Finite element simulation of the residual stresses in high strength carbon steel butt weld incorporating solid-state phase transformation

A Finite Element Simulation of Residual Stresses Induced by Thermal

and Lattice Mismatch in Thin Films

Alireza Moridi1, H.H. Ruan

2, L.C. Zhang

3*, Mei Liu

4

School of Mechanical and Manufacturing Engineering, The University of New South Wales, NSW

2052, Australia 1 [email protected], 2

[email protected], 3 [email protected], 4

[email protected]

*Corresponding Author

Abstract During the cooling process from deposition

temperature to room temperature, stresses

develop in thin silicon layer deposited on

sapphire due to thermal and lattice mismatch.

This paper used the finite element method to

analyse the thermal mismatch effect on the

stress variation in the silicon film thickness of

300nm. It was shown that the interfacial shear

stresses are apparent at the boundary and

negligible at the centre. Hence the boundary

effect must be properly treated in the finite

element modelling; otherwise results may not

be reliable. In addition, the intrinsic stresses

induced by the lattice mismatch and

dislocations are also modelled for various film

thicknesses. A superposition of stresses due to

thermal, lattice mismatch and dislocations

should render the total residual stress in the

silicon on sapphire (SOS) systems.

Keywords Silicon on Sapphire, Residual Stress, FEA,

Thermal and lattice mismatch

1. Introduction In order to generate a thin film of a

semiconductor material for electronic circuits,

e.g., a silicon film, the hetero-epitaxial growth

on an insulated substrate at high temperature is

common process. However, this brings about

residual stresses due to the different thermo-

mechanical properties of different layers. Such

stresses could induce destructive consequences

such as buckling, cracking, permanent

deformation and delamination.

Residual stresses arise mainly for two reasons.

First, the coefficients of thermal expansion

(CTE) of the thin film and the substrate are

different. Secondly, the lattice structures of

different materials are dissimilar. The residual

stresses induced by the disparate lattice

structures can partially be relaxed by forming

lattice defects in the growth process. Although

the total residual stresses can be measured by

experimental methods, it is hard to distinguish

the contributions of CTE and lattice

mismatches as well as the relaxation due to

lattice defects. Hence, a theoretical analysis is

necessary. To this end, the finite element

method is an efficient tool which can provide

the variation of local stresses in the thin film

and the stress discontinuity across the layers.

A number of studies have been carried out

using the finite element analysis to investigate

residual stresses in film on substrate systems

[1-3]. These results mainly focused on the

effect of deposition temperature and the

simulation models were two dimensional with

isotropic temperature-independent material

properties. A more comprehensive investigation

has been carried out by Pramanik and Zhang [4]

where they used anisotropic material properties

and a three dimensional finite element (FE)

model to investigate the residual stresses in the

thin film and substrate. In addition to the

thermal mismatch investigation, Subramaniam

and Ramakrishnan [5] calculated the release of

the elastic energy by dislocation nucleation.

Most of the previous FE analysis only focused

on the effect of CTE mismatch and overlooked

the effect of lattice mismatch. This study

investigates the CTE and lattice mismatches

with 3D and 2D FE models respectively. The

lattice defects were simplified as edge

dislocations, such that the stress relaxation by

these dislocations can be considered. The

superposition of these two models should give a

more accurate evaluation of residual stresses

and can be verified with the experimental

results available.

2. Finite element modelling

2.1. The geometries A schematic of the three dimensional FE model

is shown in Fig. 1. The model shape resembles

a silicon-on-sapphire system, which includes

all the possible geometrical effects of a silicon

wafer [4]. A volume of interest (VOI) was

defined in the centre of the model with the

finest mesh as shown in Fig. 1. In order to

avoid boundary effects the radial dimension of

the model is 30 times the thickness of the thin

film. Overall, the VOI contained 4,440

elements, and the whole model consisted of

34,628 elements. The finite element model is

solved by ANSYS V12.1 with the 10-noded

SOLID 98 elements which can cope with both

thermal and mechanical responses.

In order to calculate the stress induced by

lattice mismatch and dislocation, a two

dimensional model is established with the

PLANE223 plain-strain elements. The element

size at the interface is the same as the lattice

constant of sapphire. The initial strain induced

by lattice mismatch and edge dislocations were

applied to calculate the residual stress.

2.2. Properties of the SOS materials

A mono-crystalline sapphire structure has been

shown in Fig. 2. To minimize the effect of

lattice mismatch between silicon and sapphire,

the silicon layer is normally deposited on the R-

plane of sapphire [6]. Furthermore, the material

layers are assumed to be perfectly bounded to

each other. The silicon properties are

considered to be orthotropic, whereas the

sapphire is regarded as an anisotropic material

whose anisotropic elasticity is obtained from

the stiffness matrix available in the literature [7]

and is converted from C-plane to R-plane.

2.3. Boundary and loading conditions

The simulation for thermal mismatch was

performed without any external force. The

bottom of the model was fixed in z-direction

and a node at a corner bottom of the model was

fixed in x-, y- and z-axes to eliminate the rigid

body motion. The model was cooled down from

900 °C to 25 °C by only convection from all the

surrounding surfaces. The simulation was run

for one hundred sub-steps with the film

coefficient of convection being 10.45 W/m2°C

[8].

3. Results and discussion 3.1. Thermal stresses

Fig. 3 shows distribution of the normal residual

stresses in x, y and z directions for the 300nm

thin film. The vertical axis is the distance from

the interface and the horizontal axis is the

residual stress. It can be seen the in-plane

normal stresses (xx and yy) are compressive

inside the thin film and tensile within the

sapphire substrate. They reach their maxima at

the interface. The stress in zz direction is near

to zero in comparison with xx and yy direction

which is consistent with experimental findings

and theoretical calculations [9]. Moreover, the

stress gradients in the thin film is very small as

shown in Fig. 3, which is consistent with the

experimental measurements by Liu et al. [10].

The average in-plane residual stresses xx and

yy for 300nm thin film are -658.75MPa and -

609.11MPa, respectively. The small difference

between xx and yy is due to the anisotropy of

sapphire.

Fig.1. Geometry of an SOS thin film system: (a) full model (b) VOI

a

1

XY

Z

ELEMENTS

MAT NUM

b

1

X Y

Z

ELEMENTS

MAT NUM

Volume of Interest

(VOI) Crystalline Silicon

Sapphire Substrate

Interface

Fig. 2. Coordinate systems in the sapphire

crystal and in the R-plane [8]

Fig. 3 Variation of stresses in the SOS System

Similar results have been obtained for different

film thicknesses (300 nm~5 m) for the thermal

mismatch investigations. In order to justify this

independency of thickness, let us consider the

non-dimensional form of the normal residual

stress. The normal stress in the system can be

expressed as:

),,,,,,( fsfsfs ttTEEf

(1)

Invoking the Buckingham’s Pi (Π) Theorem

and constructing the non-dimensional groups

for the three fundamental variables of length

(L), mass (M), and temperature (θ), the non-

dimensional equation can be expressed as:

),,,(f

sfs

f

s

s t

tTT

E

Ef

E

(2)

This clearly indicates that if the non-

dimensional term f

s

t

t remains constant the

normal stress in the system does not change.

The variation of the non-dimensional normal

stress with the thickness ratio fs tt is shown in

Fig. 4. It is noted that if 20fs tt , the normal

stress remains constant for the same thermal

conditions. Therefore, when this ratio in an

SOS system is very large, it is safe to conclude

that the normal stress due to thermal mismatch

is independent of the film thicknesses.

In addition, the results show a significant shear

stress on the boundary as illustrated in Fig. 5.

The top surface in Fig. 5 is the interface

between the thin film and the substrate. The

shear stress on the boundary should be carefully

considered in modelling. The model should be

large enough to minimize the boundary effect.

Moreover, this shear stress may cause cracks or

delamination.

Fig. 4. Effect of f

s

t

t on normal residual stress.

-0.7

-0.5

-0.3

-0.1

0.1

0.3

-800 -600 -400 -200 0 200

Dis

tan

ce f

rom

th

e in

terf

ace

(μm

)

Stress (MPa)

σx

σy

σz

-1000

-800

-600

-400

-200

0

0 10 20 30 40

Th

erm

al R

esid

ual

Str

ess

(MP

a)

ts/tf

Fig. 5. Shear stress in (a) xy, (b) xz and (c) yz

directions on the substrate and on the VOI.

3.2. Stresses due to lattice mismatch and

dislocations

If the lattice constant of the thin film is larger

than that of the substrate (as in a Silicon-on-

Sapphire system), the film is under

compression and the substrate is under tension

due to the lattice misfit. Dislocations or other

lattice defects can nucleate at the interface to

mitigate the misfit stress. For simplicity, we

only considered the edge dislocations which

relieve the misfit stress by adding vertical atom

planes in the substrate [5], as shown in Fig. 6.

A symmetric 2D model under the plane strain

condition is adopted. The mesh size near the

interface is taken to be the same as the substrate

lattice constants which are respectively

0.479nm and 0.516nm in “a” and “c” directions

as defined in Fig. 2. The silicon film has the

lattice constant of 0.545nm [11]. Hence the

nominal misfit strain is about 13.8% and 5.6%

along “a” and “c” directions, respectively.

Fig. 6. Schematic of an edge dislocation at the

interface

Fig. 7. Tension and compression in thin film

and substrate due to the dislocation.

The edge dislocation is modelled by feeding the

strain corresponding to the introduction of an

extra plane of atoms in the substrate. The

burgers vector is equal to the lattice constant

and in the direction parallel to the interface. The

local strain corresponding to an edge

dislocation can be calculated as:

2/12/ sab

where as is the lattice constant of the substrate

and b is the burgers vector. Therefore, the

introduction of dislocations will cause the local

tension in the film and compression in the

substrate as shown in Fig. 7.

X Y

Z

-136.238

-103.278

-70.319

-37.359

-4.4

28.56

61.519

94.479

127.438

160.398

1

MN

MX

X Y

Z

-1.046

-.783751

-.52143

-.25911

.003211

.265532

.527853

.790174

1.052

1.315

NODAL SOLUTION

STEP=1

SUB =100

TIME=3600

SXY (AVG)

RSYS=0

DMX =.577E-04

SMN =-1.046

SMX =1.315

X Y

Z

-277.588

-214.138

-150.689

-87.239

-23.789

39.66

103.11

166.56

230.009

293.459

1

MN

MX

X Y

Z

-4.124

-2.889

-1.653

-.417706

.817744

2.053

3.289

4.524

5.76

6.995

NODAL SOLUTION

STEP=1

SUB =100

TIME=3600

SXZ (AVG)

RSYS=0

DMX =.577E-04

SMN =-4.124

SMX =6.995

X Y

Z

-254.954

-192.904

-130.855

-68.805

-6.756

55.294

117.343

179.393

241.442

303.492

1

MN

MX

X Y

Z

-4.599

-3.448

-2.298

-1.147

.002987

1.153

2.304

3.454

4.605

5.755

NODAL SOLUTION

STEP=1

SUB =100

TIME=3600

SYZ (AVG)

RSYS=0

DMX =.577E-04

SMN =-4.599

SMX =5.755

tension area

compression area

b

a

c

Fig. 8. A typical normal stress contours due to lattice mismatch and dislocations

The dislocation simulation has been carried out

by assuming the maximum relaxation in the

interface. Hence, dislocation happens for every

7 (1/13.8%) atomic planes in “a” direction and

18 (1/5.6%) atomic planes in “c” direction. In

addition to the stress caused by dislocations the

stress by the lattice mismatch is also calculated

by feeding the correspondent strains. A typical

result of the simulation is shown in Fig. 8.

The stress around Line A is mainly due to the

dislocations and that in Line B is mainly due to

lattice mismatch. The stress contours are

limited from -3,000MPa to 3,000MPa for a

better illustration. The stress distributions from

the interface to the top surface of the thin film

along Line A and Line B in Fig. 8 are shown in

Fig. 9 for a typical 300nm thin film SOS

system. The results show that within 70nm

form the interface the effect of dislocations and

lattice mismatch are significant and their effects

become negligible if the film thickness

becomes much larger.

Table 1. Effect of film thickness on the normal

stress due to lattice mismatch and dislocations

Film

thickness

(nm)

Stress in “a”

direction (MPa)

stress in “c”

direction (MPa)

100 -816.4 -60.2

200 -421.7 -30.0

300 -281.4 -20.1

400 -211.1 -15.0

500 -168.9 -12.0

The average normal residual stresses for

different film thickness are listed in Table 1.

The total average of the residual stress in the

thin film can be obtained by superposing the

stress due to dislocations and lattice mismatch

and the residual stress due to thermal mismatch.

It has been found that the residual stresses in a

thinner film are much larger than those in a

thick film due to the effects of lattice mismatch.

The results have been verified by experiments.

However, due to confidential agreement it is

not included in this article.

Fig. 9(a). Normal stress distribution in the

middle of a dislocation and between two

dislocations in “a” direction for a 300nm thin

film

ANSYS 12.1

-3000

-2333

-1667

-1000

-333.333

333.333

1000

1667

2333

3000

-20000

-10000

0

10000

20000

30000

0.0001 0.001 0.01 0.1 1

Str

ess

(MP

a)

Distance from the interface (µm)

in the middle of a dislocation

between two dislocations

Thin Film

B A

Substrate

Fig. 9(b). Normal stress distribution in the

middle of a dislocation and between two

dislocations in “c” direction for a 300nm thin

film

4. Conclusion It is shown through the finite element analysis

that the residual stress calculated based solely

on thermal mismatch underestimates the total

residual stress in a thin film. The intrinsic

residual stress must include the effect of lattice

mismatch and dislocations. For very thin films,

the lattice mismatch should be particularly

considered. For thicker films, the effect of

lattice mismatch fades out, such that thermal

mismatch is the main cause of the residual

stresses.

Acknowledgement This research was supported by an ARC grant.

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1457.

-10000

-5000

0

5000

10000

15000

20000

25000

0.0001 0.001 0.01 0.1 1

Str

ess

(MP

a)

Distance from the interface (µm)

in the middle of a dislocation

between two dislocations