2.6  Quantum Mechanical Effects - Confinement

Until now purely quantum mechanical effects, such as charge carrier confinement, have not been addressed. In MOS structures, considered in this work, carrier confinement emerges whenever local potential wells are formed  [50]. Deep potential wells cause discrete quantized energy levels and confine the charge carriers in one or more directions. One of the most widely recognized case of carrier confinement occurs in all MOSFETs at the semiconductor-insulator interface, whenever the MOSFET is driven into inversion as shown in Figure 2.10. In this case the charge carriers are confined in the direction perpendicular to the interface and free to move in the other two directions, thus forming a two dimensional electron gas  [50].

Figure 2.10:

The confinement of electrons and the emergence of sub-bands in a exemplary 1D nMOS structure, where x = 0nm corresponds to the insulator-semiconductor interface. Shown are the conduction band edge Ec and the first sub-band Ecs at 7.7meV as well as the normalized electron concentrations as obtained by a classical calculation and by a quantum mechanical one. The model error of a classical calculation can be well seen in the Capacitance-Voltage curve (inset). For pMOS analog results are obtained. For the calculation the Vienna Schrödinger Poisson Solver (VSP)  [51] has been used.

When deriving the BTE in Section 2.2, the wave character of the electrons was assumed to be negligible in favor of a classical description of the physical system. In MOS structures with thin insulators in the range of nanometers, e.g. t_ox 1nm, charge carrier confinement needs to be included in order to yield an accurate description of the device. For an accurate quantum mechanical description of carrier confinement the Schrödinger equation

(2.56)

for a given confining electrostatic potential V ′(x) must be solved (self-consistently) with Equation (2.27), where s is the sub-band index and ϵ^s is the energy eigenvalue. In order to solve the above equation boundary conditions need to be set for an isolated device. It is natural to assume vanishing electron wave functions at the boundaries of the semiconductor. If one is interested in the penetration of the wave function into an oxide in order to calculate direct tunneling currents  [52] this can be achieved by setting a Dirichlet boundary condition at the boundaries of the oxide not interfacing with a semiconductor, although the method itself is dependent on the oxide thickness and only justified for thick oxides.