1063-7397/05/3403- © 2005 MAIK “Nauka /Interperiodica”
Russian Microelectronics, Vol. 34, No. 3, 2005, pp. 173–180. Translated from Mikroelektronika, Vol. 34, No. 3, 2005, pp. 210–217.
Original Russian Text Copyright © 2005 by Gridchin, Grichenko, Lubimsky.
Square and rectangular membranes are widely
employed as part of the sensing element of capacitive
and piezoresistive pressure sensors. Their design
entails ﬁnding the deﬂection (for capacitive sensors) or
stress (for piezoresistive ones).
Recently, the ﬁnite-element method (FEM) was
used to calculate membrane deﬂections. Predicted data
were compared with measured ones [1–3]. In addition,
a comparison was made between FEM calculations in
two and three dimensions . Membrane deﬂections
were also treated analytically, the results being com-
pared with FEM numerical predictions [5, 6]. In partic-
 have presented formulae and a
calculation procedure for the deﬂections, stresses, and
strains of rectangular membranes in the nonlinear case
of large deﬂections. It is important to note that the mod-
els concerned have found support in adequate agree-
ment between the results compared. However, the fact
that predicted deﬂections are close to measured ones
does not imply agreement in strains or stresses, because
these are related to derivatives of displacements with
respect to coordinates .
 compared measured resis-
tances of piezoresistors placed on a square mem-
brane with FEM predictions. Gridchin
reported a comparative study of nonlinear response
by measurement and calculation according to the
above procedure .
The present paper reports on an experiment
designed to identify the validity range of the procedure
devised by Gridchin
. To this end, edge stresses
and strains and vertical deﬂections are measured on a
square membrane, and the results are compared with
predictions made by the procedure.
Membrane stresses were evaluated by measuring a
pressure-induced fractional change in the resistivity of
piezoresistors placed on the membrane (alternatively,
one may use, e.g., optical techniques to directly mea-
sure the strains).
We fabricated silicon membranes with longitudinal
and transverse polysilicon thin-ﬁlm piezoresistors
arranged in an open bridge circuit, as shown in Fig. 1.
The polysilicon ﬁlms were deposited on the planar
side of oxidized Si(100) wafers of diameter 76 mm,
with the deposition performed in a reduced-pressure
. The thicknesses of oxide and poly-
silicon ﬁlms were about 0.4 and 0.5
The ﬁlms were doped by boron implantation with sub-
Each wafer was divided into two parts. One of them
contained sensing elements formed on the nonplanar
side by anisotropic etching with 30% KOH; they were
used to examine the distribution of stress over the mem-
The chips under study were 5
5 mm in size. Mem-
brane dimensions are given in Table 1. The piezoresis-
tors were 200
m long and 20
m wide. Their centers
were positioned at predetermined distances from the
membrane edges (Fig. 1, Table 1). Preparatory to mea-
surements, each chip was bonded to a glass plate by
electrostatic diffused welding.
Measurements were made under direct-current con-
ditions, using a V2-34 voltmeter.
The other part of a wafer was cut into cantilever
beams of size
mm, with the longest edges
oriented as . They were used to determine the lon-
gitudinal and transverse piezoresistance coefﬁcients,
, relative to the membrane axis of sym-
Square-Membrane Deflection and Stress: Identifying
the Validity Range of a Calculation Procedure
V. A. Gridchin, V. V. Grichenko, and V. M. Lubimsky
Novosibirsk State Technical University, Novosibirsk, Russia
Received July 22, 2004
—The stress, vertical deﬂection, and nonlinear piezoresistive response of square membranes are mea-
sured with the aim of identifying the validity range of a calculation procedure devised in our previous study.
Maximum values of the pressure parameter are determined up to which close agreement exists between mea-
sured and calculated values.