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: 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

4. 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

5. 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

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. zero temperature limit
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:

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.