MECHANISMS OF THERMAL RF BREAKDOWN

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MECHANISMS OF THERMAL RF BREAKDOWN
Alex Gurevich ASC, University of Wisconsin
FNAL SRF Meeting, May 4, 2005
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Analytical thermal breakdown model
Instead of numerically solving this ODE, one can
solve much simpler equations for Tm and Ts
Kapitza thermal flux q ?(T,T0)(T T0)
For a general case of thermal quench, see A.
Gurevich and R. Mints, Reviews of Modern Physics
59, 941 (1987)
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Maximum temperature
Take BCS surface resistance residual
resistance R0
Since Tm T0 ltlt T0 even Hb, we may take ? and h
at T T0, and obtain the equation for Tm
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Breakdown rf field
Thermal runaway occurs at a rather weak
overheating
For ? gtgt d?, the breakdown field is limited by
the Kapitza resistance, ?(T)kT03. Thus,
?/6
is minimum at T0 ?/6
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Q-factor (linear resistance)
Q(Hb) ?Q(0)/e 0.37Q(0)
Q versus H0 for T0 2.2K and different
R0/RBCS(T0) 0, 0.2 and 0.5 (top to bottom).
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Nonlinear surface resistance in the clean limit

• RF dissipation was calculated for clean limit (l
  gtgt ?) from kinetic equations for a superconductor
  in a strong rf field superimposed on a dc field
  H(t) H0 cos?t Hdc

• Nonlinear correction due to rf pairbreaking
  increases as the temperature decreases
• At low T, the nonlinearity becomes important
  even for comparatively weak rf
• amplitudes H? (T/Tc)Hc ltlt Hc
• RF power P depends quadratically on the dc
  magnetic field. Field dependence of
• P(H0) does not necessarily indicates vortex
  contribution.

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Thermal breakdown for nonlinear BCS resistance
Bi-quadratic equation for H0(Tm)
Breakdown field
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Q-factor (nonlinear resistance)

Comparison of Q(H0) for linear and nonlinear
models for ? 20 W/mK at T0 2K and R0 0.
(b) Same as in (a), except that the Kapitza
coefficient ? is doubled, from 0.5 W/cm2K to 1
W/cm2K.
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Effect of hotspots
What happens if the surface has macroscopic
regions of radius r0 where A(x,y) is locally
enhanced (impurities, GBs, thicker oxide
patches, etc)?
Each hotspot produces a temperature disturbance
which spreads along the cavity wall of thickness
d over the distance L gtgt r0
L increases with H and diverges at the breakdown
field, H Hb
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Weak hotspots

• Dimensionless hotspot strength

• Weak hotspots ? ltlt 1. For ? 20 W/mK, T0
  2K and ? 0.5 W/cm2K, we get
• Lh ? 3mm, with Lh(T) increasing as T decreases.
  So hotspots with r0 lt 1mm are weak,
• even for strong local inhomogeneity, A0 A gt A.
  
• Maximum temperature in the hotspot Tm relative
  to the temperature Ts(H) in
• the uniform part of the cavity

For h 1, r0/Lh 1/3, A0 3A, T0 2K, we
obtain Tm Ts 0.03K
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Averaged linear BCS surface resistance

• Here Si is the area of the i-th hotspot with
  local Ai, S is the total surface area,
• Rs(h) is the uniform BCS resistance with the
  account of rf heating.
• 
  CONCLUSIONS
• Hotspots result in a strongly nonlinear
  contribution to the global surface
• resistance and can greatly increase the high
  field Q slope.
• The main contribution comes from the expansion
  of the warmer area around a hotspot

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Effect of hotspots on the Q-factor
T 2.5K
A small fraction of hotspots can significantly
change the high-field Q slope even for the
linear BCS surface resistance