1 Particle-In-Cell Monte Carlo simulations of a radiation driven plasma Marc van der Velden, Wouter...

1

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

2

Kinetic Plasma Model

• Fluid model requires equilibrium assumptions for velocity distributions,

• Kinetic model preferable when

> L or > T plasma sheath near electrode Ignition phase of lamp of low pressure lamp

3

Outline

• PIC-Monte Carlo method,

• EUV generated plasma,

• Simulation Results,

• Summary/Outlook.

4

1D3V model

Particle-In-Cell

Interpolatecharges to grid

Solve Poissonequation

Interpolate E-field at particle position

Collisions at wall

Collisions with neutralsnew velocity

Move particlesF v x

sx

xixi 11xixi 1

1 )(

i

sii x

xx

xA

q

Bi-linear interpolationPoisson equation

00

2

2

ie nne

x

V

x

VVxE iis

1)(

Bi-linear interpolation

m

qEtttvttv

ttvttxttx

)()(

)()()(

21

21

21

Leap-frog scheme




Collisions at wall


Move particlesF v x




Collisions at wall


Move particlesF v x




Collisions at wall


Move particlesF v x




Collisions at wall


Move particlesF v x




Collisions at wall


Move particlesF v x




Collisions at wall


Move particlesF v x

Particle-wall interactionMonte-Carlo Collisions

5

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,

].1,0[,)1ln(

exp1)(

rndvvN

rndt

tvvNtp

bg

bg

time to next collision:

• Probability p(t) of collision after time t:

6

Null-collision method

• Problem: Velocity dependent collision frequency: c = N (v) v• Solution: Introduce extra dummy process c = max{N (v) v}

• In case of collision: Draw random number to determine process.

0 100 200 300

0

10

20

30

Co

llisi

on

fre

qu

en

cy (

E)

[MH

z]

Electron energy [eV]

Elastic Elastic+Excitation Elastic+Excitation+Ionization Elastic+Excitation+Ionization

+Null-Collision

Null-Collision

• Processes: elastic electron scattering

e- + Ar e- + Ar collisional excitation

e- + Ar e- + Ar*

electron-impact ionizatione- + Ar 2e- + Ar+

elastic ion scatteringAr+ + Ar Ar+ + Ar

charge exchange collisionsAr+ + Ar Ar + Ar+

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0.0

0.2

0.4

0.6

0.8

1.0

0

30

60

90

120

150

1800.0

0.2

0.4

0.6

0.8

1.0

1 eV 10 eV 30 eV

Collision angle

• Collisions treated in center-of-mass-frame

Hard-sphere collisions: Forward scattering:

]1,0[,)1(81

21cos

r

r

r

r21cos

eVEE

E27, 0

0

8

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

9

Radiation driven plasma

• EUV radiation from plasma source,• Argon background gas: p = 0.01 – 1 Pa,

Photo-ionization of background gas,creating a plasma!

Atom

EUV photonh = 92 eV

Fast electronEkin = 76 eV

Slow ion

WallPlasmasheath

Bulk plasma

• Very expensive!Quasi-

neutrality

• Formation of a plasma sheath,

• Ions accelerated towards walls,

• Sputtering of optics?

• Influence of photo-electric effect?-

--

electrons

--

-ionsPhoto-

electrons

--

-Vpl

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Photo-electric effect

• Photons absorbed in mirror cause collision cascade and secondary electron emission; • Case 1) no photo-effect

• 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.

0 10 20 30 400.00

0.05

0.10

0.15

En

erg

y d

istr

ibu

tion

fun

ctio

n

Electron energy [eV]

4

2

)(

6)(

WE

EWES

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‘Numerical’ Setup

5 cm

• 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.

Multi-layer mirror

Wall

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0.0 0.2 0.4 0.6 0.8 1.0

1E14

1E15

1E16

Pla

sma

de

nsi

ty [m

-3]

Position [cm]

10 ns electrons 10 ns ions

0.0 0.2 0.4 0.6 0.8 1.0

1E14

1E15

1E16

Pla

sma

de

nsi

ty [m

-3]

Position [cm]

10 ns electrons 10 ns ions 50 ns electrons 50 ns ions

Results(1): plasma density

• 100 ns EUV pulse,

• Sheath build-up,

• Low-density, ionization degree 10-5.

0.0 0.2 0.4 0.6 0.8 1.0

1E14

1E15

1E16

Pla

sma

de

nsi

ty [m

-3]

Position [cm]

10 ns electrons 10 ns ions 50 ns electrons 50 ns ions 500 ns electrons 500 ns ions

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0.0 0.2 0.4 0.6 0.8 1.0

1014

1015

1016

Pla

sma

den

sity

[m-3]

Position [cm]


Results(1): plasma density

• 100 ns EUV pulse,

• Sheath build-up,

• Low-density, ionization degree 10-5.

No photo-effect Cold ph-e-

Hot ph-e-

0.0 0.2 0.4 0.6 0.8 1.0

1014

1015

1016

Pla

sma

de

nsi

ty [m

-3]

Position [cm]


0.0 0.2 0.4 0.6 0.8 1.0

1014

1015

1016

Pla

sma

dens

ity [m

-3]

Position [cm]


14

0 1 2 3 4 5

0

20

40

60

80

100

Mea

n el

ectr

on e

nerg

y [e

V]

Position [cm]

10 ns 50 ns 100 ns 200 ns 1000 ns

Results(2): electron energy

• Electron energy decreases:

1) Most-energetic electrons reach walls first,2) Electron-impact ionization,3) Excitation.

No photo-electrons Cold photo-electrons

Hot photo-electrons

0 1 2 3 4 5

0

20

40

60

80

100

Mea

n el

ectr

on e

nerg

y [e

V]

Position [cm]

10 ns 50 ns 100 ns 200 ns 1000 ns

0.00 0.01 0.02 0.03 0.04 0.05

0

20

40

60

80

100

Mea

n el

ectr

on e

nerg

y [e

V]

Position [cm]

10 ns 50 ns 100 ns 200 ns 1000 ns

15

0 1 2 3 4 5

0

20

40

60

Pot

entia

l [V

]

Position [cm]

10 ns 50 ns 100 ns 200 ns 1000 ns

Results(3): potential

• Initially negative potential at mirror due to photo-electrons,

• Plasma potential max 80 V.

• Photo-effect has effect on potential

0 1 2 3 4 50

20

40

60

80

Pot

entia

l [V

]

Position [cm]

10 ns 50 ns 100 ns 200 ns 1000 ns


0 1 2 3 4 5-40

-20

0

20

40

60

80

Pot

entia

l [V

]

Position [cm]

10 ns 50 ns 100 ns 200 ns 1000 ns

Hot photo-electrons

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Results(4): ion impact

• Ions accelerated by sheath potential drop,

• Ions reach wall after EUV pulse,

.

4,

1 221

21

mm

mmUU surfthr

• Maximum ion energy close to sputter threshold.

0.0 0.2 0.4 0.6 0.8 1.0

0

10

20

30

40

50 no photo-electrons hot photo-electrons cold photo-electrons

EU

V-in

ten

sity [a.u

.]

EUV off

Ave

rag

e io

nim

pa

ct e

ne

rgy

[eV

]

Time [s]

EUV on

SputterThreshold

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Results(6): Including Ar2+

• EUV-photons energetic enough for double photo-ionization of argon.

• Sputtering dominated by Ar2+.

0.0 0.2 0.4 0.6 0.8 1.0

0

20

40

60

80

100 No photo-electrons Ar+

No photo-electrons Ar2+

Cold photo-electrons Ar+

Cold photo-electrons Ar2+

EU

V-in

ten

sity [a.u

.]

EUV off

Ave

rag

e io

nim

pa

ct e

nerg

y [e

V]

Time [s]

EUV on

SputterThreshold

single double single double0

1

Sp

utt

er

Ra

te [

a.u

.]


Ar+

Ar2+

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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.

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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.

1 Particle-In-Cell Monte Carlo simulations of a radiation driven plasma Marc van der Velden, Wouter...

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