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MEMS Fabrication 3-13

width W , the effective mask width at a depth z.The relation between W and W is given by the lateral
z m z
etch rate of a {100} facet and the time that facet was exposed to the etchant at depth z:

W W 2R ∆t z (3.3)
xy
z
m
where R the lateral underetch rate (i.e., etch rate in the xy plane)
xy
∆t the etch time at depth z
z
The underetching, U ,of the effective mask opening W is given by:
z z
U TR R ∆t (3.4)
z xy xy z

where T is the total etch time so far. The width of the etched pit W tot at depth z is further given by the
sum of W and twice the underetching for that depth:
m
W W 2U W 2TR (3.5)
tot z z m xy

Or, because T can also be written as the measured total etch depth z divided by the vertical etch rate R ,
z
Equation (3.5) can be rewritten as:

W W 2z(R /R ) or since R R z
m
tot
xy
z
xy
W W 2z (3.6)
tot
m
The width of the etched recess is therefore equal to the photolithographic mask width plus twice the etch
depth — independent of that etch depth, in other words. The walls remain vertical independent of the
depth z.
For sufficiently long etch times, {111} facets eventually take over from the vertical {100} facets. These
inward sloping {111} facets are first introduced at the corners of arectangular mask and grow larger at
the expense of the vertical sidewalls until the latter ultimately disappear altogether. Alignment of mask
features with the 100 directions so as to obtain vertical sidewalls in [100] wafers, therefore, is not very
useful for the fabrication of diaphragms. However, it can be very effective for anticipating the undercut-
ting of convex corners on [100] wafers. This useful aspect will be revisited when discussing corner
compensation.

3.3.4.3 [110]–Oriented Silicon

In Figure 3.11,we show a unit cell of Si properly aligned with the surface of a [110] Si wafer. This draw-
ing will enable us to predict the shape of an anisotropically etched recess on the basis of elementary geo-
metric crystallography. Four of the eight equivalent {111} planes are perpendicular to the (110) wafer
surface. The remaining four are slanted at 35.26° with respect to the surface. Whereas the intersections of
the four vertical {111} planes with the (100) wafer surface are mutually perpendicular, they enclose an
angle γ 90° in the (110) plane. Moreover, the intersections are not parallel or perpendicular to the main
wafer flat ([110] in this case) but rather enclose angles δ or δ γ with this direction. It follows that a
mask opening that will not be undercut (i.e., oriented such that resulting feature sidewalls are exclusively
made up by {111} planes) cannot be a rectangle aligned with the wafer flat but must be a parallelogram
skewed by γ 90° and δ degrees off-axis. The angles γ and δ are calculated by calculating β first [Peeters,
1994] (see Figures 3.11 and 3.12):

1 2
a
2 2 2
tan β (3.7)
a 2
2

© 2006 by Taylor & Francis Group, LLC