Unipolar Plasma Model of RF Breakdown Z. Insepov, J. Norem 1) - - PowerPoint PPT Presentation

SLIDE 1

Unipolar Plasma Model

• f RF Breakdown
• Z. Insepov, J. Norem

1) Purdue University, 2) Nanosynergy Inc

Feb 12, 2014"RF breakdown physics" sprint

SLIDE 2

2

Outline

• Motivation - RF breakdown in cavities
• Calculation of electric field for real surfaces
• Experimental enhancement factor for dark current
• Cluster field evaporation in high electric field
• Unipolar (Schwirzke) plasma model development
• Surface sputtering by ions
• Atomistic model of non-Debye plasma
• Plasma model of RF BD
• Conclusions

SLIDE 3

RF Breakdown examples

[From the 2001 Report on the Next Linear Collider]

Severe damage Moderate damage CLIC NLC NLC

CERN, Izquierdo, 2008

SLIDE 4

Calculation of electric field for real surfaces

SLIDE 5

FE m ulti-physics sim ulation

• Comsol simulation : Solving Laplace’s equation for arbitrary geometry

V V n E

el g

1 = = = ⋅ = ∆ φ φ φ  

V

el

1 = φ

V

g

= φ

2 µm = ⋅n E  

SLIDE 6

Field Enhancem ent by tips

• Comsol simulation vs analytical theory of field enhancement

2µm

ρ=100 nm

1µm

ρ=10nm ρ=1nm

SLIDE 7

Field Enhancem ent by cones

• Comsol simulation of field enhancement at sharp cones

1µm

• G. Arnau Izquierdo, 2008

β~ 1000

SLIDE 8

Enhancem ent at crack’s edges

• Sharp tips, edges and corners of the cracks

can significantly enhance the electric field

• More exotic cracks can enhance the electric

field too

β ~ 100

SLIDE 9

9

Triple junction E-fields

• We have been modeling, cracks, junctions, edges and other shapes
• Comsol simulation of field enhancement at triple crack junction

1µm

β=140

SLIDE 10

Experim ental enhancem ent factor

• btained from dark current

m easurem ents: β = 1 8 4

SLIDE 11

11

Comparison with experiment

X rays show that cavities break down at Elocal ~ 7–10 GV/m

[Norem, PR STAB (2003)]

Fowler-Nordheim field emission (1928)

( )

2 3 2 6 2 2 3 2

MV/m(eV) 6830 A/(MV) eV 10 54 . 1 m A , exp ) ( ) ( = × =         − =

FN FN surf FN surf FN surf

B A E B E A E i β φ φ β

β – Local Field Enhancement

φ = 4.6eV

β=184

SLIDE 12

Cluster field evaporation – a result

• f a high local electric field

SLIDE 13

W hy atom istic sim ulation?

Flyura Djurabekova and Kai Nordlund, University of Helsinki

3.6 β = 6 β =

1.5 β =

CLIC RF Breakdown Workshop, CERN 2008

SLIDE 14

Cluster field evaporation

1×105 2×105 4×105 6×105 8×105 1×106

9.3 nm, 50×103 7.6 nm, 30×103 19 nm, 450×103

14 12 10 8 6 4 Voltage (KV)

[Miller, Atom Probe Tomography (2000)] [Insepov Norem, Phys Rev STAB (2004)]

Figure shows abrupt discontinuities in the voltage vs. number of ions in a DC- field evaporation system and evidence for large clusters produced at field ion microscope tips.

Single ion field evaporation : Ecr = 30 GV/m

SLIDE 15

Electric field initiates breakdow n

• Surfaces contain grain

boundaries, tips, oxides, dust particles

• A strong electrostatic field

enhancement can be generated

• Maxwell stress includes electric

forces acting on the tip

• The chunks fill the near region
• f the vacuum
• Ionization by FN-electrons and

Coulomb explosion form plasma

• Unipolar plasma model can

explain triggering of the breakdown

SLIDE 16

Crater form ation via field evaporation

• A new mechanism of crater formation –

pulling out a large area of the surface

• S. Yip, MIT 2014 (private communication)

SLIDE 17

Unipolar ( Schw irzke) Plasm a m odel developm ent

SLIDE 18

Double electric layer in plasm a

• F. R. Schwirzke,

IEEE Trans. on Plas. Sci., 19, 690 (1991)

Tokamak

SEM image of plasma damaged metal surface: Superposition of “younger” (10 µm) and “older” craters (30-40 µm).

SLIDE 19

Unipolar arc breakdow n m odel

• Neutrals accumulated in the dark space
• Ionization of neutrals by FN-current
• Percolation of dark space via ionization
• Crater formation via explosion

BD triggered by impact ionization

Dark space Cavity surface

Debye length Plasma potential Plasma

Unipolar Arc Model

2 1 2

, , 2 ln 2         = =         = e n V V E m M e T k V

e pl d d pl s e i e B pl

ε λ λ π

Surface field d Vpl z sheath φ Vpl ~ 100 V ~ 1-2 nm plasma

SLIDE 20

Unipolar Arc m odel in linac

• Schwirzke model

Plasma potential ( )

m V U E d m V U eV kT m n kT n U E e n kT m M e kT U

D f f D c D e e e e D f f e e D e i e f 10 9 f 3

• 25

2 1 2

10 5 5 . 1 , 10 2 50 , 18 10 1 12 . 5 , 2 , 2 ln 2 × ≈ ≈ × ≈ × ≈ ≈ ≈ × ≈ × ≈ ≈ =             =

−

λ β λ λ λ ε λ π

+ + + + + + + + + +

• λD~2 nm

hot spot e e e + + + +

Linac Plasma

[Insepov, Norem, 2012]

n ~ 1025 m-3 surface

Heating occurs via ion bombardment. Plasma fueling:

• Evaporation of surface atoms
• Tip explosion by high electric field

Y~10

SLIDE 21

Self-sputtering by plasm a

plasma Self-sputtering is the main mechanism of plasma fueling d ∼ 1.5λD Coulomb explosion

• f tips and fragments

SLIDE 22

Surface sputtering by ions

• Sigmund’s theory – linear cascades, not good for

heavy ions and low energies

• Monte Carlo codes: binary collisions, not accurate

at low energies

• Empirical models based on MC – good for known

materials

• Molecular dynamics – time consuming but no limit

for energies, ion masses, temperatures, dense cascades, thermal properties - can verify the OOPIC/VORPAL simulations

SLIDE 23

Sputtering theory and m odels

• Eckstein-Bohdansky’s model
• Sigmund’s theory

( )

{ }

( ) ( )

( )

. 708 . 1 882 . 6 1728 . 2288 . 1 1 ln 441 . 3 A 4685 . energy) reduced

• (

, , parameters adjustable , , 1 1 ) (

2 1 3 2 3 2 2 2 1 2 1 2 2 3 2

2 1

− + + + = + = + = −             −               − =

−

ε ε ε ε ε ε ε ε ε ε

TF n L L th th th n

s Z Z a e Z Z a M M M E E Q E E E E Qs E Y

( ) ( ) ( ) ( ) ( ) ( )

t. coefficien C power, stopping nuclear energy, binding surface

• density,

atomic energy, deposited , 0420 . 1 4 3 , ) (

1 2 2

− − − − = = = Λ Λ = E S U N E F E NS M M E F NU U NC E F E Y

n s D n D s s D

α π

Not applicable for light ion, high energy ions (no electronic stopping power). Needs adjustable parameters.

[Bohdansky, NIMB B (1984)] [P. Sigmund, Phys. Rev. B (1969)]

Not applicable for heavy ions C0, Us - adjustable parameter.

SLIDE 24

Yam am ura’s em pirical m odel

• Yamamura’s interpolation model based on Monte-Carlo code

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

. 4 . , 7 . 5 1 , , 7 . 6 , 1 042 . ) ( parameter, adjustable power, stopping nuclear energy, binding surface

• density,

atomic 1 042 . 1 042 . ) (

2 2 1 2 1 2 1 2 1 2 1 1 2 1 2

M M M M M M M M M M E E E E S s s E S U M M E Y E S U N E E NU E S M M E E NU E F E Y

th s th n n n n s n s th s n th s D

+ =        ≤ + ≥ =       − × + = − − −       − =       − = γ γ γ ε ε α α α

No temperature dependence

SLIDE 25

25

MD sim ulation of Copper self-sputtering at high T and E

• Self-sputtering is the mechanism for fueling unipolar surface plasma.
• Unipolar model requires Y > 10 typical at low ion energies.
• MD predicts very high sputtering yields for high surface T and E.
• Erosion rates on the order of ~ 1 m/s.

[Insepov et al, NIMB, 2010]

SLIDE 26

Atom istic m odel of non-Debye near-surface plasm a

SLIDE 27

n-T Diagram for plasmas

/

F kT

θ = ε

1/3 2

4 3

e

n e kT π   Γ =    

Nonideality parameter for electrons

3

4 3

D D e

r N n π =

Number of electrons in the Debye sphere Degeneracy parameter

10 10

1

10

2 10 3

10

4

10

5

10

6 10 7

10

8

10

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10 10

1

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10

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Classical gas Quantum gas

Te, K ne, cm-3

F

a e ε = /

2

1 Γ = 10

D

N =

1 θ =

MD

ionization

SLIDE 28

10 10

1

10

2 10 3

10

4

10

5

10

6 10 7

10

8

10

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10 10

1

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2 10 3

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6 10 7

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10

28

Quantum gas

Te, K ne, cm-3

F

a e ε = /

2

100

D

N =

1 θ =

Particle-in-cell

MD

ionization

/

F kT

θ = ε

1/3 2

4 3

e

n e kT π   Γ =    

Nonideality parameter for electrons

3

4 3

D D e

r N n π =

Number of electrons in the Debye sphere Degeneracy parameter

Density-Temperature Diagram

SLIDE 29

10 10

1

10

2 10 3

10

4

10

5

10

6 10 7

10

8

10

8

10

10

10

12

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10 10

1

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2 10 3

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6 10 7

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10

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10

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10

26

10

28

Quantum gas

Te, K ne, cm-3

F

a e ε = /

2

100

D

N =

1 θ =

Particle-in-cell

ionization

/

F kT

θ = ε

1/3 2

4 3

e

n e kT π   Γ =    

Nonideality parameter for electrons

3

4 3

D D e

r N n π =

Number of electrons in the Debye sphere Degeneracy parameter

MD

Density-Temperature Diagram

SLIDE 30

• Classical molecular dynamics (MD) simulations with a

pseudopotential to account for quantum effects

• Two component plasma of electrons and copper ions
• Long range Coulomb interactions (N-body problem)
• Nearest image method (periodical boundary conditions)

for the transversal dimensions

• Absorption of electrons to the surface with generation of

the surface electrostatic field

• Simulation of the relaxation process
• Averaging over an ensemble of initial states

Simulation Features

SLIDE 31

Ionization potential for Copper Umin = Uei(0) = - 7.73 eV (σ = 0.21nm)

Electron-electron and ion-ion potentials are pure Coulomb. The erf-like electron-ion interaction potential given above was used e.g. for simulations of sodium clusters in T. Raitza, H. Reinholz, G. Röpke, I. Morozov,

• E. Suraud, Contrib. Plasma Phys 49 496 (2009).

Electron-ion interaction potential

0.4 0.8 1.2 1.6 2

• 10
• 8
• 6
• 4
• 2

Coulomb Erf, Umin = – 7.7 eV

r, nm Uei, eV

2

( ) erf

ei

Ze r U r r   = −   σ  

Interaction Potentials

SLIDE 32

2

( ) erf

ei

Ze r U r r   = −   σ  

Ionization potential for Copper Umin = Uei(0) = - 7.73 eV (σ = 0.21nm)

Electron-electron and ion-ion potentials were pure Coulomb. The erf-like electron-ion interaction potential shown above was previously used for simulations of sodium clusters in [T. Raitza et al, Contrib. Plasma Phys (2009)].

0.4 0.8 1.2 1.6 2

• 10
• 8
• 6
• 4
• 2

Coulomb Erf, Umin = – 5.1 eV Erf, Umin = – 7.7 eV

r, nm Uei, eV

Electron-ion interaction potential

Test potential Umin = Uei(0) = - 5.1 eV (σ = 0.32nm)

Interaction Potentials

SLIDE 33

Ez z = 0 x y Lz Ly Lx

/2 /2 2 2 2 /2 /2 2 2 2 2 2 2 2

1 ( ) 4 2 arctan log 2 2

y x x y

L L L L

z dx dy x y z a a z a z a z a z a z a

− −

= − + +       + −   = − +       + + +        

∫ ∫

σ φ πε σ πε

2 2 2

( ) arctan 2

z

a E z z z a z   ∂ = − =   ∂ +   φ σ πε 2 2

y x

L L a = =

x y

q L L = σ

(charge density) Electric potential on z axis Longitudinal component of the electric field

Surface electric field in box

SLIDE 34

10 20 30 40 50 0.001 0.01 0.1 1 10

t, fs Esurf, GV/m

ne = 1027 m-3 ne = 1025 m-3 ne = 1023 m-3

0.5 1 1.5 2 2.5 0.2 0.4 0.6 0.8 1 1.2

ions electrons, t = 1.7fs electrons, t = 17fs electrons, t = 170fs

n/n0 z, nm

E-field and density vs time

SLIDE 35

Te = 1eV, n0 = 1027m-3, Γ = 2.32

1 2 3 4 0.2 0.4 0.6 0.8 1 1.2

ions electrons

n/n0 z, nm

1 2 3 4 0.01 0.1 1

z, nm σ, int. units

width at 1/e level λ = 1.04nm exponential fit σ ~ exp(-z/λexp) λexp = 0.43nm statistical fluctuations

Stationary plasma sheath

SLIDE 36

50 100 150 200 250

• 16
• 12
• 8
• 4

t/τe Esurf, GV/m

The parameters used in this paper Umin = - 7.73 eV (σ = 0.21nm) Test potential: Umin = - 5.1 eV (σ = 0.32nm)

Relaxation of the surface field Charge distribution in the sheath area

0.4 0.8 1.2 1.6 2

• 10
• 8
• 6
• 4
• 2

Coulomb Erf, Umin = – 5.1 eV Erf, Umin = – 7.7 eV

r, nm Uei, eV

2

( ) erf

ei

Ze r U r r   = −   σ  

1 2 3 0.01 0.1 1

σ (Umin = -7.73eV) λexp = 1.5λD σ (Umin = -5.1eV) λexp = 1.8λD

z, nm σ, int. units

Effect of Interaction potential

SLIDE 37

40 80 120 160 200 0.2 0.4 0.6 0.8 1 1.2

ions electrons

n/n0 r, nm

1 2 3 4

ions electrons

n/n0 r, nm

10 20 30

ions electrons

n/n0 r, nm n0 = 1017cm-3 Te = 1 eV

1 2 3 4 2 4 6 8 10 2 4 6 8 10

r, nm E, GV/m φ, V

10 20 30 0.2 0.4 0.6 0.8 2 4 6 8

r, nm E, GV/m φ, V

40 80 120 160 200 0.01 0.02 0.03 0.04 0.05 1 2 3 4

r, nm E, GV/m φ, V

n0 = 1019cm-3 Te = 1 eV n0 = 1021cm-3 Te = 1 eV z, nm z, nm z, nm

z, nm z, nm z, nm

Sheath for stationary state

SLIDE 38

40 80 120 160 200 0.2 0.4 0.6 0.8 1 1.2

ions electrons

n/n0 r, nm

1 2 3 4

ions electrons

n/n0 r, nm

10 20 30

ions electrons

n/n0 r, nm

n0 = 1017cm-3 Te = 1 eV n0 = 1019cm-3 Te = 1 eV n0 = 1021cm-3 Te = 1 eV z, nm z, nm z, nm

10 20 30 0.001 0.01

z, nm σ, int. units

40 80 120 160 200 0.0001 0.001

z, nm

1 2 3 4 0.01 0.1 1

z, nm σ, int. units

width at 1/e level λ = 1.04nm exponential fit σ ~ exp(-z/λexp) λexp = 0.43nm statistical fluctuations

Sheath for stationary state

SLIDE 39

10

25

10

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28

10

29

0.1 1 10 100

λexp (σ ~ exp{-r/λexp}) 1.7λD λ (width at 1/e level)

power fit: λ ~ n-0.4

λ, nm ne, m-3

2

4

e D e

kT n e λ = π

Te = 10eV

10

23

10

24

10

25

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28

0.1 1 10 100

λexp (σ ~ exp{-r/λexp}) 1.7λD λ (width at 1/e level)

power fit: λ ~ n-0.4

λ, nm ne, m-3

2

4

e D e

kT n e λ = π

Te = 1eV

Screening length vs density

SLIDE 40

λ/λD (width at 1/e level), T=1eV

power fit: λexp ~ n0.1

λ/λD (width at 1/e level), T=10eV

power fit: λexp ~ n0.05

λexp/λD (σ ~ exp{-r/λexp})

power fit: λ ~ n0.01

10

23

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24

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25

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10

29

1 2 3 4 5 6 7 8

λ/λD ne, m-3

Screening length vs density

SLIDE 41

10

• 1

10 1

2 3 4 5 6 7 8

λ/λD (width at 1/e level), T=1eV λ/λD (width at 1/e level), T=10eV

power fit: λexp ~ n0.3

λ/λD Γ

Screening length vs G

1/3 2

4 3

e

n e kT π   Γ =    

SLIDE 42

10

23

10

24

10

25

10

26

10

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10

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0.1 1 10 100

MD Power fit for MD Schwirzke Corrected Schwirzke

E, GV/m ne, m-3

T = 1eV 10

24

10

25

10

26

10

27

10

28

10

29

0.1 1 10 100

MD Power fit for MD Schwirzke Corrected Schwirzke

E, GV/m ne, m-3

T = 10eV

E-field vs plasma density

SLIDE 43

Plasma model of RF breakdown

(1) Fowler-Nordheim equation for electrons (β = 100, 200) (2) Langmuir-Child equation for ion current from plasma to the tip (d=1 µm) (3) Richardson-Dushman equation for thermionic emission of electrons from liquid Cu (T=1300K) (4) Sputtering Flux was calculated from Bohm current (plasma ion fluxes) times the sputtering yield at T=1300K

SLIDE 44

44

OOPIC Pro 2.5D modeling

Simulation showing how rf arcs start (805 MHz)

SLIDE 45

45

Sum m ary of the Arc m odel

SLIDE 46

46

Tonks-Frenkel instability

• Dimensions of structures imply Esurface ~ 1 GV/m, if Psurface tension = PElectrostatic.
• T. Proslier, ANL

Capillary waves can measure surface fields

SLIDE 47

Theory of Thonks instability

MC

f t r h k t r h t r h dt t r dh + ∇ − ∆ + ∇ = ) , ( ) , ( ) , ( ) , (

4

ν η

The dynamics of a non-equilibrium surface profile in contact with plasma can be determined from an surface dynamics (Kuramoto-Sivashinski) equation: Here, h – is the heights at a 2d- position r {x,y}, at time t. The coefficients in this equation have the meanings: η – the viscosity coefficient,  – the surface tension term, k - the diffusion coefficient, fMC – the sputter by plasma ions.

SLIDE 48

48

Other applications of arcing

• We are beginning to develop parameter sets for these cases:
• Tokamak edge plasmas
• Large surface area and long DC pulses.

This model predicts that breakdown will occur when the Elocal >5 – 6 GV/m.

• (φ/λD)β ~ 6 GV/m
• With a 100 eV sheath potential, and λD ~ 6 µm gives,
• β ~ (6 GV/m)(6E-6m)/(100 eV) ~ 400,
• Laser Ablation, micrometeorite impacts
• Tiny areas and very short DC pulses.
• Dense plasmas can appear and arcs must trigger more quickly.

With λD ∼ 0.1 µm,

• (φ/λD)β ~ 11 GV/m,
• φ ~ (11 GV/m)(1E-7m)/30 ~ 40 eV
• These arcs would have similar parameters and would develop as

described above

SLIDE 49

49

Conclusions

• Our picture of arcs is becoming simpler and

more general.

• We find electrostatic fields can both trigger

and drive arcs.

• Materials properties are the clue for

understanding of unipolar arc formation and rf breakdown.

• We are exploring new mechanisms and news

models, with a number of papers underway.