# Mathematical Models for Transport in Macroscopic and Mesoscopic Systems - Abstract

**Wulf, Ulrich**

*Quantum scaling behavior of nanotransistors*

In our previous papers on ballistic field effect transistors [J. Appl. Phys. 96, 596 (2004), and ibid. 98, 84308 (2005)] it was demonstrated that under certain conditions ('SMAT' approximation) it is possible to reduce the fully three-dimensional transport problem to an effectively one-dimensional one. It was found that already a simple, piecewise linear ansatz for the effective potential leads to a qualitative description of the output characteristics of ultra-small transistors with channel lengths of several tens of nanometers. Here we want to demonstrate that for an effectively one-dimensional potential a scale invariant description of the transistor-transport-problem is possible. For the particular piecewise linear ansatz it is found that apart from a properly normalized temperature the output characteristics depend only on a single unitless barrier parameter $beta^th = 2m mu d^2 hbar^-2$ which depends on the effective mass $m$ of the electrons, the channel length $d$, and the chemical potential $mu$ in the source contact. At channel lengths between ten to thirty nanometers and ultra-high n-doping in the source contact $beta^th$ takes values between a few hundreds ('weak barrier') up to a few thousands ('strong barrier'). At weak barriers relatively high source-drain leakage currents may result so that instead of a clear 'OFF'-state regime a 'quasi-OFF-state' tunneling transistor regime arises. The features of the output- and of the transfer characteristics in this regime are structurally very similar to the 'short-channel effects' resulting in a drift-diffusion model for the transistor. At large $beta^th$ these undesirable quasi-short-channel effects are significantly reduced.