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Residual stresses in dielectrics caused by metallization lines and pads
Acta mater.Vol. 44, No. 6,pp. 2353.-2359, 1996
Pergamon 0956-7151(95)00341-X
RESIDUAL STRESSES IN DIELECTRICS METALLIZATION LINES AND M. Y. HE’, J. LIPKIN’,
D. R. CLARKE’,
Elsevier Science Ltd Copyright 0 1996 Acta Metallurgica Inc. Printed in Great Britain. All rights reserved 1359~6454196$15.00 + 0.00
CAUSED PADS
BY
A. G. EVANS’ and M. TENHOVER
‘Materials Department, University of California, Santa Barbara, CA 93106-5050, ‘Division of Applied Sciences, Harvard University, Pierce Hall, Cambridge, MA 02138 and jWendel1 Research Center, Carborundum Company, Niagara Falls, NY, U.S.A. (Received 12 May 1995; in revised form 25 August 1995) Abstract-Residual stresses in dielectrics and semiconductors induced by metal lines, pads and vias can have detrimental effects on the performance of devices and electronic packages. Analytical and numerical calculations of these stresses have been performed for two purposes. (1) To illustrate how these stresses
relate to the residual stress in the metallization and its geometry; (2) to calibrate a piezo-spectroscopic method for measuring these stresses with high spatial resolution. The results of the calculations have been presented using non-dimensional parameters that both facilitate scaling and provide connections to the stresses in the metal, with or without yielding. Preliminary experimental results obtained for Au/Ge eutectic pads illustrate the potential of the method and the role of the stress analysis.
1. INTRODUCTION Residual stresses have an important (and sometimes, crucial) influence on the fabrication and reliability of many devices [l-3]. Both intrinsic and thermal expansion mismatch stresses are involved [4]. A knowledge base has been established that provides some understanding about the origin of these stresses [4], as well as their effect on decohesion at interfaces and on cracking [S, 61. However, a continuing problem is the quantitative prediction of the magnitude of these stresses and of the incidence of decohesion and cracking. The principal problem concerns the stresses induced by the metallization. These are caused either by interconnects or vias or by the brazes used to attach the Si to the dielectric [l, 21. Residual stresses associated with metallization have previously been measured by one of two methods: beam bending and X-rays [7,8]. These methods have been used with continuous metal films on either dielectric or semiconductor substrates. Beam curvature and X-ray line shift measurements give the average stress in the metal. In special cases, X-ray methods can measure the stress gradient as a function of depth normal to the interface [9]. The greater challenge is to address the spatial distribution of stress in the substrate when the metallization is patterned to form interconnects, vias or pads. These stresses dominate interface decohesion and cracking [5,6]. So far, measurements have been sparse and calculations limited. Laser-based piezospectroscopic methods have the requisite resolution [lo-121. These methods include fluorescence [lo, 1 l]
and Raman spectroscopy [12]. Either method is capable of obtaining stress information in the substrate around the metallization. One purpose of the present study is to demonstrate the fluorescence spectroscopy method for obtaining stresses in a dielectric substrate with high spatial resolution. The other is to inter-relate these stresses to those in the metal. Numerical calculations and preliminary experiments performed on eutectic metal pads are used for these purposes. 2. STRESS MEASUREMENT
METHOD
2.1. Approach A gold-germanium eutectic on Cr-doped sapphire (ruby) was chosen as the model metal/dielectric system, such that stress measurements could be made using a piezo-spectroscopic technique, based on the systematic shift of fluorescence lines with stress [IO]. Electrons excited from the chromium ions in the ruby by a laser decay back to the ground state. emitting photons in the process. In stress-free ruby. these photons occur at two characteristic frequencies. Application of stress perturbs the local environment. of the chromium ion, shifting the fluorescence lines to higher or lower frequencies, depending on the stress state [l 1, 131. This frequency shift Av can be related to the stresses in the sapphire lattice oiir by the following constitutive relation [13]: Av = H,a,,
(1)
where Hi is the first-order piezo-spectroscopic coefficient. Second-order effects are small for the configurations used in the present study and are
2353
HE et al.:
2354
RESIDUAL
neglected. However, Av is affected by the temperature [ 131, such that laser-induced temperature changes can be problematical, unless independently measured. 2.2. Experimental
procedures
Stress measurements have been made in the substrate surface by employing a chromium “intechnique [14], wherein only the top diffusion” (micron-thick) layer of the substrate fluoresces. For this purpose, a basal plane oriented sapphire substrate was prepared by evaporating 2OOA of chromium onto one surface, followed by heating in a vacuum furnace for 2 h at 1500°C to diffuse the chromium into the sapphire. The metal features were created by DC Magnetron sputtering. The first two layers comprised an adhesion layer of W/Ti and a diffusion barrier of Pt/Rh, each - 100 nm thick. The final 40 pm thick Au/Ge layer had the eutectic composition. The features comprised square pads, 2.1 mm on edge, having rounded corners (Fig. 1). A typical surface profile is shown in Fig. l(a). A magnification of the edge is shown in Fig. l(b). However, there was significant variability in the edge profile from pad to pad. Stress measurements were made using an Optical Microprobe in which a laser beam could be focused
STRESSES
to a spot - 1 pm diameter. Samples were probed across the center of each feature by translating in 1 pm steps using a linear actuator. Argon reference spectra were used to correct for mechanical drift in the optics. The temperature was monitored to account for peak shifts due to temperature changes. The spectra were analyzed using a commercial software package, wherein each peak was fit to a mixed GaussianLorentzian function. The central position of each peak was used to determine the stress by taking a point far from the feature as a stress-free reference. 2.3. Spatial convolution When the stresses exhibit large spatial variability over short distances, the finite size of the optical probe convolutes the stress function, e(x) with the probe response function, p(x). In order to deconvolute the measured fluorescence profile, p(x) must be experimentally calibrated. This was done by using a step-source. The function C(x) measured at such a source is simply related to p(x), by, p(.~) rz dC;‘dr.
Experimentally, a step function has been created by placing a 200 8, thick line of Au onto a Cr3+ doped sapphire substrate. Scanning in 1 pm steps across the edge has given the convolution of the probe function with a step function shown in Fig. 2(a). Upon fitting the derivatives to a Lorentzian, [Fig. 2(b)], the probe
1200
??
? ?
,,,,,,,/,,11,,~,,,,,,,,,,,,,,,,,,,,,,,,
3
? ? ? ?
-5
Dv%wxx
(2)
(vm)
Fig. 1. (a) Atomic force microscopy scan of Au/Ge eutectic pad on sapphire substrate; (b) magnification of the edge profile.
0
Position,
5
10
15
x (wm)
Fig. 2. (a) Signal for Au strip C(l) that convolutes the probe function p(x) with the step function. (b) The derivative of C(x) that gives the probe width.
HE et al.:
RESIDUAL
STRESSES
2355
Problem @
Problem @
Problem @
Problem @
Fig. 3. The metallization
geometries
width at half maximum was found to be 3 pm. The high spatial resolution of the fluorescence probing is thus affirmed. 3. STRESS ANALYSIS 3.1. Procedure A finite element method has been used to calculate the stresses in the dielectric. Four representative metallization geometries have been used: a line having
used for the numerical
calculations.
uniform thickness [Fig. 3(a)], a line having constant curvature [Fig. 3(b)], a circular pad [Fig. 3(c)] and a square pad [Fig. 3(d)] both with constant thickness. The dielectric is considered to be elastic and the metal elastic-perfectly plastic, with yield strength, crO. Residual strain in the system has been motivated by a mismatch in thermal expansion coefficient between the metal, TV,, and the dielectric, cld (Aa = a, -Q), and a cooling temperature, AT (Fig. 4). 3.2. Analytical approximations Useful principles are established by means of approximate analytical calculations (Fig. 5). A pair of line forces in the x-axis, P per unit thickness,
dielectric
initial state (stress-free)
unattached: cooling by AT
(a)
(4
B 0
(a)
6
Stresses Induced by Surface Forces, P
h
unattached: apply 6 to achieve displacement continuity at interface
attach: surface forces 5 applied for stress free requirements
ti (b)
(4 Fig. 4. The differential
(d) strain that causes the stresses in the substrate.
BendIng Induced Stresses
Fig. 5. The forces and moments imposed by the metallization used to obtain analytical estimates for the stress in the dielectric.
HE et al.:
2356
is applied terminates
to the dielectric surface (J’ = +d), such that,
where
P = -31
RESIDUAL
STRESSES
the film
(3)
where 0 is the average misfit (residual) stress in the film and h is the film thickness. The stresses e)?. in the dielectric around thin metal strips, width 2U: are then obtained from the following formulae [Fig. 5(a)]. The stresses along the dielectric surface (Z = 0), are [15] g,d 41~ C% - [(y/d)2 - l] and along the symmetry
(4a) 0
plane (y = 0),
[(z/d)’ + 11
and along the symmetry
Misfit Stress,
2
25
3
E,AaAT/(l-v)Oo
All subsequent results are for, I?,/&,, = 6 and Q,/E, = 0.001. The system geometries were varied within the ranges 0.04 < h/d 2 0.1 and 1 2 h,/d s 10. A typical finite element mesh includes 760 elements and 2409 nodes. A convergence study revealed that these meshes provided accurate results. fij Narrow; Lines
3d h, plane (_r = 0),
15
Fig. 6. Effect of misfit strain on the stress induced in the dielectric beneath a thin metal line (i = 1’= 0).
The large tensile stress concentrations near the edge are a particular concern for fracture in the dielectric and for interface debonding. For finite substrates thickness h,, bending occurs and redistributes the stress. The stresses induced by bending are given by beam theory [Fig. 5(b)]. The stresses near the surface (Z z 0) are [15] g,>d oh
1
Normalwed
4/n
0y.vd -= gh
05
The normalized u,, stresses in the dielectric at the symmetry plane, near the interface (pi = 0, : = 0), are plotted in Fig. 6 as functions of the normalized misfit stress. R = E,,,AaAT/(l -v)e,. The o,, stress increases linearly with increase in fi until the metal line has fully yielded, at n z 1. Thereafter, the e,., remains constant. The stress distributions along the symmetry plane (y = 0) after the metal line has fully yielded are summarized in Figs 7 and 8 (for h/d = 0.025, 0.04, 0.1 and h,/d = 1, 10). Note that the same normalized stresses would obtain before yielding, if the yield strength ca were replaced by the misfit stress in the metal. For a relatively thin substrate (h,/d = 1). the stresses vary linearly with the distance from the surface, because the stresses are dominated by bending (Fig. 7). For thicker substrates, the
(5b)
The total stresses in the dielectric substrate are the sum of equations (4) and (5). For plane strain (E,, = 0), the err stresses can be readily obtained from 0,). because e._: x 0, giving, or, = W,,
(6a)
0X.Y + 01) =(T,,,(l +v).
(6b)
such that
3.3. Numerical results The finite element method was used to obtain detailed results for the substrate stresses induced by metal lines or pads (Fig. 3). Problems 1 and 2 (metal lines) were treated as plane strain and problem 3 (metal pad) as axisymmetric. For comparison, a 3-D finite element analysis has been performed for a square pad, problem 4. A general purpose finite element code, ABAQUS, with eight-node biquadratic elements is used for the plane strain and axisymmetric calculations and 20-node quadratic brick elements for the 3-D calculations. 3.3.1. Uniform thickness metallizations. The analytical approximations have established that the normalized stresses in the dielectric, cr,d/cr,h, are only dependent on the system geometry. They are independent of the material properties. This was first affirmed by finite element results calculated for narrow lines with a range of properties: Ed/E,,, = l-6 and 5,/E,,, = 0.0005-0.007 (subscript d for dielectric and m for metallization).
surface
4 , , ,,,,
-5
L
,//
,, ,
Fig. 7. Stresses in the dielectric beneath the metal 0: = 0) for a fully yielded narrow metal line, h,/d = 1. The analytical results are also shown.
HE et al.:
4, ,,,,, ,, , ,, , ,
!
,
,
RESIDUAL
2351
STRESSES
,
h,id = 10
Distance
from Surface,
z/d
8. Stresses in the dielectric beneath the metal 0, = 0) for a fully yielded narrow, thin metal line, h,/d = 10.
B
-
h/d = 0.025
:; ii
-10
I
i
!I
I O5
I
I
k
Ck?tllW
I -;d-
I
I
I ’
Edge
Fig. 9. Stresses in the dielectric, near the surface (z = 0) for a fully yielded narrow, thin metal line (h,/d = 10). The analytical
Fig. 10. Stresses in the substrate beneath a circular pad, at the symmetry location 0, = 0): (a) thin substrate, h,/d = 1; (b) thick substrate, (h,/d = 10).
result is also shown.
the numerical results approach at smaller h/d (Fig. 9). stresses vary rapidly near the metal strip (Fig. 8) and are given with good accuracy by the analytical formula [equation (411.The large stress concentrations near the edge (z = 0, y = d) are confirmed (Fig. 9). Moreover,
the analytical
(ii) Circular Pad The equivalent results for the circular pad are summarized in Figs 10 and 11. As for the line. the
Edge v
Distance
Fig. 11. Stresses
from Center,
r/d
near the surface
results
Distance
of the dielectric (z = 0) beneath a circular stresses; (b) tangential stress.
from Center,
rid
pad, (h,/d = 10): (a) radial
HE et al.:
2358
,Symmetry
Plane
RESIDUAL
\
Fig. 12. The finite element mesh used for analysis of the square pad. For symmetry reasons, only l/4 of the pad/ substructure need be analyzed. The metal pad is at the top right outlined by ABCD. It is located on the face projected onto the plane of the figure.
film becomes fully plastic when Q 2 1. The stress distributions along the symmetry plane (y = 0) after the metal pad has fully yielded are summarized in Fig. 10 (for h/d = 0.025, 0.05, 0.1 and h,/d = 1, 10). For a relatively thin substrate (h,/d = l), the stresses again vary linearly with the distance from the surface [Fig. 10(a)]: whereas, for thicker substrates, the stresses vary rapidly near the metal strip [Fig. 10(b)]. The numerical results for the radial or, and tangential uBu stresses near the edge and the interface (Fig. 11) indicate higher stresses at smaller h/d.
STRESSES
nodes (Fig. 12). The computation required about 17 h of CPU time to complete 10 increments, to ensure that the metal pad had fully yielded. The crV,,stresses along the symmetry plane (s = 0, z = 0) near the interface, after the metal pad has fully yielded (Fig. 13) coincide closely with the axisymmetric values, except in the region near the edge. This result suggests that the results for the circular pads can be used to a good approximation for square pads. 3.3.2. Curved lines. For a line with constant curvature, the u,, stress distributions along the interface (Z = 0) after the metal line has fully yielded are summarized in Fig. 14 (for h/d = 0.1, 0.2, 1.0 and h,/d = 10). These stresses are compressive at the center of the line, but become tensile near the edge and reach a maximum at the edge. A comparison with the solution for a line of uniform thickness [Fig. 14(b)] illustrates one similarity and two differences. The stresses along the symmetry plane (I; = 0) are similar. However, when the line has a curved surface, the compressive edge stress concentration is absent and the tensile concentration has substantially reduced magnitude. A rounded edge having the measured profile indicated in Fig. l(b) causes similar effects [Fig. 14(b)] but the stress reductions are much smaller.
1 _i &b----__________ j 2 6
(iii) Square Pad
z
c
.*.A
-2
____---
__-*
i
The 3-D finite element analysis performed for a square pad, with h/d = 0.025, h,/d = 10, includes 2900 20-node quadratic brick element and 13923
Edge -10
R Center of pad
si‘
~
y/d -
Edge of pad
Fig. 13. The aY> stresses near the surface (z = 0) for the square pad, compared with those for the circular pad (/q/d = IO).
; CWM
8
I
I
I
1 O5
t
b j
1 i
h
1
1 ‘5.
j
c
*
y,d&,L
Edge
Fig. 14. (a) The CT~., stresses near the surface (2 = 0) of a line having constant curvature (h,/d = IO); (b) a comparison of the G,, stresses for lines having different profiles.
HE et al.:
I
I.
1.5
R
%
Position
(mm)
RESIDUAL
I
225
4. PRELIMINARY EXPERIMENTAL RESULTS
Fluorescence scans made across the center of the flat, square Au/Ge pads are indicated in Fig. 15. The line shifts exhibit the same four features evident in the stresses: (i) sharp maxima at the edges of the pads; (ii) a minimum at the pad center; (iii) a rapid approach to zero beyond the pads; and (iv) a sign change from regions beneath the pad to those outside. The line shifts beneath the pads were quite reproducible, but there was appreciable pad-to-pad variability in the magnitude of the pads at the pad termination. A convenient means for comparing the measured peak shifts with the calculated values is to jrst use the results near the pad center, where the stresses are relatively uniform and essentially unaffected by the probe width. Subsequently, the full spatial distribution can be correlated. For circular, flat pads (h,/d z 10, the stresses near the center) are (Figs 10 and ll), (7)
* 0. CT;?z rJzxz Is;) N The line shift [equation
(l)] is thus
Av z 5.2II,(cr,h/d) enabling as
the stress in the metallization
00
2359
temperature induced by the laser, which has not been accounted for in the analysis. Further study would be needed to address this effect. The probe used in these studies has sufficientl) narrow width (3pm) that the convolution of the calculated stresses with the probe function p(x), is essentially the same as the stress itself. Hence, it should be possible to superpose the measured stresses directly onto the values calculated for the pad with the rounded edges (Fig. 15). The measured values., though having extensive variability at the edges., appear to reproduce the calculations.
Fig. 15. Two experimental results for the cyy stresses beneath a Au/Ge square pad on sapphire. Also plotted is the stress calculated for a pad having the measured profiled indicated in Fig. l(c).
crrr z co0 = -2.6rs,h/d
STRESSES
(84 to be expressed
Av (d/h)
73i=iy.
Inserting the measured line shift gives a stress (TVz 600 MPa. This stress is comparable with the tensile yield strength for this eutectic, estimated by hardness indentation as 550 MPa. The implication is that the thermal expansion mismatch, upon cooling after deposition, has been sufficient to cause yielding of the eutectic. However, there is an elevation in the
5. CONCLUDING
REMARKS
Stresses induced in a dielectric substrate by surface metallizations have been calculated for a range of geometric configurations. The stresses attain maximum values dictated by the yield strength of the metal. These arise whenever the misfit strains caused b) thermal expansion, etc. are large enough to exceed the yield strain. There are large stress concentrations near the edges of the lines. These stress peaks can be diminished by altering the profile of the metallization, toward a more rounded configuration. Piezo-spectroscopic measurements of the stress in sapphire induced by metal pads have been made. The principal finding is that the stresses measured beneath the pad (in the sapphire), where edge effects have a minimal influence, coincide closely with the calculated values.
REFERENCES 1. A. T. English and C. M. Melliar-Smith, Ann. Rev. Muter. Sci. 8, 459 (1978). 2. R. R. Tummala. J. Am. Ceram. Sot. 74. 894 (1991). 3. R. R. Tummala and E. J. Rymaszewski, Microelectronic, PackaPinP Handbook. Van Nostrand-Reinhold. New York (1989). 4. W. D. Nix, Muter. Trans. ZOA, 2217 (1989). 5. J. W. Hutchinson and Z. Suo, Adv. uppl. Mech. 29, 63 (1992). Actu metall. mater. 6. A. G. Evans and J. W. Hutchinson, 43, 2507 (1985). 7. M. E. Thomas, M. P. Harnett and J. E. McKay, J. Vuc. Sci. Technol. A6, 2570 (1988). 8. I. C. Novan . and J. B. Cohen. Residual Stress-Measure ment by D@uction and Interpretation. Springer-Verlag, NY (1987). 9. B. Eigenmann, B. Scholtes and E. Macherauch, Surj: Engng 7, 221 (1993). 10. Q. Ma and D. R. Clarke, AMD, 181, American Society of Mechanical Engineers, 27 (1993). Il. D. M. LiDkin and D. R. Clarke. Fract. Mech. 25. 250 (1995). 12. Q. Ma, S. Chiras, D. R. Clarke and Z. Suo, J. uppl. Phys. 78, 1614 (1995). 13. J. He and D. R. Clarke, J. Am. Ceram. Sot. 78, 1347 (1995). 14. Q. Wen, D. R. Clarke, N. Yu and M. Nastasi, Appl. Phys. Lett. 66, 293 (1995). 15. S. Timoshenko and J. N. Goodier, Theory of_E/asticity. McGraw-Hill, NY (1950).
Pergamon 0956-7151(95)00341-X
RESIDUAL STRESSES IN DIELECTRICS METALLIZATION LINES AND M. Y. HE’, J. LIPKIN’,
D. R. CLARKE’,
Elsevier Science Ltd Copyright 0 1996 Acta Metallurgica Inc. Printed in Great Britain. All rights reserved 1359~6454196$15.00 + 0.00
CAUSED PADS
BY
A. G. EVANS’ and M. TENHOVER
‘Materials Department, University of California, Santa Barbara, CA 93106-5050, ‘Division of Applied Sciences, Harvard University, Pierce Hall, Cambridge, MA 02138 and jWendel1 Research Center, Carborundum Company, Niagara Falls, NY, U.S.A. (Received 12 May 1995; in revised form 25 August 1995) Abstract-Residual stresses in dielectrics and semiconductors induced by metal lines, pads and vias can have detrimental effects on the performance of devices and electronic packages. Analytical and numerical calculations of these stresses have been performed for two purposes. (1) To illustrate how these stresses
relate to the residual stress in the metallization and its geometry; (2) to calibrate a piezo-spectroscopic method for measuring these stresses with high spatial resolution. The results of the calculations have been presented using non-dimensional parameters that both facilitate scaling and provide connections to the stresses in the metal, with or without yielding. Preliminary experimental results obtained for Au/Ge eutectic pads illustrate the potential of the method and the role of the stress analysis.
1. INTRODUCTION Residual stresses have an important (and sometimes, crucial) influence on the fabrication and reliability of many devices [l-3]. Both intrinsic and thermal expansion mismatch stresses are involved [4]. A knowledge base has been established that provides some understanding about the origin of these stresses [4], as well as their effect on decohesion at interfaces and on cracking [S, 61. However, a continuing problem is the quantitative prediction of the magnitude of these stresses and of the incidence of decohesion and cracking. The principal problem concerns the stresses induced by the metallization. These are caused either by interconnects or vias or by the brazes used to attach the Si to the dielectric [l, 21. Residual stresses associated with metallization have previously been measured by one of two methods: beam bending and X-rays [7,8]. These methods have been used with continuous metal films on either dielectric or semiconductor substrates. Beam curvature and X-ray line shift measurements give the average stress in the metal. In special cases, X-ray methods can measure the stress gradient as a function of depth normal to the interface [9]. The greater challenge is to address the spatial distribution of stress in the substrate when the metallization is patterned to form interconnects, vias or pads. These stresses dominate interface decohesion and cracking [5,6]. So far, measurements have been sparse and calculations limited. Laser-based piezospectroscopic methods have the requisite resolution [lo-121. These methods include fluorescence [lo, 1 l]
and Raman spectroscopy [12]. Either method is capable of obtaining stress information in the substrate around the metallization. One purpose of the present study is to demonstrate the fluorescence spectroscopy method for obtaining stresses in a dielectric substrate with high spatial resolution. The other is to inter-relate these stresses to those in the metal. Numerical calculations and preliminary experiments performed on eutectic metal pads are used for these purposes. 2. STRESS MEASUREMENT
METHOD
2.1. Approach A gold-germanium eutectic on Cr-doped sapphire (ruby) was chosen as the model metal/dielectric system, such that stress measurements could be made using a piezo-spectroscopic technique, based on the systematic shift of fluorescence lines with stress [IO]. Electrons excited from the chromium ions in the ruby by a laser decay back to the ground state. emitting photons in the process. In stress-free ruby. these photons occur at two characteristic frequencies. Application of stress perturbs the local environment. of the chromium ion, shifting the fluorescence lines to higher or lower frequencies, depending on the stress state [l 1, 131. This frequency shift Av can be related to the stresses in the sapphire lattice oiir by the following constitutive relation [13]: Av = H,a,,
(1)
where Hi is the first-order piezo-spectroscopic coefficient. Second-order effects are small for the configurations used in the present study and are
2353
HE et al.:
2354
RESIDUAL
neglected. However, Av is affected by the temperature [ 131, such that laser-induced temperature changes can be problematical, unless independently measured. 2.2. Experimental
procedures
Stress measurements have been made in the substrate surface by employing a chromium “intechnique [14], wherein only the top diffusion” (micron-thick) layer of the substrate fluoresces. For this purpose, a basal plane oriented sapphire substrate was prepared by evaporating 2OOA of chromium onto one surface, followed by heating in a vacuum furnace for 2 h at 1500°C to diffuse the chromium into the sapphire. The metal features were created by DC Magnetron sputtering. The first two layers comprised an adhesion layer of W/Ti and a diffusion barrier of Pt/Rh, each - 100 nm thick. The final 40 pm thick Au/Ge layer had the eutectic composition. The features comprised square pads, 2.1 mm on edge, having rounded corners (Fig. 1). A typical surface profile is shown in Fig. l(a). A magnification of the edge is shown in Fig. l(b). However, there was significant variability in the edge profile from pad to pad. Stress measurements were made using an Optical Microprobe in which a laser beam could be focused
STRESSES
to a spot - 1 pm diameter. Samples were probed across the center of each feature by translating in 1 pm steps using a linear actuator. Argon reference spectra were used to correct for mechanical drift in the optics. The temperature was monitored to account for peak shifts due to temperature changes. The spectra were analyzed using a commercial software package, wherein each peak was fit to a mixed GaussianLorentzian function. The central position of each peak was used to determine the stress by taking a point far from the feature as a stress-free reference. 2.3. Spatial convolution When the stresses exhibit large spatial variability over short distances, the finite size of the optical probe convolutes the stress function, e(x) with the probe response function, p(x). In order to deconvolute the measured fluorescence profile, p(x) must be experimentally calibrated. This was done by using a step-source. The function C(x) measured at such a source is simply related to p(x), by, p(.~) rz dC;‘dr.
Experimentally, a step function has been created by placing a 200 8, thick line of Au onto a Cr3+ doped sapphire substrate. Scanning in 1 pm steps across the edge has given the convolution of the probe function with a step function shown in Fig. 2(a). Upon fitting the derivatives to a Lorentzian, [Fig. 2(b)], the probe
1200
??
? ?
,,,,,,,/,,11,,~,,,,,,,,,,,,,,,,,,,,,,,,
3
? ? ? ?
-5
Dv%wxx
(2)
(vm)
Fig. 1. (a) Atomic force microscopy scan of Au/Ge eutectic pad on sapphire substrate; (b) magnification of the edge profile.
0
Position,
5
10
15
x (wm)
Fig. 2. (a) Signal for Au strip C(l) that convolutes the probe function p(x) with the step function. (b) The derivative of C(x) that gives the probe width.
HE et al.:
RESIDUAL
STRESSES
2355
Problem @
Problem @
Problem @
Problem @
Fig. 3. The metallization
geometries
width at half maximum was found to be 3 pm. The high spatial resolution of the fluorescence probing is thus affirmed. 3. STRESS ANALYSIS 3.1. Procedure A finite element method has been used to calculate the stresses in the dielectric. Four representative metallization geometries have been used: a line having
used for the numerical
calculations.
uniform thickness [Fig. 3(a)], a line having constant curvature [Fig. 3(b)], a circular pad [Fig. 3(c)] and a square pad [Fig. 3(d)] both with constant thickness. The dielectric is considered to be elastic and the metal elastic-perfectly plastic, with yield strength, crO. Residual strain in the system has been motivated by a mismatch in thermal expansion coefficient between the metal, TV,, and the dielectric, cld (Aa = a, -Q), and a cooling temperature, AT (Fig. 4). 3.2. Analytical approximations Useful principles are established by means of approximate analytical calculations (Fig. 5). A pair of line forces in the x-axis, P per unit thickness,
dielectric
initial state (stress-free)
unattached: cooling by AT
(a)
(4
B 0
(a)
6
Stresses Induced by Surface Forces, P
h
unattached: apply 6 to achieve displacement continuity at interface
attach: surface forces 5 applied for stress free requirements
ti (b)
(4 Fig. 4. The differential
(d) strain that causes the stresses in the substrate.
BendIng Induced Stresses
Fig. 5. The forces and moments imposed by the metallization used to obtain analytical estimates for the stress in the dielectric.
HE et al.:
2356
is applied terminates
to the dielectric surface (J’ = +d), such that,
where
P = -31
RESIDUAL
STRESSES
the film
(3)
where 0 is the average misfit (residual) stress in the film and h is the film thickness. The stresses e)?. in the dielectric around thin metal strips, width 2U: are then obtained from the following formulae [Fig. 5(a)]. The stresses along the dielectric surface (Z = 0), are [15] g,d 41~ C% - [(y/d)2 - l] and along the symmetry
(4a) 0
plane (y = 0),
[(z/d)’ + 11
and along the symmetry
Misfit Stress,
2
25
3
E,AaAT/(l-v)Oo
All subsequent results are for, I?,/&,, = 6 and Q,/E, = 0.001. The system geometries were varied within the ranges 0.04 < h/d 2 0.1 and 1 2 h,/d s 10. A typical finite element mesh includes 760 elements and 2409 nodes. A convergence study revealed that these meshes provided accurate results. fij Narrow; Lines
3d h, plane (_r = 0),
15
Fig. 6. Effect of misfit strain on the stress induced in the dielectric beneath a thin metal line (i = 1’= 0).
The large tensile stress concentrations near the edge are a particular concern for fracture in the dielectric and for interface debonding. For finite substrates thickness h,, bending occurs and redistributes the stress. The stresses induced by bending are given by beam theory [Fig. 5(b)]. The stresses near the surface (Z z 0) are [15] g,>d oh
1
Normalwed
4/n
0y.vd -= gh
05
The normalized u,, stresses in the dielectric at the symmetry plane, near the interface (pi = 0, : = 0), are plotted in Fig. 6 as functions of the normalized misfit stress. R = E,,,AaAT/(l -v)e,. The o,, stress increases linearly with increase in fi until the metal line has fully yielded, at n z 1. Thereafter, the e,., remains constant. The stress distributions along the symmetry plane (y = 0) after the metal line has fully yielded are summarized in Figs 7 and 8 (for h/d = 0.025, 0.04, 0.1 and h,/d = 1, 10). Note that the same normalized stresses would obtain before yielding, if the yield strength ca were replaced by the misfit stress in the metal. For a relatively thin substrate (h,/d = 1). the stresses vary linearly with the distance from the surface, because the stresses are dominated by bending (Fig. 7). For thicker substrates, the
(5b)
The total stresses in the dielectric substrate are the sum of equations (4) and (5). For plane strain (E,, = 0), the err stresses can be readily obtained from 0,). because e._: x 0, giving, or, = W,,
(6a)
0X.Y + 01) =(T,,,(l +v).
(6b)
such that
3.3. Numerical results The finite element method was used to obtain detailed results for the substrate stresses induced by metal lines or pads (Fig. 3). Problems 1 and 2 (metal lines) were treated as plane strain and problem 3 (metal pad) as axisymmetric. For comparison, a 3-D finite element analysis has been performed for a square pad, problem 4. A general purpose finite element code, ABAQUS, with eight-node biquadratic elements is used for the plane strain and axisymmetric calculations and 20-node quadratic brick elements for the 3-D calculations. 3.3.1. Uniform thickness metallizations. The analytical approximations have established that the normalized stresses in the dielectric, cr,d/cr,h, are only dependent on the system geometry. They are independent of the material properties. This was first affirmed by finite element results calculated for narrow lines with a range of properties: Ed/E,,, = l-6 and 5,/E,,, = 0.0005-0.007 (subscript d for dielectric and m for metallization).
surface
4 , , ,,,,
-5
L
,//
,, ,
Fig. 7. Stresses in the dielectric beneath the metal 0: = 0) for a fully yielded narrow metal line, h,/d = 1. The analytical results are also shown.
HE et al.:
4, ,,,,, ,, , ,, , ,
!
,
,
RESIDUAL
2351
STRESSES
,
h,id = 10
Distance
from Surface,
z/d
8. Stresses in the dielectric beneath the metal 0, = 0) for a fully yielded narrow, thin metal line, h,/d = 10.
B
-
h/d = 0.025
:; ii
-10
I
i
!I
I O5
I
I
k
Ck?tllW
I -;d-
I
I
I ’
Edge
Fig. 9. Stresses in the dielectric, near the surface (z = 0) for a fully yielded narrow, thin metal line (h,/d = 10). The analytical
Fig. 10. Stresses in the substrate beneath a circular pad, at the symmetry location 0, = 0): (a) thin substrate, h,/d = 1; (b) thick substrate, (h,/d = 10).
result is also shown.
the numerical results approach at smaller h/d (Fig. 9). stresses vary rapidly near the metal strip (Fig. 8) and are given with good accuracy by the analytical formula [equation (411.The large stress concentrations near the edge (z = 0, y = d) are confirmed (Fig. 9). Moreover,
the analytical
(ii) Circular Pad The equivalent results for the circular pad are summarized in Figs 10 and 11. As for the line. the
Edge v
Distance
Fig. 11. Stresses
from Center,
r/d
near the surface
results
Distance
of the dielectric (z = 0) beneath a circular stresses; (b) tangential stress.
from Center,
rid
pad, (h,/d = 10): (a) radial
HE et al.:
2358
,Symmetry
Plane
RESIDUAL
\
Fig. 12. The finite element mesh used for analysis of the square pad. For symmetry reasons, only l/4 of the pad/ substructure need be analyzed. The metal pad is at the top right outlined by ABCD. It is located on the face projected onto the plane of the figure.
film becomes fully plastic when Q 2 1. The stress distributions along the symmetry plane (y = 0) after the metal pad has fully yielded are summarized in Fig. 10 (for h/d = 0.025, 0.05, 0.1 and h,/d = 1, 10). For a relatively thin substrate (h,/d = l), the stresses again vary linearly with the distance from the surface [Fig. 10(a)]: whereas, for thicker substrates, the stresses vary rapidly near the metal strip [Fig. 10(b)]. The numerical results for the radial or, and tangential uBu stresses near the edge and the interface (Fig. 11) indicate higher stresses at smaller h/d.
STRESSES
nodes (Fig. 12). The computation required about 17 h of CPU time to complete 10 increments, to ensure that the metal pad had fully yielded. The crV,,stresses along the symmetry plane (s = 0, z = 0) near the interface, after the metal pad has fully yielded (Fig. 13) coincide closely with the axisymmetric values, except in the region near the edge. This result suggests that the results for the circular pads can be used to a good approximation for square pads. 3.3.2. Curved lines. For a line with constant curvature, the u,, stress distributions along the interface (Z = 0) after the metal line has fully yielded are summarized in Fig. 14 (for h/d = 0.1, 0.2, 1.0 and h,/d = 10). These stresses are compressive at the center of the line, but become tensile near the edge and reach a maximum at the edge. A comparison with the solution for a line of uniform thickness [Fig. 14(b)] illustrates one similarity and two differences. The stresses along the symmetry plane (I; = 0) are similar. However, when the line has a curved surface, the compressive edge stress concentration is absent and the tensile concentration has substantially reduced magnitude. A rounded edge having the measured profile indicated in Fig. l(b) causes similar effects [Fig. 14(b)] but the stress reductions are much smaller.
1 _i &b----__________ j 2 6
(iii) Square Pad
z
c
.*.A
-2
____---
__-*
i
The 3-D finite element analysis performed for a square pad, with h/d = 0.025, h,/d = 10, includes 2900 20-node quadratic brick element and 13923
Edge -10
R Center of pad
si‘
~
y/d -
Edge of pad
Fig. 13. The aY> stresses near the surface (z = 0) for the square pad, compared with those for the circular pad (/q/d = IO).
; CWM
8
I
I
I
1 O5
t
b j
1 i
h
1
1 ‘5.
j
c
*
y,d&,L
Edge
Fig. 14. (a) The CT~., stresses near the surface (2 = 0) of a line having constant curvature (h,/d = IO); (b) a comparison of the G,, stresses for lines having different profiles.
HE et al.:
I
I.
1.5
R
%
Position
(mm)
RESIDUAL
I
225
4. PRELIMINARY EXPERIMENTAL RESULTS
Fluorescence scans made across the center of the flat, square Au/Ge pads are indicated in Fig. 15. The line shifts exhibit the same four features evident in the stresses: (i) sharp maxima at the edges of the pads; (ii) a minimum at the pad center; (iii) a rapid approach to zero beyond the pads; and (iv) a sign change from regions beneath the pad to those outside. The line shifts beneath the pads were quite reproducible, but there was appreciable pad-to-pad variability in the magnitude of the pads at the pad termination. A convenient means for comparing the measured peak shifts with the calculated values is to jrst use the results near the pad center, where the stresses are relatively uniform and essentially unaffected by the probe width. Subsequently, the full spatial distribution can be correlated. For circular, flat pads (h,/d z 10, the stresses near the center) are (Figs 10 and ll), (7)
* 0. CT;?z rJzxz Is;) N The line shift [equation
(l)] is thus
Av z 5.2II,(cr,h/d) enabling as
the stress in the metallization
00
2359
temperature induced by the laser, which has not been accounted for in the analysis. Further study would be needed to address this effect. The probe used in these studies has sufficientl) narrow width (3pm) that the convolution of the calculated stresses with the probe function p(x), is essentially the same as the stress itself. Hence, it should be possible to superpose the measured stresses directly onto the values calculated for the pad with the rounded edges (Fig. 15). The measured values., though having extensive variability at the edges., appear to reproduce the calculations.
Fig. 15. Two experimental results for the cyy stresses beneath a Au/Ge square pad on sapphire. Also plotted is the stress calculated for a pad having the measured profiled indicated in Fig. l(c).
crrr z co0 = -2.6rs,h/d
STRESSES
(84 to be expressed
Av (d/h)
73i=iy.
Inserting the measured line shift gives a stress (TVz 600 MPa. This stress is comparable with the tensile yield strength for this eutectic, estimated by hardness indentation as 550 MPa. The implication is that the thermal expansion mismatch, upon cooling after deposition, has been sufficient to cause yielding of the eutectic. However, there is an elevation in the
5. CONCLUDING
REMARKS
Stresses induced in a dielectric substrate by surface metallizations have been calculated for a range of geometric configurations. The stresses attain maximum values dictated by the yield strength of the metal. These arise whenever the misfit strains caused b) thermal expansion, etc. are large enough to exceed the yield strain. There are large stress concentrations near the edges of the lines. These stress peaks can be diminished by altering the profile of the metallization, toward a more rounded configuration. Piezo-spectroscopic measurements of the stress in sapphire induced by metal pads have been made. The principal finding is that the stresses measured beneath the pad (in the sapphire), where edge effects have a minimal influence, coincide closely with the calculated values.
REFERENCES 1. A. T. English and C. M. Melliar-Smith, Ann. Rev. Muter. Sci. 8, 459 (1978). 2. R. R. Tummala. J. Am. Ceram. Sot. 74. 894 (1991). 3. R. R. Tummala and E. J. Rymaszewski, Microelectronic, PackaPinP Handbook. Van Nostrand-Reinhold. New York (1989). 4. W. D. Nix, Muter. Trans. ZOA, 2217 (1989). 5. J. W. Hutchinson and Z. Suo, Adv. uppl. Mech. 29, 63 (1992). Actu metall. mater. 6. A. G. Evans and J. W. Hutchinson, 43, 2507 (1985). 7. M. E. Thomas, M. P. Harnett and J. E. McKay, J. Vuc. Sci. Technol. A6, 2570 (1988). 8. I. C. Novan . and J. B. Cohen. Residual Stress-Measure ment by D@uction and Interpretation. Springer-Verlag, NY (1987). 9. B. Eigenmann, B. Scholtes and E. Macherauch, Surj: Engng 7, 221 (1993). 10. Q. Ma and D. R. Clarke, AMD, 181, American Society of Mechanical Engineers, 27 (1993). Il. D. M. LiDkin and D. R. Clarke. Fract. Mech. 25. 250 (1995). 12. Q. Ma, S. Chiras, D. R. Clarke and Z. Suo, J. uppl. Phys. 78, 1614 (1995). 13. J. He and D. R. Clarke, J. Am. Ceram. Sot. 78, 1347 (1995). 14. Q. Wen, D. R. Clarke, N. Yu and M. Nastasi, Appl. Phys. Lett. 66, 293 (1995). 15. S. Timoshenko and J. N. Goodier, Theory of_E/asticity. McGraw-Hill, NY (1950).