Mathematical Physics Research Group

Research

Mathematical models for charge transport in low-dimensional structures

Relevant modeling and computational issues relating to charge transport in electronic devices with characteristic nanometric dimensions, such as MOSFETs, double-gates and nanowires, are addressed. The most used approach to simulate such devices is to introduce a subband structure obtained by solving a Schroedinger-Poisson system, under appropriate assumptions on the wave function, coupled with a system of semiclassical Boltzmann equations in the non-quantized directions. From this system it is possible to derive physics-based hydrodynamic models using the Maximum Entropy Principle, which also take into account the heating of the device, including the transport of phonons. However, the Boltzmann equation is no longer valid if highly variable potentials are present in the structure which can induce tunneling phenomena. In this case it is necessary to use other models including the Wigner transport equation, which is the natural generalization of the Boltzmann equation to the quantum case. The numerical solution of the Wigner equation, which is highly non-local, however, presents considerable difficulties. Monte Carlo simulations based on “signed-particle” schemes have been recently introduced and appear very promising from the point of view of calculation and accuracy.

A second topic that is addressed is the transport of charges in graphene, an innovative material, also promising for new generation electronic devices. While there are efficient solvers on the market, which use mathematical models of varying degrees of complexity, for the simulation of semiconductor devices built with materials traditionally used in the electronics industry, such as silicon or GaAs, the same is not yet available for graphene. We are therefore helping to overcome this deficiency with the formulation of appropriate mathematical models of mesoscopic and macroscopic transport - drift-diffusion, energy-transport and hydrodynamic - starting at the kinetic level from the Boltzmann equations by applying the method of moments and obtaining the necessary closure relations via the maximum entropy method. At the same time, appropriate numerical schemes for these models are studied which will be validated through comparisons with Monte Carlo simulations and with the direct simulation of the Boltzmann equation using WENO-type finite difference schemes and with discontinuous Galerkin-type finite elements.