While well established for larger gaps, Paschen's law (PL) fails to accurately predict breakdown for microscale gaps, where field emission becomes important. This deviation from PL is characterized by the absence of a minimum breakdown voltage as a function of the product of pressure and gap distance, which has been demonstrated analytically for microscale and smaller gaps with no secondary emission at atmospheric pressure [A. M. Loveless and A. L. Garner, IEEE Trans. Plasma Sci. 45, 574–583 (2017)]. We extend these previous results by deriving analytic expressions that incorporate the nonzero secondary emission coefficient, , that are valid for gap distances larger than those at which quantum effects become important (∼100 nm) while remaining below those at which streamers arise. We demonstrate the validity of this model by benchmarking to particle-in-cell simulations with γSE = 0 and comparing numerical results to an experiment with argon, while additionally predicting a minimum voltage that was masked by fixing the gap pressure in previous analyses. Incorporating γSE demonstrates the smooth transition from field emission dominated breakdown to the classical PL once the combination of electric field, pressure, and gap distance satisfies the conventional criterion for the Townsend avalanche; however, such a condition generally requires supra-atmospheric pressures for breakdown at the microscale. Therefore, this study provides a single universal breakdown theory for any gas at any pressure dominated by field emission or Townsend avalanche to guide engineers in avoiding breakdown when designing microscale and larger devices, or inducing breakdown for generating microplasmas.
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