Finite-element implementation for electron transport in nanostructures

We have modeled transport properties of nanostructures using Green’s-function method within the framework of the density-functional theory. The scheme is computationally demanding, so numerical methods have to be chosen carefully. A typical solution to the numerical burden is to use a special basis-function set, which is tailored to the problem in question, for example, the atomic-orbital basis. In this paper we present our solution to the problem. We have used the finite-element method with a hierarchical high-order polynomial basis, the so-called $p$ elements. This method allows the discretation error to be controlled in a systematic way. The $p$ elements work so efficiently that they can be used to solve interesting nanosystems described by nonlocal pseudopotentials. We demonstrate the potential of the implementation with two different systems. As a test system a simple Na-atom chain between two leads is modeled and the results are compared with several previous calculations. Secondly, we consider a thin hafnium dioxide $(HfO2)$ layer on a silicon surface as a model for a gate structure of the next generation of microelectronics.