1. 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. 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. 3 Outline PIC-Monte Carlo method, EUV generated plasma, Simulation Results, Summary/Outlook.

4. 4 1D3V model Particle-In-Cell Interpolate charges to grid Solve Poisson equation Interpolate E-field at particle position Collisions at wall Collisions with neutrals new velocity Move particles F v x Bi-linear interpolation Poisson equation Bi-linear interpolationLeap-frog scheme Interpolate charges to grid Solve Poisson equation Interpolate E-field at particle position Collisions at wall Collisions with neutrals new velocity Move particles F v x Interpolate charges to grid Solve Poisson equation Interpolate E-field at particle position Collisions at wall Collisions with neutrals new velocity Move particles F v x Interpolate charges to grid Solve Poisson equation Interpolate E-field at particle position Collisions at wall Collisions with neutrals new velocity Move particles F v x Interpolate charges to grid Solve Poisson equation Interpolate E-field at particle position Collisions at wall Collisions with neutrals new velocity Move particles F v x Interpolate charges to grid Solve Poisson equation Interpolate E-field at particle position Collisions at wall Collisions with neutrals new velocity Move particles F v x Interpolate charges to grid Solve Poisson equation Interpolate E-field at particle position Collisions at wall Collisions with neutrals new velocity Move particles F v x Particle-wall interaction Monte-Carlo Collisions

5. 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, time to next collision: Probability p(t) of collision after time t:

6. 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. 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 +

7. 7 Collision angle Collisions treated in center-of-mass-frame Hard-sphere collisions:Forward scattering:

8. 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. 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 photon h = 92 eV Fast electron E kin = 76 eV Slow ion Wall Plasma sheath Bulk plasma Very expensive! Quasi- neutrality Formation of a plasma sheath, Ions accelerated towards walls, Sputtering of optics? Influence of photo-electric effect? - - - electrons - - - ions Photo- electrons - - - V pl

10. 10 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: E e = 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.

11. 11 ‘Numerical’ Setup 5 cm 1-D equidistant grid, 300 grid points:  x < D.  10 5 super particles, one super particle represents 10 9 real particles. Time steps of 1 ps:  t « (2 /  e ),  t ). Boundary Conditions: mirror and wall are grounded. Multi-layer mirror Wall

12. 12 Results(1): plasma density 100 ns EUV pulse, Sheath build-up, Low-density, ionization degree  10 -5.

13. 13 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 -

14. 14 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

15. 15 Results(3): potential Initially negative potential at mirror due to photo-electrons, Plasma potential max 80 V. Photo-effect has effect on potential No photo-electrons Cold photo-electrons Hot photo-electrons

16. 16 Results(4): ion impact Ions accelerated by sheath potential drop, Ions reach wall after EUV pulse, Maximum ion energy close to sputter threshold.

17. 17 Results(6): Including Ar 2+ EUV-photons energetic enough for double photo-ionization of argon. Sputtering dominated by Ar 2+.

18. 18 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.

19. 19 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.