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CCP 2006
(S05-I22: Invited Talk)
Modeling of RF Window Breakdown
Transition of window breakdown
from vacuum multipactor discharge
to rf plasma
2006. 08. 29
H. C. Kim, Y. Chen, and J. P. Verboncoeur
Dept. of Nuclear Engineering, UC Berkeley
Topic 0.
0. Introduction and Models
I. Vacuum Multipactor Discharge
II. Transition to RF Plasma
Undesirable Discharge in HPMs
(High-Power Microwave)
Conductor
Peak power  1 MW
~ 100 ns pulse
RF Window (Dielectric)
1GHz ~ 100' s GHz
RF generator
(e.g. Magnetrons,
Linear beam
tubes, Gyrotrons,
Free-Electron
Lasers, and so on)
Either Vacuum
or Background Gas
y
Incoming EM wave > Outgoing EM wave
 Discharge can degrade device performance or even
damage devices, including catastrophic window failure.
x
z
z : direction of wave
propagation
In Vacuum (Multipactor Discharge)
(TE or TEM mode)
• Single-surface multipactor on a dielectric
+ +
+
-
Vacuum
 Multipactor discharge* is an avalanche
caused by secondary electron emission.
E y  Erf 0 sin( t ) : leads to electron energy gain.
-
 transit  2m v z , 0 eE z 0
(= life time)
+
-
2
m v z,0
xtransit 
(maximum distance)
2 eE z 0
+
Ez : makes electrons
return to the surface.
Eiy  f ( Erf 0  ,  transit)
* Observed in various systems
(e.g. RF windows, accelerator structures, microwave tubes and devices, and rf
satellite payloads)
Analytic Solution of Single Particle Motion
 Solution of the equation of motion for the electron in Vacuum
(TE or TEM mode)
For the constant Ez = Ez0 during the flight,
vx,0, vy,0: initial velocity of the electron emitted from the surface
0: initial phase of the rf electric field at that time (t=t0)
v z ,0 , v y ,0
-
t  tf
E y  Erf 0 sin( t )

 Erf 0 sin(  (t  t0 )  0 )
-
Ez  Ez 0
Ei
• The z- and y-components of the impact electron energy
1
1  eErf 0 
2


Eiz  mv z , 0 , Eiy 
2
2m   
2
0  t0

mv y , 0 

  2mv z , 0

cos



cos(

)

 

0
0
e
E

e
E



rf 0

  z0


f rf
 transit

 rf
f transit
2
With Background Gas (RF Plasma)
-
+
-
+ +
+
+
-
+
 Under the high-pressure
background gas, an rf plasma is
formed.
 The rf plasma is a candidate
for window breakdown on the
air side.
Discharge Sustainment
 Electron generation mechanisms in the system
• Secondary Electron Emission (SEE) on a surface
originated from electron impact to a material
: Dominant in Vacuum or under the low-pressure gas
probabilistic event
+
-
-
• Ionization in the volume
originated from ionization collisions between electrons and the background gas
: dominant under the high-pressure gas
probabilistic event
-
+
-
• Another emission mechanisms: thermionic emission, photo emission,
field emission, explosive emission, and so on.
Secondary Emission due to Electron Impact
 Energy and angular dependence of secondary emission yield
(the ratio of the incident flux to the emission flux)

- Ei
i,
i,
 1
(Electron Impact Energy)
 1
 1
[Ref] Vaughan et al, IEEE (1989);
IEEE (1993)
 (Electron Impact Angle)
1D3V Particle-In-Cell (PIC) Model
x
+ y
+
+
+
+
+ E y  Erf 0 sin( t )
Dielectric
+
+ +
+
+
+
+
• Condition of
left dielectric
 max 0  2
Emax 0  400 eV
Dielectric
(δ=0)
Eth  12.5 eV
k s  1
k sw  1
Tx 0, y 0  2 eV
L
BC : Ex ( x  0)  0
)
E x( , wall

 wall
N   N i i N e e  N i e
 e e

0
0
0 A
N e,i : number of electrons and ions in the space
 Simulation Tool :
Modified XPDP1 from PTSG,
UC Berkeley
[Ref] J.P. Verboncoeur et al., J.
Comput. Phys. 104, 321 (1993)
Topic I.
0. Introduction and Models
I. Vacuum Multipactor Discharge
II. Transition to RF Plasma
* Our model is based on electrostatic fields and the magnetic field is not taken
into account.
Dynamics using Monte-Carlo Simulation*
 Susceptibility Curve for Plane Wave
Ez
Discharge on
(Positive growth rate)
Ez
Discharge off : low  due to
• Too high impact energy
• Too small impact energy
Problem: No oscillation appears
even though  trans  1
* [Ref] Ang et al, IEEE Trans. Plasma Sci. 26, 290 (1998)
Model of Monte-Carlo Simulation
• Emission of initial seed electrons from the surface
vz,0, vy,0: Maxwellian distribution
: Uniform distribution
→ Calculate the impact energy and angle
 transit  2m v z , 0 eE z 0
i
titeration  max ( transit
)
i
(from analytic solution of one particle motion)
• Update
→ Calculate the secondary electron yield
N n 1 e (from model of SEC due to electron impact)
E z 0,n 1  
2 0
Problem : titeration ~  rf
• Ejection of multiple secondary electrons (Nn+1) from the surface
vz,0, vy,0: (from the energy distribution of secondary electrons)
i
i
: nsecondary




1
n
transit( n : iteration , i : parent particle )
The phase of next injection is taken from the phase of impact for the
parent electron.
* [Ref] Ang et al, IEEE Trans. Plasma Sci. 26, 290 (1998)
Dynamics using PIC Simulation
(solving field eqn.
self-consistently)
 PIC simulation shows that the electron number and the Ez
oscillate at twice the rf frequency, saturating after 1 ns.
 Ez oscillates in and out of the susceptibility region.
[Ref] H.C. Kim and J.P. Verboncoeur, Phys. Plasmas 12, 123504 (2005)
Plane Wave
Erf 0  3 MV/m at 1 GHz (L - band)
PIC: Susceptibility Curve (Plane vs. TE10)
• Effect of transverse field structure

E y ( x, t )  E y0 sin( t ) sin(
x)
dx
E y ,dc vs. Ez 0,dc
Plane wave
TE10 mode
Ey
E y0 sin( t )
~ x 1.5
0
dx
x
TE10 mode
cf. Plane Wave : Ey ( x, t )  Ey0 sin( t )
z : direction of wave propagation
 In TE10 mode, the upper boundary of the susceptibility
diagram is nearly vertical so that only the lower boundary is
relevant.
Summary for Topic I
 In HPM systems, the time-dependent physics of the single-surface
multipactor has been investigated by using PIC simulation.
 The normal surface field and number of electrons oscillate at twice the rf
frequency.
 The effect of the transverse field structure on the discharge has
been investigated.
 In TE10, the upper boundary of the susceptibility diagram is nearly
vertical so that only the lower boundary is relevant.
Topic II.
0. Introduction and Models
I. Vacuum Multipactor Discharge
II. Transition to RF Plasma
Collision with Argon Background Gas
 The argon gas is used in this study because of its simplicity in the
chemistry (compared with air).
• Electron-Neutral Collision
• Ion-Neutral Collision
PIC: Number of Particles (I)
 Vacuum multipactor discharge
 The secondary electron emission
is the only mechanism for generating
electrons.
Vacuum
 The number of electrons still
oscillates as in the vacuum case but
increases slowly in time, as a result
of electron-impact ionization.
p  10 mTorr
Ne  Ni
# of ions ~ # of ionization events
between electrons and argon gas
Erf 0  2.82 MV/m at 2.85 GHz (S - band), Argon
PIC: Number of Particles (II)
Ne ~ Ni
p  1 atm
 transit c  1
 The numbers of electrons and ions are nearly the same and
increase abruptly in time.
 Collisional ionization becomes the dominant mechanism to
generate electrons.
Erf 0  2.82 MV/m at 2.85 GHz, Argon
PIC: Electron Mean Energy
Vacuum and p  10 mTorr
 Electrons in the multipactor
discharge gain their energy by being
accelerated from the rf electric field
during the transit time.
p  1 atm
 At high pressures, electrons suffer lots
of collisions and lose the significant
amount of energy gained from the rf
electric field.
Erf 0  2.82 MV/m at 2.85 GHz, Argon
PIC: Electron Energy Distribution
Spatially averaged
 Below 50 Torr, the EEPF is bi-Maxwellian type.
 At high pressures, the EEPF becomes Druyvesteyn type since the
electron temperature decreases with the collision frequency.  c  
Erf 0  2.82 MV/m at 2.85 GHz, Argon
PIC: Electron and Ion Densities
p  100 Torr
p  20 mTorr
 trans c  1
p  10 Torr
 At low pressures, the multipactor
discharge is formed near the dielectric
window.
 At intermediate pressures, both
multipactor discharge and rf plasma
exist.
 At high pressures, only rf plasma is
formed, away from the surface of the
window.* Time-averaged over a cycle
PIC: Electric Field Profile
 At low and intermediate pressures, the electric field is positive on
the surface, indicating that the multipactor discharge can be sustained.
 At high pressures, the electric field is negative on the surface. The
energy of electrons impacting the surface is low enough so that the
secondary electron emission yield is less than 0.5.
Erf 0  2.82 MV/m at 2.85 GHz, Argon
PIC: Secondary Electron Emission
 Secondary electron emission yield on the dielectric
 : Erf 
Transition Pressure (10~50 Torr)
  c
 transit  1
  c
 transit c  1
EEPF of rf plasma is
Druyvesteyn.
surface discharge
is collisionless.
 (2.85 GHz ) ~  c
 (5.7 GHz ) ~  c
 Below 10 Torr, the secondary yield is near unity so that multipactor
discharge can be sustained.
 As the pressure increases, collisions suppress the impact energy and
hence the secondary electron yield  decreases to less than unity.
* For particles accumulated over a cycle
Experiment for the Breakdown on the Air Side
 The HPM surface flashover experiments at Texas Tech Univ.
Incident P
Transmitted P
Reflected P
Flashover delay time
Absorbed P = Incident P – Transmitted P –
Reflected P
[Ref] G. Edmiston, J. Krile, A. Neuber, J. Dickens, and H. Krompholz, “High Power
Microwave Surface Flashover of a Gas-Dielectric Interface at 90 to 760 Torr,” IEEE Trans.
Plasma Sci. (to be published).
Experiment for the Breakdown on the Air Side
Air: 90 ~ 760 Torr
3 MW,
UV
3 MW
4.5 MW
f = 2.85 GHz
Simple theory
 Eeff p vs. pτ is universal for different
Erf0 at the given pressure range.
: L. Gould and L. W. Roberts, J.
Appl. Phys. 27, 1162 (1956).
Eeff 
Erf 0
2[1  (  c ) 2 ]
PIC: Discharge Formation Time (I)
 Simulation results of argon gas for various E-fields and frequencies
Eeff p vs. pτ
Eeff 
Erf 0
2[1  (  c ) 2 ]
 ~c
• At very high pressures,
 Eeff p vs. pτ is universal for
different Erf0 and .
• At very low pressures
   c
Erf 0  c
Erf 0


p

2 p
Eeff
PIC: Discharge Formation Time (II)
 Simulation results of argon gas for various E-fields and frequencies
 [ns]
τ vs. p
(1.43 GHz) ~  c
• At low pressures,
 τ vs. p is universal for different
Erf0 and .
 (2.85 GHz ) ~  c
Summary for Topic II
 In HPM systems, adding an argon background gas, we have
investigated the transition of window breakdown from single-surface
vacuum multipactor discharge to rf plasma.
• There is an intermediate pressure regime where both multipactor discharge
and rf plasma exist.
• In our parameter regime, the transition pressure ( less than unity) is
between 10 and 50 Torr in argon.
 The discharge formation time () has been obtained as a function of
the gas pressure.
• The normalization Eeff p vs. pτ predicted by the simple theory holds only
at very high pressures.
• At low pressures, the discharge formation time is independent of Erf0 and .
Conference on Computational Physics 2006
Thank you for your attention.
* This work was supported in part by AFOSR Cathodes and Breakdown
MURI04 grant FA9550-04-1-0369, AFOSR STTR Phase II contract FA955004-C-0069, and the Air Force Research Laboratory - Kirtland.
Case 1 : Erf 0  3 MV/m and f rf  1 GHz
Case 2 : Erf 0  0.3 MV/m and f rf  1 GHz
Case 3 : Erf 0  3 MV/m and f rf  10 GHz
Case 4 : Erf 0  0.3 MV/m and f rf  0.1 GHz
MC : E-Field Trace
 The normal electric field and the number of electrons
oscillate with time only for Case 1 in the MC model.
MC versus PIC Results
Case 1
 Like the PIC simulation result, the oscillation period in our MC
simulation is half the rf period.
 However there is still a significant discrepancy in amplitude and
phase between the MC and PIC results, which comes from the
assumptions on which the MC simulation is based.
MC versus PIC Results
 The parameter regime where the multipactor discharge develops
is also the narrower in the MC simulation than in the PIC
simulation.
PIC : Power Trace
S
Erf2 0
 1.33 101 Erf2 0 MW/cm 2
2
Erf 0 [MV m]
Case 1
( S  1.2 MW/cm 2 )
PIC : Power Trace
~ 2%
Case 1
~ 0.5%
Case 2
 In vacuum multipactor discharge, the rf phase randomization
of electrons occurs only upon the collision with the surface.
 The phase delay of the discharge power with respect to
the input power comes from the finite transit time for electrons to
interact with the surface. It means that the electrons~ are
5%not
totally in equilibrium with the local rf electric field.
 As the transit time is larger (or the electric field is smaller), the
phase difference is larger.
PIC : Scaling with Erf0/frf
Grow
Decay
Cases 1 and 4

 rf
 transit

f transit
f rf
Cases 2 and 3
• The shape of the closed curve of the trajectory depends
on the amplitude of the rf electric field normalized to the
rf frequency (Erf0/frf).
At the beginning
X (um)
X (um)
PIC: Spatial Distribution of Electrons in TE10
At transient
Time
Z (um)
Z (um)
Weak E rf
Strong E rf
Weak E rf
X (um)
Time
At the steady state
Z (um)
E 0y  5 MV/m
2.85 GHz, Vacuum
Explanation of Spatial Distribution in TE10
 Susceptibility Curve
Center
Discharge on
(Positive growth rate)
Periphery
At transient E z
At steady state
Experiment for the Breakdown on the Air Side
 The HPM surface flashover experiments at Texas Tech Univ.
• WR284 S-Band waveguide
7.21 cm X 3.40 cm
(A = 24.5 cm2)
f  2.85 GHz  0  10.52 cm
(Air)
[Ref] G. Edmiston, J. Krile, A.
Neuber, J. Dickens, and H.
Krompholz, “High Power
Microwave Surface Flashover
of a Gas-Dielectric Interface
at 90 to 760 Torr,” IEEE Trans.
Plasma Sci. (to be published).
PIC: Discharge Formation Time
g ( t t 0 )
Assuming n(t )  n(t0 )e
• Discharge formation time 
Say,
n(t  t0   )
 108
n(t  t0 )
 g  18.42
g : effective volume ionization rate
obtained by fitting the number trace
t0 : determined from the time
that mean kinetic energy reaches
steady state, assuming g also
reaches steady state.
Erf 0  1.57 MV m at 2.85 GHz, 150 Torr
Comparison
• Flashover time: Experiment
at Texas Tech Univ. (Air)
• Discharge formation time: PIC
(Argon)
3 MW,
UV
3 MW
4.5 MW
 Since the statistical delay time is not considered in the
simulation and the background gas is different, there is an order of
magnitude difference in time between experiment and simulation.
 But, the qualitative trends are similar.
τ   p , Erf 0 
PIC:2nd Order Method for Particle Collection
 The velocity and position at time the particle crosses the boundary
v np1/ 2 , xnp1
v n-p 1 / 2 , xnp
Velocity
[Ref] H.C. Kim, Y. Feng, and J.P.
Verboncoeur, “Algorithms for collection,
injection, and loading in particle
simulations ”, J. Comput. Phys. (to be
published)
tn+1/2
tn
Position
tn+1
PIC: 2nd Order Method for Particle Ejection
ven-g , xne g
g ' '  t2e t
Velocity
tn-1/2
tn-1
Position
tn