The free electron model is a simple model for electrons in a metal crystal where it is assumed that the electrons become completely separate from their metal ions. This means they can be seen as behaving like an ideal gas with all the qualities that come with it. This means the electrons electrostatic repulsion is assumed to be negligible (hopefully reduced by the positive ions), the crystal lattice is generally ignored except for the boundary. The mass of the electrons is replaced with an effective mass to assist in this analysis with even negative effective mass being possible if it is a hole moving instead of an electron.
Now when the wire being examined is only as wide as the de Broglie wavelength for the electron things start to get quite a bit more complicated. The quantum mechanical effects take over and these nanowires are, as a result, to study such fundamental effects. It is believed that the electronic states at the surface of the wire have considerable control over the current density and distribution in the wire. Results show that this at least true for indium-arsenic (InAs) wires which, when oxidised slightly, have a very high density of surface states. These surface states are very willing to liberate conduction electrons and this could explain the high conductivity even when their is no n-type doping present in the InAs nanowires. There is also a low contact resistance (resistance due to quality of connection between two conductors rather than inherent resistance) between InAs nanowires which is indicative of a layer of electrons gathering at the wire’s surface.
The electron gas formed according to the free electron model would be heavily determined by these surface features as well as the dimensions of the wire. A theoretical study has recently been done using the models to describe the electron distributions in hexagonal InAs nanowires with differing sizes and surface states. It was found that electrons liked to cluster and accumulate in the hexagon prism’s corners while flowing in a cylinder down the middle. This inhomogeneity can be alleviated through various doping techniques.