1. B.I.R.D. model HV BREAKDOWN in VACUUM
Breakdown Induced by Rupture of Dielectric layer
This is a three-step process

2. DARK CURRENT
Fowler-Nordheim

3. DARK CURRENT Fowler-Nordheim Local field-enhancement factor 
Local field-enhancement factor 
𝐸 0 →𝛽 𝐸 0
𝛽≈ 10 2 ÷ 10 3
𝐴≈ 10 −16 ÷ 10 −12 𝑚 2

4. DARK CURRENT
Fowler-Nordheim
𝐺𝑒𝑜𝑚𝑒𝑡𝑟𝑖𝑐𝑎𝑙 𝑎𝑚𝑝𝑙𝑖𝑓𝑖𝑐𝑎𝑡𝑖𝑜𝑛 𝑓𝑎𝑐𝑡𝑜𝑟 𝛽

5. DARK CURRENT
Fowler-Nordheim
𝐺𝑒𝑜𝑚𝑒𝑡𝑟𝑖𝑐𝑎𝑙 𝑎𝑚𝑝𝑙𝑖𝑓𝑖𝑐𝑎𝑡𝑖𝑜𝑛 𝑓𝑎𝑐𝑡𝑜𝑟 𝛽

6. DARK CURRENT
Fowler-Nordheim
𝐺𝑒𝑜𝑚𝑒𝑡𝑟𝑖𝑐𝑎𝑙 𝑎𝑚𝑝𝑙𝑖𝑓𝑖𝑐𝑎𝑡𝑖𝑜𝑛 𝑓𝑎𝑐𝑡𝑜𝑟 𝛽

7. DARK CURRENT
Fowler-Nordheim
𝐺𝑒𝑜𝑚𝑒𝑡𝑟𝑖𝑐𝑎𝑙 𝑎𝑚𝑝𝑙𝑖𝑓𝑖𝑐𝑎𝑡𝑖𝑜𝑛 𝑓𝑎𝑐𝑡𝑜𝑟 𝛽

8. DARK CURRENT
Fowler-Nordheim
𝐺𝑒𝑜𝑚𝑒𝑡𝑟𝑖𝑐𝑎𝑙 𝑎𝑚𝑝𝑙𝑖𝑓𝑖𝑐𝑎𝑡𝑖𝑜𝑛 𝑓𝑎𝑐𝑡𝑜𝑟 𝛽

9. DARK CURRENT Fowler-Nordheim + geometrical amplification
𝐼=𝐴 𝑘 1 𝛽 𝐸 0 − 𝑘 2 𝛽 𝐸 0
𝐸 0 →𝛽 𝐸 0
h(mm) 
A(m2)
0.50 46
4.0E-17
0.75 52
1.2E-16
1.00 58
1.9E-16
1.25 62

10. DARK CURRENT Fowler-Nordheim + different amplification mechanisms  
 (kV-)
7.4
7.2E-12
F.N.
Exper.
k1 (A/V2)
3.31E-07
2.50E-19
k2 (V/m)
6.85E+10
7.45E+08

11. DARK CURRENT Fowler-Nordheim law
Fits well all experimental data (for given  and A)
Too high -value and (often) unreliable emitting surfaces
Dependence of  and A from the electrode gap
Fowler-Nordheim-like law
Different amplification mechanism must be considered

12. BREAKDOWN Breakdown is usually interrupted by externally circuitry
Breakdown changes the microscopic surfaces
Breakdown Voltage is a function of distance
TOTAL VOLTAGE EFFECT
𝑉 𝐵𝐷 ~ ℎ 𝛼

13. Micro-Current (BURSTS)
High Voltage Holding
CURRENT-CONDITIONING
Sphere-Plane configuration – gap=30mm

14. Micro-Current (BURSTS)
High Voltage Holding
CURRENT-CONDITIONING (cleaned electrodes)
Sphere-Plane configuration – gap=30mm

15. Micro-Current (BURSTS)
High Voltage Holding
CURRENT-CONDITIONING (cleaned electrodes)
Sphere-Plane configuration – gap=30mm

16. Micro-Current (BURSTS)
High Voltage Holding
CURRENT-CONDITIONING
Involves Current and Pressure Peaks
Depends on the surface treatment
CLUMP MODEL (Detachment of material)

17. Micro-Current (BURSTS)
High Voltage Holding
CURRENT-CONDITIONING
Involves current and Pressure Peaks
Depends on the surface treatment
CLUMP MODEL (Detachment of material)
It seems to be a good answer to experimental observation
It also gives a good answer to the Total Voltage Effect of
Breakdown Phenomena
CLUMP MODEL TAKES INTO ACCOUNT THE ENERGY GAINED BY CHARGE PARTICLES THROUGH THE GAP

18. A new Model needed
Dark current seems to be a quantum-tunneling-effect, qualitatively described by Fowler-Nordheim law
Current (and pressure) random bursts are due to detachment of clumps from electrode surfaces
Breakdown could be thought as a particular intense burst that starts an avalanche-like
In the Fowler-Nordheim, Cramberg-Slivkov frameworks, the main unsolved problems are
 and A unrealistic experimental values and their dependence on the gap.
Continuous presence of clumps

19. B.I.R.D. model (Breakdown Induced by Rupture of Dielectric layer) 𝐸 0
𝐸 0
𝜎 0 −
𝜎 0 +
𝜎 p +
𝜎 p −
𝐸 0 = 𝜎 𝜎 0 − − 𝜎 𝑝 + + 𝜎 𝑝 − 2 𝜀 0
𝐸= 𝜎 𝜎 0 − − 𝜎 𝑝 + − 𝜎 𝑝 − 2 𝜀 0
Charge conservation:
𝜎 𝜎 𝑝 + = 𝜎 0 − + 𝜎 𝑝 − h h0
𝜎 𝑝 + = 𝜎 𝑝 −

20. B.I.R.D. model (Breakdown Induced by Rupture of Dielectric layer) 𝐸 0
𝐸 0
𝜎 0 −
𝜎 0 +
𝜎 p +
σ ′ = 𝜎 0 − − 𝜎 𝑒 𝜎′
𝐸 0 = 𝜎 𝜎 0 − − 𝜎 𝑝 + +𝜎′ 2 𝜀 0
𝐸= 𝜎 𝜎 0 − − 𝜎 𝑝 + −𝜎′ 2 𝜀 0
Charge conservation: h h0
𝜎 𝜎 𝑝 + = 𝜎 0 − +σ′
𝜎 𝑝 + = 𝜎 𝑝 −

21. B.I.R.D. model (Breakdown Induced by Rupture of Dielectric layer) 𝐸 0
𝐸 0
𝜎 0 −
𝜎 0 +
𝜎 p +
σ ′ = 𝜎 p − − 𝜎 𝑒 𝜎′
𝜎 𝑝 ≝𝜎 𝑝 + = 𝜎 𝑝 −
𝜎 0 ≝𝜎 0 +
𝐸 0 = 𝜎 0 𝜀 0
𝐸= 𝜎 0 −𝜎′ 𝜀 0 h h0

22. B.I.R.D. model (Breakdown Induced by Rupture of Dielectric layer) 𝐸 0
𝐸 0
𝜎 0 −
𝜎 0 +
𝜎 𝑝 = 𝜖 0 𝜒𝐸
𝜎 p + 𝜎′
𝜖 𝑟 =1+𝜒
𝜎 𝑒 = 𝜀 𝑟 𝜀 0 𝐸− 𝜖 0 𝐸 0
𝜀 𝑟 𝜀 0 𝑑𝐸 𝑑𝑡 = 𝜖 0 𝑑𝐸 0 𝑑𝑡 + 𝐽 𝑒
This equation has to be coupled with h h0
𝑅𝐼= 𝑉 0 − ℎ 0 𝐸 0 −ℎ𝐸

23. B.I.R.D. model (Breakdown Induced by Rupture of Dielectric layer)
𝜀 0 𝑑 𝐸 0 𝑑𝑡 = 𝑉 0 − ℎ 0 𝐸 0 −ℎ𝐸 𝑅𝑆 − 𝐽 𝑒 𝜀 0 𝜀 𝑟 𝑑𝐸 𝑑𝑡 = 𝑉 0 − ℎ 0 𝐸 0 −ℎ𝐸 𝑅𝑆 h h0
A model for 𝐽 𝑒 is needed.
It depends on
charge density 𝜎 ′ = 𝜖 0 𝐸 0 −𝐸
average electric field 𝐸 at surface

24. B.I.R.D. model (Breakdown Induced by Rupture of Dielectric layer)
𝜀 0 𝑑 𝐸 0 𝑑𝑡 = 𝑉 0 − ℎ 0 𝐸 0 −ℎ𝐸 𝑅𝑆 − 𝐽 𝑒 𝜀 0 𝜀 𝑟 𝑑𝐸 𝑑𝑡 = 𝑉 0 − ℎ 0 𝐸 0 −ℎ𝐸 𝑅𝑆 h h0
A model for 𝐽 𝑒 is needed.
𝐸 = 𝐸 0 +𝐸 2
𝐽 𝑒 ~ 𝐸 0 −𝐸 𝐸 − 𝐸 𝑑
𝐽 𝑒 =𝑘 𝐸 0 −𝐸 𝐸 0 +𝐸 2 − 𝐸 𝑑
𝑘=0
for
𝐸 < 𝐸 𝑑

25. B.I.R.D. model (Breakdown Induced by Rupture of Dielectric layer)h h0
𝐽 𝑒 =𝑘 𝐸 0 −𝐸 𝐸 0 +𝐸 2 − 𝐸 𝑑
𝐸 0 = 𝐸 0 (0) 𝜀 𝑟
𝐹.𝑁.−𝑙𝑖𝑘𝑒 𝑐𝑢𝑟𝑟𝑒𝑛𝑡
𝐽 𝑒 (0)= 𝑘 1 𝐸 − 𝑘 2 𝐸 0 ≈ 𝑘 1 𝐸 0 2 𝑒 − 𝑘 2 𝐸 0

26. B.I.R.D. model (Breakdown Induced by Rupture of Dielectric layer)h h0
NUMERICAL SOLUTIONS
Dark Current
Dielectric-Rupture

27. B.I.R.D. model Rupture Time:
Time needed for a burst to happen is given by:

28. B.I.R.D. model Rupture Time:
Time needed for a burst to happen is given by:
Each voltage V explores a given range of  values

29. B.I.R.D. model #

30. B.I.R.D. model How can you verify BIRD model?
Overlapping consecutive ranges
Delay Time

31. B.I.R.D. model

32. B.I.R.D. model
Qualitatively reproduced
by (modified) B.I.R.D. model

33. B.I.R.D. model
Thank you for the attention