1. A Review of CLIC Breakdown Studies
Helga Timkó, Flyura Djurabekova, Kai Nordlund, Stefan Parviainen, Aarne Pohjonen, Avaz Ruzibaev, Juha Samela,
Sergio Calatroni, Rocío Santiago Kern, Jan Kovermann, Walter Wuensch,
Helsinki Institute of Physics and CERN
In collaboration with the Max-Planck Institut für Plasmaphysik
Helga Timkó

2. Outline Why to model vacuum discharges? Experiments at CERN
Compact Linear Collider and others
Experiments at CERN
RF and DC breakdowns
Theory: A multiscale model
Tip growth
Plasma formation
Surface damage
Plasma build-up in detail
2D Arc-PIC code
Simulation results
Helga Timkó

3. Breakdown studies have a broad application spectrum
Fusion physics
Satellite systems
Industry
Linear collider designs

4. Main concern: Vacuum discharges in CLIC
Why to study vacuum discharges?
Going to the limits of conventional acceleration techniques  highest possible gradient
Estimated power consumption: 415 MW (LHC: 120 MW)  cost reduction by efficiency optimisation
Knowing how to lower breakdown rate is a key issue in points (1) and (2)
Detail of a CLIC accelerating structure, working at 100 MV/m

5. Another motivation: Understanding scaling laws
We’ve seen that cavity high gradient performance is governed by scaling laws
What is the physics behind these laws?
Why the dependency BDR ~ Eacc30 · τ5 ?
Why the modified Poyting vector?
A. Grudiev, S. Calatroni, and W. Wuensch, “New local field quantity describing the high gradient limit of accelerating structures”, Phys. Rev. ST Accel. Beams 12, (2009)

6. Aim: Predicting cavity breakdown behaviour in the design phase
In order to get there, we will have to solve many things...
Lowering breakdown rate (BDR)
How to prevent breakdowns? = How are they triggered?
Better understanding BDR to predict structure behaviour
Statistical or deterministic, independent events or ‘memory’?
Influence of material properties, surface treatments?
Interpreting measurements – develop a theory:
Involves many areas of physics
Different phenomena are interacting in a complicated way, involving time scales ~fs – h and length scales ~nm – m
Non-linear evolution of processes

7. Part I: Breakdown Experiments

8. Breakdowns in RF cavities – and how to diagnose them
Normal waveform
Breakdown waveform
― Incident
― Transmitted ― Reflected
Reflected
Transmitted
Incident
accelerating structure

9. Some open questions... How does BDR vs. gradient scale and why?
What is the physics behind conditioning?
How does BDR vs. gradient scale and why?

10. Modelling DC discharges
First we have to under- stand breakdowns in DC, before generalising to RF
Simple and cost-efficient testing with two DC setups
DC, and theory parameters adjusted to RF conditions
How do we know, whether and how results translate to RF?
Limited energy from the circuit
e.g. Cu
r=1 mm
d=20 μm
~ 4-6 kV

11. Connection between DC and RF?
Connection indicated both by theory and experiments; but how to relate them?
Optical spectro-scopy, RF
BDR vs gradient in DC and RF:
Despite all differences in the experimental setup, slopes are almost the same
Optical spectro-scopy, DC
A. Descoeudres, T. Ramsvik, S. Calatroni, M. Taborelli, and W. Wuensch, “dc breakdown conditioning and breakdown rate of metals and metallic alloys under ultrahigh vacuum”, Phys. Rev. ST Accel. Beams 12, (2009)
Courtesy of J. Kovermann

12. RF scaling laws holding in DC?
Energy scaling in DC vs. pulse length scaling in RF RF DC
H. Timko, M. Aicheler, P. Alknes, A. Oltedal, A. Toerklep, M. Taborelli, S. Calatroni, W. Wuensch, F. Djurabekova ,and K. Nordlund, “Energy Dependence of Processing and Breakdown Properties of Cu and Mo”, Phys. Rev. ST Accel. Beams, submitted for publication

13. Conditioning and ranking of materials
Also in DC, materials exhibit conditioning, although differently as in RF. Connection between them?
Ranking of materials according to their breakdown field reached after
Determined by
lattice structure?
A. Descoeudres, F. Djurabekova, and K. Nordlund, “DC breakdown experiments with Cobalt electrodes”, CLIC Note, (2009)

14. An evolving field enhancement?
Does repeated application of the field modify the surface?
Is the field enhancement factor in fact β = β(E) ?
A. Descoeudres, Y. Levinsen, S. Calatroni, M. Taborelli, and W. Wuensch , “Investigation of the dc vacuum breakdown mechanism”, Phys. Rev. ST Accel. Beams 12, (2009)

15. Part II: Multi-scale Model of Breakdowns

16. Electrical breakdown in multi-scale modeling approach
Kai Nordlund, Flyura Djurabekova
Stage 1: Charge distribution at the surface
Method: DFT with external electric field
Stage 4: Plasma evolution, burning of arc
Method: Particle-in-Cell (PIC)
Stage 5: Surface damage due to the intense ion bombardment from plasma
Method: Arc MD
~few fs
~few ns
~ sec/hours
~10s ns
~ sec/min
~100s ns P L A S M O N E T
Stage 2: Atomic motion & evaporation ;
Joule heating (electron dynamics)
Method: Hybrid ED&MD model (includes Laplace and heat equation solutions)
Stage 3b: Evolution of surface morphology due to the given charge distribution
Method: Kinetic Monte Carlo
Stage 3a: Onset of tip growth;
Dislocation mechanism
Method: MD, analysis of dislocations
Avaz Ruzibaev
Flyura Djurabekova
Aarne Pohjonen
Stefan Parviainen
The work has been started in parallel for all the stages, except for stage 3b. In the following slides the key stones of each stage will be described in details.
Helga Timkó
Juha Samela
Leila Costelle

17. Stage 1: DFT Method for charge distribution in Cu crystal
Writing the total energy as a functional of the electron density we can obtain the ground state energy by minimising it.
This information will give us the properties of Cu surface
Total energy, charge states (as defect energy levels)
The calculations are done by SIESTA (Spanish initiative for electronic structure with thousands of atom)
The code allows for including an external electric field
The surface charges under the field are analyzed using the Mulliken and Bader charge analysis
In the present study we perform ab initio density-functional calculations to investigate the properties of copper surface under sufficiently high external electric fields.  Lattice constants, cohesive energies, surface energies and work functions are  strongly affected by the applied field. In the same time, this effect can be different on the different Cu surfaces due to their different crystallographic orientations. In order to find these differences we calculate these quantities for the most common crystal surfaces, (100), (111), (110), and (211) to define the most and the least resistant surface orientation to the field effect. After implementation of external electric field with strength from 0.1 V/Ang up to 1 V/Ang the charge transfer on surface atoms due to the presence of an external field is analysed by Mulliken and Bader charge analysis technique [1,2]. Structure relaxation effects are studied for a flat surface, a surface with a single adatom in different lattice sites and a surface with step edges.         For our calculations we use Spanish initiative for electronic structure with thousands of atoms (SIESTA) [3,4]  ab initio simulation package, which uses linear combination of atomic orbitals (LCAO) type basis functions and norm conserving pseudopotentials.  Perdew, Burke & Ernzerhof (PBE) scheme of Generalized gradient approximation (GGA) was used for the exchange and correlation functionals. The Cu surface has been modelled by using a super-cell with 256 atoms. The bulk lattice constant and some properties of the surface, such as the cohesive energy, surface relaxation, and  surface energy were calculated prior to the application of an external electric field.

18. (conductive material)‏
Stage 2: Hybrid ED&MD – Partial surface charge induced by an external electric field
Standard MD solving Newton’s eqs.
Gauss’ law  charge of surface atoms
Laplace eq.  local field
 Motion of surface atoms corrected; pulling effect of the field
Laplace solver
Laplace solution
φ=const
(conductive material)‏
Verification of the charge assessment
Verification is done by comparing the potential energy evolution of an adatom on the charged surface while it is dragged up from the surface statically. Comparison with Finnis-Sinclair potentials.
F. Djurabekova, S. Parviainen, A. Pohjonen, and K. Nordlund, “Atomistic modelling of metal surfaces under electric fields: direct coupling of electric fields to a molecular dynamics algorithm”, Phys. Rev. E 83, (2011)
[DFT] results from T. Ono et al. Surf.Sci., 577,2005, 42

19. Short tip on Cu (100) surface at the electric field 10 V nm-1 (Temperature 500 K)
If you want to show the dynamics. It is an old slide, but the new results are very similar.

20. Stage 2: Dynamics of electrons for temperature account
At such high electric fields, field emission is a non-negligible phenomenon
Electrons escaping from the surface with significant current will heat the sharp features on the surface, causing eventually their melting.
The change of temperature (kinetic energy) due to Joule heating and heat conduction is calculated by the 1D heat equation
Emax
↑E0 Je
We also account for the size effect as due to the size of the tip comparable to mean free path of conductive electrons the resistivity is increased signifcantly!
S. Parviainen, F. Djurabekova, H. Timko, and K. Nordlund, “Implementation of electronic processes into MD simulations of nanoscale metal tips under electric fields“, Comp. Mater. Sci., (2011), accepted for publication

21. Stage 3a: Onset of tip growth
Force
Fixed atoms →
Fel
STRESS
=0E2/Y +
Presence of an electric field exerts a tensile stress on the surface
Presence of a near-to-surface void may trigger the growth of a protrusion
A. Pohjonen, F. Djurabekova, A. Kuronen, K. Nordlund, S. Fitzgerald, “Dislocation
nucleation from near surface void under static tensile stress on surface in Cu”,
Jour. Appl. Phys. (2010), submitted for publication

22. Stage 4: Plasma evolution
Corresponding to experiment...
Stage 4: Plasma evolution
First approach: 1d3v electrostatic PIC-MCC code; the new 2D-model is discussed in Part III
Provides us with a link between
Micro- & macroscopic surface processes: Triggering (nano- scale)  plasma  crater formation (visible effect)
Theory & experiments: Using reasonable physical assumptions (theory), the aim is to predict the evolution of measurable quantities (experiment)
jFE
H. Timko, K. Matyash, R. Schneider, F. Djurabekova, K. Nordlund, A. Hansen, A. Descoeudres, J. Kovermann, A. Grudiev, W. Wuensch, S. Calatroni, and M. Taborelli , “A One-Dimensional Particle-in-Cell Model of Plasma Build-up in Vacuum Arcs”, Contrib. Plasma Phys. 51, 5-21 (2011)

23. Under what conditions will an arc form?
Two conditions need to be fulfilled: ( scaling btw. DC and RF)
High enough initial local field to have growing FE current
Reaching a critical neutral density  ionisation avalanche
The sequence of events leading to plasma formation:
‘Point of no return’: lmfp < lsys – corresponding to a critical neutral density ~ /cm3 in our case  ionisation avalanche
High electric field
Electron emission, neutral evaporation
Ionisation  e–, Cu and Cu+ densities build up
Sputtering neutrals

24. Stage 5: Cathode damage due to ion bombardment
Knowing flux & energy distribution of incident ions, erosion and sputtering was simulated with MD
Flux of ~1025 cm-2s-1 on e.g. r = 15 nm circle  1 ion/20 fs
H. Timko, F. Djurabekova, K. Nordlund, L. Costelle, K. Matyash, R. Schneider, A. Toerklep, G. Arnau-Izquierdo, A. Descoeudres, S. Calatroni, M. Taborelli , and W. Wuensch, “Mechanism of surface modification in the plasma-surface interaction in electrical arcs”, Phys. Rev. B 81, (2010)

25. Mechanism of surface modification

26. Comparison to experiment
Self-similarity:
Crater depth to width ratio remains constant over several orders of magnitude, and is the same for experiment and simulation
50 nm

27. Part III: 2D Particle-in-Cell Simulations of Plasma Build-up

28. 2D Arc-PIC code
The new 2D Arc-PIC code has been developed and benchmarked last year
Serial 2d3v electrostatic particle-in-cell code
Cylindrical symmetry
Monte-Carlo Collision routines adapted from K. Matyash, IPP
Optional external magnetic field
Several options for potential boundary conditions and particle injection schemes
Simple options to carry out different tests
Discharge option (physics model described later)
Will be accessible for CERN and HIP
Automatic analysis scripts
Extensive code documentation to be published as a CLIC note

29. On the PIC method... Ideal for discharge modelling:
Kinetic description of the plasma
Necessary for collisionless plasmas that are far from a Maxwell- Boltzmann distribution
Self-consistent electrostatic(-magnetic) field description
Crucial for this high-current phenomenon close to an absorbing wall, giving rise to a plasma sheath
ϕ(j+1,k) n(j+1,k)
ϕ(j+1,k+1)
n(j+1,k+1)
ϕ(j,k) n(j,k)
ϕ(j,k+1)
n(j,k+1)
(z,r)
(vz,vr,vt)
Particles: Lagrangian, continuous frame
Plasma macro-quantities: Newtonian, discrete frame, on “grid points”

30. ...and its limitations Why discharge modelling is so hard:
Stability requirements
Non-linear problem, while discretisation is linear  prone to numerical instabilities
Stability diagnostics is built-in
Limited dynamic range
Can’t cover 1 nA – 100 A
Memory-heavy simulations
High density only in a few simulation cells, while memory allocation is homogeneous

31. Geometry: 2D cylindrical
Particles have 2 coordinate and 3 velocity components
Particles are projected into a 2D-slice of a cylinder
Difference from 3D: In 2D, no turbulent phenomena and associated transport/diffusion can occur

32. How to calculate density in a 2D (r,z) world?
j=0
j=nr
k=0
k=nz
One cell represents a 3D ring of the cylinder, therefore
Cell volume V = V(r) ~ r
A particle in an outer cell corresponds to less density!!

33. 2D Arc-PIC: An efficient code
Some details for those who are familiar with PIC...
Efficient: All variables are dimensionless
No unnecessary multiplications
A well-optimised finite difference method Poisson solver
Using the ‘SuperLU’ lower-upper triangular matrix factorisation and inversion package
An accurate implicit particle mover using the Boris method
Charge assignment and field interpolation: Via 1st order cloud-in-cell scheme
Guarantees momentum conservation on a uniform grid

34. Benchmarking via code-to-code comparisons
1D Arc-PIC
Codes are usually tested via simple test problems that have a known analytic solution
If the code is then run in a non-linear regime as in the case of breakdowns, subtle, earlier undiscovered flaws may lead to a wrong solution of the problem, and we wouldn’t even necessarily notice it!
Code-to-code comparisons are essential for such problems (if comparison to theory is difficult)
A collaboration on this with Sandia National Labs is ongoing since 1.5 years now;
Obtained good agreement with a simple 1D discharge model
Aleph

35. Overview of vacuum discharge boundary conditions
e- FE from a flat surface
Cathode, 0 V
Anode, 5.8 kV Cu e-
Cu+ Cu
‘Flat’ cathode surface, e- emission enhanced by βf
Cu+, E > 100 eV e-
Cu+, Φ > Φth Cu Cu
Cu+
FE tip, e- emission enhanced by β0
e- FE, Cu evaporation
β0 can be ‘eroded’ and ‘molten’ to βf above a given jmelte-
SEY=const., enhanced Y above Φth based on MD simulations
Experimental sputtering Y (Yamamura & Tawara)
20 μm, 290 MV/m

36. Circuit model – Inspired by the DC setup
EXPERIMENT
A vacuum arc in steady-state has typically a burning voltage of 10 V, we apply 5 – 6 kV over 20 μm! Isn‘t this a contradiction?
No: A vacuum discharge is a transient phenomenon; during the plasma build-up, the external potential will drop (the available energy gets consumed)
One thing we would like to predict is how it’s current- voltage characteristic looks like
e.g. Cu
r=1 mm
d=20 μm
~ 4-6 kV
SIMULATION

37. Circuit model – Where does the charge come from?
Another fact: Vacuum arcs carry 10 – 100 A current at a maximum; what is the physics behind charge depletion and how to implement it?
We simulate an idealistic ‘pool’ of charges that is immediately accessible, but much can happen in a fraction of a ns!
One may interpret the capacitance as the capacitance of the discharge gap itself
Charge depletion is modelled implicitly via a reasonable electron FE cut-off at 12 GV/m, which is the range of applicability of pure field emission

38. From FE to discharge currents
In real life we can observe the full dynamic range of a vacuum discharge:
> 10s pA in ‘weak’ FE phase
Space charge limited ‘strong’ FE phase, typically ~ nA – μA
Discharge current, up to 10 – 100 A
At the same time, the involved area changes:
Typically – m2 for weak FE  Rem ~ 0.1 – 100 nm
During the discharge, the bombarded area has R ~ 10 – 100 μm
Up to 12 orders of magnitude difference
Up to 12 orders of magnitude difference
Discharge e-
10s μm
Cu+ FE e-
10s nm

39. From FE to discharge currents – How to model it?
Cathode
How can we model this with a method that can cover 3 – 4 orders of magnitude dynamic range at the most?
Start from strong FE (0.01 – 0.1 A)
Try to model the expansion of the spot through reasonable physical assumptions
The region outside the FE tip (from the ‘flat’ surface) can be involved in field emitting electrons through a combination of βf ~ 10 – 15 and a sheath
Motivation: We never measured β < 10 – 15; also a ‘flat’ surface will have asperities that can lead to a spreading FE
e- FE from a flat surface
e- FE, Cu evaporation

40. An example simulation β0 = 35, βf = 12 12 × 20 μm 120 × 200 grids
290 MV/m
C = 1 pF
rCu/e = 0.012
N.B. mirrored image; particles in outer cells correspond to less density

41. Coordinates – zoomed in

42. Observations Fully cathode dominated phenomenon
Although FE starts from a small area, the discharge plasma can involve a macroscopic area on the cathode
Transitions seen:
Transition from strong FE to a small discharge plasma
Sudden ionisation avalanche
A plasma sheath forms, the plasma becomes quasi-neutral
Focusing effect
Transition from a surface-defined phase to a volume-defined phase
When neutrals fill the whole system
Self-maintaining
Macroscopic damage 1.
1.06 ns 2.
1.60 ns

43. Potential
Plasma: Sheath
FE: Space charge

44. Current-voltage characteristic
Limited FN: imitates current depletion! Growing current: expanding spot
Current-voltage characteristic
C = 1 pF, U0 = 5.8 kV
 W= μJ

45. Current-voltage characteristic For C = 0.3 pF (~ 5 μJ)

46. Conclusions on plasma modelling
The 2D Arc-PIC code has been developed and its simulations give us new type of insight into the physics of breakdown plasmas
The code is available for CERN and HIP, but obtaining reliable results is an art of itself
Extremely time-consuming, because the result is highly sensitive to input parameters (exponential dependencies)
Requires a deep knowledge of the PIC method, in order to filter out unphysical effects

47. Summary and Outlook
We have established a ranking of materials and understood many bits and pieces of the puzzle already
Still many open questions remain. To answer them, we need a close interaction between theory and experiment
Future of DC experiments: To test more basic physics
Multi-scale model: Give predictions to their outcome
jFE

48. Interested in more? Come to our MeVArc Breakdown workshop 27-30th June 2011, in Helsinki! Thank you!