An Integral Equation Approach to Model the Drastic Change in Depletion Width From Bulk to Nanoscale Junctions

Abstract

The space-charge region (SCR) in semiconductor bulk and nanoscale junctions is mostly analyzed using the differential equation, namely, Poisson’s equation. We explore an alternate analysis using the integral equation, namely, Coulomb’s law, and show its superiority to the differential equation approach in revealing the drastic change in SCR electrostatics from bulk to single nanofilm (NF) to single nanowire/nanotube (NW/NT) junction. The advantages of the integral equation approach are as follows. First, it helps us to view the junction SCR in bulk as a series of sheet charges, that in thin NFs as a series of line charges, and that in thin NWs/NTs as a series of point charges; such a view logically yields the different functional forms of the depletion width dependence on dielectric constant, potential drop, and inverse doping, namely, squareroot in bulk, linear in NFs, and exponential in NWs/NTs. Second, it shows that the partially depleted space-charge tails in NFs/NWs/NTs must extend to infinity, and that these tails rather than the completely depleted regions set up the required potential difference across the junction; it also reveals that for large distance, ${z}$ away from the junction, the field varies as $1/{z}^{2}$ in NF and $1/{z}^{3}$ in NW/NT, and the potential/charge vary as $1/{z}$ in NF and $1/{z}^{2}$ in NW/NT. To ensure accuracy of the depletion width model, we capture the effect of space-charge tails in NFs/NWs/NTs by employing a periodic charge distribution. We also show how prior works overestimated the depletion width by orders of magnitude.