of 9
Competing failure mechanisms in thin films: Application to layer transfer
L. Ponson,
a

K. Diest, H. A. Atwater, G. Ravichandran, and K. Bhattacharya
Division of Engineering and Applied Science, California Institute of Technology, Pasadena, California
91125, USA

Received 22 October 2008; accepted 29 December 2008; published online 7 April 2009

We investigate the origin of transverse cracks often observed in thin films obtained by the layer
transfer technique. During this process, two crystals bonded to each other containing a weak plane
produced by ion implantation are heated to let a thin layer of one of the material on the other. The
level of stress imposed on the film during the heating phase due to the mismatch of thermal
expansion coefficients of the substrate and the film is shown to be the dominent factor in
determining the quality of the transferred layer. In particular, it is shown that if the film is submitted
to a tensile stress, the microcracks produced by ion implantation are not stable and deviate from the
plane of implantation making the layer transfer process impossible. However, if the compressive
stress exceeds a threshold value, after layer transfer, the film can buckle and delaminate, leading to
transverse cracks induced by bending. As a result, we show that the imposed stress

m
—or
equivalently the heating temperature—must be within the range −

c


m

0 to produce an intact
thin film where

c
depends on the interfacial fracture energy and the size of defects at the interface
between film and substrate. ©
2009 American Institute of Physics
.

DOI:
10.1063/1.3078801

I. INTRODUCTION
Various applications in electronics and optics require the
synthesis of high quality, defect-free single crystals on a sub-
strate of a different material. Diverse heteroepitaxial growth
processes have been proposed

e.g., Ref.
1

, but these meth-
ods impose severe restrictions on the film/substrate combina-
tions. Recently, the
layer transfer process
has been proposed
and shows promise as an alternative when the film/substrate
pair is very different.
2
,
3
Layer transfer is accomplished by
implanting hydrogen or helium ions into a bulk crystal of the
film to be transferred and then bonding it to a substrate.
Acting as damage precursors, these ions induce nucleation
and growth of cavities when the specimen is heated at a
sufficiently high temperature, transferring onto the substrate
a single crystal thin film whose thickness corresponds to the
depth of ion implantation. However, for some systems and
heating conditions, undesirable transverse cracks are also
produced in the thin film during the splitting process. This
phenomenon renders the transferred thin film useless for ap-
plications in microelectronics and others. Therefore, under-
standing the origin of such cracks is crucial to avoid their
formation. Identifying quantitatively the conditions and the
systems that are advantageous to nucleate these undesirable
cracks will help to define the limitations of the layer transfer
process and to design possible solutions to overcome these
limitations. This motivates the present analysis and the
mechanism of formation of these undesirable cracks is the
central point of this study.
In Sec. II, the geometry used during the layer transfer
process as well as the state of stress in the film are described.
Then, a first possible origin of thin film failure is investigated
in Sec. III: the stability of cracks nucleating from defects
introduced by ion implantation is analyzed, and we show that
these cracks propagate parallel to the film/substrate interface
only for a compressive state of stress in the film. In Sec. IV,
we show that a compressive stress in thin film can also lead
to cracking by buckling, delamination, and then failure of the
film. This analysis provides an acceptable range for the com-
pressive stress and therefore limitations of the heating tem-
perature for a given system with fixed film thickness that will
lead to a continuous thin film. In Sec. V, these theoretical
predictions are combined with experimental observations
made on a lithium niobate film bonded to a silicon substrate.
The two failure mechanisms previously proposed are clearly
identified in an analysis of the specimen after layer transfer.
The theoretical criterion for good layer transfer


c


m

0

is found to agree with experimental observations.
II. GEOMETRY OF THE SYSTEM AND STRESS STATE
OF THE FILM
To perform layer transfer, the material to be cut is
bonded on a substrate as shown in Fig.
1
. A bonding layer,
observed to improve adhesion and avoid undesirable crack-
ing for some systems, is also shown. Its influence on the
whole system is limited to the interface properties between
film and substrate

fracture energy and defect size

so that
this interlayer can be neglected in the following analysis
without loss of generality. Such a layered system is then
submitted to an elevated temperature

T
and microcracks
can nucleate in the plane of the film where hydrogen and/or
helium has been previously implanted

dashed plane in Fig.
1

. When these microcracks coalesce, the bulk single crystal
is separated from the transferred thin film with thickness
h
.
During the heating phase of the process, the film is sub-
mitted to a homogeneous biaxial stress

m
caused by the
mismatch in thermal expansion between the film and the
substrate. Noting


=

s

f
, the difference between the lin-
ear thermal expansion coefficients of the substrate and the
a

E-mail: ponson@caltech.edu.
JOURNAL OF APPLIED PHYSICS
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, 073514

2009

0021-8979/2009/105

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film, one can show that irrespective of the thickness and
thermal properties of the bonding layer, the stress in the film
is given by
4

m
=
E
1−


T


,

1

where
E
and

are the Young’s modulus and the Poisson’s
ratio of the film, respectively. We will see that to ensure
transfer of a thin film without undesirable transverse cracks,
the stress imposed on the film must be within a certain range
of values to be determined in Secs. III–VI.
III. STABILITY OF MICROCRACKS IN THE FILM
Let us focus first on the trajectory of microcracks that
initiate from the defects induced by the presence of hydrogen
and/or helium in the specimen. To result in layer transfer,
these microcracks are expected to propagate in a relatively
straight manner, i.e., parallel to the interface between the film
and the substrate. The stability analysis of a one-dimensional

1D

crack propagating in a two-dimensional

2D

elastic
medium submitted to an internal stress was performed by
Cotterell and Rice.
5
To apply this result to the layer transfer
process, we should make the hypothesis that the behavior of
the three-dimensional

3D

system as represented in Fig.
1
is
analogous to that of a cut of the full system along a plane
perpendicular to the film/substrate interface, e.g., the plane

OYZ

. In other words, we should suppose that the 2D
penny-shaped microcracks propagating in the plane of ion
implantation of the film can be approximated by 1D crack
lines. This simplification is
a priori
not obvious, and in the
following, we will study the propagation of a 2D crack in a
3D elastic medium. Figure
2
represents a part of the crack
front of a 2D penny-shaped microcrack when observed at a
sufficiently small scale so that the crack front appears
roughly straight. The average front is taken parallel to the
z
axis of the local coordinates

Oxyz

and propagates along the
x
axis. As a result, the crack propagates in a plane parallel to

x
,
z

, i.e., parallel to the interface between the film and the
substrate. The question we address here is whether the crack
will go on propagating within a plane parallel to

x
,
z

or will
deviate from the straight trajectory because of the deflections
generated by local heterogeneities in the film, as, e.g., those
induced by ion implantation. To proceed to such a stability
analysis, we slightly perturb the crack front with respect to
the straight geometry and study if those perturbations will
tend to zero or will diverge while the crack is propagating.
We define both out-of-plane perturbations
f

x
,
z

along the
y
axis

and in-plane perturbations
g

z
,
t

along the
x
axis

that
are represented in Fig.
2
. One can show that for small de-
flections, only the out-of-plane perturbations
f

x
,
z

are rel-
evant to determine the local shearing
K
II

x
,
z

at the crack tip
and, hence, the trajectory of the crack.
6
Let us note that the
following analysis is independent of the shape of the small
crack perturbations and remains valid whatever the choice of
g
and
f
are. To predict the crack trajectory, we apply the
principle of local symmetry:
5
,
7
,
8
locally, at every point of the
front
M

x
,
f

x
,
z

,
z

, the crack propagates in a pure mode I

opening

state of stress. This condition is written as
K
II

M

x
,
f
,
z

=0.

2

Movchan
et al.
9
calculated the mode II stress intensity factor
of a slightly perturbed crack propagating in an infinite 3D
elastic medium for any perturbation
f

x

. Using their result,
the local mode II stress intensity factor of cracks propagating
in the ion implanted plane of the specimen can be expressed
as
K
II
=

K
I
0
2

f

x


x
,
z

K
I
0
2

2−3

2−



+

f

x
,
z


f

x
,
z


z

z

2
dz

+

K
II
memory
,

3

where the
memory
term
K
II
memory
is given by

K
II
memory

x
,
z

=−


x


+


w
x
II

x
x

,
z
z







fT
xx


x


x

,
z


+



fT
xz


z


x

,
z


+
w
z
II

x
x

,
z
z






fT
xz


x


x

,
z


FIG. 1. Geometry and stress field of the layered system. The dashed plane
coincides with the plane of ion implantation.
FIG. 2. Geometry of a slightly perturbed crack propagating in the film.
073514-2 Ponson
etal.
J. Appl. Phys.
105
, 073514

2009

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+



fT
zz


z


x

,
z


dx

dz

,

4

with
w
x
II

x
,
z

=
−2
xH

x


3
/
2


x
2
+
z
2


1+
2

2−

1−

z
/
x

2
1+

z
/
x

2
,
w
z
II

x
,
z

=
−2
xH

x


3
/
2


x
2
+
z
2

2

2−

2
z
/
x
1+

z
/
x

2
,

5

where
H

x

is the Heaviside function. In the preceding ex-
pressions,
K
I
0
represents the average mode I stress intensity
factor applied to the crack by the heated gas in the micro-
cavities, while
T
xx
,
T
zz
, and
T
xz
are the
T
-stress terms or
constant stresses imposed on the film in the absence of
cracks. This implies that
T
xx
=

m
,
T
zz
=

m
, and
T
xz
=0. Equa-
tion

3

provides the different contributions to the mode II
shearing at a point
M
of the crack front induced by the per-
turbations of the fracture surface. The first term in Eq.

3

corresponds to the contribution of the local slope along the
propagation direction, while the second term provides the
shearing induced by perturbations of the crack front. The
third term, also referred to as the memory term, gives, as
indicated by its name, the mode II contribution induced by
the out-of-plane deviations of the crack line between its point
of initiation and current position. This term is expressed as a
function of the internal stress

m
in the film, using the full
expression of Eq.

4

and changing
T
xx
,
T
zz
, and
T
xz
by their
relevant expressions. Isolating the first term proportional to
the local slope of the crack surface, the expression of the
crack path, as given by the principle of local symmetry of
Eq.

2

, can be rewritten as


f

x


x
,
z

=
1

2−3

2−



+

f

x
,
z


f

x
,
z


z

z

2
dz

+

m
2
K
I
0


x


+


w
x
II

x
x

,
z
z





f

x


x

,
z


+
w
z
II

x
x

,
z
z





f

z


x

,
z


dx

dz

.

6

This equation predicts the crack path and so the stability
of the failure process: let us take a perturbation of the crack
oriented along the positive
y
axis so that
f
0. If

f

x

0, the
local perturbation
f

x
,
z

is rapidly suppressed during crack
propagation, and the crack surface is on average flat. On the
contrary, if

f

x
0, even a small perturbation will grow and
will lead to a macroscopic deviation of the crack from the

x
,
z

plane parallel to the interface. In the latter case, crack
propagation trajectory is referred to as unstable. This situa-
tion will clearly lead to catastrophic transverse cracks in the
thin film during the layer transfer process.
Next, we assess the relevance of each term of the right-
hand side of Eq.

6

that determines the stability of microc-
racks in the film during heating. The first term acts as a
nonlocal restoring force along the crack front that tries to
maintain it straight. However, this term does not prevent the
crack from deviating away from the mean crack plane
10
and
therefore does not contribute directly to the stability of the
crack. The second term is composed of a part proportional to

f

x
and another proportional to

f

z
. To assess the relative im-
portance of each term, one can compare their two prefactors,
w
x
II
and
w
z
II
, respectively. According to Eq.

5

,
w
z
II
is smaller
than
w
x
II
,
11
and for most values of

z
,
x

, one gets
w
x
II
w
z
II
1. In
other words, the stability of the crack is mainly dictated by
the term proportional to

f

x
, leading to the approximation


f

x


x
,
z


m
2
K
I
0


x


+

w
x
II

x
x

,
z
z





f

x


x

,
z


dx

dz

.

7

From this equation, one can assess the evolution of the local
slope of the crack surface. From Eq.

5

, one notes that
w
x
II
0. Therefore, the sign of

m
will determine the evolution
of the solution of Eq.

7

.If

m
0, then

f

x
is expected to
increase when the crack propagates, while with

m

0,

f

x
will tend to zero after a finite distance.
12
From analysis of the stability of a crack propagating dur-
ing heating, one obtains:

i

If the thin film is in a state of tensile stress


m
0

,
then the microcracks nucleated from the damage in-
duced by ion implantation during the heating phase
will deviate from the plane of implantation. One can
therefore expect transverse cracks within the trans-
ferred thin film from systematic deviations of these
microcracks.

ii

If the film is in a state of compressive stress


m

0

, then the microcracks are expected to propagate
along a straight trajectory within the plane of ion im-
plantation and will result in the transfer of a crack-
free single crystal thin film. This compressive stress
state is obtained if the thermal expansion coefficient
of the film if larger than that of the substrate

see Eq.

1


.
As a result, the condition

m

0 is necessary to obtain
straight crack propagation and therefore an intact thin film.
Let us note that this result is not limited to multilayer sys-
tems and can be extended to other systems where the crack
trajectory needs to be analyzed: a 2D crack will remain con-
fined to a plane perpendicular to the external tensile loading
if the stress is in compression along all the directions of this
plane, while it will deviate from the straight trajectory if the
stress is tensile along the mean plane of the crack. This
analysis shows that the stability criterion shown for 2D elas-
tic solids under mode I loading
5
remains valid for 3D sys-
tems. In Sec. IV, we will investigate another possible origin
of film cracking and show that there is a limit to the amount
of compressive stress the film can support, and an exces-
sively high compressive stress in the film can also lead to
poor quality transferred thin films.
073514-3 Ponson
etal.
J. Appl. Phys.
105
, 073514

2009

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IV. BUCKLING, DELAMINATION, AND FAILURE OF
THE FILM
Here, another possible mechanism for film cracking dur-
ing layer transfer process is investigated. Previously, we have
shown that a state of tensile stress in the crystal containing
the implanted plane must be avoided to ensure proper layer
transfer. Therefore, systems with negative mismatch


=

s

f
between thermal expansion coefficients of the sub-
strate and the film will be advantageously chosen. As an
indirect consequence, the thin film freshly obtained may un-
dergo a high compressive stress

m

0, as given by Eq.

1

.
It is well known that thin films under compression can
buckle and delaminate.
4
,
13
We will see that these processes
can have catastrophic consequences because it can lead to
film failure by bending. The conditions leading to buckling,
delamination, and failure of the film produced by layer trans-
fer and subjected to a compressive stress

m
are investigated
in detail in Secs. IV B, IV C and IV D, respectively. The film
is supposed to be perfectly brittle so that the equations of
elasticity for thin plates can be used. In addition, in first
approximation, the fracture energy
G
c
of the film/substrate
interface is assumed to be constant and independent of the
phase angle
=arctan

K
II
K
I

of the stress acting on the
interface.
4
A. Delamination of a film with a semi-infinite defect
Under compressive stress, a film bonded to a substrate
can delaminate in order to release its internal stress. For an
infinite film bonded to an infinite rigid substrate with a
straight delamination front separating the film into two semi-
infinite bonded and debonded parts, the elastic energy re-
leased during the propagation over a unit area is given by
4
G
del
=

m
2
h
2
1−

2
E
,

8

where
h
,
E
, and

are the thickness, the Young’s modulus,
and the Poisson’s ratio of the film, respectively. Let us note
that we do not consider cases where the substrate modulus is
appreciably smaller than that of the film. This situation has
been considered elsewhere.
14
Noting
G
c
the interfacial frac-
ture energy between the film and the substrate

or the bond-
ing layer if it has been added to the system

Fig.
1

, one can
use the Griffith criteria
G
del
=
G
c
providing the onset of crack
propagation to get an expression of the critical stress

del
for
delamination,

del
=
2
EG
c
h

1−

2

.

9

It must be emphasized that the initial condition taken here
with a semi-infinite debonded zone favors interfacial crack
propagation. In more realistic systems with defects or deb-
onded zones of finite size at the interface between film and
substrate

or bonding layer

, such a level of compressive
stress might not induce delamination. In addition, another
mechanism must be taken into account to describe the
delamination of films: buckling, frequently observed in thin
film under compression, leads to modifications of the expres-
sion of the energy release rate
G
as given in Eq.

8

.Inthe
following section, we focus on this process and the condi-
tions for film buckling. The out of plane displacements of the
film are then taken into consideration in order to predict
propagation of the delamination crack. In all the following,
we limit our analysis to a 2D geometry of the specimen

e.g.
plane

Oxy

in Fig.
1

. We consider defects of length 2
a
at
the interface between film and substrate and determine if
these debonded zones can grow and lead to catastrophic con-
sequences for layer transfer.
B. Buckling of the film
We consider the situation represented in Fig.
3

a

where
an initial defect or debonded zone of size 2
a
is present at the
interface between the film and the substrate. Under a suffi-
ciently high compressive stress, the film can buckle as shown
in Fig.
3

b

, and a stability analysis of the film provides
expression for the critical stress.
4
Consider now that the film is submitted to a given com-
pressive stress

m
. One can use this expression to show that
buckling will occur if the delamination zone is larger than a
critical size
a
b
, where
a
b
=

h
2
E
3

1−

2


m
.

10

This process is energetically favorable because, in essence, it
increases the effective length of the film.
C. Propagation of the delamination front induced by
film buckling
As mentioned previously, buckling of the film affects the
energy release rate of the interfacial crack so that the buck-
ling pattern must be taken into consideration when predicting
the onset of delamination. In particular, the stress concentra-
tion at the edge of a debonded zone changes drastically with
the size of the buckling zone. This effect is represented in
Fig.
4
where the variations in the energy release rate
G
are
represented as a function of the half-length
a
of the deb-
onded zone. For sufficiently large buckling zones,
G
might
reach
G
c
and the interfacial crack can propagate. To assess
the critical size
a
p
that allows a buckling pattern to propa-
gate, one can derive the value of the energy release rate for a
buckled zone of length 2
a
,
4
FIG. 3. Two-dimensional profile of a debonded part of a film

a

without
and

b

with buckling.
073514-4 Ponson
etal.
J. Appl. Phys.
105
, 073514

2009

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G

a

=

m
2

1−

2

h
2
E

1−
a
b
2
a
2

1+3
a
b
2
a
2
,

11

that is represented in Fig.
3

b

. It is interesting to note that at
the onset of film buckling

a
=
a
b

, there is no driving force
for delamination

G
=0

. However, if the compressive stress
in the film is increased, the value of
a
b


m

will decrease
leading finally to a net increase in the delamination driving
force. As a result, propagation is possible at a certain stress
level when the condition
G

a

=
G
c
is satisfied. Solving the
previous equation with respect to
a
, one obtains the critical
length
a
p
above which the buckling zone will propagate,
a
p
=

h
2
E
1−

2
1

m
1
1+
4−
6
EG
c
h

1−

2


m
2
.

12

a
p
being a decreasing function of

m
, it is also clear in this
representation that a sufficiently large compressive stress
will induce delamination. Note that Eq.

12

is only valid for

m
3
2

del
where

del
has been introduced in Eq.

9

. For
smaller values of compressive stress

m
in the film, the
buckling zone remains stable regardless of the initial size of
the debonded zone.
However, the previous analysis is limited to crack initia-
tion and to predict the full evolution of the system beyond
initiation, it is important to separate two cases, as illustrated
in Fig.
4
:

1

If the critical length
a
p
for interfacial crack propagation
is smaller than
3
2
a
b
, where
a
b
is the critical length for
buckling

Eq.

10


, the equation
G
c
=
G

a

has only one
solution
a
p
given by Eq.

12

, corresponding to the size
of the smallest defect leading to crack initiation. The
condition
G
c
G

a

for crack propagation being satis-
fied whenever
a
a
p
, this situations corresponds to
crack propagation without arrest.

2

If the length
a
p
is larger than
3
2
a
b
, the equilibrium
equation for debonding is satisfied for two crack lengths,
a
p
and
a
a
. The elastic energy released is larger than the
fracture energy only for crack extensions between these
two length scales so that initiation and crack arrest occur
successively for
a
=
a
p

Eq.

12


and
a
=
a
a
, with
a
a
=

h
2
E
1−

2
1

m
1
1−
4−3


del

m

2
.

13

The conditions for both situations can be rewritten in
terms of stress, and unstable crack propagation corre-
sponds to

m

del
, while crack arrest will be observed
if

del

m
3
2

del
. The value of the defect length cor-
responding to
a
p
=
a
a
is noted
a
del
, where
a
del
=

h
2
4
2
hE
3
G
c

1−

2

.

14

In both cases, the propagation of these interfacial cracks
may adversely affect the quality of the transferred thin film.
In particular, for sufficiently large buckled patterns, i.e., large
enough interfacial crack extension, a transverse crack in-
duced by the bending generated in the film can fracture the
crystal layer. It is worth noting that this process may not
occur for an interfacial failure with a small extension. The
conditions to obtain such transverse cracks are now dis-
cussed in detail.
D. Failure of the thin film induced by bending
When buckling occurs, the delaminated zone undergoes
bending. If the original debonded zone is sufficiently small,
bending increases while the size of the buckling zone in-
creases. For a sufficiently large buckling zone, the film is not
strong enough to support the tensile stress induced by bend-
ing in the film and a crack initiating from the upper surface
of the film in
x
=0 will propagate parallel to the
y
axis toward
lower surface

Fig.
3

b


. In this geometry, crack propagation
is expected to be without arrest, and propagation will occur
throughout the crystal layer.
To predict the onset of crack initiation, we use a criterion
based on the value of the curvature of the film

akin to criti-
cal strain

, as e.g. in Ref.
15
: failure occurs when the curva-
ture
d
2
w
dx
2
at some point of the film exceeds the critical value
1
R
c
, where
R
c
is a constant depending not only on the intrinsic
strength of the material but also on the state of surface of the
freshly cut crystal. As seen in Fig.
3

b

, a possible transverse
crack will initiate around
x
=0 where the local curvature of
the film is maximum. The deflection
w

x

of the film is then
expressed in terms of the delaminated zone size 2
a
and the
compressive stress

m

e.g., Ref.
4

, providing an expression
for the maximum curvature
d
2
w
dx
2
x
=0
of the film. From this
expression and the curvature based failure criterion intro-
duced previously, one can show that transverse failure occurs
for buckled thin film larger than
a
f
, with
a
f
=

4
3

1−

2

hR
c

m


1−
1−




m
2
,

15

where


=
E
2
3

1−

2

h
R
c
.

16

Note that film failure is impossible if

m



, regardless of
the size of the debonded zone. For

m
=


, we introduce the
FIG. 4.

Color online

Variations in the energy release rate of an interfacial
crack at the edge of a buckled zone of length 2
a

Fig.
3

b


.2
a
b
corre-
sponds to the minimum length for a debonded zone in a film of the same
thickness under the same compressive stress to buckle.
073514-5 Ponson
etal.
J. Appl. Phys.
105
, 073514

2009

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