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Nonequilibrium statistical mechanics of electrons in a diode

Nonequilibrium statistical mechanics of electrons in a diode. Renato Pakter Collaboration: Felipe Rizzato, Yan Levin, and Samuel Marini Instituto de Física, Universidade Federal do Rio Grande do Sul Porto Alegre, Brazil. *Work supported by CNPq, FAPERGS, Brazil, and US-AFOSR. Introduction.

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Nonequilibrium statistical mechanics of electrons in a diode

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  1. Nonequilibrium statistical mechanics of electrons in a diode Renato Pakter Collaboration: Felipe Rizzato, Yan Levin, and Samuel Marini Instituto de Física, Universidade Federal do Rio Grande do Sul Porto Alegre, Brazil *Work supported by CNPq, FAPERGS, Brazil, and US-AFOSR

  2. Introduction • Magnetized and unmagnetized diodes are employed in many different areas: microwave sources, space propulsion, semiconductor industry, accelerators • Most of the theoretical investigations assume that the electron flow is either cold or is a fluid with a postulated equation of state (isothermal) • However, electrons interact through long-range forces such that the collision duration time diverges and the system does not reach equilibrium but becomes trapped in QSS • Here we present results of fully kinetic description of the electron flow based on the collisionless Boltzman (Vlasov) Equation for both magnetized and unmagnetized diodes

  3. Outline • Unmagnetized diode: • space charge limited regime • kinetic stationary states • verify validity of eq. state • Magnetized diode: • rich behavior in the regarding the transition to space-charge limited regime

  4. Model: unmagnetized diode L • Electrons injected from the cathode with known velocity distribution f0(v) • 1D motion: planes of charge moving along x direction • Nonrelativistic motion • Particles self-fields are taken into account Electrons emitted with given velocity distribution f0(v) e- flow x E V

  5. Space-charge limited regime L • electron cloud completely screens the accelerating potential at injection, cathode electric field vanishes • For cold fluid f0(v)=d(v): Child-Langmuir Law Electrons emitted with given velocity distribution f0(v) e- flow x Eself E V

  6. Kinetic description L Electrons emitted with given velocity distribution f0(v) e- flow x E V Kinetic description (Vlasov-Poisson Equations): with

  7. Determining the stationary state single particle constants of motion • by Jean’s theorem, in the SS: • in the SS the potential j(x) becomes time independent and the single particle energy is conserved • At x = 0: • Hence, satisfies boundary and SS condition • Substituting in the Poisson equation leads to a closed equation for j(x)

  8. Unidirectional Maxwellian distribution • We assume: • Then: • And: with

  9. Molecular-dynamics simulation • Ns particles are distributed along 0 < x < L • They evolve according to • Particles exiting at x=L are reinjected at x=0 • Particles near the cathode are continuously reshuffled to satisfy

  10. Comparisons local temperature: density System is not isothermal!

  11. Comparisons space charge limit curve: Ec 0 continuously zero temperature limit Rizzato, Pakter, Levin (2009)

  12. Magnetized diode (a) noninsulated: V0 > (b) insulated: V0 < Kinetic description (Vlasov-Poisson Equations): with

  13. Stationary state • For the sake of simplicity we consider insulated cases with a waterbag distribution at injection: • Using the same procedure as before we obtain: v y/L

  14. Theoretical prediction: low temperature (fixed vmin, vmax, and V0/B0) accelerating branch space-charge limited branch multiple stationary solutions: first order nonequilibrium phase transition

  15. Molecular-dynamics simulation • To model charge buildup in real devices, we start with empty system • Particles are continuously injected at x=0 with prescribed f0(v) • They evolve according to: • Particles that reach the cathode or the anode leave the simulation • Compute the evolution of the cathode electric field: 0

  16. Simulation example

  17. Comparisons h0=0.835 h0=0.836 Marini, Rizzato, Pakter (2014)

  18. Higher temperature

  19. Parameter space: critical point

  20. Cold space-charge limited case • Lau et. al. extensively studied the cold case and found that it does not reach a stationary state in the space-charge limited case; • Turbulent regime

  21. Cold space-charge limited case

  22. Finite injection temperature • flow becomes laminar • true stationary states • Turbulent-laminar transition? Mechanism?

  23. Gaussian distribution lower temperture: higher temperture:

  24. Conclusions • We derived a fully kinetic theory for the electron flow in diodes. • The electrons do not relax to an equilibrium with a known equation of state, precluding the use of conventional isothermal assumptions. • The magnetized diode presents a rich behavior in the parameter space regarding the transition to space-charge limited regime. • Can be used as a test bed for the study of phase transitions in long-range interacting driven systems.

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