- Email: [email protected]
Calibration procedure for piezoresistance coefficients of polysilicon sheets and application to a stress test chip
ACIWORS A ELSEVIER
Sensors and ActuatorsA 62 (1997) 475--479
Calibration procedure for piezoresistance coefficients of polysilicon sheets and application to a stress test chip A. Bossche *, J.R. Mollinger Electronic Instrumentation Laboratory Faculty of Electrical Engineering~DIMES De!ft University of Technology Mekelweg 4, 2628 CD, Delft, Netherlands
Abstract This paper describes a general calibration procedure for polysilicon piezoresistors for two cases; the case when the crystaUites have sufficiently random orientations to assume isotropic behaviour of the polysilicon sheet and the case in which the crystalIites show a preferential orientation which requires a anisotropic treatment of the polysilicon layer. Finally, the theory is applied to calibrate a polysilicon resistor array used for package stress measurements. For this specific case, where not all stress components are of significance, it is shown that the procedures can be significantly simplified. © 1997 Elsevier Science S.A.
Keywnrds: Calibration;Piezoresistancecoefficients;Polysiliconsheets; Stress test chips
1. Introduction The procedure for the calibration of polysilicon piezoresistors depends on the structure and the (planar) orientation of the polysilicon sheet. In general it can be assumed that the crystallite sizes resulting from different polysilieon technologies (0.4-2 Ixm) are sufficiently small compared to the resistor sizes (approx. 100 b~m X 100 Izm) to neglect the influence of individual crystallite orientation. This means that the polysilicon layer can be assumed uniform for all resistors, However, the crystallites (and thus the polysilicon sheet) may still show a preferential orientation. Therefore, two different (macro) polysilicon structures can be distinguished: one having a net orientation of the crystallites (non-isotropic structure), and one having no net orientation (isotropic structure). These cases require different approaches for interpreting the resistance changes for both calibration and stress measurements. The following sections describe general calibration procedures for both isotropic and non-isotropic polysilicon sheets as well as specific procedures for a package stress test chip.
E~ = ~uii+ ~:k,ok~ij+ c~UA Tij Po For monocrystalline silicon the crystal symmetry and the mechanical equilibrium conditions require that the electrical and mechanical indices (in any orthogonal base) are symmetrical as well. This means that 7Tijkl = qrjikl = qTui k = ~ttk.For this reason usually a simpler notation is used in the literature on the basis of only two indices running from one through six according to the following scheme: [ ij (k/): 11, 22, 33, 23, 13, 12] ~ [a(b) : I...6]. This means that the tensor equation can be rewritten in the new indices as
F-'i= 8aij-k qTal,o'bij-b oLSaA Tij Po For monocrystalline silicon (or more general crystals having a diamond-like lattice structure) it was proven [ 1 ] that piezoresistivity (and any other phenomena that can be
7.
2. Calibration of anisotropic polysilicon piezoresistors In general piezoresistivity can be described by a fourthrank tensor equation as: * Correspondingauthor. Tel.: + 31 15 278 6049. Fax: + 31 15 278 5755. E-mail: [email protected]
Fig. 1. Waferand arbitrary resistor orientation.
A. Bossche J.R. Mollinger / Sensors and Actuatmw A 62 (1997) 475-479
476
Fig. 2. Stress test chip layout.
described by a fourth-rank tensor) can be described in relation to the crystallographic axes by only three independent coefficients as:
~12
0
0
0
7~12
0
0
0
g12
%11
0
0
0
0
0
0
'1~44 0
0
0
0
0
0
%44
0
0
0
0
0
0
're.
ll
f
~12
Fig. 3. Four-point bending bridge.
12 ~11
12
Bending Mode
For any other orthogonal orientation the ~' coefficients can be described in linear combinations of 7r~ t, ~'~z, and 7r44. However, in polysilicon the crystallites show many different orientations even when they show a preferential orientation. Although for each individual crystallite the same symmetry rules hold as for monocrystalline silicon, the macro piezo-
resistivity of the polysilicon cannot be expressed in terms of the principal coefficients. This means that all 36 coefficients should be regarded as independent. However, for planar resistors in the xy-plane (wafer plane) the E. and i: contributions can be ignored, leaving only 18 coefficients. Since for each arbitrarily oriented individual crystallite (and so for the resistor as a whole) it holds that "/rl2~.~-Tral, 77"16=27/'61 and ~26 =2~62, a further reduction to 15 independent equations is possible. This means that the full calibration of an anisotropic polysilicon layer requires (at least) 15 independent linear equations and so 15 resistors. The relative resistance change of each resistor at an arbitrary angle ~0with respect to the x-axis (see Fig. 1) can then be written as
477
A. Bossche, J.R. Mollinger / Sensors and Actuatom A 62 (t997) 475-479
A R / R = 0-oAE/i= (A e,cos&+ A E~.sin~/,)/i : COS~[ ( '/TI 10. I -[" 7/'120" 2 -[- "77"130"3"}- "7/'140-4 "+ 7T]50"5 "It- 7/'160"6) COSl/Y+ ( 1 /2rr160. ,
+ 1/ 2 7r2~,0"2+ 7r630.~ + rr640.4 + rr~50-~+ 7r660"6)sin6] -{- [ ( ""~"120.I + 7/'220-2 + 51"230-3 + 77"9-40"4+ "]T250-5 + '7'r26O'6) sin ~b+ ( 1/2Irl60" 1 + 1/277"9_60" 2 Jr- 7T630 3 -If-"TJ'640-4 "1- %50-5 -1- "7T660"6)COSILY]
or in matrix form: A R / R = [0-vector × Transrrpoly] X [ vrvector] with 0.vector= [cr~ 0-2 0-3 0-4 0-s 0.6], 7rvector = [rq~ .z',~ rq3 %4 rr~s rrt6 %2 rr23 %4 was rr26 %3 7r64 %5 7r66]T and with Transrrpoly as
cos~-qs sin2~p
0
0
0
cosqlsin*
0
0
0
0
0
0
0
0
0
0
cos2ql
0
0
0
0
sin:tls
0
0
0
cos~sinq~
0
0
0
0
0
0
cos-~
0
0
0
0
sin2qJ
0
0
0
2cos~sin~l
0
0
0
0
0
0
cosZqs
0
0
0
0
sin:qJ
0
0
0
2cos~sint~
0
0
0
0
0
0
cos:qJ
0
0
0
0
sin~qJ
0
0
0
2costltsin~
0
0
0
0
0
0
cos2q~
0
0
0
0
sin"¢
0
0
0
2cos~sin,
3, Calibration of isotropic polysilicon piezoresistors For planar resistors in an isotropic polysilicon layer with all coefficients and stresses given with respect to the longitudinal (1), transverse (t) and perpendicular (p) directions of the resistor, the piezoresistance relation can be written as
A R / R = 0-oAE/i = ~rl0.1+ 7rto-t + %0" 0 + TrtpO'tp-~ 'TrlpOelp-[- qTltO'lt independent of the direction (~/s) of the resistor. In order to express the relation with respect to the x-, y- and z-axes of the wafer, the stresses 05...0-, should be expressed in terms of crt...o'~ so that the piezoresistance equation becomes AR/ R = [0.vector ×Trans'rfisotrop] × [7rvector] with 0.vector= [cr~ o'2 0-3 0-4 0-5 0"6J and vrvector= [% vrt % 7rtp rr~p 7r~ iv and with Transrrisotrop given by
cosZqJ
sin2qS
0
0
0
-costpsin~
sin~
cos:~
0
0
0
cos~,~sinq*
0
0
1
0
0
0
0
o
0
cos,
si~
o
0
0
0
-simp
cos~
0
0
0
0
2 cost~sinq* -2 cost~sin~
cos~tg-sin2q~
4. Application to a stress test chip The drive for this research was the development of a test chip for package stress measurements, see Fig. 2. The lower right corner of this chip carries an array of 256 polysilicon piezoresistors in six different orientations. The calibration procedure requires a well-defined stress pattern to be exerted on the chip. The best and most common way to do this is the so-called fourpoint bending bridge, see Fig. 3. For this purpose the largest possible square is cut out of the 4 inch wafer and stressed on the bridge. Since this bridge cannot exert the vertical stress component (0.3), the sensitivity for this stress cannot be calibrated. For piezoresistors in monocrystalline Si this is no problem, since there a simple relationship exists between the principal coefficients and the sensitivities to the distinct stress components. For polysilicon resistors such a relationship does not exist. Fortunately, however, finite-element (FE) simulations on encapsulated chips show that 0-3 is usually much smaller than the other components and can be neglected. Even the peaks in 0-3 at the die corners are likely to be considerably smaller than simulated because of
478
A. Bossche. J.R. Mollinger / Sensolw and Actuatom A 62 (1997) 4 75-47g
epoxy relaxation. Therefore, it is possible in package stress measurements [2] to neglect the vertical stress. For this specific application it is easy to see that the piezoresistance relations above reduce to: A n i s o t r o p i c case A R / R = [ o-vector X Trans,n'poly] × [ 7rvector] with
0-vector= [o'j 0-2 0-4 °'5 °-6] qrvector= [Tr~ "n'~, 7r~4 7r~.~~'J6 rr22 7r24 "rr_~57r,(, %4 %.~ 7r~6]~ and
cos2~ sin2,
0
0
costllsintlJ
0
0
0
0
0
0
0
0
cos=*
0
0
0
sin=k0
0
0
cos,sin*
0
0
0
0
0
cos~qJ
0
0
0
sin:~
0
0
2costltsintlJ
0
0
0
0
0
cos~
0
0
0
sin~
0
0
0
0
0
0
cos"q:
0
0
0
sinaqt
0
Trans~:poly=
2costllsinqJ 0
0 2cosqJsinq~
l s o t r o p i c case A R / R = [ 0-vector × Trans'rrpoly] × [ 7rvector] with
o-vector=
[0-1 (21'2 O"4 O"5 0"6] qrvector= [7r~ 7rt ~'tp 7rip % ] T
and
Transxisotrop=
cos2~
sin=,
0
0
~os,sin~
sinaqj
cos2,
0
0
cos,sin~
0
0
costl/
sint~
0
0
0
-sinqJ
cosq~
0
2 cos~sint~
-2 cos~sin~
0
0
cos2~-sin2~
The calibration procedure proceeds as follows. The wafer is positioned on the bending bridge and all (4 X 256) piezoresistors are scanned and measured in the unstressed situation and in a well-defined stress situation. For each quadrant the readings are split into six groups (con'esponding to the six resistor orientations) and for each group a best-fit interpolation is made, resulting in resistance values at all 256 raster points (resistor cell centres). The simulated stress values are also interpolated at these points. This means that at each cell centre we have six equations. For the isotropic case only live coefllcients are to be solved and a solution can be found at each cell. For the anisotropic case 12 coefficients have to be solved, which means that at least two cells (2 X 6 equations) should be involved. It will be clear that the cells should be chosen with sufficient spacing in order to ensure different stress patterns. If not, the system of equations may be insolvable. Another (practical) problem that may arise is that, clue to inevitable variations in material properties along the wafer, it may be difficult to solve 12 coefficients with sufficient accuracy. If so, it may be more suitable to solve piezocoefficients for each resistor orientation apart using the readings for five identical resistors at different locations. It should be noted that this latter approach characterizes piezoresistivity for specific resistor orientations only and n o t for the polysilicon sheet as a whole. A further simplification is possible when the piezoresistors are placed at sufficient ( > 200 Ixm) distance from the die edges. Simulations show that, with the exception of the outer die edges, the out-of-plane shear-stress components (o- 4 and o-s) are much smaller than the in-plane components. Even the peaks at the die comers are likely to be considerably smaller than simulated
479
A. Bossche, J.R. Mollinger/ Sensors and Actuators A 62 (1997) 475-479
because of epoxy relaxation. Therefore, when the above restriction holds, it is possible in package stress measurements [2] to neglect the out-of-plane stresses as well and the procedures above simplify to:
A n i s o t r o N c case
IsotroDic case
o v e c t o r = [0~ o 2 06]
o v e c t o r = [0~ 0 2 o6]
xvector=
x v e c t o r = [x I n t glt] T
[Tc~t x~2 "hi6 x22 rc26 r~66]T
and
and
Transxisotmp:
Transxpoly:
cos:0 0
sin"* c°s2tlJ
cos0sin* 0
0 sin~
0 costllsintlJ
i
0
0
cos~ql
0
sin"qJ
2 costpsint[I
5. Conclusions It has been shown that the piezoresistive behaviour of polysilicon layers can be described by either 15 (anisotropic polysilicon) or six (isotropic polysilicon) independent coefficients. This means that in the calibration procedures at least 15 (respectively six) independent resistor measurements are required. However, when only the in-plane stress components are of significance only six (anisotropic polysilicon) or three (isotropic polysilicon) independent coefficients have to be resolved. The vectors and matrices used to resolve the piezoresistance coefficients from the equations AR/R= [ crvector × Trans 7rxxxx] X [ ~'vector] can be used as well for resolving the stress components in the actual stress measurements by rewriting the relations as AR/R= [ ~-vectorv X TransTrxxxT] X [ o-vectorT],
Acknowledgements This project is funded by the Dutch Foundation of Applied Research (STW).
References
[ I ] J.F. Nye, Physical Properties 1985.
of Co'stals,
Clarendon Press, Oxford,
I
I cofltlJ sin~
sinhl/ coslq/
-cos~sin~ cos~sint[/
2 eos~sintll -2 costltsint) cos2~-sinhlt
[2] H.C.J.M.van Gestel,ReliabilityrelatedxesearehonplasticIC-packages: a test chip approach, Ph.D. Dissertation, Delft University of Technology,The Netherlands,1994,ISBN90-6275-960-2.
Biographies Andre Bossche was born in Rotterdam, The Netherlands, on August 27, 1956. He received the M.Sc degree in electrical engineering with honours in 1983, and the Ph.D. degree in electrical engineering in 1988, both from the Delft University of Technology, The Netherlands. He is currently engaged in research work on the subjects of reliability and packaging of sensors and microsystems at the Electronic Instrumentation Laboratory of the Delft University of Technology. JeffMollinger was born in Jakarta, Indonesia, on September 9, 1951. After finishing his studies in electronics in 1975, he joined the TNO Department of Groundwater Survey. In 1980 he joined the Volker Stevin Dredging Company, and in 1981 Delft University of Technology, The Netherlands, at the Department of Mechanical Engineering. Currently he is working as a senior technician at the Electronic Instrumentation Laboratory of Delft University of Technology, where he supports scientific research on the subject of reliability and electronic measurement systems.
Sensors and ActuatorsA 62 (1997) 475--479
Calibration procedure for piezoresistance coefficients of polysilicon sheets and application to a stress test chip A. Bossche *, J.R. Mollinger Electronic Instrumentation Laboratory Faculty of Electrical Engineering~DIMES De!ft University of Technology Mekelweg 4, 2628 CD, Delft, Netherlands
Abstract This paper describes a general calibration procedure for polysilicon piezoresistors for two cases; the case when the crystaUites have sufficiently random orientations to assume isotropic behaviour of the polysilicon sheet and the case in which the crystalIites show a preferential orientation which requires a anisotropic treatment of the polysilicon layer. Finally, the theory is applied to calibrate a polysilicon resistor array used for package stress measurements. For this specific case, where not all stress components are of significance, it is shown that the procedures can be significantly simplified. © 1997 Elsevier Science S.A.
Keywnrds: Calibration;Piezoresistancecoefficients;Polysiliconsheets; Stress test chips
1. Introduction The procedure for the calibration of polysilicon piezoresistors depends on the structure and the (planar) orientation of the polysilicon sheet. In general it can be assumed that the crystallite sizes resulting from different polysilieon technologies (0.4-2 Ixm) are sufficiently small compared to the resistor sizes (approx. 100 b~m X 100 Izm) to neglect the influence of individual crystallite orientation. This means that the polysilicon layer can be assumed uniform for all resistors, However, the crystallites (and thus the polysilicon sheet) may still show a preferential orientation. Therefore, two different (macro) polysilicon structures can be distinguished: one having a net orientation of the crystallites (non-isotropic structure), and one having no net orientation (isotropic structure). These cases require different approaches for interpreting the resistance changes for both calibration and stress measurements. The following sections describe general calibration procedures for both isotropic and non-isotropic polysilicon sheets as well as specific procedures for a package stress test chip.
E~ = ~uii+ ~:k,ok~ij+ c~UA Tij Po For monocrystalline silicon the crystal symmetry and the mechanical equilibrium conditions require that the electrical and mechanical indices (in any orthogonal base) are symmetrical as well. This means that 7Tijkl = qrjikl = qTui k = ~ttk.For this reason usually a simpler notation is used in the literature on the basis of only two indices running from one through six according to the following scheme: [ ij (k/): 11, 22, 33, 23, 13, 12] ~ [a(b) : I...6]. This means that the tensor equation can be rewritten in the new indices as
F-'i= 8aij-k qTal,o'bij-b oLSaA Tij Po For monocrystalline silicon (or more general crystals having a diamond-like lattice structure) it was proven [ 1 ] that piezoresistivity (and any other phenomena that can be
7.
2. Calibration of anisotropic polysilicon piezoresistors In general piezoresistivity can be described by a fourthrank tensor equation as: * Correspondingauthor. Tel.: + 31 15 278 6049. Fax: + 31 15 278 5755. E-mail: [email protected]
Fig. 1. Waferand arbitrary resistor orientation.
A. Bossche J.R. Mollinger / Sensors and Actuatmw A 62 (1997) 475-479
476
Fig. 2. Stress test chip layout.
described by a fourth-rank tensor) can be described in relation to the crystallographic axes by only three independent coefficients as:
~12
0
0
0
7~12
0
0
0
g12
%11
0
0
0
0
0
0
'1~44 0
0
0
0
0
0
%44
0
0
0
0
0
0
're.
ll
f
~12
Fig. 3. Four-point bending bridge.
12 ~11
12
Bending Mode
For any other orthogonal orientation the ~' coefficients can be described in linear combinations of 7r~ t, ~'~z, and 7r44. However, in polysilicon the crystallites show many different orientations even when they show a preferential orientation. Although for each individual crystallite the same symmetry rules hold as for monocrystalline silicon, the macro piezo-
resistivity of the polysilicon cannot be expressed in terms of the principal coefficients. This means that all 36 coefficients should be regarded as independent. However, for planar resistors in the xy-plane (wafer plane) the E. and i: contributions can be ignored, leaving only 18 coefficients. Since for each arbitrarily oriented individual crystallite (and so for the resistor as a whole) it holds that "/rl2~.~-Tral, 77"16=27/'61 and ~26 =2~62, a further reduction to 15 independent equations is possible. This means that the full calibration of an anisotropic polysilicon layer requires (at least) 15 independent linear equations and so 15 resistors. The relative resistance change of each resistor at an arbitrary angle ~0with respect to the x-axis (see Fig. 1) can then be written as
477
A. Bossche, J.R. Mollinger / Sensors and Actuatom A 62 (t997) 475-479
A R / R = 0-oAE/i= (A e,cos&+ A E~.sin~/,)/i : COS~[ ( '/TI 10. I -[" 7/'120" 2 -[- "77"130"3"}- "7/'140-4 "+ 7T]50"5 "It- 7/'160"6) COSl/Y+ ( 1 /2rr160. ,
+ 1/ 2 7r2~,0"2+ 7r630.~ + rr640.4 + rr~50-~+ 7r660"6)sin6] -{- [ ( ""~"120.I + 7/'220-2 + 51"230-3 + 77"9-40"4+ "]T250-5 + '7'r26O'6) sin ~b+ ( 1/2Irl60" 1 + 1/277"9_60" 2 Jr- 7T630 3 -If-"TJ'640-4 "1- %50-5 -1- "7T660"6)COSILY]
or in matrix form: A R / R = [0-vector × Transrrpoly] X [ vrvector] with 0.vector= [cr~ 0-2 0-3 0-4 0-s 0.6], 7rvector = [rq~ .z',~ rq3 %4 rr~s rrt6 %2 rr23 %4 was rr26 %3 7r64 %5 7r66]T and with Transrrpoly as
cos~-qs sin2~p
0
0
0
cosqlsin*
0
0
0
0
0
0
0
0
0
0
cos2ql
0
0
0
0
sin:tls
0
0
0
cos~sinq~
0
0
0
0
0
0
cos-~
0
0
0
0
sin2qJ
0
0
0
2cos~sin~l
0
0
0
0
0
0
cosZqs
0
0
0
0
sin:qJ
0
0
0
2cos~sint~
0
0
0
0
0
0
cos:qJ
0
0
0
0
sin~qJ
0
0
0
2costltsin~
0
0
0
0
0
0
cos2q~
0
0
0
0
sin"¢
0
0
0
2cos~sin,
3, Calibration of isotropic polysilicon piezoresistors For planar resistors in an isotropic polysilicon layer with all coefficients and stresses given with respect to the longitudinal (1), transverse (t) and perpendicular (p) directions of the resistor, the piezoresistance relation can be written as
A R / R = 0-oAE/i = ~rl0.1+ 7rto-t + %0" 0 + TrtpO'tp-~ 'TrlpOelp-[- qTltO'lt independent of the direction (~/s) of the resistor. In order to express the relation with respect to the x-, y- and z-axes of the wafer, the stresses 05...0-, should be expressed in terms of crt...o'~ so that the piezoresistance equation becomes AR/ R = [0.vector ×Trans'rfisotrop] × [7rvector] with 0.vector= [cr~ o'2 0-3 0-4 0-5 0"6J and vrvector= [% vrt % 7rtp rr~p 7r~ iv and with Transrrisotrop given by
cosZqJ
sin2qS
0
0
0
-costpsin~
sin~
cos:~
0
0
0
cos~,~sinq*
0
0
1
0
0
0
0
o
0
cos,
si~
o
0
0
0
-simp
cos~
0
0
0
0
2 cost~sinq* -2 cost~sin~
cos~tg-sin2q~
4. Application to a stress test chip The drive for this research was the development of a test chip for package stress measurements, see Fig. 2. The lower right corner of this chip carries an array of 256 polysilicon piezoresistors in six different orientations. The calibration procedure requires a well-defined stress pattern to be exerted on the chip. The best and most common way to do this is the so-called fourpoint bending bridge, see Fig. 3. For this purpose the largest possible square is cut out of the 4 inch wafer and stressed on the bridge. Since this bridge cannot exert the vertical stress component (0.3), the sensitivity for this stress cannot be calibrated. For piezoresistors in monocrystalline Si this is no problem, since there a simple relationship exists between the principal coefficients and the sensitivities to the distinct stress components. For polysilicon resistors such a relationship does not exist. Fortunately, however, finite-element (FE) simulations on encapsulated chips show that 0-3 is usually much smaller than the other components and can be neglected. Even the peaks in 0-3 at the die corners are likely to be considerably smaller than simulated because of
478
A. Bossche. J.R. Mollinger / Sensolw and Actuatom A 62 (1997) 4 75-47g
epoxy relaxation. Therefore, it is possible in package stress measurements [2] to neglect the vertical stress. For this specific application it is easy to see that the piezoresistance relations above reduce to: A n i s o t r o p i c case A R / R = [ o-vector X Trans,n'poly] × [ 7rvector] with
0-vector= [o'j 0-2 0-4 °'5 °-6] qrvector= [Tr~ "n'~, 7r~4 7r~.~~'J6 rr22 7r24 "rr_~57r,(, %4 %.~ 7r~6]~ and
cos2~ sin2,
0
0
costllsintlJ
0
0
0
0
0
0
0
0
cos=*
0
0
0
sin=k0
0
0
cos,sin*
0
0
0
0
0
cos~qJ
0
0
0
sin:~
0
0
2costltsintlJ
0
0
0
0
0
cos~
0
0
0
sin~
0
0
0
0
0
0
cos"q:
0
0
0
sinaqt
0
Trans~:poly=
2costllsinqJ 0
0 2cosqJsinq~
l s o t r o p i c case A R / R = [ 0-vector × Trans'rrpoly] × [ 7rvector] with
o-vector=
[0-1 (21'2 O"4 O"5 0"6] qrvector= [7r~ 7rt ~'tp 7rip % ] T
and
Transxisotrop=
cos2~
sin=,
0
0
~os,sin~
sinaqj
cos2,
0
0
cos,sin~
0
0
costl/
sint~
0
0
0
-sinqJ
cosq~
0
2 cos~sint~
-2 cos~sin~
0
0
cos2~-sin2~
The calibration procedure proceeds as follows. The wafer is positioned on the bending bridge and all (4 X 256) piezoresistors are scanned and measured in the unstressed situation and in a well-defined stress situation. For each quadrant the readings are split into six groups (con'esponding to the six resistor orientations) and for each group a best-fit interpolation is made, resulting in resistance values at all 256 raster points (resistor cell centres). The simulated stress values are also interpolated at these points. This means that at each cell centre we have six equations. For the isotropic case only live coefllcients are to be solved and a solution can be found at each cell. For the anisotropic case 12 coefficients have to be solved, which means that at least two cells (2 X 6 equations) should be involved. It will be clear that the cells should be chosen with sufficient spacing in order to ensure different stress patterns. If not, the system of equations may be insolvable. Another (practical) problem that may arise is that, clue to inevitable variations in material properties along the wafer, it may be difficult to solve 12 coefficients with sufficient accuracy. If so, it may be more suitable to solve piezocoefficients for each resistor orientation apart using the readings for five identical resistors at different locations. It should be noted that this latter approach characterizes piezoresistivity for specific resistor orientations only and n o t for the polysilicon sheet as a whole. A further simplification is possible when the piezoresistors are placed at sufficient ( > 200 Ixm) distance from the die edges. Simulations show that, with the exception of the outer die edges, the out-of-plane shear-stress components (o- 4 and o-s) are much smaller than the in-plane components. Even the peaks at the die comers are likely to be considerably smaller than simulated
479
A. Bossche, J.R. Mollinger/ Sensors and Actuators A 62 (1997) 475-479
because of epoxy relaxation. Therefore, when the above restriction holds, it is possible in package stress measurements [2] to neglect the out-of-plane stresses as well and the procedures above simplify to:
A n i s o t r o N c case
IsotroDic case
o v e c t o r = [0~ o 2 06]
o v e c t o r = [0~ 0 2 o6]
xvector=
x v e c t o r = [x I n t glt] T
[Tc~t x~2 "hi6 x22 rc26 r~66]T
and
and
Transxisotmp:
Transxpoly:
cos:0 0
sin"* c°s2tlJ
cos0sin* 0
0 sin~
0 costllsintlJ
i
0
0
cos~ql
0
sin"qJ
2 costpsint[I
5. Conclusions It has been shown that the piezoresistive behaviour of polysilicon layers can be described by either 15 (anisotropic polysilicon) or six (isotropic polysilicon) independent coefficients. This means that in the calibration procedures at least 15 (respectively six) independent resistor measurements are required. However, when only the in-plane stress components are of significance only six (anisotropic polysilicon) or three (isotropic polysilicon) independent coefficients have to be resolved. The vectors and matrices used to resolve the piezoresistance coefficients from the equations AR/R= [ crvector × Trans 7rxxxx] X [ ~'vector] can be used as well for resolving the stress components in the actual stress measurements by rewriting the relations as AR/R= [ ~-vectorv X TransTrxxxT] X [ o-vectorT],
Acknowledgements This project is funded by the Dutch Foundation of Applied Research (STW).
References
[ I ] J.F. Nye, Physical Properties 1985.
of Co'stals,
Clarendon Press, Oxford,
I
I cofltlJ sin~
sinhl/ coslq/
-cos~sin~ cos~sint[/
2 eos~sintll -2 costltsint) cos2~-sinhlt
[2] H.C.J.M.van Gestel,ReliabilityrelatedxesearehonplasticIC-packages: a test chip approach, Ph.D. Dissertation, Delft University of Technology,The Netherlands,1994,ISBN90-6275-960-2.
Biographies Andre Bossche was born in Rotterdam, The Netherlands, on August 27, 1956. He received the M.Sc degree in electrical engineering with honours in 1983, and the Ph.D. degree in electrical engineering in 1988, both from the Delft University of Technology, The Netherlands. He is currently engaged in research work on the subjects of reliability and packaging of sensors and microsystems at the Electronic Instrumentation Laboratory of the Delft University of Technology. JeffMollinger was born in Jakarta, Indonesia, on September 9, 1951. After finishing his studies in electronics in 1975, he joined the TNO Department of Groundwater Survey. In 1980 he joined the Volker Stevin Dredging Company, and in 1981 Delft University of Technology, The Netherlands, at the Department of Mechanical Engineering. Currently he is working as a senior technician at the Electronic Instrumentation Laboratory of Delft University of Technology, where he supports scientific research on the subject of reliability and electronic measurement systems.