ParticleInCell Monte Carlo simulations of a radiation driven

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 9/10/2020 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 9/10/2020 or >T Ignition phase of lamp 2

Outline • PIC-Monte Carlo method, • EUV generated plasma, • Simulation Results, • Summary/Outlook. 9/10/2020 3

Particle-In-Cell Leap-frog scheme Monte-Carlo 1 D 3 V model Poisson equation Particle-wall Bi-linear interpolation Collisions interaction 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 9/10/2020 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: time to next collision: 9/10/2020 5

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 collisional excitation e- + Ar* electron-impact ionization e- + Ar 2 e- + Ar+ elastic ion scattering Ar+ + Ar charge exchange collisions Ar+ + Ar+ 9/10/2020 6

Collision angle • Collisions treated in center-of-mass-frame Hard-sphere collisions: 9/10/2020 Forward scattering: 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 9/10/2020 reflective optics 8

Radiation driven plasma EUV photon h = 92 e. V • EUV radiation from plasma source, • Argon background gas: p = 0. 01 – 1 Pa, Fast electron Ekin = 76 e. V Photo-ionization of background gas, creating a plasma! Atom Wall Plasma sheath Bulk plasma Photoelectrons ions electrons Vpl - 9/10/2020 -- • 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 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. 9/10/2020 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 9/10/2020 11

Results(1): plasma density • 100 ns EUV pulse, • Sheath build-up, • Low-density, ionization degree 10 -5. 9/10/2020 12

Results(1): plasma density • 100 ns EUV pulse, Hot ph-e- • Sheath build-up, • Low-density, ionization degree 10 -5. No photo-effect 9/10/2020 Cold ph-e- 13

Results(2): electron energy • Electron energy decreases: Hot photo-electrons 1) Most-energetic electrons reach walls first, 2) Electron-impact ionization, 3) Excitation. No photo-electrons 9/10/2020 Cold photo-electrons 14

Results(3): potential • Initially negative potential at mirror due to photo-electrons, Hot photo-electrons • Plasma potential max 80 V. • Photo-effect has effect on potential No photo-electrons 9/10/2020 Cold photo-electrons 15

Results(4): ion impact • Ions accelerated by sheath potential drop, • Ions reach wall after EUV pulse, • Maximum ion energy close to sputter threshold. 9/10/2020 16

Results(6): Including Ar 2+ • EUV-photons energetic enough for double photo-ionization of argon. • Sputtering dominated by Ar 2+. 9/10/2020 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. 9/10/2020 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. 9/10/2020 19