SRF2019, Dresden, July 4, 2019 Effect of Inhomogeneous Disorder on the Superheating Field of SRF Cavities James A. Sauls Department of Physics Center for Applied Physics & Superconducting Technologies Northwestern University & Fermilab • Wave Ngampruetikorn • Mehdi Zarea • Fermilab Collaborators: Anna Grassellino, Mattia Checchin, Martina Martinello, Sam Posen, Alex Romanenko NSF-PHY 01734332
V. Chandrasekhar W. Halperin J. Ketterson J. Koch D. Seidman N. Stern Anna Grassellino Mattia Checchin Martina Martinello Sam Posen Alex Romanenko
Electrodynamics of Superconductor-Vacuum Interfaces ◮ Program: First-Principles + Materials Inputs: Vacuum Nb Current Response & Local EM Fields for ↔ R J ( q , ω ) = − 1 q � ( q , ω ; � A ) · � A ( q , ω ) K Superconducting-Vacuum Interfaces c
Electrodynamics of Superconductor-Vacuum Interfaces ◮ Program: First-Principles + Materials Inputs: Vacuum Nb Current Response & Local EM Fields for ↔ R J ( q , ω ) = − 1 q � ( q , ω ; � A ) · � A ( q , ω ) K Superconducting-Vacuum Interfaces c ◮ Material Inputs: ◮ Fermi Surfaces - DFT & dHvA ◮ Pairing/Decoherence via Electron-Phonon Coupling
Electrodynamics of Superconductor-Vacuum Interfaces ◮ Program: First-Principles + Materials Inputs: Vacuum Nb Current Response & Local EM Fields for ↔ R J ( q , ω ) = − 1 q � ( q , ω ; � A ) · � A ( q , ω ) K Superconducting-Vacuum Interfaces c ◮ Material Inputs: ◮ Fermi Surfaces - DFT & dHvA ◮ Pairing/Decoherence via Electron-Phonon Coupling ◮ Impurity & Structural Disorder ◮ Surface Scattering: S surf ( p , p ′ ) ◮ surface structure factor ◮ mesoscopic roughness � backscattering � Andreev scattering � sub-gap dissipation
Electrodynamics of Superconductor-Vacuum Interfaces ◮ Program: First-Principles + Materials Inputs: Vacuum Nb Current Response & Local EM Fields for ↔ R J ( q , ω ) = − 1 q � ( q , ω ; � A ) · � A ( q , ω ) K Superconducting-Vacuum Interfaces c ◮ Material Inputs: ◮ Theoretical & Analytical Tools ◮ Fermi Surfaces - DFT & dHvA ◮ Migdal-Eliashberg: electron-phonon ◮ Pairing/Decoherence via ◮ Asymptotic Expansions: Electron-Phonon Coupling k B T c / E f , ¯ h / τ E f , ¯ h / p f ξ , ¯ h ω / E f ... ◮ Impurity & Structural Disorder ◮ Surface Scattering: S surf ( p , p ′ ) ◮ surface structure factor ◮ mesoscopic roughness ◮ Selection Rules & Scattering Theory � backscattering � Andreev scattering ◮ Keldysh Transport Equations � sub-gap dissipation
Electrodynamics of Superconductor-Vacuum Interfaces ◮ Program: First-Principles + Materials Inputs: Vacuum Nb Current Response & Local EM Fields for ↔ R J ( q , ω ) = − 1 q � ( q , ω ; � A ) · � A ( q , ω ) K Superconducting-Vacuum Interfaces c ◮ Material Inputs: ◮ Theoretical & Analytical Tools ◮ Fermi Surfaces - DFT & dHvA ◮ Migdal-Eliashberg: electron-phonon ◮ Pairing/Decoherence via ◮ Asymptotic Expansions: Electron-Phonon Coupling k B T c / E f , ¯ h / τ E f , ¯ h / p f ξ , ¯ h ω / E f ... ◮ Impurity & Structural Disorder ◮ Surface Scattering: S surf ( p , p ′ ) ◮ surface structure factor ◮ mesoscopic roughness ◮ Selection Rules & Scattering Theory � backscattering � Andreev scattering ◮ Keldysh Transport Equations � sub-gap dissipation ◮ Developing Methods & Codes to Compute the Nonlinear A.C. Surface Impedance ◮ Nonequilibrium Quasiparticle, Cooper Pair & Vortex Dynamics D. Rainer & J. A. Sauls, Strong-Coupling Theory of Superconductivity, World Scientific (1995); arXiv:1809.05264
Electronic band structure of Niobium DFT Calculation of the Electronic Band Strucuture 60 Ef= 18.1 eV Nb=[Kr] 4 d 4 5 s 1 50 24 bands 40 E[eV] Fermi Energy = 18.1 eV 30 20 2 bands cross the Fermi energy 10 G H G P N ◮ P . Giannozzi et al., J. Phys. Cond. Mat. 29 465901 (2017)
Phonons in Niobium 250 Theory - Phonon dispersion Inelastic Neutron Scattering 200 150 h ω [1/cm] 100 - 50 0 G H N G P N – DFT Perturbation Theory • Inelastic Neutron Scattering; ◮ Baroni, S., de Gironcoli, S., Dal Corso, A. & Giannozzi, P ., Rev. Mod. Phys. 73, 515562 (2001), Phonons and related crystal properties from density-functional perturbation theory ◮ B.M. Powell, et al., Phonon properties of niobium ..., Can. J. Phys. 55, 1601 (1977)
Electron-Phonon Spectral Function α 2 F ( ω ) α 2 F 1 Exp ◮ Maxium Phonon Frequency: 0 . 8 h ω max = 27 . 0 meV ¯ � ∞ 0 d ω α 2 F ( ω ) 0 . 6 ◮ λ = 2 = 1 . 18 ω ◮ Electron-Electron Repulsion: 0 . 4 µ ∗ = 0 . 30 ◮ Eliashberg Theory: 0 . 2 T c = 9 . 47 K ◮ Tunneling Inversion 0 G. Arnold et al., JLTP 40, 225 (1980). 0 5 10 15 20 25 30 h ω [meV] ¯ ◮ DFT Perturbation theory fails for Nb ? ◮ Inversion of dI / dV from PETS does not yield bulk α 2 F ( ω ) ? ◮ Nb surface has defects that suppress the high- ω spectrum ? G. Schierning et al., Phys. Stat. Solidi RRL 9, 431 (2015)
Eliashberg Equations ✂ ˆ D k − k ′ ν ( iω j − iω j ′ ) V k − k ′ ( iω j − iω j ′ ) ✁ ˆ g nm,ν ( k ′ , k ) g mn,ν ( k , k ′ ) + Σ pa n k ( iω j ) = ˆ ˆ G m k ′ ( iω j ′ ) G m k ′ ( iω j ′ ) π T ω j ′ λ ( n k , m k ′ , ω j − ω j ′ ) δ ( ε m k ′ − ε f ) N ( ε F ) ω j ∑ Z n k ( i ω j ) = 1 + � ω 2 j ′ + ∆ 2 m k ′ j ′ m k ′ ( i ω j ′ ) ∆ m k ′ ( i ω j ′ ) π T [ λ ( n k , m k ′ , ω j − ω j ′ ) − µ ∗ N ( ε F ) ∑ Z n k ( i ω j ) ∆ n k ( i ω j ) = � c ] δ ( ε m k ′ − ε f ) ω 2 j ′ + ∆ 2 m k ′ ( i ω j ′ ) m k ′ j ′
Strong coupling superconducting gap ∆ 0 = 1 . 56 meV 1 . 5 1 ∆ [meV] 0 . 5 Eliashberg Theory T c = 9 . 45 K Weak coupling BCS 0 0 2 4 6 8 T [K]
Anisotropy of the Gap and Fermi Velocity speed [ 10 6 m / s ] ∆ [meV] T = 0 . 5 K ◮ Gap Anisotropy: ∆ max = 2 . 54 meV ∆ min = 1 . 38 meV ∆ av = 1 . 56 meV = 1 . 3 × 10 6 m/s = 0 . 2 × 10 6 m/s ◮ Velocity Anisotropy: v max v min f f ◮ Strong Anisotropy of the Fermi Velocity - Impact on Critical Currents?
Theoretical Program ◮ Develop Computational Code & Tools for Electronic Structure of Nb - Phonon Spectra & Density of States - DFT Perturbation Theory - Electron-Phonon Coupling - Eliashberg Theory - Strong-Coupling Superconducting Gap on the Fermi Surface ◮ Incorporate Disorder and Surface Scattering - Constraints from Surface and Materials characterization ⇓ ◮ Develop computational transport theory - charge and heat response under strong EM field conditions at the superconductor-vacuum interface
SRF Performance Goals - What Can Theory Provide? ◮ Push to high Q-factor & Reduce a.c. dissipation up to high E acc ◮ Understand physics of current response at f � GHz at high B s ◮ Push E max acc - processes determining Meissner stability/breakdown ⇓ Materials based theory of Non-Equilibrium Superconductivity under Strong Nonlinear A.C. Conditions
Meissner state is metastable up to the superheating fi eld T T Type-I Type-II T c T c metastable metastable Meissner Meissner Meissner Meissner B B B c B c 1 B c 2 B sh Superheating fi eld (Max Field Gradient)
Superheating field is determined from local critical current • We solve simultaneously surface field superconductor 1. Eilenberger equation - for quasiparticle spectrum B 2. Gap equation Magnetic field B sh - for excitation gap 3. Impurity T-matrix equation x - for the effect of disorder p s Superfluid momentum (Vector potential) 4. Maxwell’s equation - for B -field and current profiles j s / j c Supercurrent density • To obtain superheating field, 1 increase surface field until current reaches critical value 6
Nonlinear D.C. Current Response � � � d ε tanh ε � j s ( x ) = − eN f v f A ( ˆ p , ε , x ) p ˆ 2 T p , ε ; x ) ≡ − 1 ◮ Spectral Function: A ( ˆ π Im G ( ˆ p , ε ; x ) ◮ local impurity self-energies, � Σ imp ( x ) = γ ( x ) � � G � ◮ local superconducting order parameter: ∆ ( x ) ◮ local condensate momentum, p s = ¯ h 2 ∇ r ϑ − e c A ◮ perturbation expansion in ε ∈ { ξ / λ L , ξ / ζ } Propagator for Quasiparticles and Cooper Pairs: τ 3 − ˜ − 1 p , ε , x )=[ ˜ ε ( ε , x ) − v f · p s ( x )] � ∆ ( ε , x )( i σ y � τ 1 ) � � ≡ [ G � τ 3 − F ( i σ y � G ( ˆ τ 1 )] π ∆ ( ε , x ) | 2 − [ ˜ | ˜ ε ( ε , x ) − v f · p s ( x )] 2 ˜ ε ( ε , x ) = ε + γ ( x ) � G ( ˆ ˜ p , ε , x ) � ˆ ∆ ( ε , x ) = ∆ ( x )+ γ ( x ) � F ( ˆ p , ε , x ) � ˆ p p � d ε tanh ε ∆ ( x ) = g 2 T Im � f ( ˆ p , ε , x ) � ˆ p , 2 x p s ( x ) − 4 π e c 2 � ∂ 2 j s [ p s ( x ) , γ ( x )] = 0
Disorder Suppresses Supercurrents ∞ − λ ≡ B sh is affected via 2 mechanisms ➤ more screening current ✗ Lower critical supercurrent ✓ Longer penetration depth ➤ less screening current γ Effective penetration depth Critical current density Impurity scattering rate, γ Impurity scattering rate, γ B sh / B 0 For uniform disorder Current suppression dominates 0.84 Penetration depth dominates Impurity scattering rate, γ 0.80 ◮ F. P .-J. Lin and A. Gurevich, Effect of impurities on the superheating field of type-II superconductors, PRB 85, 054513 (2012). • G. Catelani and J. P . Sethna, The superheating field for superconductors in the high- κ London limit, PRB 78, 224509 (2008).
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