J Comput Electron (2008) 7: 213–216 DOI 10.1007/s10825-008-0193-7 Boundary conditions with Pauli exclusion and charge neutrality: application to the Monte Carlo simulation of ballistic nanoscale devices H. López · G. Albareda · X. Cartoixà · J. Suñé · X. Oriols Published online: 2 February 2008 © Springer Science+Business Media LLC 2008 Abstract Electron transport becomes (quasi-) ballistic for nanoscale devices with active regions smaller than 20 nm. Under these conditions, the current and the noise are mainly determined by the electron injection process. Thus, the nu- merical simulation of these small devices can be very sen- sible to the boundary conditions (BC). In this work, we present a novel BC for (time-dependent) particle simulators that fulfill Fermi statistics and charge neutrality at the con- tacts. Monte Carlo simulations of a nanometric two-terminal device using a traditional injection model and the novel model presented in this work are compared. Keywords Monte Carlo method · Ballistic transport · Boundary conditions · Injection model 1 Introduction For advanced nanoscale devices, where the electron trans- port tends to become ballistic, the device performance is mainly determined by the injection process, rather than by the transport inside the active region. Therefore, the devel- opment of accurate boundary conditions (BC) for the (time- dependent) simulators of nanoscale devices is mandatory. The BC used in most quantum simulators of nanoscale devices are conceptually identical to those used in the Lan- dauer formalism: the injection of electrons is determined by the occupation function at the contact. For ballistic systems, H. López · G. Albareda · X. Cartoixà · J. Suñé · X. Oriols ( ) Universitat Autònoma de Barcelona, 08192 Bellaterra, Spain e-mail: xavier.oriols@uab.es H. López e-mail: hender.lopez@campus.uab.cat when the bias that fall in the active region is fixed to that of the external battery, this injection process implies a charge imbalance at the contacts. On the other hand, the standard in- jection model for the semi-classical Monte Carlo (MC) sim- ulation is based on adapting the injection rate of electrons in order to achieve charge neutrality at the contacts. This latter injection model is not accurate for degenerate condi- tions, because it neglects the Fermi correlations between the injection of consecutives electrons. In this work, we specify the rate of injection of electrons and the boundary conditions of the Poisson equation for the simulation of nanoscale devices that takes into account, at the contacts, the Pauli exclusion among electrons and the requirement of charge neutrality. The injection model pre- sented here can be adapted to semi-classical MC simula- tors [1], quantum algorithms that are implicitly time depen- dent [2] or to hybrid Boltzmann–Schrödinger solvers [3]. 2 Boundary conditions for ballistic systems 2.1 Rate of injection of electrons In the Landauer approach, the key assumption that deter- mines the BC is that electrons entering the open system de- pend only upon the occupation function and density of states of the reservoir, and that the electrons leaving the device de- pend only upon the active region. We define the rate of in- jection following the same assumptions. The Pauli principle determines that two electrons (fermions) cannot be associ- ated to the same wave function. Therefore, a spatial sepa- ration between two electrons with identical wave vectors, k x , k y , k z , is necessary. Such spatial separation translates into a time interval separation at a particular position. As