4.3.1 Local Oxide Capacitance

All considered methods employ the oxide capacitance C_ox as a key parameter of the extraction procedure. In the methods derived by Mahapatra [66] and by Li [179] C_ox is treated as a constant

This approach is applicable only for transistors with a fixed oxide thickness and leads to erroneous results when t_ox depends on interface coordinate x. Moreover, even in the case of uniform thickness distribution, such a scheme leads to spurious results. In fact, in such an approach the capacitor is considered ideal i.e. the electric field is assumed uniform. In work of Chim et al. [177] it was possible to find an extension of (4.14) when t_ox depends on x (t_ox = t_ox(x)). In practice, however, a substantial distortion occurs near the source/drain ends of the gate. The aforementioned simplification does not strongly affect the transistor characteristics. However, the electric field non-uniformity is of particular importance for the extraction of the interface state density profile after HC stress because N_it(x) peaks near the drain end of the gate where the capacitor non-ideality is most pronounced [41]. Additionally, Lee et al. [178] have already demonstrated that the coordinate dependence of the capacitance C_ox(x) due to the fringing effect is essential.

4.3.1.1 Conformal-Mapping Method

**Figure 4.11:** The conformal transformation used to solve the gate/drain fringing problem. The complicated case of the corner gate form (red dashed line) is reduced to a ray.

The conformal-mapping method is most helpful for the calculation of the fringing electric field in simple two-dimensional boundary problems (which is our case) by transforming the boundary to a soluble form [184]. A series of simulations based on the Lee et al. approach [178] (described below) allow us to draw the conclusion that, for a local oxide capacitance consideration, a simplified structure can be used. Namely, the gate contact can be interpreted geometrically as a ray instead of more complicated corner variants. The problem with the coordinate system is shown in Figure 4.11. Here, the origin of the considered system is placed at the drain end of the gate. The z = x+iy plane is mapped into the ω = φ+iψ plane with the functional relationship between z and ω described by

The suggested conformal transformation shown in Figure 4.11 reduces the pristine problem to the Laplace problem between two parallel infinitely long metallic plates at different potentials [184]. The analytical solution for the potential and the electric field distribution in the oxide near the gate end following from the conformal map and that obtained by MiniMOS-NT are presented in Figure 4.12. One can see a rather good agreement between the obtained results which means that the chosen geometrical system interpretation is correct.

**Figure 4.12:** The comparison of equipotential and field-lines in the oxide near the gate corner as the analytical solution and simulated by means of device simulator MiniMOS-NT.

The local oxide capacitance can be defined as the ratio between the surface charge density σ_surface and the interface potential U₀, written as

That is, for the determination of the local oxide capacitance it is first of all necessary to find the surface charge density σ_surface. According to [185]

It should be mentioned that in (4.23) the parameter t_ox has been changed to t_ox(x). As reported in [185], this substitution is legitimate until dt_ox(x)/dx which is correct for devices with a real topology of the gate contact. The value of C_ox(x) in (4.23) has a maximum with C_ox(= ε_ox/t_ox) in the middle of the gate and decreases gradually toward a much lower value outside the gate edge due to the fringing effect. At x→-∞ (or φ→-∞) the obtained equation (4.23) asymptotically turns into an expression for the parallel-plate capacitance (4.14).

4.3.1.2 Estimation of Approach Importance Area

It is rather important to clarify at which distance from the drain edge of the gate contact the electric field becomes uniform. In other words, where the fringing effect is negligible and the parallel-plate capacitor approximation is applicable. For this purpose let us find a position on the x-axis where the electric field intensity differs by 1% from the uniform one, i.e F₀ = U₀/t_ox [184]. For any position of consideration in the Figure 4.11 system, the electric field intensity can be defined using the employed conformal transformation as F = |dω/dz|. Therefore, from (4.21) one may conclude

Assuming F/F₀ = 0.99 we obtain exp(φ) = 0.0101 and, consequently, φ = -4.61. A substitution of the obtained values in the expression for the x coordinate in (4.16) results in

Here, x = L_g is the position of the drain end of the gate. Thus, one cane consider the local oxide capacitance as constant (i.e. use the parallel-plate capacitor approximation) already at a distance of 1.47 of the oxide thickness at the drain end of the gate contact. As a result, it is possible to operate with a simple expression (4.14) if the region of interest is outside of the mentioned area.

4.3.1.3 Lee's Approach for Modeling of Local Oxide Capacitance

For evaluating the local oxide capacitance using device modeling we employ the method suggested by Lee et al. [178]. To determine the C_ox(x) by means of simulations, a careful calculation of the local flatband V_fb(x) and threshold V_th(x) voltage distributions is performed. Due to the symmetry of the source and drain for the fresh device, one presents results only for the drain half of the device [178]. As mentioned in Section 4.1, for an unstressed transistor the V_th(x) and V_fb(x) profiles can be obtained by the device simulator MiniMOS-NT. For any point on the device interface the post-stress local threshold V_th(x) and local flatband V_fb(x) values are related to the pre-stressed ones by (4.13) which considers the change between pre- and post-stress concentrations of interface traps and bulk oxide charges. The presence of a uniform probe oxide charge N_ot,uniform leads to a local threshold voltage shift ΔV_th,uniform(x) = V_{th,s(uniform)}(x) - V_th(x) [178]. Therefore, the local oxide capacitance can be found using

A typical example of ΔV_th,uniform(x)) induced by a uniform oxide charge density of 5×10¹¹cm^-2 is shown in Figure 4.13a.

**Figure 4.13:** (a) Local threshold and flatband voltage distributions with uniform oxide charge profiles. Inset: the dependence of the oxide thickness vs. the lateral coordinate. (b) The local oxide capacitance calculated using the approach of Lee *et al.*, compared with the newly developed analytical model. Here L_g is the position of the drain end of the gate.
(a) (b)