Binping xiao superconducting surface impedance under radiofrequency field
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This document presents a new first-principles calculation of field-dependent RF surface impedance for BCS superconductors, particularly focusing on SRF cavities. The study expands upon the Mattis-Bardeen theory, providing insights into how cavity performance can be improved and indicating that the reduction in resistance with increased field is an intrinsic effect. Overall, the findings align well with experimental results without the need for additional fitting parameters.
![Cavity Performance
Quality Factor
4
1E+11
1E+10
1E+09
All measured at 2.0K
Superheating
Field
0 50 100 150 200
Peak Magnetic Field [mT]
1.5 GHz 7-cell CEBAF
cavity with 230 μm BCP
230 μm BCP + 34 μm
EP
1.5GHz single cell
CEBAF cavity with 3 h
1400 C baking
• P. Dhakal, G.
Ciovati, G. R.
Myneni, in: IPAC
2012.
• C. E. Reece et al.,
in PAC2005
• C. E. Reece, and
H. Tian, in
LINAC2010
H. Pandamsee: “Two Major Open Physics Topic in RF Superconductivity”](https://image.slidesharecdn.com/binpingxiao-superconductingsurfaceimpedanceunderradiofrequencyfield-141009041828-conversion-gate02/85/Binping-xiao-superconducting-surface-impedance-under-radiofrequency-field-4-320.jpg)
![Calculation result and explanation (2)
In M-B theory, mathematically, the scattering between
any two k and k’ with photon interaction equals to the
scattering between E and E+ħ흎.
With moving Cooper pairs, mathematically, the
scattering between any two k and k’ with photon
interaction equals to the scattering between E+εext and
E+ε’ext+ ħ흎.
14
Absorb/release a photon
푹 ∝ [풇 푬 − 풇 푬 + ℏω ]품풅푬
E E+ħ흎
The “golden rule” in extreme anomalous limit
and low temperature approximation
Note that 푷푭푽풔>>ħ흎 could happen, the overlap
between red and purple could be significant.
Net effect: release energy, cause Rs
Term 1 Term 2 Term 3
푹 ∝ [풇 푬ퟏ − 풇 푬ퟐ ][풇 εext +풇 −εext ]품(풉)풅푬
Rs decreasing?
• Source: angle between 푽푭 (any direction) and 푽풔 cause
energy split with angle dependence.
• Consequence: Energy split and modified single
particle distribution function cause an overall
reduction effect in scattering.
Any E Any E’
Net effect: release energy, cause Rs
Absorb a photon
푷푭푽풔
Absorb
a
photon
E+ε’extE+ε + ħ흎 ext
E+εext E+ε’ext+ ħ흎](https://image.slidesharecdn.com/binpingxiao-superconductingsurfaceimpedanceunderradiofrequencyfield-141009041828-conversion-gate02/85/Binping-xiao-superconducting-surface-impedance-under-radiofrequency-field-14-320.jpg)
Binping xiao superconducting surface impedance under radiofrequency field
- 1. How Good Can an SRF Cavity Be? A New First-Principle Calculation of Field-Dependent RF Surface Impedance of BCS Superconductor Binping Xiao Collider-Accelerator Department, Brookhaven National Lab Based on the Ph.D dissertation supervised by C. E. Reece and M. J. Kelley College of William and Mary, Jefferson Lab Thin films and new ideas for SRF, 10/7/2014 1
- 2. Outline 2 Cavity performance BCS theory, Mattis-Bardeen (M-B) theory and extension Calculation results and explanation Theory vs. experiments Summary J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Physical Review 108, 1175 (1957) D. C. Mattis and J. Bardeen, Physical Review 111, 412 (1958) B.P. Xiao, C.E. Reece, M.J. Kelley, Physica C 490 (2013) 26–31
- 3. RF surface impedance in cavity 3 A simplified diagram of SRF device. Introduction to SRF Linac Technology by Rong-li Geng http://en.wikipedia.org/wiki/Superconducting_Radio_Freque ncy current d a a Zs = Rs + iXs
- 4. Cavity Performance Quality Factor 4 1E+11 1E+10 1E+09 All measured at 2.0K Superheating Field 0 50 100 150 200 Peak Magnetic Field [mT] 1.5 GHz 7-cell CEBAF cavity with 230 μm BCP 230 μm BCP + 34 μm EP 1.5GHz single cell CEBAF cavity with 3 h 1400 C baking • P. Dhakal, G. Ciovati, G. R. Myneni, in: IPAC 2012. • C. E. Reece et al., in PAC2005 • C. E. Reece, and H. Tian, in LINAC2010 H. Pandamsee: “Two Major Open Physics Topic in RF Superconductivity”
- 5. BCS, M-B and extension Each Cooper pair was assumed to have zero net momentum. By minimizing 5 the free energy one get a single particle distribution function f and Density of State (DoS), represented by h. Considering that a single particle scattering from one energy state to another one, by considering the energy of each state (with one photon added to either the initial state or the final state.), the probability of initial state and the possibility of scattering, a scattering form was calculated using f and h. This scattering form was then applied to the anomalous skin effect to get the surface impedance. We assume that all Cooper pairs move with the same velocity Vs, and apply this assumption to BCS theory to get new f and h.* These new f and h are applied to get a new scattering form, thus a new surface impedance. We average these effects over both RF cycle and depth into the surface to get the effective surface resistances (Rs) under different fields.** *J. Bardeen, Reviews of Modern Physics 34 (1962) 667. **I.O. Kulik, V. Palmieri, Particle Accelerators 60 (1998) 257.
- 6. Justifications of this extension 6 A mean (phonon) field theory, same as BCS theory, differs from Eliashberg. Consider single photon absorption only, same as M-B theory, differs from de Visser*. Dealing with moving Cooper pairs, same as Bardeen**. The way to deal with mean free path, same as M-B theory. Calculates the “ideal” case, which means the best performance a cavity can achieve. A cavity with short mean free path is not necessary to be a “non-ideal” cavity. No extra parameters needed except those in M-B theory + residual How good it can explain experimental results? *de Visser, P.J., et al., physical review letters, 2014. 112: p. 047004. **Bardeen, J., Reviews of Modern Physics, 1962. 34(4): p. 667.
- 7. 7 Cooper pair with zero momentum Two particles can be attractive to each other with electron-phonon interaction, if they have: • momentums in opposite direction • same energy state k (one ↑ the other ↓) • k in a shell nearby Fermi level kF, with its thickness determined by Debye energy Consequence of attraction: • Bonded particles become Bosons and get condensed • Excited particles obey Fermi-Dirac distribution • A “forbidden zone” appears nearby kF, with its thickness to be 2휟 (energy gap) • Energy of excited particle (hole below kF and electron above kF) changes from|εk|=|1/2mvk 2- εF| before condensation to 푬풌 = 휺풌 ퟐ + 휟ퟐ after condensation A macroscopic quantum effect! kεFF=225meV k↑ 2- εF -k↓ εk=1/2mvk 2kTD=47.4me V (Energies are based on Nb with selected parameters)
- 8. 8 Cooper pair with a net momentum “States with a net current flow can be obtained by taking a pairing (k1↑, k2↓) with k1+k2 =2q, and 2q the same for all virtual pairs” – quoted from BCS theory A small net momentum appears Consequence: • Energy split appears for ↑ and ↓ • The energy split is angle dependent • This angle can be any number 풗푭 풗풔 α k+q↑ 2q 2εs<0.0048meV -k+q↓ εk+q=1/2m(푣푘+ 푣푠)2 - εF= εk + εs + εext ε-k+q=1/2m(푣푘- 푣푠)2 - εF= εk + εs - εext εext = mvFvs cos α = pFvsx
- 9. Energy split caused by moving Cooper pairs BCS theory extension Before condensation 휀푘 휀−풌+풒 = 휀푘 + 휀푠 − 휀푒푥푡 , 휀풌+풒 = 휀푘 + 휀푠 + 휀푒푥푡 After condensation 퐸푘 = 휀풌 9 2 + 훥2 1 2 퐸−풌+풒↓ = 퐸푘 − 휀푒푥푡 , 퐸풌+풒↑ = 퐸푘 + 휀푒푥푡 푬풌+풒↑ with 퐸푘 = 휀풌 + 휀푠 2 + 훥2 1 2 5 4 3 2 1 0 푬풌 |휺풌| -5 0 5 5 4 3 2 1 0 푬−풌+풒↓ Energy split -6 -4 -2 0 2 4 6 휺풌 휺풌 풓풆풇: 휺푭 Energy needed to break a Cooper pair, 휟 for ↑, and the same for ↓ 휟 − 휺풆풙풕 for ↑, 휟 + 휺풆풙풕 for ↓
- 10. f and h modified by moving Cooper pairs Modified density of states and probability of single occupation at T<Tc: Low field limit density of states and distribution function 10 For electron For hole Also appeared in I.O. Kulik, V. ퟏ. ퟐΔ Palmieri, Particle Accelerators 60 (1998) 257. Plots with 푷푭푽풔= 0.4Δ and T/Tc=0.97 - ퟏ. ퟐΔ Below EF Above EF Density of states and distribution function with moving cooper pairs, angle averaged For electron For hole -- α = π -- α = π/2 -- α = 0 - ퟐΔ Density of states and distribution function with moving cooper pairs, angle-dependent 1 푁(0) 푑푁(퐸) 푑퐸 = 푑휀 푑퐸 J. P. Turneaure, J. Halbritter, and H. A. Schwettman, Journal of Superconductivity 4, 341 (1991)
- 11. Modification of M-B theory by moving Cooper pairs Net momentum 0 2q Initial X0 Scattering happens between any two k and k’ 11 BCS/M-B Extension any two k and k’ with q, 푬풌 + 휺풆풙풕 and 푬풌′ + 휺풆풙풕′ 00 (k↑, -k↓) (k’↑,-k’↓) (k+q↑, -k+q↓) (k’+q↑,-k’+q↓) Final 00 X0 (k↑, -k↓) (k’↑,-k’↓) (k+q↑, -k+q↓) (k’+q↑,-k’+q↓) Ground(+) or excited(-) + + (k↑, -k↓) (k’↑,-k’↓) (k+q↑, -k+q↓) (k’+q↑,-k’+q↓) Energy difference Wi-Wf Ek↑-Ek’↑ Ek+q↑-Ek’+q↑ Probability of initial state f dependent Modified f dependent Scattering matrix elements h dependent Modified h dependent Absorbing/releasing one photon: additional energy difference ±ℏ(ω-is), s→0
- 12. Final expression • The final expression is a quadruple integration, besides the integrations in energy and in reciprocal space shown in M-B theory, the extension has two additional integrations in angles, related to k and k’. • The averaging over both RF cycle and depth into the surface requires two additional integrations. • A MathematicaTM script was developed to calculate the Rs vs Bpk. It is slow, but it works. • No parameter fittings can be done using current script due to the slow calculation speed. 12
- 13. Calculation result and explanation (1) 13 Magnetic flux density (mT) 0 50 100 150 608 604 600 596 592 588 584 30 25 20 Rs (nΩ) 15 10 5 0 * M-B Why decreasing? 0 200 400 600 800 1000 Xs (μΩ) vs (m/s) Surface resistance, Rs, (red line) and reactance, Xs, (blue dashed line) versus Cooper pair velocity and corresponding magnetic field for Nb at 2 K and 1.5 GHz.
- 14. Calculation result and explanation (2) In M-B theory, mathematically, the scattering between any two k and k’ with photon interaction equals to the scattering between E and E+ħ흎. With moving Cooper pairs, mathematically, the scattering between any two k and k’ with photon interaction equals to the scattering between E+εext and E+ε’ext+ ħ흎. 14 Absorb/release a photon 푹 ∝ [풇 푬 − 풇 푬 + ℏω ]품풅푬 E E+ħ흎 The “golden rule” in extreme anomalous limit and low temperature approximation Note that 푷푭푽풔>>ħ흎 could happen, the overlap between red and purple could be significant. Net effect: release energy, cause Rs Term 1 Term 2 Term 3 푹 ∝ [풇 푬ퟏ − 풇 푬ퟐ ][풇 εext +풇 −εext ]품(풉)풅푬 Rs decreasing? • Source: angle between 푽푭 (any direction) and 푽풔 cause energy split with angle dependence. • Consequence: Energy split and modified single particle distribution function cause an overall reduction effect in scattering. Any E Any E’ Net effect: release energy, cause Rs Absorb a photon 푷푭푽풔 Absorb a photon E+ε’extE+ε + ħ흎 ext E+εext E+ε’ext+ ħ흎
- 15. Theory vs Experiments 7E+10 6E+10 5E+10 4E+10 3E+10 2E+10 1E+10 0E+00 0 20 40 60 80 100 120 Q0 Bpk (mT) 2.0 K, 1.5 GHz - Theory + 1.7 nohm G1G2 1400C (LG), 1.5 GHz, 2.0 K 2.0 K, 1.3 GHz - Theory + 3.0 nohm TE1AES005 800C (FG), 1.3 GHz, 2.0 K TE1AES003 800C (FG), 1.3 GHz, 2.0 K Calc for: = 32 nm = 40 nm /Tc = 1.85 mfp = 50 nm 15 Mattis-Bardeen P. Dhakal, et al., PRST-AB, 2013. 16(4): p. 042001. A. Grassellino, et al., Supercon. Sci.and Tech., 2013. 26(10): p. 102001. No need for extra parameters!
- 16. More data… 16 Palczewski et al., LINAC2014 “Textbook” values About the inconsistency at the beginning, there are several possibilities: 1, The choose of parameters 2, Measurement errors 3, Cavity performance could be further improved 4, Some facts that are not considered in this model: phonon distribution, multiple photo absorption, additional non-linear effects, etc. We actually predicted the behavior at low temperatures
- 17. Exciting? Let’s be honest Eichhorn et al. 17 5E+10 5E+10 4E+10 4E+10 3E+10 3E+10 2E+10 2E+10 1E+10 5E+09 0E+00 A. Grassellino et al. +5 microns EP +7 microns EP +10 microns EP 0 5 10 15 20 25 30 35 Dhakal et al.
- 18. Summary 18 Previous surface impedance calculations are available only for the low field limit. A field-dependent derivation of the Mattis-Bardeen theory of SRF surface impedance has been developed. The extended range of gradients is treated for the first time. Without any extra parameters except those from original M-B theory, field-dependent Rs agreement with experiment with recent heat-treated/ Nb-doping Nb with unusual surface loading is excellent at different temperatures, with residual resistance to be constant. The reduction in resistance with increasing field is seen to be an intrinsic effect. For type-I, and type-II under Hc1. What is going to happen between Hc1 and Hc2?
- 19. 19 I would like to thank Drs. Ilan Ben-Zvi and Sergey Belomestnykh for their comments and suggestions during the preparation of this talk. Thank you for your attention!