**2.1 Experimental details**

The hemispherical specimen enables us to obtain the etch rate for all range of crystallographic orientations under the same etching conditions simultaneously, because all the orientations are placed on its surface. The P-type single-crystal hemispherical specimen with a diameter of 44 mm (resistivity: 6–12 Ω-cm) is used in the evaluation. The ingot of hemispheres is provided by Sumitomo Sitix Corporation. In order to produce hemispheres, it is mechanically ground, lapped and polished into mirrored surfaces with a sphericity of less than 10 µm, latitude from 0 to 90° and surface roughness of 0.005–0.007 µm in the arithmetical average by Okamoto Kogakukosakusho Corporation. The etch rate at each orientation is calculated by measuring the shape change before and after etching. The optimized etching depth should be in the range of 100–150 µm in order to avoid interference between neighboring orientations while maintaining the resolution of the geometry measurement. The shape is measured using a 3D profile machine UPMC550-CARAT (Carl Zeiss Co.) with an accuracy of less than 1.0 µm. Fig. 3 shows the locations of crystallographic orientations on the silicon hemispherical sample and a schematic view of the surface profile measurement. Area 'A' corresponds to the measurable area of the hemispherical silicon sample. The place outside the measurement zone and the bottom area are protected by a thermally grown oxide layer. The surface profile is probed every 2° of latitude ranging from 0° to 70°, and every 2° of longitude ranging from 0° to 360°. Supplements of de-ionized (DI) water into an etching bath every 2 h control the tolerance of etching temperature within 1 °C. The total numbers of probe points to be measured are 6480.

TMAH (Toyo Gosei Co. Ltd) and Triton X-100 (Amersham Biosciences) are used as the main etchant and surfactant, respectively. The Triton solution is used to prepare the surfactantadded TMAH solution. The fresh etchant is employed in every subsequent experiment.
