A.4 Aspect Ratio

Figure A.2: Differently shaped trenches with their characteristic dimensions which are used to determine the aspect ration (AR).

The aspect ratio (AR) of a trench is defined as the ratio between the height $H$ and the width or length $L$ of a trench and is used as a characteristic quantity especially for chemical processes like material etching and depositing. For chemical material deposition, the aspect ratio calculation for rectangular trenches has been introduced in [332] (cf. Figure A.2a) using

$$\textsf{\sc ar}{=}\frac{H}{L}$$ (A.12)

However, fabrication processes in general do not provide rectangular-shaped trenches because etching processes always produce tilted side walls and in the entry section of the trench at the top there is often an additional rounding according to surface diffusion during material deposition. Nevertheless, the impact of the tilted side walls on the width and the height of the particular trench can often be neglected with respect to the overall dimensions of the trench. Then, a rectangular shape can be justified. However, if the trench can not sufficiently be approximated by a rectangle, a different metric has to be applied to provide a characteristic criterion to compare different trenches using a different calculation of AR

$$\textsf{\sc ar}{=}\frac{H_{\mathrm{m}}}{L_{\mathrm{m}}},$$ (A.13)

where the height and width are approximated by the mean value $H_{\mathrm{m}}$ and $L_{\mathrm{m}}$ , respectively. For instance, the height and the width can be approximated by their mean values as

$$H_{\mathrm{m}} {=}{{\frac{1}{2}}}\left(H_1 + H_2\right),$$ (A.14)

$$L_{\mathrm{m}} {=}{{\frac{1}{2}}}\left({{\frac{1}{2}}}\left(L_{\m... ...\right) + {{\frac{1}{2}}}\left(L_{\mathrm{H1}} + L_{\mathrm{H2}}\right)\right).$$ (A.15)