# Thermal-field emission flicker (1/f) noise and diffusive equilibrium density fluctuations

## Abstract

A model of diffusive equilibrium density fluctuations in a grand-canonical ensemble is constructed for systems of finite size. The particle number autocorrelation is developed from a Langevin-type bounded-diffusion equation. Both probe and sample geometries affect its spectrum, which factors into two terms representing the particle creation rate and diffusion according to a multidimensional Carsons theorem. The spatial decay of the kernel in the spectrums integral equation is measured by a frequency-dependent correlation length that depends on particle lifetime, diffusivity, and probe resolution. The kernel and its transform, the mutual coherence function, collapse to the Ornstein-Zernike spatial distribution but with the new result that the classical correlation length is given by a ratio of diffusive and thermodynamic variables. For the limiting case of an unbounded system with infinite particle lifetime, Voss and Clarkes spatially correlated spectrum is rederived. However, for this ensemble a finite particle lifetime is a necessary equilibrium condition. Littles theorem is generalized when particle interactions are included. Noise-power integrals converge in all cases. Frequency exponents characterize the spectra and, when a small region is probed in a quasi-two-dimensional system, broadband 1/f noise occurs. A Lorentzian spectrum results in the limit of no diffusion. A lower length limit introduced to avoid the breakdown of the diffusion approximation at small time and space intervals can in some cases be identified with probe resolution and is measurable when a certain crossover in frequency exponents is identified. The analysis is then applied to fluctuations in the electron current, thermal field emitted from a single-crystal tungsten cathode. These are coupled to self-diffusion of surface defect adatoms on the cathode by the Fowler-Nordheim equation. Other frequency crossovers yield surface diffusivities and their activation energies, which for comparable W(h k l) planes are consistent with field-ion microscopy measurements. The defect vacancy activation energy is estimated from the temperature dependence of the adatom creation rate and is similar to that obtained from emitter surface-tension measurements. For the projection optics of field emission systems spatial resolution is mapped to the Gaussian source diameter dg. Using this new measurement method dg20 AI is obtained for 1000 K thermal-field emission from tungsten. © 1988 The American Physical Society.