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On-chip metal deformation measurements: A capacitive approach
Sensors and Actuators A, 25-27
801
(1991) 801-807
On-chip Piezoresistive Stress Measurement in Three Directions H. C. J. M. VAN GESTEL,
A. BOSSCHE
and J. R. MOLLINGER
Electronic hstrumentation Laboratory, Deportment of Electrical Engineering, Derft University of Technology, P.O. Box 5031 2600 GA Derfr (The Netherlands)
Abstract
2a*E. . +&r,(T-T,)+z(T-To)* 2
J
A sensor design has been developed and realized in order to determine the mechanical stresses on the surface of a silicon die. The purpose of this sensor chip is to examine the stresses on the surface of the chip caused by chip encapsulation and package temperaturecycle tests. Our approach uses the piezoresistive effect. However, the piezoresistive strain gauges not only measure the in-plane stresses but also the stress perpendicular to the chip surface. In this article we give the outline for the design of our stress sensor and show how this sensor is capable of separating the different mechanical stress components.
1 (2)
+.-.
Since all tensors of odd rank vanish when there is a centre of symmetry [2], we conclude that for a cubic crystal lattice such as silicon the first-, third- and fifth- (etc.)rank tensors must equal zero. The first-, third- and fifthorder terms in eqn. (2) are i?E, E
3.
i3Ei
aT* ’
a&’
’
a*E. -2;
ai,ai,
2a2Ei aT,$T
a2Ei
a &a Tqr
Since these expressions equal zero, the following expression remains: 1. Basic Theory of Piezoresistance E,=!$+;
By applying an electric field in a semiconductor we have electric field components Eti which depend on the current density 4, on the temperature T and on the mechanical stresses Tg. Hence we may write [l] Ei = E(b, T,, T)
(1)
Writing eqn. (1) as a McLaurin series around zero current, zero stress and temperature TO, we get aEi aEi , aEi 4 = a~,, TM + ai, lj + a~ CT - T,)
+
O924-4247/91/$3.50
TO) +-$i, J
m
kl
2d2E. ----!i.(T-TO) ai,ar J
.I
1
+...
(3)
aE.
--! = pij = specific resistivity ai, a*E. L = xGk!= piezoresistance coefficients aTk,ai,
J
kl
.[
and we now designate the remaining derivatives at zero current, zero stress and temperature TO as follows:
-$$
+s;T,(T-
zT,,r; J
= +T(p,) = a = temperature coefficient of resistivity
Hence we may now express the resulting equation for the components Ei as 0 Elsevier Sequoia/Printed in The Netherlands
802
follows:
shortened, the form
Ei = piiij + niik,Tk/ii+ aATij,
(4)
where AT=T-To In eqn. (4) we have the fundamental tensor equation, in first-order approximation, which relates the electric field with the current density, the temperature and the mechanical stresses. To obtain a much more common expression for eqn. (4) we reduce the four indices of the piezoresistance coefficients to two. Therefore we use the fact that silicon is a cubic crystal with a centre of symmetry, so we can write a second-rank tensor as pii in reference to the crystallographic ( = principal) axes pij = p0& (here 6 denotes the Kronecker delta). Mechanical equilibrium projects [ 31 yield Tkl = Tlk. Therefore we may also interchange the indices for the piezoresistive coefficients, z&l = shorten the notation for the rcs and factor out the zero stress resistivity p. using njikl
nnijkl+
pO%b?
and
=
nijlk
=
njilk,
CI+~~CL
(5)
According to the scheme 11+1
23=32-+4
22-2
13=31+5
33+3
12=21+6
we
get ifa=1,2,3
or
b=l,2,3
/,0&b = 2,$k,,
if a = 4, 5, 6
Or
b = 4, 5, 6
PO&b
if a = 4, 5, 6
and b = 4, 5, 6
PO%b
=
=
~$A
4%jkl>
Hence, by rewriting the tensor eqn. (4) by means of the preceding scheme, we obtain El/p0 = c&i,+ Q,T&
+ aATi,
(6)
0 0
0 0
0 0
n&$0 0 7TM
From eqn. (7) we draw the conclusion that there are three independent constants, in reference to the crystallographic axes. We may then rewrite eqn. (6) in full with respect to the crystallographic axes as follows, ooE, =il+nllT,il+n12il(T2+ + MT&
T3)
+ T&)
o&z = iz + ~11Tziz + 7t,&(T, + T3) + MT&
+ T&)
aoE3 = i3 + zll T3i3+ n,,i,(T, + T2) + nM(T5il + T,i,)
(8)
To express the piezoresistance equation referred to arbitrary oriented axes, we use primed coefficients to indicate that these coefficients are the ones that belong to the new coordinate system. To express eqn. (8) relative to an arbitrary set of axes, we may transform the current density (ii), the stress (TV) and also the electric field (Ei) to the new arbitrarily oriented system of axes, and then by substituting these expressions back into eqn. (8) we obtain the piezoresistance equations referred to the new axes but in which only the piezoresistance coefficients referred to the crystallographic axes appear. Thus, in the new set of equations the primed piezoresistance coefficients (n&) appear as coefficients of various density and stress components and can be expressed in terms of the unprimed rcs. Let quantities without a prime be referred to a new Cartesian system whose orientation is specified by the transformation matrix Old
where and kl+b and 6, = 1 for a = 1,2,3. Note also that a fourth-rank tensor for a material having a cubic lattice and a centre of symmetry possesses, after the indices are
0 0
x1
x2
x3
11 & I3
ml m2
n1 n2
m3
n3
ij+a
d
New
xi
xi
(9)
803 TABLE 1. The piezoresistance coefficients for rotated axes (I[* = q, - n,, - A~) Analogous coefficients
Coefficient
Expression
XII l-c;2= ?r;, lr;c = 2&
x,,+n,(1,*1,*+m,*m,*+n,*n,*) 2rc,(I,‘i, + m13m2 + nj3n2)
7-c&= 2& R& = TM =66
27r,(I, 1,1,*+ m, mIm3* + n, nzn3’) 2nA(i112=13 + m, m22mj + n, n2*n3) rcu + 27~,(1,~1,~+ m,*rn2=+ tq2nz2)
*,, -2n,(l,zm,2+m,*n,*+n,*I,*)
That is, the xi axis has the direction cosines ii, mi, ni with respect to the crystallographic axes (i = 1,2,3). The result of these substitutions in tensor form may be written as o,,E’,
=i’,(l
+n’,,TI)
CJ,,J!?; = i’, x&TI
+ i;(
+i~~&Tp+i~a$,Tp
1 + n;,T~)i;n&TI,
for p = 1,2,. . . ,6 aoE; =
ii 7r&Tp + i;7-c&Tp
(10)
+ i;( 1 + 7c&Tp)
where the expressions for the primed piezoresistive constants expand according to Table 1. The values of the piezoresistive constants with reference to the crystallographic axes for a particular type of doped silicon are given in Table 2. TABLE2.
PieZOreSiStnCC
COeffiCientS
at
rOOm
the current as a pure longitudinal one also in the xi direction. To calculate the piezoresistivity for our resistor we need to transform our axis system. The original orientation may be described by the unit vectors x, = (lOO), x2= (010) and x,= (001). The new axes (x’,, x;, xi) can be found as follows. Since the resistor lies in the (100) plane we assign to xj the (100) direction which is perpendicular to the silicon surface. The x’, and xi axes lie in the (100) plane and may be rotated by an angle 4 (see Fig. 1). Since we are interested in the voltage drop over a piezoresistor, we expand the first equation of (8) and obtain +n;,T;
a,,~?‘, =i;(l
+K;~T;+~;~T;
+ l&T& + $5 T; + ?rifjT!)
(11)
tmpm-
ture [l] Doping-type
p (ohm cm)
HI2 =[I1 ( x IO-l2 dynes/$)
p-type n-type
7.8 11.8
+ 6.6 ~ 102.2
-1.1 + 53.4
$138.1 - 13.6
2. Design of the Silicon Strain Sensor Old Having determined an expression for the piezoresistivity, let us now express the piezoresistance for a resistor on a silicon chip processed on a ( 100) wafer. Consider a resistor with the current direction along the x’, axis. We neglect any effects that occur due to the diffusion/implantation process and regard
New:!
1
Fig. 1. Rotation of the x’, and x; axes in the (100) plane and the corresponding direction cosines matrix.
804
TTp
By means of the direction cosines matrix, we may express the primed piezoresistance coefficients in terms of the original piezoresistance coefficients. We also rename our piezoresistance coefficients and stresses with new suffixes: 7Z;,= It, = n,, - 2n, ( cos2q sin2q)
T; = T,
7~;~= 7ct= ‘11,2 + 2n,(cos2~ sin2q)
r; = T,
7c;, = ret= n;z
T; = T,,
71;4= 0
(12)
n;s = q, = 2lrA x (cos q sin3q - cos3cp sintp) T5 = T, 11;6= n,, = n;s
r6 = T,,
Rewriting eqn. (11) with the suffixes just introduced and also giving the current density and the electric field the suffix 1, we have o&, = i,( 1 + 71,T, + x,(T, + TP) + nr,,(T,,+ T,,))
(13)
Considering that R gp,,i AR E poi(qT, + R,(T, + T,) + MT,t + T,,N
(14)
we may express AR/R as AR/R = 111 T + n,(r, + T,) + ME, + T,,) Here the suffix 1 indicates longitudinal to the current direction, t transverse to the current direction, p perpendicular to the chip surface; It is in-plane shear whereas lp denotes shear on the surface of the chip along the current direction. It is essential to use these indices instead of, e.g. x, y and z because we would like to express the stresses relative to the current direction. To illustrate the meaning of these stresses, we have pictured a resistor in Fig. 2. Our further intention is not to calculate the piezoresistance coefficients but merely to indicate that the equation above validates a linear approach or even a higher-order approach to the change in resistivity due to
Fig. 2. Stresses relative to the current direction
mechanical stress plus that different stress components have a different influence on the change in specific resistivity. The values for the piezoresistive coefficients are being determined by a calibration method. However, since the values for the unprimed piezoresistive coefficients are known for particular doping concentrations, this equation is useful to determine the directions with optimal sensitivity for the piezoresistors. Let us now first calculate the piezoresistance coefficients as a function of q for both n- and p-type resistors using the data of Table 2 and the expressions given in eqn. (12). The results are plotted in Figs. 3 and 4. From these Figures it is obvious that the best longitudinal direction for n-type resistors’ is along a crystallographic axis, whereas the p-type resistors are more sensitive for longitudinal stresses in the directions with a 45” offset from crystallographic axes. Note that two piezoresistors rotated 90” from each other show the same piezoresistive
-1.0-e-10
1
I co1
410> -57
I
___.n
t
I>
___
7T
It
Fig. 3. Piezoresistance coefficients for p-type silicon in the ( 100) plane.
805
Fig. 6. Piezoresistors
rotated
by O”, 45” and 90”.
stresses mentioned above. For a resistor whose longitudinal axis is rotated by an angle cp from the x-direction we may write [3] co1
<010> -7T
I
Fig. 4. Piezoresistance the ( 100) plane.
___.n
12
t
coefficients
-=OOl=-
___7T
for n-type
T: = $(T, + TJ + f(T, - T,) x cos 2~p+ TX.,,sin 2~
It silicon in
T;, = ;(Ty + TJ + $(T,t - T,) x cos 250- r,, sin 2~
behaviour. Hence, we are able to construct a strain rosette with four resistors, two p-type and two n-type, that shows an optimal stress sensitivity for stresses longitudinal and transverse to the current direction. Such a strain gauge rosette is pictures in Fig. 5. Having established the rosette design, we are now able to express the change in resistance according to uniform stresses that are applied to these four resistors. Since the piezoresistors are rotated towards each other, longitudinal, transverse and shear stresses are not equal for each of them. We have to express these stress components for each resistor by means of the uniform stress components. To do so, let us assume we have uniform stresses T,, T,, T,, and shear stresses Trvr TX,, TYZ(see Fig. 6). We are now describing the longitudinal, transverse and shear stresses in terms of the
T; =
(15)
T,
T;,,, = Ts_ = i( T, - T,)sin 2q + TX, cos 2~ T:,= = TI, = r,= cos cp + Tyz sin cp T;, = Tl& = -T .,.=sin cp + T,,=cos cp
In our design the p-type resistors are longitudinally and transversely placed to the die edges, so we can write by means of eqns. (15) and (14) AR,, 0) ho
) = np,T, + nptCTy + T-1 + q,,(T,y
mt
90) 4,
+ TA
w , = np,TV + n,,(Tx + TJ + np,,Kr
- Txy)
The n-type resistors are rotated 45” and 135” from the chip’s longitudinal axis. Filling in the stresses again, we obtain A& 45, & + ~&T,
45, = n&V,
+ Tv) + Tx.v)
+ Ty) - Txy + TJ
+ G,,(:(T~ - T.1 + ;fiCL A% 135,& 135 , = n&T, + nn,(#‘x
rosette
with four resistors.
+ T,z))
+ Ty) - TX,>
+ T,J + Tx, + TJ
+ 7~&%Tyz
Fig. 5. Piezoresistive
(16)
- Ty;) - i(Ty - TxN
(17)
With four resistors, two p-type and two ntype, each in a different direction, we have found equations with six stress components. However, as we may observe from Figs. 3
806
and 4, the shear coefficient xl, almost equals zero. Hence, these six stress components may be reduced to four if we regard xl, to be zero. So if we determine the values of the piezoresistance coefficients as accurately as possible, we are able by measuring the relative changes in resistance of the four piezoresistors to determine the in-plane stresses T,, Ty and Txy and the perpendicular stress T, as well. If we observe a slight deviation of 7cltfrom zero we have to disregard this contribution to the relative change in resistance. Now having established the theoretical description of measuring mechanical stress components and showing that our measurements can be achieved with four resistors, we discuss the design as it is realized on a chip. It is essential that the layout uses both n- and p-type resistors. This may be illustrated by eqns. (16) and ( 17). From these equations the stresses have to be determined by solving four equations with four unknown stresses (setting njl to zero). This set of equations can only be solved when at least one of the four resistors has a different doping type to prevent the set of equations becoming singular. Another important layout consideration is the fact that when one solves the stresses for four resistors, it is assumed that stresses are uniform over all four resistors. Therefore it is important to design the resistors to be as small as possible, to indeed achieve uniform stresses over four resistors. A third problem that may arise is the chance that a resistor is not perfectly implanted during IC production. Its behaviour as a result of the stresses will not be as we expected. Therefore we must find some way to get redundant information from a large number of resistors. This has been achieved by not making single rosettes, but designing one large matrix of resistors. Hence, in this way, almost every resistor may be used in four groups of four resistors, delivering us more information. A final remark on our design is that we wanted to use high-resistivity resistors for our sensor. This implies that the stress sensitivity of our piezoresistors is large but that we have
Fig. 7. Part of the chip layout of the three-dimensional stress sensor array. Note that the matrix elements are connected in x and y directions.
to deal with both the temperature dependence of the resistivity itself and the temperature sensitivity of the piezoresistive effect! Therefore we have throughout substituted the matrix resistors by diodes which are used for temperature measurement. Putting all this together, it may be clear that a lot of parameters have to be determined before we are able to use this sensor. In Fig. 7 part of the chip layout is pictured. The matrix elements are connected in x and y directions and their values are determined by means of the four-point measurement method. The desired switching electronics is realized externally with discrete components.
3. Calibration It may be clear from the previous Section that calibration is very important in order to be able to use the three-dimensional stress sensor. To be able to calibrate the stress sensor, we have constructed a four-point bending bridge [4] in an environmental chamber, see Fig. 8. By pushing the outer rods downwards, we are able to bend the test beam very precisely. An automatic calibration program has been written that controls the temperature of our environmental chamber and the bending of the four-point bending bridge. This pro-
807 x =II,+II,T+~~T~.
..
Also important during calibration is the temperature coefficient of the resistivity tl. We may also use a higher-order approach here if necessary. Fig. 8. Illustration of the four-point bending bridge.
4. Discussion gram makes use of the relation between the downward movement from the outer rods and the stresses on the chip. Calculations of beam material and the anisotropic silicon chip were made using finite elements (ANSYS). The crystal plane that is being used for the chips is also important for calibration. In our case we have a (100) plane. This means that the piezoresistive constants change as the longitudinal current direction ( = resistor direction) changes. If one uses, for example, the (111) plane, the piezoresistive constants remain constant for every current direction in the plane. Hence, in our case we have an extended calibration procedure since we have to determine three piezoresistive constants for a particular resistor in a particular direction. Since the piezoresistive coefficients are periodic with 90”, see Figs. 3 and 4, we only have to add one more equation for solving the piezoresistive coefficients when the stresses are known. Therefore we need to calibrate our sensors in two steps. One calibration must be performed on a chip placed on the bending beam with its die edge along the edge of the bending beam, and the other must be performed on a chip rotated over 45”. To incorporate the temperature effects and possible non-linear piezoresistive effect, we may extend the piezoresistive constants to a higher order. Naturally all piezoresistive coefficients are written as a polynomial as a function of temperature. Hence, the goal of calibration is to achieve piezoresistive constants like:
The sensor design explained above is particularly mentioned for our packaging research. Therefore we have a large matrix of resistors and diodes, since we want to monitor the stresses over the surface of the silicon die. We also want to measure the stresses involving the temperature as far as possible. In this way the complete calibration procedure becomes quite complicated and demands an automatic measurement set-up. This is achieved by using an IBM-compatible PC and ASYST programming language. The use of this piezoresistive sensor array becomes complicated mainly because we are dealing with a large number of sensors. However, the idea of measuring different mechanical stress components that cannot be measured with metal foil strain gauges might be attractive for several purposes. Experimental data with reference to the packaging measurements will be presented later.
References W. P. Masonand R. N. Thurston, Use of piezoresistive materials in the measurement of displacement, force and torque, J. Acoust. Sot. Am., 29 (1957) 1096-1101. J. F. Nye, Physical Properties of Crystals, Clarendon Press, Oxford, 1985. J. M. Gere and S. P. Timoshenko, Mechanics of Materials, PWS Engineering, 2nd SI edn., 1984. S. A. Gee, V. R. Akylas and F. W. van den Bogert, The design and calibration of a semiconductor strain gauge array, IEEE Int. Conf Microelecrronic Test Structures, Long Beach, 1988, pp. 185-191.
CA,
U.S.A.,
Feb. 22-23,
801
(1991) 801-807
On-chip Piezoresistive Stress Measurement in Three Directions H. C. J. M. VAN GESTEL,
A. BOSSCHE
and J. R. MOLLINGER
Electronic hstrumentation Laboratory, Deportment of Electrical Engineering, Derft University of Technology, P.O. Box 5031 2600 GA Derfr (The Netherlands)
Abstract
2a*E. . +&r,(T-T,)+z(T-To)* 2
J
A sensor design has been developed and realized in order to determine the mechanical stresses on the surface of a silicon die. The purpose of this sensor chip is to examine the stresses on the surface of the chip caused by chip encapsulation and package temperaturecycle tests. Our approach uses the piezoresistive effect. However, the piezoresistive strain gauges not only measure the in-plane stresses but also the stress perpendicular to the chip surface. In this article we give the outline for the design of our stress sensor and show how this sensor is capable of separating the different mechanical stress components.
1 (2)
+.-.
Since all tensors of odd rank vanish when there is a centre of symmetry [2], we conclude that for a cubic crystal lattice such as silicon the first-, third- and fifth- (etc.)rank tensors must equal zero. The first-, third- and fifthorder terms in eqn. (2) are i?E, E
3.
i3Ei
aT* ’
a&’
’
a*E. -2;
ai,ai,
2a2Ei aT,$T
a2Ei
a &a Tqr
Since these expressions equal zero, the following expression remains: 1. Basic Theory of Piezoresistance E,=!$+;
By applying an electric field in a semiconductor we have electric field components Eti which depend on the current density 4, on the temperature T and on the mechanical stresses Tg. Hence we may write [l] Ei = E(b, T,, T)
(1)
Writing eqn. (1) as a McLaurin series around zero current, zero stress and temperature TO, we get aEi aEi , aEi 4 = a~,, TM + ai, lj + a~ CT - T,)
+
O924-4247/91/$3.50
TO) +-$i, J
m
kl
2d2E. ----!i.(T-TO) ai,ar J
.I
1
+...
(3)
aE.
--! = pij = specific resistivity ai, a*E. L = xGk!= piezoresistance coefficients aTk,ai,
J
kl
.[
and we now designate the remaining derivatives at zero current, zero stress and temperature TO as follows:
-$$
+s;T,(T-
zT,,r; J
= +T(p,) = a = temperature coefficient of resistivity
Hence we may now express the resulting equation for the components Ei as 0 Elsevier Sequoia/Printed in The Netherlands
802
follows:
shortened, the form
Ei = piiij + niik,Tk/ii+ aATij,
(4)
where AT=T-To In eqn. (4) we have the fundamental tensor equation, in first-order approximation, which relates the electric field with the current density, the temperature and the mechanical stresses. To obtain a much more common expression for eqn. (4) we reduce the four indices of the piezoresistance coefficients to two. Therefore we use the fact that silicon is a cubic crystal with a centre of symmetry, so we can write a second-rank tensor as pii in reference to the crystallographic ( = principal) axes pij = p0& (here 6 denotes the Kronecker delta). Mechanical equilibrium projects [ 31 yield Tkl = Tlk. Therefore we may also interchange the indices for the piezoresistive coefficients, z&l = shorten the notation for the rcs and factor out the zero stress resistivity p. using njikl
nnijkl+
pO%b?
and
=
nijlk
=
njilk,
CI+~~CL
(5)
According to the scheme 11+1
23=32-+4
22-2
13=31+5
33+3
12=21+6
we
get ifa=1,2,3
or
b=l,2,3
/,0&b = 2,$k,,
if a = 4, 5, 6
Or
b = 4, 5, 6
PO&b
if a = 4, 5, 6
and b = 4, 5, 6
PO%b
=
=
~$A
4%jkl>
Hence, by rewriting the tensor eqn. (4) by means of the preceding scheme, we obtain El/p0 = c&i,+ Q,T&
+ aATi,
(6)
0 0
0 0
0 0
n&$0 0 7TM
From eqn. (7) we draw the conclusion that there are three independent constants, in reference to the crystallographic axes. We may then rewrite eqn. (6) in full with respect to the crystallographic axes as follows, ooE, =il+nllT,il+n12il(T2+ + MT&
T3)
+ T&)
o&z = iz + ~11Tziz + 7t,&(T, + T3) + MT&
+ T&)
aoE3 = i3 + zll T3i3+ n,,i,(T, + T2) + nM(T5il + T,i,)
(8)
To express the piezoresistance equation referred to arbitrary oriented axes, we use primed coefficients to indicate that these coefficients are the ones that belong to the new coordinate system. To express eqn. (8) relative to an arbitrary set of axes, we may transform the current density (ii), the stress (TV) and also the electric field (Ei) to the new arbitrarily oriented system of axes, and then by substituting these expressions back into eqn. (8) we obtain the piezoresistance equations referred to the new axes but in which only the piezoresistance coefficients referred to the crystallographic axes appear. Thus, in the new set of equations the primed piezoresistance coefficients (n&) appear as coefficients of various density and stress components and can be expressed in terms of the unprimed rcs. Let quantities without a prime be referred to a new Cartesian system whose orientation is specified by the transformation matrix Old
where and kl+b and 6, = 1 for a = 1,2,3. Note also that a fourth-rank tensor for a material having a cubic lattice and a centre of symmetry possesses, after the indices are
0 0
x1
x2
x3
11 & I3
ml m2
n1 n2
m3
n3
ij+a
d
New
xi
xi
(9)
803 TABLE 1. The piezoresistance coefficients for rotated axes (I[* = q, - n,, - A~) Analogous coefficients
Coefficient
Expression
XII l-c;2= ?r;, lr;c = 2&
x,,+n,(1,*1,*+m,*m,*+n,*n,*) 2rc,(I,‘i, + m13m2 + nj3n2)
7-c&= 2& R& = TM =66
27r,(I, 1,1,*+ m, mIm3* + n, nzn3’) 2nA(i112=13 + m, m22mj + n, n2*n3) rcu + 27~,(1,~1,~+ m,*rn2=+ tq2nz2)
*,, -2n,(l,zm,2+m,*n,*+n,*I,*)
That is, the xi axis has the direction cosines ii, mi, ni with respect to the crystallographic axes (i = 1,2,3). The result of these substitutions in tensor form may be written as o,,E’,
=i’,(l
+n’,,TI)
CJ,,J!?; = i’, x&TI
+ i;(
+i~~&Tp+i~a$,Tp
1 + n;,T~)i;n&TI,
for p = 1,2,. . . ,6 aoE; =
ii 7r&Tp + i;7-c&Tp
(10)
+ i;( 1 + 7c&Tp)
where the expressions for the primed piezoresistive constants expand according to Table 1. The values of the piezoresistive constants with reference to the crystallographic axes for a particular type of doped silicon are given in Table 2. TABLE2.
PieZOreSiStnCC
COeffiCientS
at
rOOm
the current as a pure longitudinal one also in the xi direction. To calculate the piezoresistivity for our resistor we need to transform our axis system. The original orientation may be described by the unit vectors x, = (lOO), x2= (010) and x,= (001). The new axes (x’,, x;, xi) can be found as follows. Since the resistor lies in the (100) plane we assign to xj the (100) direction which is perpendicular to the silicon surface. The x’, and xi axes lie in the (100) plane and may be rotated by an angle 4 (see Fig. 1). Since we are interested in the voltage drop over a piezoresistor, we expand the first equation of (8) and obtain +n;,T;
a,,~?‘, =i;(l
+K;~T;+~;~T;
+ l&T& + $5 T; + ?rifjT!)
(11)
tmpm-
ture [l] Doping-type
p (ohm cm)
HI2 =[I1 ( x IO-l2 dynes/$)
p-type n-type
7.8 11.8
+ 6.6 ~ 102.2
-1.1 + 53.4
$138.1 - 13.6
2. Design of the Silicon Strain Sensor Old Having determined an expression for the piezoresistivity, let us now express the piezoresistance for a resistor on a silicon chip processed on a ( 100) wafer. Consider a resistor with the current direction along the x’, axis. We neglect any effects that occur due to the diffusion/implantation process and regard
New:!
1
Fig. 1. Rotation of the x’, and x; axes in the (100) plane and the corresponding direction cosines matrix.
804
TTp
By means of the direction cosines matrix, we may express the primed piezoresistance coefficients in terms of the original piezoresistance coefficients. We also rename our piezoresistance coefficients and stresses with new suffixes: 7Z;,= It, = n,, - 2n, ( cos2q sin2q)
T; = T,
7~;~= 7ct= ‘11,2 + 2n,(cos2~ sin2q)
r; = T,
7c;, = ret= n;z
T; = T,,
71;4= 0
(12)
n;s = q, = 2lrA x (cos q sin3q - cos3cp sintp) T5 = T, 11;6= n,, = n;s
r6 = T,,
Rewriting eqn. (11) with the suffixes just introduced and also giving the current density and the electric field the suffix 1, we have o&, = i,( 1 + 71,T, + x,(T, + TP) + nr,,(T,,+ T,,))
(13)
Considering that R gp,,i AR E poi(qT, + R,(T, + T,) + MT,t + T,,N
(14)
we may express AR/R as AR/R = 111 T + n,(r, + T,) + ME, + T,,) Here the suffix 1 indicates longitudinal to the current direction, t transverse to the current direction, p perpendicular to the chip surface; It is in-plane shear whereas lp denotes shear on the surface of the chip along the current direction. It is essential to use these indices instead of, e.g. x, y and z because we would like to express the stresses relative to the current direction. To illustrate the meaning of these stresses, we have pictured a resistor in Fig. 2. Our further intention is not to calculate the piezoresistance coefficients but merely to indicate that the equation above validates a linear approach or even a higher-order approach to the change in resistivity due to
Fig. 2. Stresses relative to the current direction
mechanical stress plus that different stress components have a different influence on the change in specific resistivity. The values for the piezoresistive coefficients are being determined by a calibration method. However, since the values for the unprimed piezoresistive coefficients are known for particular doping concentrations, this equation is useful to determine the directions with optimal sensitivity for the piezoresistors. Let us now first calculate the piezoresistance coefficients as a function of q for both n- and p-type resistors using the data of Table 2 and the expressions given in eqn. (12). The results are plotted in Figs. 3 and 4. From these Figures it is obvious that the best longitudinal direction for n-type resistors’ is along a crystallographic axis, whereas the p-type resistors are more sensitive for longitudinal stresses in the directions with a 45” offset from crystallographic axes. Note that two piezoresistors rotated 90” from each other show the same piezoresistive
-1.0-e-10
1
I co1
410> -57
I
___.n
t
I>
___
7T
It
Fig. 3. Piezoresistance coefficients for p-type silicon in the ( 100) plane.
805
Fig. 6. Piezoresistors
rotated
by O”, 45” and 90”.
stresses mentioned above. For a resistor whose longitudinal axis is rotated by an angle cp from the x-direction we may write [3] co1
<010> -7T
I
Fig. 4. Piezoresistance the ( 100) plane.
___.n
12
t
coefficients
-=OOl=-
___7T
for n-type
T: = $(T, + TJ + f(T, - T,) x cos 2~p+ TX.,,sin 2~
It silicon in
T;, = ;(Ty + TJ + $(T,t - T,) x cos 250- r,, sin 2~
behaviour. Hence, we are able to construct a strain rosette with four resistors, two p-type and two n-type, that shows an optimal stress sensitivity for stresses longitudinal and transverse to the current direction. Such a strain gauge rosette is pictures in Fig. 5. Having established the rosette design, we are now able to express the change in resistance according to uniform stresses that are applied to these four resistors. Since the piezoresistors are rotated towards each other, longitudinal, transverse and shear stresses are not equal for each of them. We have to express these stress components for each resistor by means of the uniform stress components. To do so, let us assume we have uniform stresses T,, T,, T,, and shear stresses Trvr TX,, TYZ(see Fig. 6). We are now describing the longitudinal, transverse and shear stresses in terms of the
T; =
(15)
T,
T;,,, = Ts_ = i( T, - T,)sin 2q + TX, cos 2~ T:,= = TI, = r,= cos cp + Tyz sin cp T;, = Tl& = -T .,.=sin cp + T,,=cos cp
In our design the p-type resistors are longitudinally and transversely placed to the die edges, so we can write by means of eqns. (15) and (14) AR,, 0) ho
) = np,T, + nptCTy + T-1 + q,,(T,y
mt
90) 4,
+ TA
w , = np,TV + n,,(Tx + TJ + np,,Kr
- Txy)
The n-type resistors are rotated 45” and 135” from the chip’s longitudinal axis. Filling in the stresses again, we obtain A& 45, & + ~&T,
45, = n&V,
+ Tv) + Tx.v)
+ Ty) - Txy + TJ
+ G,,(:(T~ - T.1 + ;fiCL A% 135,& 135 , = n&T, + nn,(#‘x
rosette
with four resistors.
+ T,z))
+ Ty) - TX,>
+ T,J + Tx, + TJ
+ 7~&%Tyz
Fig. 5. Piezoresistive
(16)
- Ty;) - i(Ty - TxN
(17)
With four resistors, two p-type and two ntype, each in a different direction, we have found equations with six stress components. However, as we may observe from Figs. 3
806
and 4, the shear coefficient xl, almost equals zero. Hence, these six stress components may be reduced to four if we regard xl, to be zero. So if we determine the values of the piezoresistance coefficients as accurately as possible, we are able by measuring the relative changes in resistance of the four piezoresistors to determine the in-plane stresses T,, Ty and Txy and the perpendicular stress T, as well. If we observe a slight deviation of 7cltfrom zero we have to disregard this contribution to the relative change in resistance. Now having established the theoretical description of measuring mechanical stress components and showing that our measurements can be achieved with four resistors, we discuss the design as it is realized on a chip. It is essential that the layout uses both n- and p-type resistors. This may be illustrated by eqns. (16) and ( 17). From these equations the stresses have to be determined by solving four equations with four unknown stresses (setting njl to zero). This set of equations can only be solved when at least one of the four resistors has a different doping type to prevent the set of equations becoming singular. Another important layout consideration is the fact that when one solves the stresses for four resistors, it is assumed that stresses are uniform over all four resistors. Therefore it is important to design the resistors to be as small as possible, to indeed achieve uniform stresses over four resistors. A third problem that may arise is the chance that a resistor is not perfectly implanted during IC production. Its behaviour as a result of the stresses will not be as we expected. Therefore we must find some way to get redundant information from a large number of resistors. This has been achieved by not making single rosettes, but designing one large matrix of resistors. Hence, in this way, almost every resistor may be used in four groups of four resistors, delivering us more information. A final remark on our design is that we wanted to use high-resistivity resistors for our sensor. This implies that the stress sensitivity of our piezoresistors is large but that we have
Fig. 7. Part of the chip layout of the three-dimensional stress sensor array. Note that the matrix elements are connected in x and y directions.
to deal with both the temperature dependence of the resistivity itself and the temperature sensitivity of the piezoresistive effect! Therefore we have throughout substituted the matrix resistors by diodes which are used for temperature measurement. Putting all this together, it may be clear that a lot of parameters have to be determined before we are able to use this sensor. In Fig. 7 part of the chip layout is pictured. The matrix elements are connected in x and y directions and their values are determined by means of the four-point measurement method. The desired switching electronics is realized externally with discrete components.
3. Calibration It may be clear from the previous Section that calibration is very important in order to be able to use the three-dimensional stress sensor. To be able to calibrate the stress sensor, we have constructed a four-point bending bridge [4] in an environmental chamber, see Fig. 8. By pushing the outer rods downwards, we are able to bend the test beam very precisely. An automatic calibration program has been written that controls the temperature of our environmental chamber and the bending of the four-point bending bridge. This pro-
807 x =II,+II,T+~~T~.
..
Also important during calibration is the temperature coefficient of the resistivity tl. We may also use a higher-order approach here if necessary. Fig. 8. Illustration of the four-point bending bridge.
4. Discussion gram makes use of the relation between the downward movement from the outer rods and the stresses on the chip. Calculations of beam material and the anisotropic silicon chip were made using finite elements (ANSYS). The crystal plane that is being used for the chips is also important for calibration. In our case we have a (100) plane. This means that the piezoresistive constants change as the longitudinal current direction ( = resistor direction) changes. If one uses, for example, the (111) plane, the piezoresistive constants remain constant for every current direction in the plane. Hence, in our case we have an extended calibration procedure since we have to determine three piezoresistive constants for a particular resistor in a particular direction. Since the piezoresistive coefficients are periodic with 90”, see Figs. 3 and 4, we only have to add one more equation for solving the piezoresistive coefficients when the stresses are known. Therefore we need to calibrate our sensors in two steps. One calibration must be performed on a chip placed on the bending beam with its die edge along the edge of the bending beam, and the other must be performed on a chip rotated over 45”. To incorporate the temperature effects and possible non-linear piezoresistive effect, we may extend the piezoresistive constants to a higher order. Naturally all piezoresistive coefficients are written as a polynomial as a function of temperature. Hence, the goal of calibration is to achieve piezoresistive constants like:
The sensor design explained above is particularly mentioned for our packaging research. Therefore we have a large matrix of resistors and diodes, since we want to monitor the stresses over the surface of the silicon die. We also want to measure the stresses involving the temperature as far as possible. In this way the complete calibration procedure becomes quite complicated and demands an automatic measurement set-up. This is achieved by using an IBM-compatible PC and ASYST programming language. The use of this piezoresistive sensor array becomes complicated mainly because we are dealing with a large number of sensors. However, the idea of measuring different mechanical stress components that cannot be measured with metal foil strain gauges might be attractive for several purposes. Experimental data with reference to the packaging measurements will be presented later.
References W. P. Masonand R. N. Thurston, Use of piezoresistive materials in the measurement of displacement, force and torque, J. Acoust. Sot. Am., 29 (1957) 1096-1101. J. F. Nye, Physical Properties of Crystals, Clarendon Press, Oxford, 1985. J. M. Gere and S. P. Timoshenko, Mechanics of Materials, PWS Engineering, 2nd SI edn., 1984. S. A. Gee, V. R. Akylas and F. W. van den Bogert, The design and calibration of a semiconductor strain gauge array, IEEE Int. Conf Microelecrronic Test Structures, Long Beach, 1988, pp. 185-191.
CA,
U.S.A.,
Feb. 22-23,