1. Title Greg Werner Hasan Padamsee Alexander Romanenko Modeling and Simulating Voltage Breakdown

2. - Enhanced field emission leads to catastrophic growth of electron current at an emitter. 26389 56350 C 0.0 3133 F - The plasma cloud expands, while ion bombardment removes surface adsorbates (such as hydrocarbons and fluorine) in a starburst pattern around the original microemitter. - The plasma dissipates and current stops. - Electrons ionize desorbed gas or metal vapor around emitter, forming a plasma “cloud,” which conducts discharge current. A Voltage Breakdown Event at the Field Emitter

3. Stages of Breakdown: 1. Enhanced field emission What is the local surface electric field? 2. Neutral vapor production Why so much so quickly? 3. Evolution of arc Expansion of plasma cloud Formation of starbursts and craters 4. Extinguishing of discharge

4. Enhanced field emission What is the local surface electric field? Field emission should occur at 5GV/m, not 50MV/m. Does pointy geometry enhance the field? Yes (to some extent) Is pointy geometry responsible for the field emission commonly seen at 50MV/m? Probably not...

5. Is the local electric field really 5GV/m? Fowler-Nordheim parameters for almost 700 emitters, from Thomas Habermann’s dissertation, Wuppertal, 1999 (see also Philipp Niedermann’s dissertation, Geneva, 1986) Emitter Area (cm 2 ) Field Enhancement Factor 10um 2 ! Geometry can enhance the local electric field, but geometry alone cannot explain this wide variation.

6. Given enhanced FE,why is breakdown so sudden? Background pressure Field Emission Current 1 min 100nA 1uA 10uA Breakdown Current 100ns 40A 5ntorr 100ntorr

7. A possible source of gas burst: Can chance ions start a neutral avalanche by sputtering in the presence of high FE current? electron beam Electric Field 1. “Chance” ion appears 2. Ion sputters two neutrals 3. Neutral is ionized 4. Ion sputters more neutrals, etc. The electron beam must be very intense to ionize sputtered neutrals with high probability. Assuming “reasonable” values (including: each ion sputters 1.1 neutrals), a crude estimate of the minimum electron current I necessary to cause an ion avalanche for given electron beam radius r is: rI 1um100mA 100nm10mA 10nm1mA 1nm100uA

8. A possible source of gas burst: Can chance ions start a neutral avalanche by sputtering in the presence of high FE current? rI 1um100mA 100nm10mA 10nm1mA 1nm100uA 1. The necessary electron currents are too high. 2. Electrons ionize best at tens of eV, but ions require hundreds of eV to sputter more than one neutral (and the ion will gain the same energy the electron had when it created the ion). No

9. Evolution of the Plasma Starburst and Craters Look at the pictures...

10. SEM (Secondary Emission) Images 0.0 23550 Nb 33078 81800 C 0.0 35533 V 0.0 5900 F AES (Auger) images BEFORE AFTER Original particle was V The Cathode After Breakdown on a typical Nb surface 50um Surface sputtered clean Particle mostly gone (only faint traces left)

11. “Starbursts” on Nb anodized to different oxide thicknesses 600-650Å400-450Å100-150Å200-250Å 30-70Å

12. Simulation the initiation of breakdown with OOPIC Open-source C++ code for Linux 2D square (x-y) and cylindrically symmetric (z-r) geometry Electrostatic and Electrodynamic field solvers (self-consistent, on mesh) Simulates particle collisions (including ionization) with Monte Carlo method Program: OOPICpro (object-oriented particle-in-cell code) maintained by TechX Corporation (www.txcorp.com)

13. Our Breakdown Simulation z r Simulation Space Our simulations are inspired by those done by Jens Knobloch using MASK: Geometry: Cylinder, r=8um, z=32um, 64  128 grid Field emission:  =250, A=0.034um 2 Electric Field: 30MV/m (DC or RF) Magnetic Field: ignored Neutral gas: effuses from emitter region as if heated to 2000K Timestep: 5 fs field emission and gas infusion Applied Field

14. Aside: Simulating Field Emission Is Difficult The basic problem: FN emission is extremely sensitive to the surface electric field. The practical problem: The space-charge of emitted current reduces the electric field at the emitter (negative feedback); the finite mesh-size and time-step cause a delay in the feedback loop, resulting in unphysical instability above a certain current threshold Solutions: 1. Emit electrons 1 mesh cell from emitter, where (due to finite mesh) they have the largest affect on surface field 2. Limit the amount of current emitted in a single time-step to something reasonable (depending on surface field)

15. z (m) r (m) Trajectories of electrons from a protrusion (  =12) (Aside on simulating FE continued) Minor point: To the extent that you believe geometry is responsible for field enhancement, you should give field- emitted electrons appropriate transverse (and perhaps normal) velocities. Tranverse velocity imparted to particle emitted from protrusion: where E is the field, enhanced by factor  at the protrusion with tip radius R tip, and r is the particle’s initial radial coordinate. Electric Field

16. Our Breakdown Simulation z r Simulation Space Our simulations are inspired by those done by Jens Knobloch using MASK: Geometry: Cylinder, r=8um, z=32um, 64  128 grid Field emission:  =250, A=0.034um 2 Electric Field: 30MV/m (DC or RF) Magnetic Field: ignored Neutral gas: effuses from emitter region as if heated to 2000K Timestep: 5 fs field emission and gas infusion Applied Field

17. 5GHz, mass 1 5 GHz, 30MV/m, atoms/ions of mass 1 (H)

18. 5 GHz, 30MV/m, atoms/ions of mass 39 (Ar)

19. 11 GHz, 30MV/m, atoms/ions of mass 39 (Ar)

20. DC, 30MV/m, atoms/ions of mass 39 (Ar)

21. Ball Lightning Crossing a Kitchen and a Barn woodcut from "L'Atmosphere" by C. Flammarion, 2nd. ed., Paris, 1873. Hasan Padamsee Alexander Romanenko Jerry Shipman, Dan Lundberg, Laurel Ying Ball Lightning