4.6  Calibration

Whether or not DG reproduces confinement correctly largely depends on the boundary conditions, the mesh spacing and the parameters λ_n for electrons and λ_p for holes. As stated in the last section, Robin boundary conditions at insulator-semiconductor interfaces have proven to be best suited since they deliver the best fit with the carrier concentrations obtained from a solution of the Schrödinger equation (cf. Section 2.6).

4.6.1  Calibration for the Drift Diffusion Model

In Figure 4.1 the results of a fit, with a density gradient quantum corrected drift diffusion model, for electrons are shown for various grid spacings. This figure clarifies the need for a finely spaced grid for simulation, when using a quantum correction model. In Figure 4.4 Capacitance-Voltage curves, obtained by density gradient and a Schrödinger-Poisson solver, for n- and p-channel 1D MOS structures for various dopings are shown and compared  [90]. Since density gradient does not exhibit a free, doping dependent parameter, a set of parameters only works for a certain bulk doping  [5]. This is the main disadvantage of density gradient. Another requirement of density gradient, as for any Schrödinger-Poisson solver, is that the grid needs to be in the sub-nanometer regime in order to fully refine the 2D electron gas in a MOS structure, as demonstrated in Figure 4.1.

4.6.2  Calibration for a SHE of the BTE

This section is devoted to the calibration of density gradient in ViennaSHE  [65]. To this end Robin-Boundary conditions and the simple scheme have been implemented and a comparison with VSP  [51] has been carried out in weak inversion. The parameters of the fit are given in Table 4.1 and the resulting calibration is shown in Figure 4.3.

Figure 4.3:

Results of the calibration of the DG model to a first-order SHE  [65] using a solution of the Schrödinger-Poisson equation obtained via VSP  [51]. Left: The fit for electrons has been obtained using a 1D NMOS structure, Robin boundary conditions at the silicon-silicon-dioxide-interface, an acceptor doping of 3 × 1017cm-3, an oxide thickness of 1nm and a uniform grid spacing (orthonormal grid) of 0.1nm. Right: The fit for holes has been obtained using exactly the same device as for electrons but instead of an acceptor doping a donor doping of 3 × 1017cm-3 has been used. The parameters are given in Table 4.1

Figure 4.4:

Capacitance-Voltage curves for various channel dopings in 1D MOS structures, obtained by a DG solution compared to a solution obtained using VSP  [51]. Well visible is the increasing discrepancy between DG and a real Schrödinger-Poisson solution for dopings above and below ND = 1017cm-3, for which the Robin boundary condition coefficients have been fitted to the Schrödinger-Poisson solution. For comparison the CV-curves obtained using plain drift diffusion are shown in black. The fitting parameters used with MinimosNT have been previously published in  [90].