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 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

Outline Unmagnetized diode: Magnetized 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

Model: unmagnetized diode Electrons emitted with given velocity distribution f0(v) e- flow x E V 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

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

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

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)

Unidirectional Maxwellian distribution We assume: Then: And: with

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

Comparisons local temperature: density System is not isothermal!

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

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

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

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

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:

Simulation example

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

Higher temperature

Parameter space: critical point

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

Cold space-charge limited case

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

Gaussian distribution lower temperture: higher temperture:

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.