  Contact resistance and current crowding are important to nanoscale electrical
contacts. In this paper, we present a self-consistent model to characterize
partially overlapped parallel contacts with varying specific contact
resistivity along the contact length. For parallel tunneling contacts formed
between contacting members separated by a thin insulating gap, we examine the
local voltage-dependent variation of potential barrier height and tunneling
current along the contact length, by solving the lumped circuit transmission
line model (TLM) equations coupled with the tunneling current self
consistently. The current and voltage distribution along the parallel tunneling
contacts and their overall contact resistance are analyzed in detail, for
various input voltage, electrical contact dimension, and material properties
(i.e. work function, sheet resistance of the contact members, and permittivity
of the insulating layer). It is found the existing one-dimensional (1D)
tunneling junction models become less reliable when the tunneling layer
thickness becomes smaller or the applied voltage becomes larger. In these
regimes, the proposed self-consistent model may provide a more accurate
evaluation of the parallel tunneling contacts. This work provides insights on
the design, and potential engineering, of nanoscale electrical contacts with
controlled current distribution and contact resistance via engineered spatially
varying contact layer properties and geometry.
