Particle-In-Cell%20Monte%20Carlo%20simulations%20of%20a%20radiation%20driven%20plasma

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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

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Kinetic Plasma Model

• Fluid model requires equilibrium assumptions for
  
• velocity distributions,
• Kinetic model preferable when
• ? gt L or ? gt T
• plasma sheath near electrode Ignition
  phase of lamp
• of low pressure lamp

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Outline

• PIC-Monte Carlo method,
• EUV generated plasma,
• Simulation Results,
• Summary/Outlook.

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Particle-In-Cell
1D3V model
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Monte Carlo Collisions

• Charged particles collide with background gas,
• Collision event that instantaneously changes
• the velocity, in both magnitude and direction,

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Null-collision method

• Problem Velocity dependent collision frequency
  ?c N ?(v) v
• Solution Introduce extra dummy process ? ?c
  maxN ?(v) v
• In case of collision Draw random number to
  determine process.

• 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

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Collision angle

• Collisions treated in center-of-mass-frame
• Hard-sphere collisions Forward scattering

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Next generation lithography

• Diffraction limited Smaller wavelength is
  smaller features!
• EUV-radiation 13,5 nm wavelength,

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Radiation driven plasma

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

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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.

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Numerical Setup
Multi-layer mirror
Wall

• 1-D equidistant grid,
• 300 grid points ?x lt ?D.
• ? 105 super particles,
• one super particle represents
• 109 real particles.
• Time steps of 1 ps ?t (2 / ?e),
• ?t lt (?x / ltvgt).
• Boundary Conditions
• mirror and wall are grounded.

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Results(1) plasma density

• 100 ns EUV pulse,
• Sheath build-up,
• Low-density,
• ionization degree ? 10-5.

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Results(1) plasma density

• 100 ns EUV pulse,
• Sheath build-up,
• Low-density,
• ionization degree ? 10-5.

Hot ph-e-
No photo-effect
Cold ph-e-
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Results(2) electron energy

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

Hot photo-electrons
No photo-electrons
Cold photo-electrons
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Results(3) potential

• Initially negative potential
• at mirror due to photo-electrons,
• Plasma potential max 80 V.
• Photo-effect has effect on potential

Hot photo-electrons
No photo-electrons
Cold photo-electrons
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Results(4) ion impact

• Ions accelerated
• by sheath potential drop,
• Ions reach wall
• after EUV pulse,

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Results(6) Including Ar2

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