In the course of the derivation of Boltzmann’s transport equation, it was assumed that the wave nature of the electrons can be neglected. The concept of the BTE, as shown in Figure 2.4, is that during free streaming the electrons are treated as classical particles embedded in the lattice of fixed positively charged ions. As long as the electrons scatter frequently while traveling, the phase of their wave function and initial speed is lost, which is true for channels longer than the coherence length and slowly spatially varying electrostatic potentials. This is for example not the case when the channel only consists of a few atoms or a single molecule (biomolecule) [53, 54]. In this case a non-equilibrium Schrödinger equation for charge transport needs to be solved. The formalism most commonly used to solve the Schrödinger equation in non-equilibrium is refered to as Non-Equilibrium Greens Functions Approach (NEGF) [55]. The channel, of a MOSFET for example, is then modelled as consisting of a few slices (2D case) of atoms with contacts in thermal equilibrium on the left and right of the channel (cf. Figure 2.11).
| Figure 2.11: | Schematic of a 2D nanoscale MOSFET, consisting of a few atoms, simulated using NEGF. For every 1D slice (yellow atoms) a Schrödinger equation is assembled and solved. It is assumed that the contact (red atoms) are in thermal equilibrium. |
The one-particle Schrödinger equation in NEGF formalism then reads
 | (2.57) |
where E is the electron energy, HC is the Hamiltonian for the channel, ΣL and ΣR are the self-energies accounting for the left and right contacts, ΣS is the self-energy due to scattering, B is to account for the boundary conditions at the contacts, G is the retarded Green’s function, G< is the lesser Green’s function and GA is the advanced Green’s function [55, 56]. From G< macroscopic quantities, such as the electron concentration and the current density can be obtained. The main advantage of NEGF is that the wave character of the electrons is preserved, which leads to a highly accurate description of nanoscale (~ 10nm) devices. But as soon as the devices become larger, the matrices in Equation (2.57) become too large for any numerical solver. In large devices, scattering occurs more frequently, which increases the number of off-diagonals in Equation (2.57). In addition it is not possible to rigorously consider electron-electron scattering in the NEGF method. Thus special attention has to be paid to the calculation of the scattering rates. Nevertheless, NEGF is one of the most accurate methods to describe charge carrier transport in nanoscale devices as well as direct tunneling in MOS structures [56]. In the course of this thesis NEGF was used to estimate direct tunneling currents, where only phonon scattering was considered.