Prediction of microelectronic substrate warpage using homogenized finite element models

Microelectronic Engineering 83 (2006) 557–569 www.elsevier.com/locate/mee

Prediction of microelectronic substrate warpage using homogenized finite element models Parsaoran Hutapea a,*, Joachim L. Grenestedt b, Mitul Modi c, Michael Mello c, Kristopher Frutschy c a

Department of Mechanical Engineering, Temple University, 1947 N 12th Street, Philadelphia, PA 19122, United States b Department of Mechanical Engineering and Mechanics, Lehigh University, Bethlehem, PA 18015, United States c Assembly Technology Development, Intel Corporation, Chandler, AZ 85226, United States Received 26 August 2005; received in revised form 20 November 2005; accepted 16 December 2005 Available online 10 January 2006

Abstract The focus of this study was to numerically predict effective thermo-mechanical properties and substrate warpage of high-density microelectronic substrates used in organic CPU packages. Microelectronic substrates are typically composed of several polymer, fiber-weave, and copper layers and are filled with a variety of complex features, such as electric traces, plated-through-holes, micro-vias, and adhesion holes. When subjected to temperature changes, these substrates may warp, driven by the mismatch in coefficients of thermal expansion (CTE) of the constituent materials. This study focused on predicting substrate warpage in an isothermal condition. The numerical approach consisted of three major tasks: estimating homogenized (effective) thermo-mechanical properties of the features; calculating effective properties of discretized layers using the effective properties of the features; and assembling the layers to create twodimensional (2D) finite element (FE) plate models and to calculate warpage of the substrates. The effective properties of the features were extracted from three-dimensional (3D) unit cell FE models, and closed-form approximate expressions were developed using the numerical results, curve fitting, and some simple bounds. The numerical approach was applied to predict warpage of production substrates, analyzed, and validated against experimentally measured stiffness and CTEs. In this paper, the homogenization approach, numerical predictions, and experimental validation are discussed.
2005 Elsevier B.V. All rights reserved. Keywords: Microelectronic substrates; Reliability; Thermo-mechanical; Finite element

1. Introduction The objective of the present work was to predict effective thermo-mechanical properties and isothermal warpage of high-density microelectronic substrates used in organic CPU packages (Fig. 1(a) and (b)). In principle, the numerical method used in this work is similar to the method that was previously developed by Grenestedt and Hutapea [1–4] for predicting isothermal warpage of a production printed circuit board (PCB) fabricated using X-Y routing method

*

Corresponding author. Tel.: +1 2152047805; fax: +1 2152044956. E-mail address: [email protected] (P. Hutapea).

0167-9317/$ - see front matter
2005 Elsevier B.V. All rights reserved. doi:10.1016/j.mee.2005.12.009

[5]. The objectives of the previous work were to estimate the effective properties of the copper layers and, using the effective properties, to develop simplified 2D. Finite element (FE) plate models of the PCB to be used simultaneously with an optimization procedure to reduce the PCB warpage. The classical lamination theory [6] was used to model the PCB. However, in the previous work, only electric trace was considered as the main substrate feature by neglecting other PCB features, such as vias, copper pads, adhesion holes, etc. The optimization of the PCB electric artworks was done by changing only electric trace widths and/or spacing to find a PCB with the least warpage. It should be mentioned that 3D FE analyses may be used for such modeling and tuning, but the

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P. Hutapea et al. / Microelectronic Engineering 83 (2006) 557–569

Fig. 1. (a) A microelectronic substrate, (b) a cross-sectional area, and (c)–(f) cross-sectional photos of substrate features.

computational effort would be too overwhelming for such an optimization procedure. The numerical predictions were confirmed with some experimental work and showed reasonable agreements [3,4]. Microelectronic substrates, in this case, were more complex and much smaller than the PCBs. In addition to electric traces, other substrate features, such as micro-vias, plated-through-holes (PTH), and adhesion holes (Fig. 1(c)–(f)) must be considered in the numerical homogenization procedure. From some experiments, these additional features were found to significantly influence the warpage of the substrates as much as the electric trace [7,8]. Thus, the major task of this work was to calculate effective thermo-mechanical properties of substrate fea-

tures using three-dimensional (3D) FE unit cell analyses [1–4]. In this paper, analytical models of effective thermomechanical properties of microelectronic substrate features are presented. In summary, the numerical approach was performed in three steps: estimating effective thermomechanical properties of the features; calculating effective properties of a discretized layer using the effective properties of the features; and assembling the layers to create 2D FE plate models and to calculate warpage of the whole substrates. These steps were illustrated in Fig. 2. The details of the determination of the effective thermomechanical properties are presented in Section 2. Using the effective properties, simplified 2D FE analyses to predict substrate warpage are discussed in Section 3. Finally,

Fig. 2. Steps for the substrate homogenization method.

P. Hutapea et al. / Microelectronic Engineering 83 (2006) 557–569

experimental verifications of the numerical prediction were performed to confirm the predictive results and these are presented in Section 4. 2. Analytical models of effective thermo-mechanical properties of substrate features The basic concept was to derive analytical models of effective properties of a lamina in a substrate and use the effective properties in conjunction with classical lamination theory to estimate thermo-mechanical properties of the substrate. The effective properties were extracted from full 3D unit cell models of substrate features, for example, as shown in Figs. 3–6. The microelectronic substrates used were composed of eight copper layers, seven dielectric layers as the intermediate layers, and a core layer consisting of PTHs as shown in Fig. 1(b). The substrate consisted of materials, such as woven glass/epoxy (FR4) typically used for the core, copper for the electric traces, micro-via and PTH plating, a built-up-film (BUF) polymer material for the copper and dielectric layers, and a plug material for the PTH. The properties of these materials were given in Table 1 and assumed to be independent of temperature changes. Note that this assumption should be sufficient for the current work, since the working temperature was the glass transition temperature of the materials [9]. 2.1. Effective properties obtained from FE unit cell analysis Electric trace FE models, as shown in Fig. 3, had eight layers with copper traces in alternating direction. The

559

copper layers were separated by dielectric layers modeled using BUF materials, and a core layer consisting of PTHs. The stack, as illustrated in Fig. 1(b), consisted of eight 0.015-mm thick copper layers, separated by 0.03-mm thick dielectric layers and PTH layer, resulting in a total substrate thickness of 1.12 mm. The in-plane size of the model was w · w, where w was varied from 0.0254 to 2.54 mm for the case, where the trace width, h, was 0.0254 mm and from 0.0834 to 83.4 mm for the case, where h was 0.0834 mm. The effective properties of the lamina with copper traces were extracted as a function of h/w, which was varied between h/w = 0% and h/w = 100%. The PTH filled in the core layer which occupies 80% of the volume. The height of the PTH was 0.8 mm and enclosed with a circular caps of 0.45 mm in diameter on the top and the bottom of the PTH. The PTH was filled with a plug material. The detail of the PTH geometry is shown in Fig. 4. The effective properties were extracted as a function of d/w, where d was the cap diameter, 0.45 mm. Two types of micro-via were assumed: ‘‘filled’’ and ‘‘unfilled’’ micro-vias. The 3D FE unit cell models can be seen in Fig. 5. The effective properties were extracted as a function of d/w, where d was the diameter of the top part of the micro-via, that was 0.138 mm. Finally, the 3D FE unit cell model of adhesion hole with a diameter of 0.2 mm is shown in Fig. 6. Similarly, the effective properties were extracted as a function of d/w, where d was the hole diameter. The periodic unit cell FE models were subjected to the boundary conditions:

Fig. 3. 3D FE unit cell model of electric trace. Black indicates copper trace, white indicates a BUF material, and dark gray represents FR4 core material. There are no PTHs, micro-vias, or adhesion holes in this model.

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Fig. 4. 3D FE unit cell model of PTH. Black indicates copper trace, white indicates a BUF material, light gray represents plug material, and dark gray represents FR4 core material. There are no traces, micro-vias, or adhesion holes in this model.

B B ab ÞðxA uA e ab þ x3 j a
ua ¼ ð b
xb Þ;

ab ÞðxCb
xD uCa
uD e ab þ x3 j a ¼ ð b Þ;

ð1Þ

where the tensorial summation convention was used with Greek letters ranging from 1 to 2; x1, x2, x3 are coordinates B C D and uA a ,ua ,ua ,ua are in-plane displacements on surfaces A, B, C, and D, respectively, as illustrated in Fig. 7. CoordiB C D nate pairs ðxA b ; xb Þ and ðxb ; xb Þ correspond to equivalent points with respect to periodicity on the vertical boundary ab are homogenized (volsurfaces A, B and C, D. e ab and j ume average) in-plane strains and out-of-plane curvatures, respectively. In ordinary FE programs, these kinematic boundary conditions are sufficient to guarantee periodicity also in the dynamic boundary conditions. The boundary conditions were enforced by multi-point constraints in the ANSYS 8.0 finite element package, using six degrees of freedom of nodes not physically connected to the finite

ab . Lamelement model for the six components of e ab and j inate stiffnesses (Aabcd, Babcd, Dabcd) introduced below were computed from an evaluation of the total strain energy resulting from the applied generalized strains, whereas laminate thermal strains (e ab ) and curvatures ( jab ) were calculated from an evaluation of the average generalized strains resulting from application of a unit change in temperature. For each copper volume fraction for each model, eighteen FE analyses were performed to compute the laminate stiffnesses and one FE analysis was conducted to calculate the laminate thermal strains and curvatures. From classical lamination theory, the plate forces Nab and moments Mab are defined as cd
N Tab ; N ab ¼ Aabcde cd þ Babcd j cd
M Tab ; M ab ¼ Babcde cd þ Dabcd j where the laminate stiffnesses are

ð2Þ

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Fig. 5. 3D FE unit cell model of micro-via. Black indicates copper trace, white indicates a BUF material, and dark gray represents FR4 core material. There were no traces, PTHs, or adhesion holes in this model.

Fig. 6. 3D FE unit cell model of adhesion hole. Black indicates copper trace, white indicates a BUF material, and dark gray represents FR4 core material. There were no traces, PTHs, or micro-vias in this model.

Aabcd ¼ Babcd ¼

Z Z

h=2


h=2

ð3Þ

Qkabcd x3 dx3 ;

ð4Þ

h=2


h=2

Dabcd ¼

Qkabcd dx3 ;

Z

where Qkabcd are the lamina plane stress stiffnesses. The forces and moments due to a temperature change DT are Z h=2 N Tab ¼ DT Qkabcd akcd dx3 ; ð6Þ
h=2

h=2


h=2

Qkabcd x23 dx3 ;

ð5Þ

M Tab ¼ DT

Z

h=2


h=2

Qkabcd akcd x3 dx3 ;

ð7Þ

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Table 1 Material properties of the constituent materials Properties

FR4

Copper

BUF

Plug

Ex, Ey, Ez (GPa) Gxy (GPa) Gxz, Gyz (GPa) mxy, mxz, myz ax, ay (ppm/C)

21.70 3.53 1.72 0.08 15.40

120.00 – – 0.33 17.60

3.50 – – 0.27 60.00

3.30 – – 0.37 19.00

where akcd are the lamina CTEs and superscript k corresponds to different lamina through the thickness. For a laminate subjected to pure thermal loading, the applied forces per unit length (Nab) and moments per unit length (Mab) are zero. Therefore, the forces and moments caused by a temperature change become: cd ; N Tab ¼ Aabcde cd þ Babcd j cd . M Tab ¼ Babcde cd þ Dabcd j

ð8Þ

For example, the stiffness determination of the electric trace, let QTr abcd be the effective stiffnesses of a lamina with copper traces. Given the laminate stiffnesses (Aabcd, Babcd, Dabcd) computed from FE analyses and the known stiffnesses of the BUF and core lamina, the stiffnesses QTr abcd can be (non-uniquely) extracted using Eqs. (3)–(5). The computed QTr abcd were plotted in Fig. 8(a)–(d) as a function of h/w. Similarly, for PTH, micro-via, and adhesion hole, the stiffnesses were plotted in Figs. 9(a)–(d), 10(a)–(d), and 11(a)–(d), respectively. Let aTr ab be the effective CTEs of a lamina with traces. Given the laminate strains (e ab ), curvatures ( jab ), and stiffnesses (Aabcd,Babcd, Dabcd) obtained from the FE analyses and a temperature change, the plate forces (N Tab ) and moments (M Tab ) due to the temperature change can be

extracted using Eq. (8). Applying N Tab , M Tab as well as lamina stiffnesses and CTEs of the BUF and the core laminate, Tr and the computed QTr abcd into Eqs. (6), (7), the CTEs aab can Tr be (non-uniquely) determined. The computed aab are plotted in Fig. 8(e) and (f) as a function of h/w. Similarly, for PTH, micro-via, and adhesion holes, the CTEs were plotted in Figs. 9(e) and (f), 10(e) and (f), and 11(e) and (f), respectively. 2.2. Effective properties obtained from Voigt and Reuss estimates The effective properties of a lamina consisting of copper and composite or polymer were derived from FE analyses in the previous section. In this section, approximate analytical expressions will be derived using the Voigt constant strain [10] and the Reuss constant stress [11] predictions. Plane stress through the thickness was assumed, i.e., ri3 = 0 for i = 1, 2, 3. The constitutive relation is rab ¼ Qabcd ecd
bab T

ð9Þ

and its inverse, eab ¼ S abcd rcd þ aab T .

ð10Þ

Volume average (overhead bar) of stress from the Voigt theory is Z 1 ab ¼ r ðQ ecd
bab T ÞdV V V abcd N N X X ¼ ecd vk Qkabcd
T vk bkab ¼ QVabcd ecd
bVab T ; ð11Þ k¼1

k¼1

where the last equality defines the effective properties according to the Voigt assumption (superscript V).

Fig. 7. An illustration of boundary conditions for the 3D FE unit cell analyses.

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563

Electric Trace

Electric Trace 160

160 Voigt Reuss FE (h = 0.0254 mm) FE (h = 0.0834 mm)

120 100

Voigt Reuss FE (h = 0.0254 mm) FE (h = 0.0834 mm)

140

QTr2222 (GPa)

QTr1111 (GPa)

140

80 60

120 100 80 60

40

40

20

20

0

0

0

0.2

0.4

a

0.6

0.8

0

1

0.2

0.4

0.6

50 Voigt Reuss FE (h = 0.0254 mm) FE (h = 0.0834 mm)

30

Voigt Reuss FE (h = 0.0254 mm) FE (h = 0.0834 mm)

40

QTr1212 (GPa)

QTr1122 (GPa)

40

20

30 20 10

10

0

0 0

0.2

0.4

c

0.6

0.8

0

1

0.2

0.4

d

h/w

0.6

0.8

1

h/w

Electric Trace

Electric Trace 80

Lamina CTE (ppm/ C)

80 Reuss

60

Voigt

o

Voigt

o

Lamina CTE (ppm/ C)

1

Electric Trace

Electric Trace 50

FE (h = 0.0254 mm) FE (h = 0.0834 mm)

40

Reuss

60

FE (h = 0.0254 mm) FE (h = 0.0834 mm)

40

20

α

Tr 22,

20

α

Tr 11,

0.8

h/w

b

h/w

0

0

0.0

0.2

e

0.4

0.6

0.8

1.0

h/w

0.0

0.2

f

0.4

0.6

0.8

1.0

h/w

Tr Fig. 8. Stiffness (QTr abcd ) and CTEs (aab ) of electric traces.

The volume average of strain from the Reuss theory is Z 1 eab ¼ ðS abcd rcd þ aab T ÞdV V V N N X X ¼ rcd vk S abcd þ T vk akab ¼ S Rabcd rcd þ aRab T ; ð12Þ k¼1

k¼1

where the last equality defines the effective properties according to the Reuss assumption (superscript R). 2.3. Analytical models for effective thermo-mechanical properties The analytical models of the effective stiffnesses and CTEs of substrate features was based on the effective properties calculated from the FE analyses of Section 2.1 and

the Voigt and Reuss estimates of Section 2.2, plus a few constants. As shown in Fig. 8, stiffnesses QTr 1111 followed Tr Tr the Voigt estimate while stiffnesses QTr , Q 2222 1122 , Q1222 folTr Tr lowed the Reuss estimate. CTEs a11 and a22 followed the Voigt and Reuss estimates, respectively. The analytical models for electric traces are listed in Table 2. For PTH, all lamina stiffnesses were between the Voigt and Reuss estimates as shown in Fig. 9(a)–(d). However, both lamina CTEs followed the Reuss estimate, Fig. 9(e) and (f). The analytical models for PTHs are listed in Table 3. lvia For micro-via, Qlvia 1111 and Q2222 were between the Voigt and Reuss estimates as shown in Fig. 10(a) and (b). However, lvia Qlvia 1122 and Q1212 were close to the Reuss estimate as shown in Fig. 10(c) and (d). Both lamina CTEs were between the Voigt and Reuss estimates, Fig. 10(e) and (f). The analytical models for the micro-vias are listed in Table 4.

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P. Hutapea et al. / Microelectronic Engineering 83 (2006) 557–569

PTH 35

30

30

25

QPTH2222 (GPa)

QPTH1111 (GP a)

PTH 35

20 15 Voigt Reuss FE

10 5

25 20 15 Voigt Reuss FE

10 5 0

0 0.0

0.2

0.4

a

0.6

0.8

0.0

1.0

0.2

0.4

0.6

10 Voigt Reuss FE

Voigt Reuss FE

8

QPTH1212 (GPa)

8

QPTH1122 (GPa)

1.0

PTH

PTH 10

6 4

6 4 2

2 0

0

0.0

0.2

0.4

c

0.6

0.8

0.0

1.0

0.2

0.4

d

d/w

0.6

0.8

1.0

d/w

PTH

PTH 17

17

C)

Voigt Reuss FE

o

22 (ppm/

o

C)

Voigt Reuss FE

16

16

α

α

PTH

PTH

11 (ppm/

0.8

d/w

b

d/w

15

15

0.0

0.2

e

0.4

0.6

0.8

1.0

0.0

0.2

0.4

f

d/w

0.6

0.8

1.0

d/w

PTH Fig. 9. Stiffness (QPTH abcd ) and CTEs (aab ) of PTHs.

Table 2 Analytical models for electric traces

Table 3 Analytical models for PTHs

Lamina stiffness V QTr 1111 ¼ Q1111 Tr Q2222 ¼ QR2222 R QTr 1122 ¼ Q1122 Tr Q1212 ¼ QR1212

Lamina stiffness V R d A A A QPTH 1111 ¼ Q1111 ð1
u Þ þ Q1111 u Þ, where u ¼ 0:274 þ 0:101 w PTH V R B B B d Q2222 ¼ Q2222 ð1
u Þ þ Q2222 u Þ, where u ¼ 0:274 þ 0:101 w V R C C C d QPTH 1122 ¼ Q1122 ð1
u Þ þ Q2222 u Þ, where u ¼ 0:693
0:351 w PTH V R D D D Q1212 ¼ Q1212 ð1
u Þ þ Q1212 u Þ, where u ¼ 0:739
0:140 wd

Lamina CTEs V aTr 11 ¼ a11 R aTr ¼ a 22 22

Lamina CTEs aPTH ¼ aR11 11 aPTH ¼ aR22 22

For adhesion hole, all lamina stiffnesses were between the Voigt and Reuss estimates as shown in Fig. 11(a)–(d). Both lamina CTEs followed the Reuss estimate, Fig. 11(e) and (f). The analytical models for the adhesion holes are listed in Table 5.

3. Simplified 2D FE analyses of substrates to predict warpage The 2D FE models of the substrate were generated by discretizing the substrate into 10 · 10 cells. The size of

P. Hutapea et al. / Microelectronic Engineering 83 (2006) 557–569

565

Micro-via

Micro-via 70

70 Voigt Reuss FE (Filled) FE (Unfilled)

50

Voigt Reuss FE (Filled) FE (Unfilled)

60

Q µvia2222 (GPa)

Q µvia1111 (GP a)

60

40 30 20

50 40 30 20 10

10 0

0

0.0

0 .2

0.4

a

0 .6

0.8

0.0

1.0

0.2

0.4

b

d/w

0.6

Micro-via 25 Voigt Reuss FE (Filled) FE (Unfilled)

15

Voigt Reuss FE (Filled) FE (Unfilled)

20

Q µvia1212 (GPa)

20

Q µvia1122 (GPa)

1.0

Micro-via

25

10 5

15 10 5

0

0

0.0

0.2

0.4

0.6

0.8

1.0

d/w

c

0.0

0.2

0.4

d

0.6

60

60 o α µvia22 (ppm / C)

70

50 40 30 Voigt Reuss FE (Filled) FE (Unfilled)

10

1.0

Micro-via

70

20

0.8

d/w

Micro-via

o α µvia11 (ppm / C)

0.8

d/w

50 40 30 Voigt Reuss FE (Filled) FE (Unfilled)

20 10 0

0 0.0

0.2

e

0.4

0.6

0.8

0.0

1.0

d/w

0.2

f

0.4

0.6

0.8

1.0

d/w

lvia Fig. 10. Stiffness (Qlvia abcd ) and CTEs (aab ) of micro-vias.

each cell was 3.8 · 3.8 mm2. In each discretized cell (100 cells · 15 layers), the characteristics of the features were identified. For example, in the copper layers, the copper volume fraction (h/w) of traces and their orientations

Table 4 Analytical models for micro-vias Lamina stiffness V R A A A d Qlvia 1111 ¼ Q1111 ð1
u Þ þ Q1111 u Þ, where u ¼ 0:854
0:083 w V R d B B B Qlvia 2222 ¼ Q2222 ð1
u Þ þ Q2222 u Þ, where u ¼ 0:854
0:083 w V R C C C d Qlvia 1122 ¼ Q1122 ð1
u Þ þ Q2222 u Þ, where u ¼ 0:654 þ 0:293 w V R D D D d Qlvia 1212 ¼ Q1212 ð1
u Þ þ Q1212 u Þ, where u ¼ 1:217
0:323 w

Lamina CTEs ¼ aR11 aPTH 11 aPTH ¼ aR22 22

were determined. In the dielectric and core layers, quantities and locations of PTHs and micro-vias were assessed. A program was used to interface with the substrate database to estimate the copper volume fractions in the copper layers and locations of the PTHs, microvias, and adhesion holes. For example, in the case of electric traces, using the estimated copper volume fractions and the trace orientations of each layer in each cell, the effective properties (Qijkl and aij) were estimated using the empirical analytical models in Tables 2–5. Using these analytical models, the laminate properties (Aijkl, Bijkl, Dijkl, N Tij , M Tij ) of the 100 cells of the substrate were calculated. Two-dimensional FE models constructed with the cells using the respective laminate properties were developed. Each cell was meshed using four elements. Commercial FE software ANSYS 8.0 was used. The

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Adhesion Hole 140

120

120

100

Voigt Reuss

80

100

QHole2222 (GPa)

QHole1111 (GP a)

Adhesion Hole 140

FE

60 40

Voigt Reuss

80

FE

60 40 20

20 0

0

0.0

0.2

0.4

0.6

0.8

d/w

a

0.0

1.0

0.2

0.4

0.6

Adhesion Hole

Adhesion Hole

Voigt Reuss

30

FE

40

QHol e1212 (GP a)

QHol e1122 (GPa)

40

20

Voigt Reuss

30

FE

20 10

10

0

0 0.0

0.2

0.4

0.6

0.8

0.0

1.0

d/w

c

0.2

0.4

0.6

0.8

1.0

d/w

d

Adhesion Hole

Adhesion Hole 60

60 Voigt Reuss

Voigt Reuss

50

C)

C)

50

FE

o

FE

o

22 (ppm/

40

Hole

30

40 30

α

α

11 (ppm/

1.0

50

50

Hol e

0.8

d/w

b

20

20

10

10

0.0

0.2

e

0.4

0.6

0.8

1.0

d/w

0.0

0.2

0.4

f

0.6

0.8

1.0

d/w

Hole Fig. 11. Stiffness (QHole abcd ) and CTEs (aab ) of adhesion holes.

substrate was modeled using a linear layer material (Shell99 element). Displacement boundary conditions were appl- ied to prevent the rigid body rotation. A unit

Table 5 Analytical models for adhesion holes Lamina stiffness V R d A A A Qhole 1111 ¼ Q1111 ð1
u Þ þ Q1111 u Þ, where u ¼
0:069 þ 0:577 w V R B B B d Qhole ¼ Q ð1
u Þ þ Q u Þ, where u ¼
0:069 þ 0:577 2222 2222 2222 w V R C C C d Qhole 1122 ¼ Q1122 ð1
u Þ þ Q2222 u Þ, where u ¼ 0:697 þ 0:139 w V R D D D d Qhole 1212 ¼ Q1212 ð1
u Þ þ Q1212 u Þ, where u ¼
0:174 þ 0:981 w Lamina CTEs V ahole 11 ¼ a11 V ahole ¼ a 22 22

temperature increase (DT = 1 C) was applied to the FE model as the thermal loading. A summary of the steps used in this homogenization method is illustrated in Fig. 2. Warpage of two similar microelectronic substrates were predicted.

4. Experimental verifications Effective CTE, stiffness, and warpage measurements were used to validate the substrate homogenization method. Effective CTE and warpage measurements were performed through the use of digital image correlation (DIC). The DIC measurement technique is a digital image processing tool capable of measuring in-plane surface deformations. Surface preparation is generally not

P. Hutapea et al. / Microelectronic Engineering 83 (2006) 557–569

567

Fig. 12. CTE contour maps of substrate 2 measured using digital image correlation.

Table 6 Numerical and experimental correlations (CTEs, ppm/C) Areas

A B C D

ax

ay

Experiment

Prediction

Experiment

Prediction

18.72 18.33 20.57 18.16

18.53 18.00 20.15 18.03

16.99 17.14 19.46 17.57

17.64 17.56 19.71 17.66

Table 7 Numerical and experimental correlations (warpage, lm/C)

Prediction Experiment

Substrate 1 ðU max
U min z z Þ=DT

Substrate 2 ðU max
U min z z Þ=DT

0.126 0.139

0.106 0.108

required so long as the surface of interest contains a sufficient amount of spatial detail or gray level variation. In the case of organic substrates, the high-density solder bump array was leveraged along with the appropriate choice of magnification. Contrast enhancement of the substrate regions lying outside of the central bump array was achieved through the implementation of a thin graphite layer. A telecentric imaging lens imaging lens and 8 bit CCD camera were used to obtain images at two different temperatures. Thermal loading was achieved through the use of Peltier thermal head within a partial vacuum environment. The 40 Torr partial vacuum environment served to mitigate any optical distortion effects induced by thermal convection. Images were digitally stored as an array of grey scale values which may be treated as intensity distribution functions. At the heart of the DIC technique lies a robust algorithm which serves to correlate correspond-

ing surface features between deformed and undeformed subset regions. Once a correlation is established, resulting in-plane displacements are deduced from the pixel coordinate transformation between correlated images. Dimensionally scaled horizontal (U) and vertical (V) displacement fields are obtained by incorporating the imaging system magnification. Strain field maps are obtained through numerical differentiation of calculated displacement fields. Last, and perhaps most useful here, strain field plots may be divided by the imposed temperature change in order to obtain whole field CTE plots. Fig. 12, Tables 6 and 7 show the DIC effective CTE contour maps, the average CTEs for each of the four regions, and the substrate warpage, respectively. In Fig. 12, regions A, B, and D are outside the die area and region C is the die area. The results show a substantial difference in the effective CTEs between the die and outside die region. Predictions of the effective CTEs were made from the homogenized substrate by subjecting the FE model to a small temperature change. Strain extracted from the FE model was divided by the temperature change to yield the effective CTEs – this method is essentially the same as what was performed in DIC. It should also be mentioned that the values in Table 7 are the slopes of the warpage change as a function of temperature changes. Tables 6 and 7 shows the comparisons of the experimental and predicted results – predictions matched experimental trends and magnitudes well. Effective extensional stiffness measurements were made using uniaxial tensile tests performed on strips cut from the substrate, as shown in Fig. 13. An MTS load frame was used to apply the uniaxial load and axial strain was measured using extensometers (locations indicated by horizontal lines in Fig. 13). The stress–strain response was generally linear up to approximately 0.2% strain, thereafter a non-linear response occurred, Fig. 14. Effective

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Fig. 13. Location of strips cut from substrates for effective stiffness measurements. Blue lines indicate the location of the extensometer clips. Specimen 3 was used to capture the die shadow regions. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)

the previous section. Predictions match the experimental trends very well (lower stiffness in the die region), however there was some error in the magnitude in the die region, which could be due to some of the complexity in the die region that was not captured by the four features examined.

200

Stress (MPa)

160

Specimens 1, 2, 4, 5

120

80 Die shadow regions, Specimens 3

40

0 0.000

0.002

0.004

0.006

0.008

0.010

1A

1B

2B

2A

3A

3B

4A

4B

5A

5B

0.012

5. Results and discussion 0.014

Strain Fig. 14. Measurements of stiffness using tensile test method. The lowest stiffness was observed in specimen 3. The stiffness is listed in Table 8.

secant modulus was calculated at 0.2% strain and the results are shown in Table 8. As in the case of the effective CTEs, the results show a substantial difference in the die versus outside die region. Table 8 shows the comparison of the experimental and predicted results. The stiffnesses of regions A, B, C and D was the average of the stiffness of the discretized areas calculated using the plate constitutive equation from the classical lamination theory and the predicted laminate stiffnesses (Aijkl, Bijkl, Dijkl) obtained from

A numerical procedure to predict effective thermomechanical properties and substrate warpage under isothermal condition has been developed. Substrate features - electric trace, PTH, micro-via, and adhesion hole – have strong influence on the warpage. The effective thermomechanical properties of these features were predicted through 3D FE unit cell analyses. Empirical analytical models for the effective properties were developed using the FE data and Voigt and Reuss estimations. Using the empirical models, the simplified 2D FE plate models were developed. Additionally, the effective CTE and stiffness predictions matched very well with the experimental results as given in Tables 6–8. The results show a substantial difference in the effective CTEs and stiffness between the die and outside the die region.

Table 8 Numerical and experimental correlations (stiffness, GPa)

Prediction Experiment

Specimen 1

Specimen 2

Specimen 3

Specimen 4

Specimen 5

26.20 26.61

24.87 25.45

21.05 18.42

24.40 25.17

25.79 24.06

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6. Conclusions

References

The present numerical method has the potential to be used as a numerical tool to predict the warpage and mechanical properties of electronic substrates. Such a capability is critical in assessing substrates before fabrication to ensure that the complexity of the substrate design produces stiffnesses, CTEs, and warpage consistent with expectations. Future work will include the development of tuning method to reduce the microelectronic substrate warpage.

[1] J.L. Grenestedt, P. Hutapea, Appl. Phys. Lett. 81 (21) (2002) 4079– 4081. [2] J.L. Grenestedt, P. Hutapea, J. Appl. Phys. 94 (1) (2003) 686–696. [3] P. Hutapea, J.L. Grenestedt, 9th International Symposium and Exhibition on Advanced Packaging Materials, IEEE CPMT, Atlanta, GA, March 24–26, 2004. [4] P. Hutapea, J.L. Grenestedt, ASME J. Electron. Packag. 126 (3) (2004) 282–287. [5] L.W. Ritchey, Printed Circuit Design 17 (2) (2000) 26–29. [6] Y. Stavsky, N.J. Hoff, in: A.G.H. Dietz (Ed.), Composite Engineering Laminates, MIT Press, Cambridge, MA, 1969. [7] J.L. Grenestedt, P. Hutapea, K. Frutschy, Mechanics and Materials Conference, Scottsdale, AZ, June 17–20, 2003. [8] P. Hutapea, J.L. Grenestedt, M. Modi, M. Mello, K. Frutschy, ASME Interpack 05, San Francisco, California, July 17–22, 2005. [9] P. Hutapea, J.L. Grenestedt, J. Electron. Mater. 32 (4) (2003) 221– 227. [10] W. Voigt, Wied. Ann. 38 (1889) 573–587. [11] A. Reuss, Z. Angew. Math. Mech. 9 (1929) 49–58.

Acknowledgments This research was supported by Intel Corporation ATD, Chandler, Arizona. In particular, we would like to thank Ibrahim Bekar and Josh Tor for their help in measuring the substrate stiffness and effective CTEs, respectively.