Simulation of the plasma propagation along the ferroelectrics plasma source'

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Simulation of the plasma propagation along the
ferroelectrics plasma source.
Dmitry Yarmolich Physics Department,
Technion, 32000 Haifa, Israel.
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Ferroelectric plasma source (FPS)
FPS is the dielectric with high permittivity

• 1800 for BaTiO3

The FPS front metal electrode is not continuous,
but striped so at the front surface alternates
the metal and dielectric strips.
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Plasma formation in triple points (0-30 ns after
voltage application)
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Plasma propagations (30-100ns after voltage
application)
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Plasma propagations (100-200 ns after voltage
application)
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Dense plasma(gt200 ns after voltage application)
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Electric field in triple point
E, 106 V/m
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5
0
The electric field strength E at the point
between metal, dielectric and vacuum (triple
point) achieves 106 V/m and causes to
breakdown.So the current from metal electrode is
charging the virtual capacity to compensate the
dielectric surface charge.
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The equivalent scheme of FPS
The capacity C1 is formatted by the metal
electrodes.
The capacity C2(t) is formed by the rear metal
electrode and the plasma which propagating
along the FPS surface.
The capacity C4 have to be charged to permit the
plasma propagation.
The capacity C3 is sequential circuit for the
capacity C4.
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How the plasma appears.

• The Ferroelectric could be represented as a bulk
  of dipole domains.The electric field applied to
  its electrodes redirect those dipole moments
  along the electric field lines. Hence at the
  front surface of the dielectric a surface charges
  are appeared, that under the metal strips
  electrode are compensated by the free electrons
  of metal electrode, but at free dielectric
  surface strips are not compensated. So between
  the metal strips and free dielectric surface the
  potential difference are arisen from the charge
  density irregularity and this potential
  difference are depending on the dielectric
  permittivity (e) of the FPS (for big e potential
  difference is bigger) and at the triple point
  potential difference causes to electric field up
  to 106 V/m. So, the high field magnitude causes
  to electrons field emissions and as a result to
  a spark breakdown.The spark nature is the plasma,
  through that a current is flowing to compensate
  the ferroelectrics surface charges.This plasma
  tends to propagate, but until the surface charges
  under the plasma are not compensated the plasma
  could not expand toward the new dielectric area
  due to ambipolar diffusion process inside the
  plasma. So after the compensation of surface
  charges under the plasma it may to expand to new
  ferroelectric area and supplier has to provide
  current to compensate the surface charge again.
  But if at the definite potential bias the
  supplier provide more current than the existing
  plasma spots may to make use of the new plasma
  spots have to appear.

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Algorithm of processing.
After the calculation demanded plasma and
supplier parameters one may to calculate plasma
velocity, but if it was exceeded the maximal
plasma velocities which determined by plasma
ambipolar diffusion velocity, a conditions for
new spots appearance are arising. So, in other
words, if the suppler provided too much energy
for plasma propagation abilities for existing
spots , a new spots have to be produced.
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How does routine work?

• The routine principle is a recursive obtaining
  plasma parameters from former plasma parameters
  and actual current-voltage parameters of
  supplier. For example, let us take any stage of
  plasma propagation.At this stage, plasma area is
  S(n) and potential bias is Un-1 which
  represents by itself a capacitor Cn-1 so to
  move the plasma forward at a new area S(n1)
  with new potential bias Un the supplier have to
  provide a current isupp to charge a new
  capacitor dCn up to the potential bias Un and
  iold to recharge the old capacitor Cn up to the
  new potential Un .

iold Cn-1 ( Un - Un-1 ) / ?t inew dCn Un /
?t isupp Un / R inew isup - iold
The next step is obtaining addition to the spot
radius dr and plasma propagation velocity Vn
     dCn Un / ?t Un / R- Cn(Un - Un-1 ) /
?t      dCn ?t / R - Cn ( 1 - Un-1 / Un)     
dr dC / ( 4pee0prn-1 )     Vn dr / ?t
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How does routine work II ?
But if Vn gt Vmax the plasma could not propagate
at a new area so fast as supplier provide a new
portion of energy, and it gives rise to new
plasma spots at ferroelectric-metal strips
contacts irregularities, so the plasma velocity
doesnt exceed Vmax But new spots able to make
use of suppliers energy.
Vn Vmax N dr / ( Vmax ?t) dr dr / N Cn
Cn-1 dCn rn rn-1 dr
So, now we have received all necessary parameters
to calculate next step of plasma propagations
along the FPS surface.
Initial parameters of plasma spots are
determined by the breakdown voltage UBr ,by
the mean size rirr of irregularities at
ferroelectric-metal strips contacts at the triple
points, and by initial capacity of FPS C0 .
U00 U1 UBr r1rirr dC1( 4pee0pr0 /l )
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Starting and using the routine

• To start the routine you have to download the all
  files from folder projectCPH and to open with
  Matlab file GuiPlot.M, after this you have to
  run it ( F5) . User interface will be open in a
  new window.
• Tag Waveform file name is containing the .CSV
  file of supplier voltage-time dependence (you may
  to replace it by your own .CSV file) by default
  Ive attached waveform of Blumleine pulse forming
  line.
• Tag Voltage scale is containing the voltage
  scale (volt by divisions) of .CSV file .
• Tags Permittivity of FPS ,Width of FPS,
  Capacity of FPS , Irregularities size are
  containing the parameters of FPS.
• Tags Plasma relaxation time and max Plasma
  velocity are containing time of plasma living
  without energy supplying and maximal plasma
  velocity(velocity of ambipolar diffusion)
  respectively.
• Tags Supplier resistance and Breakdown
  potential are containing Supplier resistance
  and Breakdown potential (by default all
  parameters taken from real BaTiO3 cathode and
  Blumleine pulse forming line.)
• By pushing the button Calculate you will
  receive three dimensional graph and you may to
  receive two dimensional graphs by changing check
  box and clicking on button Replot .