Transcript Particle-In-Cell Monte Carlo simulations of a radiation

Particle-In-Cell Monte Carlo simulations
of a
radiation driven plasma
Marc van der Velden, Wouter Brok, Vadim Banine,
Joost van der Mullen, Gerrit Kroesen.
COST Model Inventory Workshop, April 2005
1
Kinetic Plasma Model
• Fluid model requires equilibrium assumptions for
velocity distributions,
• Kinetic model preferable when
>L
plasma sheath near electrode
of low pressure lamp
or
>T
Ignition phase of lamp
2
Outline
• PIC-Monte Carlo method,
• EUV generated plasma,
• Simulation Results,
• Summary/Outlook.
3
Particle-In-Cell
Leap-frog
scheme
Monte-Carlo
1D3V
model
Poisson
equation
Particle-wall
Bi-linear
Bi-linear
interpolation
interpolation
x(2t  qinteraction
t )Collisions
( xxi (1t )x
t v(t  12 t )
)
s
nei  ni 
Ei(xV )Vi1 eV
 x

A

x
 qE 
i

1
v(xt  12 t ) 0 v(tx 12 0t )  t 

 m
2s
xs
xi  i x
xi 1  i  1x
Interpolate
charges to grid
Collisions with neutrals
new velocity
Solve Poisson
equation
Collisions at wall
Interpolate E-field
at particle position
Move particles
F
v
x
4
Monte Carlo Collisions
• Charged particles collide with background gas,
• Collision: event that instantaneously changes
the velocity, in both magnitude and direction,
• Super particle represents many real particles, but
has charge and mass of real electron/ion,
• Probability p(t) of collision after time t:
p (t )  1  exp N bg  v  v t  

time to next collision:
t
ln(1  rnd )
, rnd  [0,1].
N bg  v  v
5
Null-collision method
• Problem: Velocity dependent collision frequency: c = N (v) v
• Solution: Introduce extra dummy process  c = max{N (v) v}
• Processes:
elastic electron scattering
e- + Ar  e- + Ar
collisional excitation
e- + Ar  e- + Ar*
electron-impact ionization
e- + Ar  2e- + Ar+
elastic ion scattering
Ar+ + Ar  Ar+ + Ar
charge exchange collisions
Ar+ + Ar  Ar + Ar+
Collision frequency (E) [MHz]
• In case of collision: Draw random number to determine process.
30
Null-Collision
20
10
Elastic
Elastic+Excitation
Elastic+Excitation+Ionization
Elastic+Excitation+Ionization
+Null-Collision
0
0
100
200
300
Electron energy [eV]
6
Collision angle
• Collisions treated in center-of-mass-frame
Hard-sphere collisions:
Forward scattering:
cos   1  2r
cos   1 
90
120
1.0
1 eV
10 eV
30 eV
60
0.8
0.6
150
2r
, r  [0,1]
1  8 (1  r )

E
, E0  27eV
E0
30
0.4
0.2
0.0 180
0.0
0
0.2
0.4
0.6
0.8
7
Next generation lithography
•
•
Diffraction limited: Smaller wavelength is smaller features!
EUV-radiation: 13,5 nm wavelength,
•
Very small absorption lengths (typically 0.1 mm):
1)
Optical path contained within vacuum setup,
p = 0.01 – 1 Pa,
2)
refractive optics
reflective optics
8
Radiation driven plasma
EUV photon
h = 92 eV
• EUV radiation from plasma source,
• Argon background gas: p = 0.01 – 1 Pa,
Fast electron
Ekin = 76 eV
Photo-ionization of background gas,
creating a plasma!
Atom
Wall
Plasma
sheath
Bulk
plasma
Photoelectrons
ions
electrons
Vpl
-
--
• Formation of a plasma sheath,
-
• Very expensive!
Quasineutrality
Slow ion
-
-
• Ions accelerated towards walls,
• Sputtering of optics?
• Influence of photo-electric effect?
9
Photo-electric effect
• Photons absorbed in mirror cause
collision cascade and secondary
electron emission;
• Case 2) hot photo-electrons
Inelastic reflection: Ee= h - W
• Case 3) cold photo-electrons
Electron scattering inside mirror:
distribution of electron energies S(E).
Above certain energy S(E)
independent of photon energy.
Energy distribution function
• Case 1) no photo-effect
0.15
0.10
S (E) 
6W 2 E
(E  W )4
0.05
0.00
0
10
20
30
40
Electron energy [eV]
10
‘Numerical’ Setup
Multi-layer
mirror
Wall
• 1-D equidistant grid,
300 grid points: x < D.
•  105 super particles,
one super particle represents
109 real particles.
• Time steps of 1 ps: t « (2 / e),
t < (x / <v>).
• Boundary Conditions:
mirror and wall are grounded.
5 cm
11
Results(1): plasma density
• Sheath build-up,
• Low-density,
ionization degree  10-5.
10 ns electrons
10 ns ions
50 ns electrons
50 ns ions
500 ns electrons
500 ns ions
-3
• 100 ns EUV pulse,
Plasma density [m ]
1E16
1E15
1E14
0.0
0.2
0.4
0.6
0.8
1.0
Position [cm]
12
Results(1): plasma density
16
Plasma density [m ]
10
-3
• 100 ns EUV pulse,
• Sheath build-up,
• Low-density,
ionization degree  10-5.
10 ns electrons
10 ns ions
50 ns electrons
50 ns ions
500 ns electrons
500 ns ions
Hot ph-e15
10
14
10
0.0
0.2
0.4
0.6
0.8
1.0
Position [cm]
16
16
15
10
14
10
0.0
0.2
0.4
0.6
Position [cm]
0.8
1.0
10
10 ns electrons
10 ns ions
50 ns electrons
50 ns ions
500 ns electrons
500 ns ions
Cold ph-e-
-3
10 ns electrons
10 ns ions
50 ns electrons
50 ns ions
500 ns electrons
500 ns ions
Plasma density [m ]
No photo-effect
-3
Plasma density [m ]
10
15
10
14
10
0.0
0.2
0.4
0.6
0.8
1.0
Position [cm]
13
Results(2): electron energy
• Electron energy decreases:
1) Most-energetic electrons
reach walls first,
2) Electron-impact
ionization,
3) Excitation.
Mean electron energy [eV]
100
Hot photo-electrons
80
60
40
10 ns
50 ns
100 ns
200 ns
1000 ns
20
0
0
1
2
3
4
5
Position [cm]
100
Mean electron energy [eV]
Mean electron energy [eV]
100
No photo-electrons
80
60
40
10 ns
50 ns
100 ns
200 ns
1000 ns
20
0
0
1
2
3
Position [cm]
4
5
Cold photo-electrons
80
60
40
10 ns
50 ns
100 ns
200 ns
1000 ns
20
0
0.00
0.01
0.02
0.03
0.04
0.05
Position [cm]
14
Results(3): potential
Hot photo-electrons
80
• Plasma potential max 80 V.
60
Potential [V]
• Initially negative potential
at mirror due to photo-electrons,
40
20
10 ns
50 ns
100 ns
200 ns
1000 ns
0
-20
• Photo-effect has effect on potential
-40
0
1
2
3
4
5
Position [cm]
No photo-electrons
60
40
10 ns
50 ns
100 ns
200 ns
1000 ns
20
Cold photo-electrons
60
Potential [V]
Potential [V]
80
40
20
10 ns
50 ns
100 ns
200 ns
1000 ns
0
0
0
1
2
3
Position [cm]
4
5
0
1
2
3
4
5
Position [cm]
15
Results(4): ion impact
• Ions accelerated
by sheath potential drop,
• Maximum ion energy close
to sputter threshold.
Average ion
impact energy [eV]
• Ions reach wall
after EUV pulse,
Sputter
Threshold
40
30
EUV
on
EUV-intensity [a.u.]
no photo-electrons
hot photo-electrons
cold photo-electrons
50
EUV
off
20
10
0
0.0
U thr
4m1m2 

, 
.
 1   
m1  m2 2
U surf
0.2
0.4
0.6
0.8
1.0
Time [s]
16
Results(6): Including Ar2+
1
• EUV-photons energetic enough for
double photo-ionization of argon.
+
• Sputtering dominated by Ar2+.
+
Average ion
impact energy [eV]
80
60
EUV
on
EUV
off
40
Sputter
Threshold
20
EUV-intensity [a.u.]
No photo-electrons Ar
2+
No photo-electrons Ar
+
Cold photo-electrons Ar
2+
Cold photo-electrons Ar
100
Sputter Rate [a.u.]
Ar
2+
Ar
0
single
double
No photo-electrons
single
double
Cold photo-electrons
0
0.0
0.2
0.4
0.6
0.8
1.0
Time [s]
17
Summary
• With PIC-MCC it is possible to simulate a plasma far from
equilibrium.
• Photo-effect has influence on sputter rate.
• Sputtering will be modest as kinetic energy of most ions will be
below sputtering threshold.
18
Outlook
• Experimental verification:
Energy sensitive mass-spectrometry,
Absolute Line Intensity measurements,
Sputter yield and sputter rate measurements.
Thompson scattering (?)
Energy resolved Secondary electron yield measurements.
19