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Thermo-elastic analysis of thin film multilayer structure with a crack
c
004%7949/87 13.00 + 0.00 1987 Pcrgunon Journals Ltd
THERMO-ELASTIC ANALYSIS OF THIN FILM MULTILAYER STRUCTURE WITH A CRACK HIROAKI DOI,SHINJISMUTA,TATSU~SAJCAMOTU and TASUKU SHIMIZU Mechanical Engineering Research Laboratory, Hitachi Ltd., 502 Kandatsu-machi, Tsuchiura-shi, Ibaraki 300, Japan Abetract-A post-processor which calculates the f-integral from the results of ADINA is developed and
a plied to cracks in thin film multilayer circuit boards for large-size computers. The relation between the je-mtegral and the material properties (Young’s modulus and the thermal expansion eeelllcient) of the
conductor and the insulator of the circuit boardis investigated.
1. INTRODlJCZ1ON
where .& is the potential energy release rate corresponding to the crack extension per unit area For a crack in an elastic body with temperature field, the j-integral is expressed as
The performance of large-size computers has been improved through an increase in their speed of calculation and decrease in their size. This improvement is due to LSIs (large scale integrated circuits) which have been improved rapidly. Recently it has become an important problem to decrease the propagation delay time and increase the LSI packaging density (the number of LSIs per unit area) of the circuit board in order to fully utilize the high performance of LSIs. The circuit board must have an insulator with a low dielectric constant to decrease the propagation delay time. Also it must have a conductor with a high conductivity to increase the LSI
packaging density. The thin film multilayer circuit board has these properties (Fig. 1). This board uses copper as a conductor, polyimide as an insulator and silicon as a substrate. As these materials have different thermal expansion coeficients (TEC), cracks tend to be produced in the board by a thermal strain. Also, the extension of these cracks brings about the decay of the board electrical characteristics. The extension of the cracks must be avoided to achieve circuit reliability. To prevent crack extension, it is effective to select conductor and insulator materials with suitable properties. In this paper the relation between the fracture mechanics parameter, j-integral, and the material properties of thin film multilayer circuit boards is investigated. These results can be used as a guideline in circuit design when the strength of the circuit
where O-x,x, is an orthogonal coordinate system whose x,-axis exists on the crack surface, r is a contour enclosing the crack tip, A is the area between r and the crack surface, We is elastic strain energy, ‘I;is surface traction, ui is displacement, eI, is stress and & is thermal strain as shown in Fig. 2. When the change of temperature is constant in the elastic body, the &integral is expressed by (3) as the third term in (2) is zero.
Equation (3) cannot be applied to general multilayer structures. An equation for the &integral for general multilayer structures has been proposed [2]. However, when the bond line exists on the x,-axis,
Conductor Insulator a
Substrate
1
materials is given.
f
I 100Ltm
Fig. 1. Thin Elm multilayercircuit board. 2. &INTEGRAL FORA CRACKON A BONDLME The j-integral Cl] is a fracture mechanics parameter which means the potential energy release rate of a crack extension in an elasto-plastic body when a body force and a thermal strain exist Then, the frac-
ture criterion is expressed as I=
S,,
(1) 283
Fig. 2. .f-integral of crack in elastic body.
284
HIROAKIDOI el al. Crack
Conductor,
(Type 3) Conductor
Li
‘ype 2)
Fig. 3. j-integral of bi-material body.
(3) can be used as the equation for the j-integral. The path independence of (3) is verified as follows, by the same method which is used for the extension of Rice’s J-integral [3] to bi-material bodies [4]. Two materials, M, and M, , bonded along the x,-axis, are shown in Fig. 3. The j-integral along rI + L, is equal to zero, as there is no singular point in the area inside r, + L, :
C&k
(Type 1)
m
Fig. 5. Cracks in thin film multilayer circuit board.
and calculates the J-integral. In this post-processor, eight-node two-dimensional solid elements are used. The integration path passes through integration points. The integral is made according to (2). 4.BINTEGRAL FOR CRACKS IN nl~ -IN MULTILAYER CIRCUIT BOARD
where (5) is used. W, dx, = 0.
(5)
i LI
Equation (4) also holds for rl + L, . Adding this equation to (4) results in: s, = 0,
(6)
where r indicates r, + TZ and (7) is used as ui2, u, and dui/dx, are continuous along the bond line:
jl,7;zdr+kTzdr=o.
(7)
Two materials, Ml and Ml, with a crack on their bond line along the x,-axis, are shown in Fig. 4. The j-integral along the closed contour rA + L, - Ts + L, is equal to zero. Also, the j-integral along I.., and L, is equal to zero. Therefore the j-integrals along arbitrary contours rA and Ts are equal.
4.1 Method The types of cracks in the thin film multilayer circuit board are shown in Fig. 5. There are three types of cracks; Type 1: in the insulator; Type 2: on the bond line between the conductor and the insulator; and Type 3: on the bond line between the insulator and the conductor which is on the insulator surface. As the TEC of the substrate is suITGently low, the TEC of the substrate is assumed to be 0 K- ‘. Also the conductor and the insulator are sufficiently thin compared with the substrate, so the conductor and the insulator are assumed to be constrained in the horizontal direction by the substrate. Considering their symmetry in shape, the boundary conditions for the three types of crack are shown in Figs 6-8 for one-quarter of the structures.
3.&INTEGRAL POST-PROCESSOR
The j-integral post-processor is a program which retrieves stress, strain and deformation from ADINA
m b=3Opm
Lb1
Fig. 6. Boundary condition (Type 1). Conductor
I-A Ml L, 0
-x1
L2 -rr,
M2
@ Fig. 4.j-integral of crack on bond line in bi-material body.
FILM
Fig. 7. Boundary condition (Type 2).
285
Thermo-elastic analysis of thin film multilayer structure
a=5flm b=3Orm
c=loltm El (Pa)
Fig. 8. Boundary condition (Type 3).
Fig. 10. &integral vs Young’s modulus of insulator (Type
For these three types of cracks, the relation between the &integral and the Young’s modulus of the conductor and the insulator, and the relation between the J-integral and the TEC of the conductor and the insulator are investigated. ADINA and the j-integral post-processor are used to calculate the j-integral. The calculation is made for a uniform temperature change of - 1°C. Young’s modulus and TEC take the values 1, 1 x lo’, 1 x 10’ and 1 x IO3 GPa and 1 x 10m6, 1 x 10m5, 2 x lo-’ and 1 x lo-’ K-l, respectively. Poisson’s ratio takes the value 0.3. An example of the finite element mesh (for Type 2) used for ADINA is shown in Fig. 9. The I-integral is calculated along three contours enclosing the crack tip. As the j-integral along the different contours takes almost the same value, their average is used as the result. The finite element mesh of each type has 937 nodes and 288 elements (Type l), 1397 nodes and 432 elements (Type 2), and 1305 nodes and 400 elements (Type 3), respectively. 4.2 Results and discussion The results for Type 1 are shown in Figs 10 and 11. The relation between the f-integral and the Young’s modulus of the insulator E, is shown in Fig. 10. The relation between the j-integral and the TEC of the insulator a1 is shown in Fig. 11. The j-integral is proportional to E, and the square of a,. This relation is also verified by the relation between the .&integral and the stress intensity factor in fracture mechanics. Cracks of Type 2 and Type 3 exist on the bond line between the conductor and the insulator. For these cracks, it became clear that J^E,af (the
A-part
I
I-’ Fig. Il. f-integral vs thermal expansion coelficicnt of insulator (Type 1).
f-integral divided by E, and the square of a,) is decided by l&/E, and a,/a,, where E, and aI indicate the Young’s modulus and the TEC of the conductor, respectively. Therefore the results for Type 2 and Type 3 are shown by the relation between .f/E,af and E,IE, and the relation between .f/E,af and a2Ja1. The results for Type 2 are shown in Figs 12 and 13. When E,/E, is small, .f/EIa: is proportional to
\ Integration path
Crack Fig. 9. Example of finite element mesh (Type 2).
10-l
Fig. 12. f-integral
IO0 E,/E,
IO’
0’
vs Young’s modulus of conductor and insulator (Type 2).
HIROAKI DOI et al.
286
r
tE2/E,
=IOO
+Ez/E,
=O.l
+Ez/E,
= 10
-0-&/E,
‘0.01
-eEp/E,
= 1 I
-*
10-l
1
I
IO0 az/a,
IO’
10:
'0 -11 10-2
' 10-l
-0
1
Ez/E,
100
=0.01
1
10’
&2/a,
Fig. 13. .f-integral vs thermal expansion cocflicimt of conductor and insulator (Type 2).
Fig. 15. .f-integral vs thermal expansion coefftcient of conductor and insulator (Type 3).
E21E, and when E,/E, surpasses 1, .ffE,af takes an almost constant value (Fig. 12). J/E,af increases as oI_z/a, increases and when a,Ja, becomes large, J/E,af is proportional to the square of al/al (Fig.
5. CONCLUSIONS
13). The results for Type 3 are shown in Figs 14 and 15. The relation between fJE,a: and E,/E, is very similar to that of Type 2 (Fig. 14). f/E,af is proportional to the square of aJar for all the aj/a, (Fig. 15). As the above results are restricted to cracks of one length, the effect of length must be made clear. Also, the development of measurement methods for the length of cracks in the thin film multilayer circuit board and the strength of circuit materials is an important problem to be solved.
‘Ii5 16’ 16’ 5 e
It”
* h
16’ It’ 16’ +ffp/t7,=1
0
16’ 16’
lo-’
Fig. 14. f-integral
10-l
100
10’
102
vs Young’s modulus of conductor and insulator (Type 3).
As a guideline for the design of thin film multilayer circuit boards, the relation between the J-integral and Young’s modulus of the circuit board materials, and the relation between the &integral and the TEC of the circuit board materials were clarified for the three types of cracks in circuit boards. REFERENCES 1. K. Kishimoto, S. Aoki and M. Sakata, On the path independent integral-1. Engng Froct. Mech. 13, 841-850 (1980). 2. M. Kikuchi, H. Miyamoto and S. Sugawara, The e&t of the cladding on the J-integral of the crack in the reactor vessel. Trrurc. JSME (A)%, 795-gOO(1986). 3. J. R. Rice, A path independent integral and the approximate analysis of strain concentration by notches and cracks. Trans. ASME, J. appl. Me& X3,379-386 (1968). 4. R. E. Smeler, On the J-integral for bi-material bodies. ht. 1. Fract. 13.382-384(1977).
004%7949/87 13.00 + 0.00 1987 Pcrgunon Journals Ltd
THERMO-ELASTIC ANALYSIS OF THIN FILM MULTILAYER STRUCTURE WITH A CRACK HIROAKI DOI,SHINJISMUTA,TATSU~SAJCAMOTU and TASUKU SHIMIZU Mechanical Engineering Research Laboratory, Hitachi Ltd., 502 Kandatsu-machi, Tsuchiura-shi, Ibaraki 300, Japan Abetract-A post-processor which calculates the f-integral from the results of ADINA is developed and
a plied to cracks in thin film multilayer circuit boards for large-size computers. The relation between the je-mtegral and the material properties (Young’s modulus and the thermal expansion eeelllcient) of the
conductor and the insulator of the circuit boardis investigated.
1. INTRODlJCZ1ON
where .& is the potential energy release rate corresponding to the crack extension per unit area For a crack in an elastic body with temperature field, the j-integral is expressed as
The performance of large-size computers has been improved through an increase in their speed of calculation and decrease in their size. This improvement is due to LSIs (large scale integrated circuits) which have been improved rapidly. Recently it has become an important problem to decrease the propagation delay time and increase the LSI packaging density (the number of LSIs per unit area) of the circuit board in order to fully utilize the high performance of LSIs. The circuit board must have an insulator with a low dielectric constant to decrease the propagation delay time. Also it must have a conductor with a high conductivity to increase the LSI
packaging density. The thin film multilayer circuit board has these properties (Fig. 1). This board uses copper as a conductor, polyimide as an insulator and silicon as a substrate. As these materials have different thermal expansion coeficients (TEC), cracks tend to be produced in the board by a thermal strain. Also, the extension of these cracks brings about the decay of the board electrical characteristics. The extension of the cracks must be avoided to achieve circuit reliability. To prevent crack extension, it is effective to select conductor and insulator materials with suitable properties. In this paper the relation between the fracture mechanics parameter, j-integral, and the material properties of thin film multilayer circuit boards is investigated. These results can be used as a guideline in circuit design when the strength of the circuit
where O-x,x, is an orthogonal coordinate system whose x,-axis exists on the crack surface, r is a contour enclosing the crack tip, A is the area between r and the crack surface, We is elastic strain energy, ‘I;is surface traction, ui is displacement, eI, is stress and & is thermal strain as shown in Fig. 2. When the change of temperature is constant in the elastic body, the &integral is expressed by (3) as the third term in (2) is zero.
Equation (3) cannot be applied to general multilayer structures. An equation for the &integral for general multilayer structures has been proposed [2]. However, when the bond line exists on the x,-axis,
Conductor Insulator a
Substrate
1
materials is given.
f
I 100Ltm
Fig. 1. Thin Elm multilayercircuit board. 2. &INTEGRAL FORA CRACKON A BONDLME The j-integral Cl] is a fracture mechanics parameter which means the potential energy release rate of a crack extension in an elasto-plastic body when a body force and a thermal strain exist Then, the frac-
ture criterion is expressed as I=
S,,
(1) 283
Fig. 2. .f-integral of crack in elastic body.
284
HIROAKIDOI el al. Crack
Conductor,
(Type 3) Conductor
Li
‘ype 2)
Fig. 3. j-integral of bi-material body.
(3) can be used as the equation for the j-integral. The path independence of (3) is verified as follows, by the same method which is used for the extension of Rice’s J-integral [3] to bi-material bodies [4]. Two materials, M, and M, , bonded along the x,-axis, are shown in Fig. 3. The j-integral along rI + L, is equal to zero, as there is no singular point in the area inside r, + L, :
C&k
(Type 1)
m
Fig. 5. Cracks in thin film multilayer circuit board.
and calculates the J-integral. In this post-processor, eight-node two-dimensional solid elements are used. The integration path passes through integration points. The integral is made according to (2). 4.BINTEGRAL FOR CRACKS IN nl~ -IN MULTILAYER CIRCUIT BOARD
where (5) is used. W, dx, = 0.
(5)
i LI
Equation (4) also holds for rl + L, . Adding this equation to (4) results in: s, = 0,
(6)
where r indicates r, + TZ and (7) is used as ui2, u, and dui/dx, are continuous along the bond line:
jl,7;zdr+kTzdr=o.
(7)
Two materials, Ml and Ml, with a crack on their bond line along the x,-axis, are shown in Fig. 4. The j-integral along the closed contour rA + L, - Ts + L, is equal to zero. Also, the j-integral along I.., and L, is equal to zero. Therefore the j-integrals along arbitrary contours rA and Ts are equal.
4.1 Method The types of cracks in the thin film multilayer circuit board are shown in Fig. 5. There are three types of cracks; Type 1: in the insulator; Type 2: on the bond line between the conductor and the insulator; and Type 3: on the bond line between the insulator and the conductor which is on the insulator surface. As the TEC of the substrate is suITGently low, the TEC of the substrate is assumed to be 0 K- ‘. Also the conductor and the insulator are sufficiently thin compared with the substrate, so the conductor and the insulator are assumed to be constrained in the horizontal direction by the substrate. Considering their symmetry in shape, the boundary conditions for the three types of crack are shown in Figs 6-8 for one-quarter of the structures.
3.&INTEGRAL POST-PROCESSOR
The j-integral post-processor is a program which retrieves stress, strain and deformation from ADINA
m b=3Opm
Lb1
Fig. 6. Boundary condition (Type 1). Conductor
I-A Ml L, 0
-x1
L2 -rr,
M2
@ Fig. 4.j-integral of crack on bond line in bi-material body.
FILM
Fig. 7. Boundary condition (Type 2).
285
Thermo-elastic analysis of thin film multilayer structure
a=5flm b=3Orm
c=loltm El (Pa)
Fig. 8. Boundary condition (Type 3).
Fig. 10. &integral vs Young’s modulus of insulator (Type
For these three types of cracks, the relation between the &integral and the Young’s modulus of the conductor and the insulator, and the relation between the J-integral and the TEC of the conductor and the insulator are investigated. ADINA and the j-integral post-processor are used to calculate the j-integral. The calculation is made for a uniform temperature change of - 1°C. Young’s modulus and TEC take the values 1, 1 x lo’, 1 x 10’ and 1 x IO3 GPa and 1 x 10m6, 1 x 10m5, 2 x lo-’ and 1 x lo-’ K-l, respectively. Poisson’s ratio takes the value 0.3. An example of the finite element mesh (for Type 2) used for ADINA is shown in Fig. 9. The I-integral is calculated along three contours enclosing the crack tip. As the j-integral along the different contours takes almost the same value, their average is used as the result. The finite element mesh of each type has 937 nodes and 288 elements (Type l), 1397 nodes and 432 elements (Type 2), and 1305 nodes and 400 elements (Type 3), respectively. 4.2 Results and discussion The results for Type 1 are shown in Figs 10 and 11. The relation between the f-integral and the Young’s modulus of the insulator E, is shown in Fig. 10. The relation between the j-integral and the TEC of the insulator a1 is shown in Fig. 11. The j-integral is proportional to E, and the square of a,. This relation is also verified by the relation between the .&integral and the stress intensity factor in fracture mechanics. Cracks of Type 2 and Type 3 exist on the bond line between the conductor and the insulator. For these cracks, it became clear that J^E,af (the
A-part
I
I-’ Fig. Il. f-integral vs thermal expansion coelficicnt of insulator (Type 1).
f-integral divided by E, and the square of a,) is decided by l&/E, and a,/a,, where E, and aI indicate the Young’s modulus and the TEC of the conductor, respectively. Therefore the results for Type 2 and Type 3 are shown by the relation between .f/E,af and E,IE, and the relation between .f/E,af and a2Ja1. The results for Type 2 are shown in Figs 12 and 13. When E,/E, is small, .f/EIa: is proportional to
\ Integration path
Crack Fig. 9. Example of finite element mesh (Type 2).
10-l
Fig. 12. f-integral
IO0 E,/E,
IO’
0’
vs Young’s modulus of conductor and insulator (Type 2).
HIROAKI DOI et al.
286
r
tE2/E,
=IOO
+Ez/E,
=O.l
+Ez/E,
= 10
-0-&/E,
‘0.01
-eEp/E,
= 1 I
-*
10-l
1
I
IO0 az/a,
IO’
10:
'0 -11 10-2
' 10-l
-0
1
Ez/E,
100
=0.01
1
10’
&2/a,
Fig. 13. .f-integral vs thermal expansion cocflicimt of conductor and insulator (Type 2).
Fig. 15. .f-integral vs thermal expansion coefftcient of conductor and insulator (Type 3).
E21E, and when E,/E, surpasses 1, .ffE,af takes an almost constant value (Fig. 12). J/E,af increases as oI_z/a, increases and when a,Ja, becomes large, J/E,af is proportional to the square of al/al (Fig.
5. CONCLUSIONS
13). The results for Type 3 are shown in Figs 14 and 15. The relation between fJE,a: and E,/E, is very similar to that of Type 2 (Fig. 14). f/E,af is proportional to the square of aJar for all the aj/a, (Fig. 15). As the above results are restricted to cracks of one length, the effect of length must be made clear. Also, the development of measurement methods for the length of cracks in the thin film multilayer circuit board and the strength of circuit materials is an important problem to be solved.
‘Ii5 16’ 16’ 5 e
It”
* h
16’ It’ 16’ +ffp/t7,=1
0
16’ 16’
lo-’
Fig. 14. f-integral
10-l
100
10’
102
vs Young’s modulus of conductor and insulator (Type 3).
As a guideline for the design of thin film multilayer circuit boards, the relation between the J-integral and Young’s modulus of the circuit board materials, and the relation between the &integral and the TEC of the circuit board materials were clarified for the three types of cracks in circuit boards. REFERENCES 1. K. Kishimoto, S. Aoki and M. Sakata, On the path independent integral-1. Engng Froct. Mech. 13, 841-850 (1980). 2. M. Kikuchi, H. Miyamoto and S. Sugawara, The e&t of the cladding on the J-integral of the crack in the reactor vessel. Trrurc. JSME (A)%, 795-gOO(1986). 3. J. R. Rice, A path independent integral and the approximate analysis of strain concentration by notches and cracks. Trans. ASME, J. appl. Me& X3,379-386 (1968). 4. R. E. Smeler, On the J-integral for bi-material bodies. ht. 1. Fract. 13.382-384(1977).