 𝑧∗(𝑢) for for 𝑧\(𝑨) 𝑧V(𝑨)

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SLIDE 11

Collective weak pinning at low frequency

‘Sanity’ tests

10 20 30 40 0.1 0.5 1 5 10 50 100 Brf [mT] Lengths [λ]

10 20 30 40 0.001 0.010 0.100 1 10 100 B [mT] η |vmax| / |fpin|

Amplitudes in y and z Curvature radius at the surface Pinning length ‘Curvature radius’ at the surface At 10 MHz At 1.3GHz

Point-like force, collective weak pinning, BC… Viscous dissipation term

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SLIDE 12

Dependence on RF field

  • The linear behavior is consistent with

collective weak pinning (but not accurate)

  • There is a factor of 100 off in

comparison with the experimental

  • results. Viscous dissipation is needed.

DBL, Hall, Liepe, Sethna, in progress

𝑆" 𝐶7A5&&3E = 𝑏 𝐶A_ + 𝑐 𝑏 = 4 3 𝑔𝜇+𝜈" 𝐶1

+𝜊

𝑘@ 𝑘E ; 𝑐 = 0

Using 𝑘E~3 ×10jk𝑘@

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SLIDE 13

Collective weak pinning at low frequency

‘Sanity’ tests

1 5 10 50 100 500 1000 1021 1022 1023 1024 1025 ℓ n [cm-3]

𝑜l/k = 𝜊 𝑚 10 𝑏+ ]

𝜊 𝑚 = 0.738 𝜊" 1 + 0.882 𝜊" 𝑚 ]

  • r

Estimate density of impurities assuming experimental value for 𝑘E~3 ×10jk𝑘@

𝑏 = 1Å 𝑏 = 2Å 𝑏 = 3Å 𝑏 = 4Å

𝐺

&'(

𝑀1 = 𝐼1

+𝜊 𝑘E

𝑘@ 𝑘@ 𝑘E

k

= 256 𝜌w 𝜊x 𝑜l+𝑏l+ 𝐺

&'( ≅

𝑔

&'( + 𝑜 𝜊+ 𝑀1

  • 𝑔

&'( ≅ 𝜊jl 𝑏k𝐼1 +

8𝜌

Individual pinning force Atomic scale

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SLIDE 14

Collective weak pinning at high frequency

Simulated solution 0 = 𝑔

< + 𝑔 > + 𝑔 & ?J + 𝑔 3

viscous Lorentz MF pinning elastic At 𝐶A_ = 20mT At 𝐶A_ = 50mT

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SLIDE 15

Sensitivity of 𝑆" to trapped flux

as a function of the RF field

  • Simulation
  • Experiment (Hall)

At 𝑘E = 1 (black), 2 (red), 3 (blue), and 4mA/µm2 (green)

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SLIDE 16

Sensitivity of 𝑆" to trapped flux

as a function of frequency

  • Simulation
  • Experiment (Oseroff)

Analytical curve Square-root Plateau

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SLIDE 17

Outline

Superconductor interface Pinning centers Vortex line HRF

a) Introduction b) Collective weak pinning c) Results d) Final remarks

a) b) c)

Mean free path (nm)

100 101 102 103

R0,B/Btrapped (nΩ/mG)

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 Experimental data Gurevich theory, ℓp = 75ℓ

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SLIDE 18

Conclusions and future work

  • Hysteretic losses might explain the dependence of the residual resistance

sensitivity to trapped flux on the RF field.

  • Our approximations are consistent, though we predict dissipations larger than the

experimental ones by a factor of about eight.

  • The collective weak-pinning model predicts three distinct regimes for the

dissipation as a function of frequency: linear, square-root, and a plateau.

  • Simulations with explicit inclusion of impurities.
  • Large amplitudes, grain boundaries, and mixed (strong and weak pinning)

scenarios.

  • Experimental check: do most trapped vortices lie perpendicular to the interface?

(We have assumed vortices that are normally aligned with respect to the interface.)

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SLIDE 19

Acknowledgments

  • TTC Topical Workshop Committee, for the invitation.
  • The Sethna and the Liepe groups in Cornell.
  • The Center for Bright Beams SRF team.
  • Prof. Alex Gurevich, for useful consultation.
  • Financial support from the Center for Bright Beams.

TTC Topical Workshop - RF Superconductivity: Pushing Cavity Performance Limits