Unified Electron Emission and Gas Breakdown Theory Across Length, Pressure, and Frequency
As electronic device dimensions decrease to micro and nanoscale, Paschen’s law (PL)—the standard theory used to predict breakdown voltage (Vb) governed by Townsend avalanche (TA)—fails due to ion-enhanced field emission (FE). Analytic models to predict Vb at these scales are necessary to elucidate the underlying physics driving breakdown and electron emission in these regimes. Starting from a previously-derived breakdown criterion coupling TA and FE, this dissertation derives a universal (true for any gas) breakdown equation. Further simplifying this equation using a matched asymptotic analysis, dependent on the product of the ionization coefficient and the gap distance, yields an analytic theory for dimensionless Vb. This analytic model unifies the coupled FE/TA regime to a universal PL derived by applying scaling parameters to the standard PL. This model enables parametric analyses to assess the effects of different parameters (such as pressure, gap distance, and field enhancement factor) on breakdown and quantify the relative contribution of FE and TA to identify the transition to the universal PL. This dissertation applies this general theory to experimental cases of different gap width, gap pressure and electrode surface roughness before exploring unification across electron emission regimes, validation with molecular dynamics simulations, and extensions to alternating current (AC).
One application of this theory to experimental data used data from a collaborator at Xi’an Jiaotong University, who used an electrical-optical measurement system to measure the breakdown voltage and determine breakdown morphology as a function of gap width. An empirical fit showed that the breakdown voltage varied linearly with gap distance at smaller gaps as in vacuum breakdown. This dissertation demonstrates that applying the matched asymptotic theory in the appropriate limits recovers this scaling with the slope as a function of field emission properties.
Pressure also plays a critical role in gas breakdown behavior. This dissertation derives a new analytic equation that predicts breakdown voltage Vb within 4% of the exact numerical results of the exact theory and new experimental results at subatmospheric pressure for gap distances from 1-25 . At atmospheric pressure, Vb transitions to PL near the product of pressure and gap distance, pd, corresponding to the Paschen minimum; at lower pressures, the transition to PL occurs to the left of the minimum. We further show that the work function plays a major role in determining whether Vb transitions from the coupled FE/TA equation back to the traditional PL to the right or the left of the Paschen minimum as pressure increases, while field enhancement and the secondary emission coefficient play smaller roles. These results indicate that appropriate combinations of these parameters cause Vb to transition to PL to the left of the Paschen minimum, which would yield an extended plateau similar to some microscale gas breakdown experimental observations.
Finally, the importance of electrode surface structure on microscale gas breakdown remains poorly understand. This dissertation provides the next step at assessing this by applying the asymptotic theory to microscale gas breakdown measurements for a pin-to-plate electrode setup in air at atmospheric pressure with different cathode surface roughness. Multiple discharges created circular craters on the flat cathode up to 40 μm deep with more pronounced craters created at smaller gap sizes and greater cathode surface roughness. The theory showed that breakdown voltage and ionization coefficient for subsequent breakdown events followed our earlier breakdown theory when we replaced the gap distance d with an effective gap distance deff defined as the sum of cathode placement distance and crater depth. Moreover, the theory indicated that deff could become sufficient large to exceed the Meek criterion for streamer formation, motivating future studies to assess whether the cathode damage could drive changes in the breakdown mechanism could for a single electrode separation distance or the Meek criterion requires modification at microscale.
We next unified field emission with other electron emission mechanisms, including Mott-Gurney (MG), Child-Langmuir (CL), and quantum space-charge-limited current (QSCL) to develop a common framework for characterizing electron emission from nanoscale to the classical PL. This approach reproduced the conditions for transitions across multiple mechanisms, such as QSCL to CL, CL to FE, CL to MG to FE, and microscale gas breakdown to PL using a common nondimensional framework. Furthermore, we demonstrated the conditions for more complicated nexuses where multiple asymptotic solutions matched, such as matching QSCL, CSCL, MG, and FE to gas breakdown. A unified model for radiofrequency and microwave gas breakdown will be compared to experimental results from Purdue University to elucidate breakdown mechanism.
The results from this dissertation will have applications in microscale gas breakdown for applications including microelectromechanical system design, combustion, environmental mitigation, carbon nanotube emission for directed energy systems, and characterizing breakdown in accelerators and fusion devices.