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Transcript of Brittle-to-ductile transition temperature in indium phosphide
BRITTLE-TO-DUCTILE TRANSITION TEMPERATURE IN INDIUM PHOSPHIDE
by
LEONARDUS BIMO BAYU AJI
Submitted in partial fulfillment of the requirements
for the degree of Master of Science
Thesis Adviser: Prof. Pirouz Pirouz
Department of Materials Science and Engineering
CASE WESTERN RESERVE UNIVERSITY
January, 2006
1
TABLE OF CONTENTS
TABLE OF CONTENTS …………………………………………………… 1
LIST OF TABLES ……………………………………………………………... 3
LIST OF FIGURES ……………………………………………………………... 4
ACKNOWLEDGEMENT …………………………………………………… 10
ABSTRACT ………………………………………………………………….. 11
1. INTRODUCTION AND OBJECTIVE ………………………………… 12
1.1. Introduction ………………………………………………………… 12
1.1.1. Crystal Structure, Slip Plane and Slip System ……………… 13
1.1.2. Dislocations and Dislocation Cores in III-V Compound
Semiconductors ………………………………………………... 14
1.1.3. Stacking Faults ………………………………………………... 17
1.1.4. Dislocation Movement ………………………………………. 19
1.1.5. Surface Polarity and Symmetry in Compound Semiconductors 20
1.2. Objectives of Research ………………………………………………… 22
2. LITERATURE STUDY ……………………………………………………... 23
2.1. Griffith Theory ………………………………………………………….. 23
2.2. Dislocation - Crack Interactions ………………………………………. 27
2.3. Dislocation Velocity in Indium Phosphide …………………………….. 31
2.4. Previous Work on Plasticity of Indium Phosphide …………………… 34
2
3. EXPERIMENTAL SETUP AND SAMPLE PREPARATION …………... 40
3.1. 4-Point Bend Tests ……………………………………………………… 40
3.1.1. Sample Preparation …………………………………………….. 43
3.1.2. Technique of Introducing Precracks ……………………………… 45
3.1.3. Instron 1361 Tensile Machine …………………………………... 46
3.2. Indentation Tests …………………………………………………….... 48
3.2.1. Sample Preparation …………………………………………….. 49
3.2.2. Static Indentation Tests ………………………………………... 49
3.2.3. Dynamic Indentation Tests ………………………………………... 51
4. RESULTS AND DISCUSSION …………………………………………….. 55
4.1. 4-Point Bend Tests ……………………………………………………… 55
4.1.1. Brittle Behavior …………………………………………………. 57
4.1.2. Transition Behavior …………………………………………….. 58
4.1.3. Ductile Behavior …………………………………………………. 58
4.2. Indentation Tests ……………………………………………………… 59
4.2.1. Static Indentation Tests ……………………………………….. 59
4.2.2. Dynamic Indentation Tests ……………………………………….. 66
5. CONCLUSION …………………………………………………………… 72
APPENDIX …………………………………………………………………….. 73
REFERENCES ...………………………………………………………… 76
3
LIST OF TABLES
Table 3.1: Properties of the InP crystal used, grown by and obtained from the
Institute of Electronic Materials Technology (ITME), Warszawa, Poland
….…………………………………………………………………….. 40
Table 4.1: g b 0 analysis for Burgers vector of partial dislocations in a face-
centered cubic crystal …………………………………………….. 64
4
LIST OF FIGURES
Figure 1.1: (a) Sphalerite structure; gray and black circles are indium and phosphorus
atoms respectively (or vice versa), (b) stacking sequence of 111 planes
in the sphalerite structure. A , B , or C planes consist of indium atoms
while a , b or c planes consist of phosphorus atoms (or vice versa)
…………………………………………………………………..….... 13
Figure 1.2: Schematic showing different core structures of dislocations in compound
semiconductors……………………………………………………..... 14
Figure 1.3: Schematic of a screw dislocation viewed from top …………………. 15
Figure 1.4: Slip in face-centered cubic crystals ………………………………….. 16
Figure 1.5: Stacking sequence of 111 planes after slip of (a) a leading partial
dislocation on a plane (between C and A in this example), (b) two
partial dislocations on adjacent planes (the first between C and A , the
second between B and C ), (c) four partial dislocations on adjacent
planes (the first between C and A , the second between B and C , the
third between A and B , and the fourth between C and A )
…………………………………………………………………….… 17
Figure 1.6: Movement of dislocations by generation and motion of kinks …….. 19
Figure 1.7: (a) Polarity in the sphalerite structure, (b) the 110 directions correspond
to the interaction of 11 1 and 11 1 planes (shaded planes), which have
111 A polarity (i.e. contain all A atoms), whereas 110 directions
5
correspond to the intersection of 111 and 111 planes, which have
11 1 B polarity (i.e. contain all B atoms) …................................... 20
Figure 2.1: Plate containing an elliptical hole with semi-axes a and b subjected to a
uniform applied tension L ……………………………………..…. 24
Figure 2.2: A sharp crack with an intersecting slip plane showing the competition
between dislocation emission and cleavage propagation …………….. 28
Figure 2.3: An atomically sharp crack is blunted when a dislocation is emitted from
the tip when the Burgers vector has a component normal to the fracture
plane………………………………………………………………….... 28
Figure 2.4: Schematic illustration of dislocation nucleation at a crack tip (a) at
BDTT T , and (b) at …………………………………………… 30
Figure 2.5: (a) Velocity, v , of dislocations in undoped and S-doped n-type InP single
crystals as a function of (a) resolved shear stress, , and (b) temperature,
……………………………………………………….. 31
Figure 2.6: Velocity versus resolved shear-stress at 723 K in various indium
phosphide crystals for (a) dislocations (b) dislocations, and (c) screw
dislocations …………………………………………………….. 33
Figure 2.7: (a) Resolved shear stress versus shear strain curves for undoped InP
between 573 K and 1023 K. (b) Magnification of (a) ………………... 34
Figure 2.8: Variation of resolved shear stress at lower yield point, LYP , versus
temperature ………………………………………………………….... 36
6
Figure 2.9: Critical resolved shear stress c against temperature, . The stress denotes
the shear stress component resolved in the 101 direction on the 111
plane. Shear strain rate is 41.2 10 s-1 ……………..…….. 37
Figure 2.10: Slip lines on the side surface of InP deformed at 300 K . The slip indicates
1101 111
2 slip with frequent cross-slip …………………………. 38
Figure 2.11: TEM images of slip bands in InP deformed at 300 K . The foil was cut
parallel to the 111 slip bands. The first image is bright field exhibiting
many straight screw dislocations (shown in Fig. 2.10) are all out of
contrast in the g 202 reflection. The second image is weak beam dark
field revealing dissociation of screw dislocations…………………..… 39
Figure 3.1: Comparison of tensile stress distribution in 3-point and 4-point bend
samples. The shaded area represents tensile stress, a region ranging from
zero at the supports of the bend samples to (a) a maximum at the midspan
for the 3-point bend geometry, and (b) uniform maximum along the whole
gauge length of the samples for the 4-point bend geometry ………….. 41
Figure 3.2: Schematic of a 4-point bend test geometry. 1y and 2y are the vertical
distances between the outer rollers and inner rollers ……………….. 42
Figure 3.3: The crystallographic orientation and dimensions of a 4-point bend sample
(actual sample dimensions after polishing). …………………………... 44
Figure 3.4: A 4-point bend test sample containing five identical radial precracks,
introduced by Knoop indents spaced about 20 μm apart ……………. 45
7
Figure 3.5: (a) Photograph of the Instron 1361 tensile machine with the tensile jig and
the furnace mounted on it, and (b) schematic of the 4-point bend jig
connected to the displacement rods of the machine ………………….. 46
Figure 3.6: Schematic of indentation mechanism. P is the applied load, d is the
indentation diagonal and h is the indentation depth ……………........ 48
Figure 3.7: Schematic of indented surface in indium phosphide with a 110
orientation of indenter diagonals ……………………………………... 50
Figure 3.8: Photograph of the Nikon High Temperature Microhardness Tester model
QM ……………………………………………………………………. 50
Figure 3.9: (a) Photograph of the HTDSI apparatus, and (b) schematic diagram of the
HTDSI apparatus: (A) sample, (B) indenter, (C) sample furnace, (D)
indenter furnace, (E) capacitance displacement gauge, and (F) load
application coil ……………………………………………………….. 52
Figure 3.10: Typical load-depth plot for a material that is subject to elastic and plastic
deformation …………………………………………………………… 53
Figure 4.1: Load displacement curves for InP samples tested at , 355 C and
378 C at a strain rate of 5 12.93 10 s . At 350 C InP deforms
elastically until fracture; at 355 C InP is in the transition region, above
355 C , e.g., at 378 C , InP deforms plastically ………….………....... 56
Figure 4.2: Applied stress versus temperature for samples tested at a strain rate of
5 12.93 10 s . The brittle-to-ductile transition temperature occurs over
a very narrow temperature range of 5 , and is characterized by a sudden
increase of the applied stress ………………………………….......... 57
8
Figure 4.3: Hardness versus temperature plot for sample tested from 380 C down to
20 C , with a step size of 20 C and a 50 g indentation load ...……. 60
Figure 4.4: Indentation impressions produced by static indentation technique at (a)
123 C , (b) 263 C and (c) 320 C …….......................................... 61
Figure 4.5: Temperature dependence of the impression diameter and crack length
along 110 and 110 directions over a temperature range of 20 C to
380 C using an indentation load of 50 g . The sample was initially heated
to the highest temperature and indented. Following this, the sample was
cooled and indentations were made in a step size of 20 C until the lowest
temperature of 20 C ………………………………………………….. 62
Figure 4.6: Bright field TEM images of stacking faults in InP deformed by static
indentation technique at 320 C . Fig. 4.7(a) shows two sets of stacking
fault ribbons perpendicular to each other. One set of the stacking fault
ribbon is out of contrast in 220 and 311 reflections, as shown in figures
(b) and (c) respectively ……………………………………………... 65
Figure 4.7: Load-displacement curves obtained from dynamic indentation tests on
001 surface of InP at different temperatures: (a) 25 C , (b) 150 C , (c)
250 C , and (d) 400 C …………………………………………...... 67
Figure 4.8: Dissipated energy versus temperature plot obtained by integrating the area
under the load-displacement curves ………………………………. 68
Figure 4.9: Energy density versus temperature plot obtained by dividing the dissipated
energy with the indentation volume ……………………………..... 69
Figure 4.10: Schematic variations of stress versus temperature in a crystal .... 70
9
Figure A.1: Applied stress versus temperature for samples tested at a strain rate of
6 19.6 10 s . The brittle-to-ductile transition temperature occurs
between 374 C and 376 C ………………………………………… 73
Figure A.2: Applied stress versus temperature for samples tested at a strain rate of
5 12.88 10 s . The brittle-to-ductile transition temperature occurs over
a narrow temperature range of 10 , and is characterized by a sudden
increase of the applied stress ……………………………………...... 75
ACKNOWLEDGEMENT
10
I thank my adviser, Professor Pirouz Pirouz for his support and guidance in this work.
His vast knowledge and tremendous zeal has been a great source of inspiration for me
during this period.
I also wish to express my gratitude to Professor Peter Legerlöf and Professor John
Lewandowski for serving on my thesis committee and helping me with valuable
suggestion.
I wish to thank Paul Wesseling for training me on Nikon microhardness machine, and to
Chris Tuma for training me on INSTRON tensile machine. I wish to thank Shanling
Wang, Ming Zhang, Changrong Li and Kevin Spear for bearing my numerous questions
and their invaluable suggestions related to this work. Many thanks to Arun Reddy, Min
Huh, Meisssa, Sebastian, Yeonseop Yu and Ali Shamimi for their friendship.
I thank my parents and brother for their constant prayer and love. I dedicated this work
for them.
11
Brittle-to-Ductile Transition Temperature in Indium Phosphide
Abstract
by
LEONARDUS BIMO BAYU AJI
001 single crystals of indium phosphide were deformed by 4-point bend tests and two
types of indentation tests; static and dynamic. Temperature ranges where the material
exhibited a brittle or a ductile behavior were investigated with particular focus on the
transition from one deformation mode to another. The 4-point bend tests and the
dynamical indentation tests show that indium phosphide exhibits a sharp brittle-to-ductile
transition temperature. Static indentation tests were used to determine the dependence of
hardness on temperature, and the plastic zone produced around the indentation
impressions was characterized by transmission electron microscopy (TEM).
CHAPTER 1
12
INTRODUCTION AND OBJECTIVES
In this chapter a brief summary of the applications and properties of indium phosphide is
given, and the importance of knowing the mechanical properties of this material is
explained. Subsequently, the objectives of this thesis and the scope of the research are
discussed.
1.1 Introduction
Indium phosphide (InP) is a promising semiconducting material for a variety of practical
applications, especially for optoelectronic devices, due to its direct energy band gap. It is
used as a platform for a wide variety of fiber communication components, including
lasers, LEDs, semiconductor optical amplifiers, modulators and photo-detectors [1].
Hence, understanding the physical properties of InP, including its mechanical properties
is essential for its wider use.
A number of investigators have performed deformation tests to study plasticity and
fracture behavior of InP. It is observed that, as in other semiconductors, InP exhibits
either a brittle or a ductile behavior depending on the applied stress, temperature, strain
rate, etc. However, very few fracture studies of InP are reported in the literature
compared to plastic deformation studies. In particular, to the best of the author’s
knowledge, there has been no report on the study of the transition from brittleness to
ductility in this material.
13
1.1.1 Crystal Structure, Slip plane and Slip System
Indium Phosphide, like other tetrahedrally coordinated semiconductors, is generally
considered to be a semi-brittle material. It crystallizes in the cubic zincblende (sphalerite)
structure, which consists of two interpenetrating f.c.c. lattices, one of which is shifted by
1114
a relative to the other. The two f.c.c. lattices are occupied by two different atoms,
i.e., indium and phosphorus [1], disposed at the corners of a regular tetrahedron. The
cubic unit cell of the sphalerite structure is shown in Fig. 1.1.
(a) (b)
Figure. 1.1. (a) Sphalerite structure; grey and black circles are indium and
phosphorus atoms respectively (or vice versa), (b) stacking sequence of 111
planes in the sphalerite structure. A , B , and C planes consist of indium atoms while a , b and c planes consist of phosphorus atoms (or vice versa).
As for f.c.c. crystals, the glide system in InP is 110 1112
a, i.e., the dislocations glide
on 111 slip planes and have Burgers vectors 1
b 1102
.
1.1.2 Dislocations and Dislocation Cores in III-V Compound Semiconductors
14
There are basically three different types of dislocations, termed , , and screw
dislocations in III-V compound semiconductor crystals with a sphalerite structure. The
and dislocations can be distinguished from one another based on: (1) the species of the
atoms ending at the extra half plane, and (2) the planes between which the extra half
plane terminates (i.e., in the shuffle set or glide set). For a long time, it was believed that
slip takes place between widely-spaced planes, i.e. the shuffle set. For instance, if we
assign A , B and C planes with In atoms and a , b , and c planes with P atoms (see Fig.
1.1(b)), dislocations with their extra half plane ending with In or P atoms are called
dislocation (also denoted as A (s), B (s) or C (s)) or dislocation (also denoted as a (s), b
(s) or c (s)). However, at present most investigators [2] believe that slip actually occurs in
the narrowly-spaced planes or the glide set. In that case, dislocations previously called α
dislocation are actually a (g), b (g) or c (g) and dislocation are actually A (g), B (g),
or C (g) (see Fig. 1.2).
Figure. 1.2. Schematic showing different core structures of dislocations in compound semiconductors (After George and Rabier [3]).
15
Screw dislocations are created when the atom planes perpendicular to the dislocation are
distorted in such a way as to form a spiral ramp of atomic planes with the dislocation as
the axis of the spiral as shown in Fig. 1.3.
Figure. 1.3. Schematic of a screw dislocation viewed from top; open and closed circles are respectively atoms above and below the slip plane.
The question as to whether the slip plane in sphalerite structures is on the shuffle set or
the glide set has been rather controversial. The dangling bonds are thought to have a high
energy and are produced by movement of shuffle or glide dislocation. These dangling
bonds can lower their energy by attracting impurity atoms (e.g. hydrogen) or by
reconstruction with neighboring dangling bonds, thereby eliminating them. Considering
the position of the dangling bonds in partial dislocations in the shuffle set and glide set,
the reconstruction would be much easier for the glide set than the shuffle set, because in
the shuffle set, the dangling bonds are normal to the slip plane and the reconstruction of
the dangling bonds of neighboring atoms along dislocation line entails large bond-
bending and bond-stretching. On the other hand, in the case of glide set, the dangling
bonds make a shallow angle with respect to the slip plane and the bond-bending and
bond-stretching involved in the reconstruction is relatively small.
16
However, Shockley [5] argues that a shuffle dislocation would be preferred because its
formation would involve only one dangling bond per atom compared to three dangling
bonds per atom - a higher energy process which implies greater resistance to motion - for
a glide dislocation.
Perfect dislocations, i.e. 1
b 1102
, in the sphalerite structure can dissociate into two
Shockley partials on the close-packed 111 planes on which they glide [6,7], in this case
slip is in a 112 direction, i.e. b 1126
a . The close-packed 111 planes can be
represented by a set of hard spheres as schematically illustrated in Fig. 1.4.
Figure. 1.4. Slip in face-centered cubic crystals (After Cottrell [6]).
The atoms in the first layer are represented by full circles, A , and atoms in the second
layer rest in the sites marked B . For the two layers to shear over each other to produce a
displacement in the 110 slip direction, it is energetically more favorable for the B
atoms to move via C positions rather than over the top of A atoms. The unit dislocation
17
with Burgers vector tot
1b 110
2 therefore dissociates into two dislocations b and
according to:
totb b bt
1 1 1
101 112 2112 6 6
The subscripts and t stand for leading and trailing. The separation of the leading and
trailing partials by a distance w creates a ribbon of stacking fault of width between them
(extended dislocation).
1.1.3 Stacking Faults
Stacking faults can be described by either removal or insertion of a close-packed layer. It
can also be formed by shearing, i.e. by the propagation of partial dislocations. If ABC
represents the stacking sequence of close-packed layers of an f.c.c. structure, then any
one of the following events can take place:
1) Shear of one leading partial dislocation on one 111 plane resulting in an intrinsic
stacking fault, as represented by Fig. 1.5(a).
18
Figure. 1.5(a). Stacking sequence of 111 planes after slip of a leading partial
dislocation on a 111 plane (between C and A in this example).
2) Two shears on successive planes forming an extrinsic stacking fault, as represented
by Fig. 1.5(b).
Figure. 1.5(b). Stacking sequence of planes after slip of two partial dislocations
on adjacent 111 planes (the first between C and A , the second between B
and C ).
3) Shear with the same displacement vector on every layer forming a microtwin, as
represented by Fig. 1.5(c).
19
Figure. 1.5(c). Stacking sequence of 111 planes after slip of four partial
dislocations on adjacent 111 planes (the first between C and A , the second
between B and C , the third between A and B , and the fourth between C and A ).
1.1.4 Dislocations Movement
There are various types of obstacles to the motion of dislocations on their slip planes. The
most intrinsic is the lattice resistance and the stress required to move a dislocation
through a crystal lattice from one Peierls valley (low-energy position for the dislocation)
to the next at 0 K . This stress is defined as the Peierls stress, p . In general, the
magnitude of Peierls stress is small for materials with an f.c.c. structure compared to
directionally bonded materials such as semiconductors.
It is commonly accepted that the mobility of dislocations in semiconductor materials,
except at high temperatures, is controlled by the Peierls mechanism. There are two
processes in the Peierls mechanism: (1) formation of a kink pair, and (2) subsequent
migration of the kinks along the dislocation line. These kinks may annihilate each other
when opposite kinks meet or they may arrive at the end of the dislocation line to
constitute a unit cycle of dislocation motion [8].
20
Figure. 1.6. Movement of dislocations by generation and motion of kinks.
1.1.5 Surface Polarity and Symmetry in Polar Compound Semiconductors
Surface polarity of sphalerite and wurtzite structures materials has been reviewed by Holt
[9]. He pointed out that the 110 and 110 directions in 100 faces of III-V
compound semiconductors are not crystallographically equivalent. The asymmetry of the
two orthogonal 110 directions in 100 faces of sphalerite structures is manifested in
phenomena arising from the surface polarity differences, i.e. 111 A 1 1 1 B .
The crystallographic anisotropy in sphalerite structures is shown in Fig. 1.7. Note in Fig.
1.7(a), the top 111 surface consists of only one type of atoms (i.e. A atoms), whereas
the bottom 11 1 surface consists of only the other type of atom (i.e. B atoms).
21
(a) (b)
Figure. 1.7. (a) Polarity in the sphalerite structures, (b) the 110 directions
correspond to the intersection of 11 1 and 1 1 1 planes (shaded planes),
which have 111 A polarity (i.e. contain all A atoms), whereas the 110
directions correspond to the intersection of 111 and 1 11 planes, which
have 1 1 1 B polarity (i.e. contain all B atoms).
The A atoms are located on the lattice sites, e.g. at the origin, whereas the B atoms are at
the other sites of the bases, e.g. at 1
4,1
4,1
4 in the cubic unit cell of the sphalerite
structure. Fig. 1.7(b) schematically shows that the 11 1 and 11 1 planes (shaded
planes) contain all A atoms, whereas 111 and 111 contain all B atoms. This
anisotropy can affect the behavior of plastic deformation along 110 and 110
directions.
1.2 Objectives of Research
22
The purpose of this work is to gain a better understanding of the mechanical properties of
InP and, specifically, of temperature ranges where the material exhibits brittleness or
ductility, and the transition from one mode of deformation to another.
Focus is drawn on dislocation processes that control the brittle-to-ductile transition
(BDT) temperature in InP using different techniques. Specifically this thesis addresses
the following:
Measurement of the fracture stress and the brittle-to-ductile transition
temperature from applied stress versus displacement curves obtained by 4-
point bend tests over a temperature range of to at fixed
strain rates.
Measurement of hardness using static indentation tests over a temperature
range of to , and characterizing the plastic zone beneath the
indentation impressions produced in the tests.
Measurement of dissipated energy from load versus depth curves obtained
by dynamic indentation tests.
Finally, transmission electron microscopy (TEM) is used to study the characteristic
morphology of deformation-induced dislocations in InP.
CHAPTER 2
23
LITERATURE STUDY
2.1 Griffith Theory
An important precursor to the Griffith study [11] was the stress analysis by Inglis [12] of
an elliptical hole in a uniformly stressed plate. Inglis proposed that local stresses at a
sharp notch or a corner could rise to a level several times larger than the applied stress.
Thus it is apparent that even a small submicroscopic flaw might be a potential source of
weakness in solids resulting in its fracture and failure.
In summary, the Inglis [12] analysis assumed that Hooke’s law holds everywhere in a
plate which is under a tensile stress L , as shown in Fig. 2.1. The boundary of the hole is
stress free (a requirement for equilibrium), and the axes a and b ( a and b are crack
dimensions) are assumed to be small compared to the plate dimension. Beginning with
the equation of an ellipse,
2 2 2 2/ / 1x a y b (2.1)
The radius of curvature at point C is given by:
2 /b a (2.2)
The greatest concentration of the normal stress, , occurs at point C, and is given by:
,0 1 2 /yy La a b (2.3)
By substituting 1/2b a from Eq. 2.2 into Eq. 2.3:
24
1/2,0 1 2 /yy La a (2.4)
In the case for b a , i.e. a , 1/2/ 1a the equation reduces to
1/2,0 / 2 /yy La a (2.5)
Figure. 2.1. Plate containing an elliptical hole with semi-axes a and b
subjected to a uniform applied tension L (After Lawn and Wilshaw [11]).
The ratio in Eq. 2.5 is often referred to as the elastic stress concentration factor. It can
easily be seen that this factor can have values much larger than unity for a narrow hole.
Thus the stress concentration depends sensitively on the crack’s shape rather than its size.
Griffith’s idea was to set up a model for a crack system in terms of reversible
thermodynamic processes. Griffith recognized that a material containing a uniformly
loaded crack in equilibrium must have a maximum free energy, U . For a static crack
system Griffith proposed that the total energy is the sum of two terms:
x
y
2a
2b C
25
L E SU W U U (2.6)
The first bracketed term is the mechanical energy of the system, where LW is the work
done by the applied load and EU is the stored elastic strain energy; this term decreases as
the crack extends. On the other hand, the second term, the surface energy SU , increases
as the crack extends, since energy is required to create new fracture surfaces. Whether a
crack extends or closes up depends on whether the left-hand side of Eq. 2.6 is negative or
positive.
For a static crack, the energy would be maximum when:
/ 0dU da (2.7)
In order for the crack to extend, the free energy must decrease and therefore:
/ 0dU da (2.8)
where a is the crack length. This is known as the Griffith criteria for crack growth.
In order to calculate the energy terms in Eq. 2.6, Griffith made use of the Inglis analysis,
considering the case of a narrow elliptical `crack`. Griffith used a standard result from
linear elastic theory for a crack formed under a constant applied load given by:
2L EW U (2.9)
where, EU is calculated using the earlier work of Inglis. The related equation can be
written as:
26
2 2 /E LU a E (2.10)
where L is applied stress normal to the crack plane (see Fig. 2.1), E is Young’s
modulus, and is half of the crack length. Assuming the crack is a slit of length 2a , SU
is
simply given by:
4SU a (2.11)
where is the free surface energy per unit area. The total energy of the system thus
becomes:
2 2 / 4LU a E a (2.12)
By applying the Griffith equilibrium condition (Eq. 2.7) into Eq. 2.12, one obtains the
critical condition for fracture given by:
1/22 /F L E a (2.13)
The above equation shows that the fracture stress, F , depends both on material
properties ,E as well as the length of the crack. As it is evident from Eq. 2.13, longer
cracks will result in lower fracture stresses.
Griffith formulated his basic criterion for fracture purely in terms of thermodynamics. He
did not explicitly take into account the physical mechanism of crack extension, which is
breaking of atomic bonds. In addition, for Eq. 2.8 to be met, the theoretical strength of
the solid must be exceeded at the crack tip (thus allowing rupture of atomic bonds) as (or
before) the fracture stress given by Eq. 2.13 is reached.
27
The maximum stress, max , at the crack tip can be calculated using Eq. 2.5 as proposed
by Inglis [12]. If the crack tip is ideally sharp, the crack tip is very small 0 and
from Eq. 2.5 the stress at the crack tip, max , becomes very large. By substituting Eq.
2.13 into Eq. 2.5, the maximum stress, max , at the crack tip is given by:
1/2 1/2 1/2
max 2 2 / / 2 /c cE a a E (2.14)
From Eq. 2.14 for a sharp crack, it is found that the stress at fracture exceeds the
theoretical tensile strength and hence satisfies the Griffith criterion (as given by Eq.
2.13), i.e. max is a necessary and sufficient condition for fracture.
2.2 Dislocation – Crack Interactions
There are two general conceptual models for assessing the brittle-to-ductile transition
temperature in a crystal. The first model is based on the criterion of dislocation-emission
at the crack tip. This model proposes that if a dislocation is nucleated at the crack tip at a
lower loading level than required for brittle cleavage, the material is considered ductile.
The principle behind this model is that emitted dislocations blunt the crack, shield the
crack tip from the applied external stress and inhibit cleavage and crack propagation. The
emitted dislocations can lower the stress field around the crack and therefore increase the
critical stress for cleavage. Rice and Thomson [13] introduced the idea that competition
between dislocation emission and crack propagation determines the failure mode, as
illustrated in Fig. 2.2; if a dislocation can be emitted from the crack tip prior to the
cleavage propagation, it is considered that the material is ductile.
28
Figure. 2.2. A sharp crack with an intersecting slip plane showing the competition between dislocation emission and cleavage propagation.
Figure. 2.3. An atomically sharp crack is blunted when a dislocation is emitted from the tip when the Burgers vector has a component normal to the fracture plane (After Rice and Thomson [13]).
In order for a dislocation to blunt a crack, it must have a component of its burgers vector
normal to the crack plane, and the slip plane must intersect the crack tip along its whole
length, i.e. the crack tip must lie in a slip plane. If the crack tip intersects a dislocation, a
localized step or jog in the crack tip is formed and this jog may act as a nucleation site for
other dislocations, favoring ductile behavior [14-15]. Even if the crack tip does not fully
crack
slip plane
crack/cleavage propagation
blunt/shield dislocations
29
lie in a slip plane, it may still be blunted along its whole length either by local nucleation
events or by a screw dislocation moving around the crack profile by repeated cross-
slipping. Schematic of an atomically sharp crack blunted by one atomic plane is shown in
Fig. 2.3.
The second model is exemplified by Hirsch et al [16]. It is based on the concept of
dislocation mobility; dislocations are emitted by sources activated by high stresses near
the crack tip and the rate at which they move away from the crack tip determines the
material toughness. The emitted dislocations can also shield the crack tip and the source,
and prevent both cleavage and further emission [17]. If the emitted dislocations are
mobile, they will move away from the crack tip, and as they move away, more
dislocations can be emitted from the crack tip. The mobility of the dislocation itself is
affected by temperature; at low temperatures, dislocations tend to be less mobile, and few
or no dislocations are emitted. Therefore crack propagation is the most likely mechanism
to relieve the stress concentration at the crack tip at low temperatures. On the other hand,
at high temperatures, dislocations are generated to blunt the crack tip; these in turn move
rapidly away, and the material is ductile.
An alternative model to describe brittle-to-ductile transition temperature is proposed by
Pirouz et al [18]. This model, as illustrated in Fig. 2.4, is based on the competition
between the nucleation and propagation of leading partial dislocations versus the
nucleation and propagation of perfect (total) dislocations (i.e. trailing partial). According
to the model, if the temperature and stress are not sufficient to nucleate the trailing partial
30
(and thus the perfect) dislocations, then the increasing stress will eventually reach a
sufficient value to rupture the bonds at the crack tip, leading to its propagation [18].
Figure. 2.4. Schematic illustration of dislocation nucleation at a crack tip (a) at
BDTT T , and (b) at BDTT T (After Pirouz et al [18]).
At low temperatures, nucleation of single leading partial dislocations is not sufficient to
make the crystal ductile because, once formed, the stacking faults dragging behind the
partials prevent further nucleation events from the same sources (see Fig. 2.4(a)), i.e.,
partial dislocation multiplication from any source stops after the first leading partial is
emitted. Consequently these leading partial dislocations contribute to a very limited
extent to the straining of the crystal, and the crystal is brittle. On the other hand, at high
temperatures, full dislocations (i.e. leading and trailing partial dislocations) can be
nucleated repeatedly from the source in the crystal (see Fig. 2.4(b)) and their glide
produces large strains and causes macroscopic yielding of the crystal, and the crystal is
ductile.
crack tip
slip plane
crack tip
slip plane
(a) (b)
31
2.3 Dislocation Velocity in Indium Phosphide
Several workers have measured the dislocation velocity in InP. Nagai [19] measured
dislocation velocity in InP using the double etching technique. The author introduced
dislocations by scratching the crystal in a 110 direction and then stressing the crystal
by 3-point bend test over a range of applied resolved shear stresses, 21 kg/mm to
25 kg/mm , and over a temperature range of 523 K to 673 K . The movement of
individual dislocation was detected by the double etching technique (measuring the
distance between dislocations before and after stressing the crystal). It was found that
there is a large difference between the velocity of and dislocations. The velocity of
dislocation versus resolved stress and temperature is shown in Fig. 2.5.
. (a) (b)
Figure. 2.5. (a) Velocity, , of dislocation in undoped and S-doped n-type InP single crystals as a function of (a) resolved shear stress, , and (b) temperature, T (After Nagai [19]).
32
Yonenaga and Sumino [20] also measured the dislocation velocity in InP. In their work,
the dislocations were introduced by scratching the crystal along the 11 1 direction and
dislocation motion took place by uniaxially compressing the crystal with a constant strain
rate at various temperatures. Their results are shown in Fig. 2.6.
The temperature and stress dependence of dislocation velocity in semiconductors can be
expressed by the following equation:
/VU kTmv A e (2.15)
where is the resolved shear stress, m is the stress exponent, which depends on the
dislocation type, VU
is the activation energy for dislocation motion ( 1 2 eV for
undoped materials), and A is a pre-exponential factor which depends weakly on
temperature.
Summary from the results obtained by Nagai [19] and by Yonenaga and Sumino [20],
they concluded that, as in other semiconductors, the velocity of dislocations in InP
decreases as temperature and resolved shear stress decrease, which means that dislocation
motions becomes more difficult and sometimes immobile at low temperatures.
33
34
Fig
ure
. 2.6
. Vel
ocit
y ve
rsus
res
olve
d sh
ear
stre
ss a
t 723
K in
var
ious
indi
um p
hosp
hide
cry
stal
s fo
r (a
) α
dis
loca
tion
s (b
) β
dis
loca
tion
s, a
nd (
c) s
crew
dis
loca
tion
s. (
Aft
er Y
onen
aga
and
Sum
ino
[20]
).
(a)
(b)
(c)
2.4 Previous Works on Plasticity of Indium Phosphide
Gall et al [21] studied the plasticity of single crystal InP between 573 K and 1023 K
using uniaxial compression tests at a strain rate of 4 110 s . They compressed the
samples along the 123 axis in an Instron machine; such an orientation is most suitable
for activating a single slip system in the crystal. Their results are shown in Fig. 2.7. In
Fig. 2.7(a), the resolved shear stress, , is plotted against the plastic shear strain, p , at
various temperatures, where is the resolved shear stress in the primary 111 101
glide system. The characteristic yield drop of InP at different temperatures is observed
together with the variation of the upper yield, uys , and lower yield, lys , points. As
expected, uys lys , and uys lys decrease with temperature.
(a) (b)
35
Figure. 2.7. (a) Resolved shear stress versus shear strain curves for undoped InP between 573 K and 1023 K . (b) Magnification of (a) (After Gall et al [21]).
The magnitude of the upper yield point and the lower yield point is a sensitive function of
the strain rate, dislocation density, and temperature. The relationship between the
dislocation velocity and the magnitude of the upper yield point is given by:
mv k (2.16)
where, k is a constant of proportionality and is the stress exponent.
At low temperatures, large stresses are needed to increase the dislocation velocity.
However once the applied stress is large enough to increase the dislocation velocity (see
Eq. 2.16), the dislocation density rapidly increases due to dislocation multiplication. At
later stage, the average of dislocation velocity decreases due to dislocation-dislocation
interactions. In general, Orowan’s relation holds:
p bv (2.17)
where, p is the shear strain rate, is the dislocation density, is the Burgers vector, and is
the average dislocation velocity. Hence, there is a decrease of the applied stress down to
the lower yield stress, lys , with increasing temperature, as shown in Fig. 2.8. This is
certainly due to an increase in the dislocation mobility at higher temperatures. Therefore,
at higher temperatures smaller applied stresses are needed to create new dislocations in
order to accommodate deformation.
36
Figure. 2.8. Variation of resolved shear stress at lower yield point, , versus
temperatures (After Gall et al [21]).
From the above discussion, Gall et al concluded that:
The upper and lower yield point is a function of temperature.
In the range between 573 K to 1023 K , the p curves show three stages;
1. Dislocation multiplication.
2. Work hardening due to the interaction of dislocations.
3. Dislocation annihilation, i.e. cross-linking
However, as the temperature increases, the transition between the different stages is
indistinguishable and the curves gradually become parabolic.
InP is brittle at about 523 K .
The most important observation from their result is that at a strain rate of 4 110 s ,
InP is ductile at 573 K but is brittle at 523 K . Thus the brittle-to-ductile transition
temperature at 4 110 s lies somewhere between 523 K and 573 K .
37
Suzuki et al [22] also studied plastic deformation of single crystal InP. They investigated
plastic deformation of InP from 500 K down to 77 K under a confining pressure. Their
results are shown in Fig. 2.9. The c T relation in this figure shows that c increases to
very large values at very low temperatures.
Figure. 2.9. Critical resolved shear stress c against temperature, . The stress
denotes the shear stress component resolved in the 101 direction on the
111 plane. Shear strain rate is 4 11.2 10 s . Data of Gall et al (1987) at 4 12 10 s are shown as well (After Suzuki et al [22]).
The side surfaces of the samples deformed in the temperature range between 200 K and
400 K were subjected to observation of slip lines, as shown in Fig. 2.10. On the 143
surface, the slip lines are straight and parallel, while on the 931 surface they are not.
From the sample geometry it is obvious that slip occurs approximately on the 111
plane, but slip is not planar which indicates occurrence of cross-slipping.
38
Figure. 2.10. Slip lines on the side surfaces of InP deformed at 300 K . The
slip lines indicate 110 1 111
2 slip with frequent cross-slip (After Suzuki et al
[22]).
Transmission electron microscopy images of the sample deformed at 300 K are shown in
Fig. 2.11. The slip lines indicate 1101 111
2 slip with frequent cross-slip. The bright
field image in Fig. 2.11 shows many dislocations lying parallel to the 10 1 direction
(shown in Fig. 2.10) are out of contrast in the g 202 reflection, which means these
dislocations are predominantly screw dislocations with a Burgers vectors 1
1012 . The
weak-beam dark field image shows the dissociation of the screw dislocations. The slip
lines and transmission electron microscopy observations confirm that deformation of InP
occurs by the operation of the 1101 111
2 slip system throughout the whole
temperature range.
39
Figure. 2.11. TEM images of slip bands in InP deformed at 300 K . The foil
was cut parallel to the 111 slip bands. The first image is bright field
exhibiting many straight screw dislocations parallel to the 10 1 direction
(shown in Fig. 2.10) are all out of contrast in the g 202 reflection. The second image is weak beam dark field revealing dissociation of screw dislocations (After Suzuki et al [22]).
From these TEM observations, Suzuki et al [22] suggested that deformation of InP at low
temperatures occurs by the motion of non-dissociated shuffle screw dislocation, while at
high temperature it occurs by the motion of dissociated glide screw dislocations.
CHAPTER 3
EXPERIMENTAL SETUP AND SAMPLE PREPARATION
40
This chapter discusses the experimental setup and sample preparation techniques used in
the present work. The 001 InP crystals used in the experiments were grown by and
obtained from The Institute of Electronic Materials Technology (ITME), Warszawa,
Poland. Some data on the as-received wafer, as provided by the supplier, are shown in
Table 3.1. In this work all the tests were performed on the 001 face of InP.
UndopedType of Conductivity N
Orientation 001
Resistivity 11.89 10 cm Mobility 23300 cm / VsCarrier concentration 15 35.2 10 cm
Table. 3.1. Properties of the InP crystal grown by and obtained from the Institute of Electronic Materials Technology (ITME), Warszawa, Poland.
3.1 4-Point Bend Tests
The purpose of the present work, as described in the previous chapter, is to gain a better
understanding of dislocation processes controlling the brittle-to-ductile transition
temperature in InP by measuring its transition temperature, BDTT , as a function of the
loading rate (or, equivalently, the strain rate ). The approach is to perform a series of
identical fracture tests at different temperatures to determine the stress required for
irreversible deformation at a certain strain rate.
41
There are various techniques for measuring the brittle-to-ductile transition temperature,
BDTT , such as bend test and tension test. The bend test was selected in this work, because
the sample preparation is relatively easier and more economical compared to other
techniques. The 4-point bend test was preferred over the 3-point bend test because in the
former, the probability of a largest flaw being exposed to the peak stress is greater than
the latter; this can been seen from the stress distribution of 3-point and 4-point bend
sample shown in Fig. 3.1.
(a) (b)
Figure. 3.1. Comparison of tensile stress distribution in 3-point and 4-point bend samples. The shaded area represents tensile stress, a region ranging from zero at the supports of the bend sample to (a) a maximum at the midspan for the 3-point bend geometry (b) uniform maximum along the whole gauge length of the sample for the 4-point bend geometry.
Thus the exact position of a precrack in a 4-point bend test is not important; as long as the
precrack lie within the central portion of the beam, they will all receive an identical
tensile stress. Unlike the 4-point bend geometry, the stress distribution in the 3-point
bend geometry is very non-uniform and the central loading line should be directly above
the precrack, which means that the peak stress occurs only along a single line on the
surface of the sample. In this case, the probability of the largest flaw in the sample being
at the surface along the line of peak stress is relatively low. Therefore, it is unlikely that
42
the 3-point bend test would reveal the strength limit of the material or the flaw size that
causes fracture. The 4-point bend test geometry is shown in more detail in Fig. 3.2.
Figure. 3.2. Schematic of a 4-point bend test geometry. 1y and 2y are the
vertical distance between the outer rollers and inner rollers.
The normal stress (Pa)app applied to the end face of the sample is given by:
23 /app Pd wh (3.1)
where is the applied load (in Kg), is the sample width (perpendicular to the plane of the
paper in Fig. 3.2), h is the sample thickness, d is the bending arm, / 2d L l , is the
separation of the outer rollers, and is the separation of the inner rollers; all the spatial
dimensions are in m .
The equation for the strain in the central part of the beam is given by:
2 26 / 3 4h l L l L l h (3.2)
where 1 2y y is the vertical displacement between the outer and inner rollers; is
equal to the crosshead displacement. In general, if the separation between the inner
P/2 P/2
P/2 P/2
43
rollers is about three times the sample width 3l w , then the central portion of the
beam undergoes pure bending, and it is valid to use Eq. 3.2 [3,30,37]. However this
equation is no longer strictly valid after the onset of plasticity at higher temperatures.
An expression for the strain rate can be obtained by differentiating Eq. 3.2 with respect to
time, and is given by:
2 26 / 3 4h l L l L l h (3.3)
where is the crosshead displacement rate. In general the value of 24h in the above
equation is very small compared to the value of 23l L l L l , and thus the strain
rate is proportional and very sensitive to the sample thickness, h . Thus it is important to
ensure that all samples have the same thickness in order to have a successful series of
experiments at the same strain rate,
3.1.1 Sample Preparation
Parallelepiped samples of undoped InP single crystal with dimensions of 33 35 2 mm
were cut with a Struers Accuton- machine using a 5 in 12.7 cm diamond blade.
The samples were polished on all sides to remove surface damage and to minimize the
introduction of unwanted precracks introduced during cutting. The orientation and
dimensions of a typical sample are shown in Fig. 3.3. As mentioned in the previous
section, in order to successfully measure the brittle-to-ductile transition temperature, a
series of fracture tests at different temperatures must be performed while maintaining the
44
other parameter, such as sample geometry identical. Thus, it is essential to ensure that all
samples have the same thickness, because the applied strain rate is a sensitive function of
the sample thickness.
Figure. 3.3. The crystallographic orientation and dimensions of a 4-point bend sample (actual sample dimensions after polishing).
Batches of up to 12 samples were prepared simultaneously. All samples were
subsequently mounted on a flat glass plate using a low melting point wax and were
polished using a Buehler Automet 2 polishing machine. The polishing procedure
consisted of four steps:
Step 1. Polish with a 2400 grit SiC paper.
Step 2. Polish with a 4000 grit SiC paper.
Step 3. Polish with a 5 μm alumina slurry (alumina powder + water) on a soft
cloth.
Step 4. Polish with a 1 μm alumina slurry on a soft cloth.
The samples were polished until they showed a mirror-like surface and no sign of any
scratch on the surface under an optical microscope. On completion, the samples were
removed by soaking the glass plate in acetone for a couple of hours, until the low melting
point wax dissolved. This made sample removal quite easy without introducing scratches
45
in the samples. The samples were then cleaned using ethanol and stored, ready for
introduction of precracks.
3.1.2 Technique of Introducing Precracks
In order to ensure that fracture always initiates within the sample, precracks were
introduced in the samples and had to be the worst flaw in the samples, so that fracture
always initiates there. In this work five radial precracks were introduced in each sample
by a 50 g Knoop indentation, with the long direction of the Knoop indent parallel to
110 at the center of the 001 face of each sample, as shown in Fig. 3.4. These five
precracks were spaced far enough from each other 20 μm to ensure that there was no
interaction between the stress fields of neighboring indentations.
Figure. 3.4. A 4-point bend test sample containing five identical radial precracks, introduced by Knoop indents spaced about 20 μm apart.
A 50 g indent load was chosen because sometimes, above this load, lateral cracks
formed during indentation. If present, these lateral cracks will continue moving toward
the surface where they can break out in the form of chipping, this should be avoided in
the fracture tests.
46
3.1.3 Instron 1361 Tensile Machine
An Instron 1361 tensile machine, as shown in Fig. 3.5, was used to perform 4-point bend
experiments. In the 4-point bend tests, load-displacement curves are obtained, which can
be converted into a stress-temperature curve to determine the brittle-to-ductile transition
temperature.
Figure. 3.5. (a) Photograph of the Instron 1361 tensile machine with the tensile jig and the furnace mounted on it, and (b) schematic of the 4-point bend jig connected to the displacement rods of the machine.
Some limitations were encountered with this machine; one of them was to keep the same
constant flow rate of argon gas in every experiment, which is important because the flow
rate of argon gas can modify the temperature of the sample and the thermocouple
reading, which can cause inconsistent temperature differences between the temperature
47
controller and the thermocouple (up to ). Another parameter that can affect the
temperature reading is the position of the thermocouple with respect to the specimen
within the jig.
The 4-point bend jig is fabricated from SiC with upper and lower rollers made from
molybdenum due to its smooth surface; reliable data were not obtained using alumina
rollers. These molybdenum rollers were obtained from Goodfellow Metals, Co. In order
to achieve a good surface finish, these rollers were electropolished in a solution of 25%
sulphuric acid, 75% methanol with a stainless steel cathode with a voltage of 10 V
applied for about 5 minutes. This technique produced an optically smooth surface finish
for the rollers.
The 4-point bend jig was then placed in the furnace tube, argon gas flow started and the
jig heated up while high purity argon gas continued flowing to avoid thermal
decomposition or oxidation of the sample. In each case, the sample was annealed at
for about one hour before initiation of deformation. Following the annealing step,
the desired temperature was set and the test conducted in the displacement mode with a
constant crosshead speed (i.e., a constant strain rate, ). Each test was continued until the
sample either fractured or deformed plastically.
3.2 Indentation Tests
48
Indentation is a popular technique to characterize the mechanical properties of materials,
such as their resistance to plastic deformation (hardness) and their resistance to cracking
(fracture toughness). In indentation tests, an indenter is forced into the surface of the
sample under a fixed load, as schematically illustrated in Fig. 3.6, for certain duration of
time.
Figure. 3.6. Schematic of indentation mechanism. is the applied load, is the indentation diagonal and h is the indentation depth.
There are three processes that can take place when the indenter is forced into a crystal.
The crystal initially deforms elastically and subsequently either fractures or deforms
plastically (or a combination of the latter). At low temperatures, in the brittle regime of
the solid, the predominant mode of accommodating the indenter is by fracturing of the
sample. On the other hand, at high temperatures, in the ductile regime of the solid, the
sample deforms plastically to accommodate the indenter.
Using such an indentation test, the hardness value of the sample can be estimated. This is
determined by the length of the diagonal of indentation impression or by the indentation
depth. The indentation microhardness (HV) was calculated using the following standard
equation:
21854.4 /HV P d (3.4)
49
where P is load in g , and d is the mean diagonal of an indent in μm .
3.2.1 Sample Preparation
Parallelepiped samples of InP with dimensions of 33 5 2 mm were cut with a Struers
Accuton- machine using a 5 in 12.7 cm diamond blade such that the large
23 3 mm faces were parallel to the 001 plane. The 001 face was then polished in
order to remove surface damage and to minimize the introduction of unwanted precracks
introduced during cutting. For this purpose, the sample was mounted on a flat glass plate
using a low melting point wax and polished using a Buehler Automet 2 polishing
machine. The polishing procedure was identical to that of the 4-point bend samples (see
section 3.1.1).
3.2.2 Static Indentation Tests
Static indentation tests were performed on the 001 face of InP with a Nikon high
temperature microhardness indenter. A diamond Vickers indenter (a square-based,
pyramidal indenter with a 136 angle) was used for indentations. In these tests, the
sample was aligned such that its 110 and 110 directions were parallel to the indenter
diagonals, as shown in Fig. 3.7. Note that the 110 directions correspond to intersection
of 111 slip planes with the 001 sample surface. The indenter was left for 15 seconds
50
on the sample before it was raised up, in this way plastic impression was observed on the
sample surface.
The tests were conducted in the temperature range of to , starting from the
highest temperature and going down gradually to in a step size of . Five to
eight indentations were made at each temperature. The load, on the indenter was 50 g .
After the indentation, the sample was etched in a HCL:H3PO4 solution for about 10
seconds and the length of the diagonals of the indentation impression were measured.
Figure. 3.7. Schematic of indented surface in InP with a 110 orientation of
indenter diagonals.
Figure. 3.8. Photograph of the Nikon High Temperature Microhardness Tester model QM.
51
Specimens for transmission electron microscopy were prepared by polishing the back
side of the indented sample to a thickness of approximately using SiC paper. The
specimen was then dimpled using diamond paste to obtain a thickness of
approximately . The specimen was then loaded into a PIPS (Precision Ion Beam
Polishing System) for ion beam thinning using double modulator at energy and an
incident beam angle of to obtain a large thin area for imaging (to make the specimen
electron transparent). Transmission electron microscopy was carried out using a Philips
CM20 TEM, operated at .
3.2.3 Dynamic Indentation Tests
Dynamic indentation tests were performed on the face of InP with a high-
temperature depth-sensitive indenter. Fig. 3.9 shows a photograph and schematic diagram
of the high-temperature depth-sensitive indentation (HTDSI) equipment.
A conventional Vickers diamond indenter is used in this machine. A current-controlled
electromagnetic moving coil is used to apply the load, which enables a wide range of
displacement rates and a maximum load up to [26]. A capacitance displacement
gauge is used not only to determine the position of indenter but also to detect the instant
when the indenter touches the sample surface prior to indentation, as well as to track the
movement of the indenter during indentation. In this machine, the sample and indenter
are heated separately in vacuum to prevent oxidation of the furnace heating element and
52
the diamond indenter. During our tests the indenter and the sample were kept at the same
temperature as close as possible
Figure. 3.9. (a) Photograph of the HTDSI apparatus, and (b) schematic diagram of the apparatus: (A) sample, (B) indenter, (C) sample furnace, (D) indenter furnace, (E) capacitance displacement gauge, and (F) load application coil (After Kim and Heuer [26]).
The basis of the depth-sensitive indentation (DSI) technique is similar to the conventional
microhardness test, the major difference being that instead of measuring the impression
diagonal after indentation, this technique monitors the penetration depth as a function of
the applied load as the indenter is driven into the material and withdrawn from it. In the
DSI test, a load-depth plot is obtained, which can be used to calculate the energy
consumed in making an indent at a fixed temperature.
A typical load-depth plot obtained from a test is shown in Fig. 3.10. The displacement of
the indenter in the vertical direction is along the x-axis and the load is along the y-axis.
Such a plot typically consists of a loading curve and an unloading curve that form a loop
showing hysteresis. Thus, in Fig. 3.10, the path OA is the loading cycle, which increases
53
continuously with the applied load. At A, the load reaches its maximum value and then
stays constant while the depth increases to B. BC corresponds to the unloading cycle
where the load gradually decreases to zero. The area bound by OABD is the total energy,
, expended by the indenter to impress into the material. This energy consists of two
parts: (1) elastic energy, , and (2) work done, , to produce a permanent
impression. The latter can be plastic deformation required to produce the impression as
well as the surface energy produced by cracking. If the material was purely elastic, the
paths OA and BC would overlap and there would be no hysteresis, i.e. , since the
elastic energy is fully recovered during the unloading path. On the other hand, if the
material is purely plastic, the path would be OABD and there would be no elastic
recovery.
54
Figure. 3.10. Typical load-depth plot for a material that is subject to elastic and plastic deformation.
Fig. 3.10 is a typical plot for a material that shows an elastic/plastic behavior. As shown
in this figure, after unloading, the depth displacement does not go back to zero, because
the material has undergone some plastic deformation. Consequently, the area under
OABC is related to the plastic energy dissipated in the volume of indentation, i.e. energy
expended to create and move dislocations during indentation process and surface energy
from any crack produced. Since the area under OABD gives the total energy, , which
consists of the plastic energy, , and the elastic energy, . The elastic energy,
is the area under CBD.
Tests were performed using a load in the temperature range between and
(the machine can in fact produce indentations over the temperature range between
and ). In each case, the indentations were started from the highest
temperature gradually decreasing to the lowest temperature at in step size of .
The indenter was left for seconds on the sample before it was raised up. Moreover
eight to ten indentations were made at each temperature.
55
CHAPTER 4
RESULTS AND DISCUSSIONS
This section presents the results of mechanical testing of InP from all the experiments
conducted in the present study. First, the results of 4-point bend tests are presented and
discussed. This is followed by indentation tests including statical indentation tests with
the Nikon high temperature microhardness indenter model QM and dynamical
indentation tests with the high-temperature depth-sensitive indentation machine.
4.1 4-Point Bend Tests
The purpose of these experiments is to identify the basic characteristics of the brittle-to-
ductile transition in InP, such as the temperature range over which transition occurs and
the form of stress versus temperature curves.
A series of tests were carried out at a crosshead displacement rate of
, which correspond to strain rate of . The variation
56
of the sample thickness in these tests was . The samples were tested over a
temperature range of to . Load-displacement curves (as plotted by the
Instron chart recorder) of a sample tested at three different temperatures at a strain rate of
are shown in Fig. 4.1.
Figure. 4.1. Load displacement curves for InP samples tested at ,
and at a strain rate of . At InP deforms elastically until fracture; at InP is in the transition region; above , e.g., at , InP deforms plastically.
The complete results from a series of tests at different temperatures are presented in Fig.
4.2, in the form of applied stress to deform the crystal versus temperature. The plot
consists of two parts: (1) up to about 350°C, all the samples underwent elastic
deformation and then fractured by cleavage in a brittle fashion at an approximately
constant applied stress of 90 MPa, and (2) above 355°C all the samples exhibited
plastic deformation.
57
Figure. 4.2. Applied stress versus temperature for samples tested at a strain rate of . The brittle-to-ductile transition temperature occurs over a very narrow temperature range of , and is characterized by a sudden increase of the applied stress.
4.1.1 Brittle Behavior
At and below, the applied load versus displacement curves were linear up to the
point where the samples broke (see Fig. 4.1(a)), indicating that the material exhibited
elastic behavior until it fractured. The samples fractured catastrophically at an applied
stress at about .
Samples tested in this regime show roughly a constant fracture stress value; the
variability of fracture stress arises because most samples did not break at the introduced
precrack which resulted in higher fracture stress than if fracture had initiated at a pre-
crack. For instance at two tests were conducted, each giving a different fracture
stress value: when the sample broke catastrophically at the introduced precracks
58
and when it did not (see Fig. 4.2). Larger indentation loads, up to were
tried to produce larger precracks and to overcome this problem, but still only very few of
the samples broke at the introduced precracks.
4.1.2 Transition Behavior
At the load-displacement curve was still linear, but at the curve exhibited
a small deviation from linearity before it broke (see Fig. 4.1(b)). This deviation indicates
the start of plasticity. As shown in Fig. 4.2, there is a rapid rise in the applied stress
between and . In that stage, the dislocations start emitting from the crack
tip, thus blunt and also shield the crack tip from the applied stress. Consequently a larger
applied stress is needed to break the sample. The temperature at which the rapid rise takes
place corresponds to the brittle-to-ductile transition temperature.
4.1.3 Ductile Behavior
Above , the samples were no longer cleaving in a brittle manner but rather, they
deformed plastically until fractured in a ductile manner (i.e. by necking). Fig. 4.1(c)
shows that the samples at exhibit yielding (ductility) followed by fracture. In the
later stages of deformation, necking occurred: beginning at the maximum load point, the
sample cross section rapidly decreased at some point until it was no longer able to
support the load, and ultimately broke. The stress needed to plastically deform InP in the
ductile regime decreases as the temperature increases. The reason for this is that
dislocation mobility increases with temperature and makes plastic shearing of the crystal
easier as temperature increases.
59
From the applied stress versus temperature plot, it is certain that the brittle-to-ductile
transition is characterized by a sudden increase of the applied stress to shear the samples.
In this work the brittle-to-ductile transition temperature occurs in a very narrow
temperature range, , between the highest temperature where the sample deforms in a
completely brittle manner and the lowest temperature where the sample yields. At a strain
rate of , the highest temperature at which the sample deforms in a
brittle manner is and the lowest temperature at which the sample deforms
plastically is .
The brittle-to-ductile transition temperature range itself is also affected by variations
between individual samples. In general, despite the fact that chipping starts to appear
with higher indentation loads, larger precrack sizes result in a narrower transition
temperature range giving a sharper applied stress versus temperature plot.
4.2 Indentation Tests
4.2.1 Static Indentation Tests
Static indentation test was another technique employed in this study to understand the
mechanism of plastic deformation of InP. In this work besides measuring the temperature
dependence of the hardness, fracture and plastic behavior of InP subjected to indentation
was also investigated.
60
The hardness and the strain rate values were extracted from the measured indentation
impression diagonals (shown in Fig. 4.4). The roughly calculated strain rate in this static
indentation test was . The temperature dependence of the hardness of
undoped InP is shown in Fig. 4.3; as the temperature increases, the sample hardness
rapidly decreases until a temperature is reached where the dislocation mobility becomes
high, yield strength becomes very low and hardness approaches a constant value.
Figure. 4.3. Hardness versus temperature curve for sample tested from down to , with a step size of and a indentation load.
61
In addition to the indentation diagonals, the temperature dependence of the length of
radial cracks emanating from the indent corners (recall that the indentation diagonals
62
Fig
ure
4.4
. In
dent
atio
n im
pres
sion
s pr
oduc
ed b
y st
atic
ind
enta
tion
tec
hniq
ue a
t (a
) 12
3°C
, (b
) 26
3°C
, an
d (c
) 32
0°C
. T
hese
fig
ures
sho
w t
hat
the
sam
ple
hard
ness
dec
reas
es w
ith
incr
easi
ng t
empe
ratu
re d
ue t
o hi
gher
dis
loca
tion
mob
ilit
y at
hi
gher
tem
pera
ture
s. I
t is
als
o ca
n be
see
n th
at c
rack
sta
rt e
man
atin
g fr
om t
he i
nden
t co
rner
s at
263
°C, a
t th
is t
empe
ratu
re
crac
k st
art t
o pr
opag
ate
from
the
inde
nt c
orne
rs a
s sh
own
in (
b).
(a)
(b)
(c)
were along two orthogonal directions) was also measured. The temperature
dependence of and crack lengths are shown in Fig. 4.5.
Figure. 4.5. Temperature dependence of the impression diameter and crack
length along and directions over a temperature range of to
using an indentation load of . The sample was initially heated to the highest temperature and indented. Following this, the sample was cooled and indentations made in step size until the lowest temperature of .
The indentation brittle-to-ductile transition (IBDT) temperature was determined from the
disappearance of radial cracks from the corners of the indentation impression. Two
values were obtained: and . Note that
the indentation brittle-to-ductile transition temperature for cracks is slightly lower
than that for cracks, showing the asymmetry of the two directions in a polar
63
crystal. This asymmetry also affects its mechanical properties along and
directions, i.e. the crack length along and directions are not the same at any
temperature below (see Fig. 4.5). This behavior, previously reported for III-V
compound semiconductors, can be explained on the basis of crystallographic anisotropy
in sphalerite structures (see section 1.1.4).
The TEM images of the sample deformed by the static indentation technique at
are shown in Fig. 4.6. Fig. 4.6(a) shows two sets of stacking fault ribbons perpendicular
to each other, when viewed along the direction. These two sets are formed on
different slip planes. One set of stacking fault ribbons is out of contrast under
and reflections (as indicated in Fig. 4.6(b) and 4.6(c), respectively). Using the
analysis (see Table. 4.1), we conclude that the partial dislocations have the same Burgers
vector inclined to the surface. These TEM observations show that plastic
deformation of InP occurs by the operation of the slip system with
dislocations belonging to this system dissociating into two partial dislocations.
64
Table. 4.1. analysis for Burgers vector of partial dislocations in a
face-centered cubic crystal (black and white regions correspond to
(dislocations in contrast) and (dislocations out of contrast)).
65
66
Fig
ure
4.6
. B
righ
t fi
eld
TE
M i
mag
es o
f st
acki
ng f
ault
s in
ind
ium
pho
sphi
de d
efor
med
by
the
stat
ic i
nden
tati
on t
echn
ique
at
320˚
C.
Fig
. 4.
5(a)
sh
ows
two
sets
of
stac
king
fau
lt r
ibbo
ns p
erpe
ndic
ular
to
each
oth
er.
One
set
of
the
stac
king
fau
lt r
ibbo
ns i
s ou
t of
con
tras
t in
220
and
311
re
flec
tion
s, a
s sh
own
in f
igur
es (
b) a
nd (
c) r
espe
ctiv
ely.
(a)
(b)
(c)
4.2.2 Dynamic Indentation Tests
As mentioned in 3.2.3, this test is basically the same as the conventional static
indentation test, but instead of measuring the impression diagonal after an indenter
penetrates a crystal to obtain its hardness value, the impression depth is monitored as a
function of the applied load to obtain a load-depth plot. This plot can subsequently be
used to calculate a dynamic hardness value. The area under the load-displacement curve
is related to the total energy dissipated in the impression volume produced by the
indentation.
Load-displacement curves at different temperatures on surface of InP are shown in
Fig. 4.7. For these tests, at every temperature, a load of was gradually applied to
the indenter to penetrate into the InP crystal with a strain rate of . The
duration of each test was seconds, after which the load was maintained for seconds
before unloading. Fig. 4.7(a) shows that there is considerable elastic recovery at low
temperatures, however as temperature increases the indenter penetrates further into the
sample, indicating softening with increasing temperature; the indentation penetration
depth increases and elastic recovery reduces, e.g. Fig. 4.7(d).
67
Figure 4.7. Load-displacement curves obtained from dynamic indentation tests on
surface of indium phosphide at different temperatures: (a) , (b) , (c)
, and (d) .
68
(a)
By integrating the area under the load-displacement curves in Fig. 4.7, we obtained a plot
of energy versus temperature for the indentation load, as shown in Fig. 4.8.
Figure. 4.8. Dissipated energy versus temperature plot obtained by integrating the area under the load-displacement curves in Fig. 4.6.
The principal mechanism of plastic deformation in a crystal is by fracture or sliding (slip)
between planes of atoms and the latter occurs in an incremental manner due to dislocation
motion. These dislocations blunt the crack tip and decrease the stress concentration at the
crack tip. Thus, with increasing temperature, the contribution of fracture to the dissipated
energy decreases, and that of shear deformation increases. Since the volume of
indentation produced by shearing increases with temperature, the energy dissipated also
increases with temperature.
We assume here that the volume of indentation produced is a perfect square base pyramid
with an angle of between opposite planes. The indentation impression volume is
69
then , where is the indentation depth. We can now estimate the energy density,
by dividing the dissipated energy, by . A plot of versus indentation
temperature is shown in Fig. 4.9.
Figure. 4.9. Energy density versus temperature plot obtained by dividing the dissipated energy with indentation volume.
Fig. 4.9 shows that the energy density, , stays constant up to about and decrease
as the temperature increases. At , the sample is deformed by fracture in a
brittle manner (which involves rupturing of atomic bonds). Since the bond energy does
not vary much with temperature, the applied stress needed to break the sample is
expected to be constant. Above the energy density declines smoothly due to
increasing dislocation mobility with temperature. We define the inflection point in the
plot as the indentation brittle-to-ductile transition (IBDT) temperature. In the
70
present case, InP exhibits a transition temperature at about . Above this
temperature the sample deformed plastically by shearing and the sample deformed by
fracture in a brittle manner below transition temperature. Fig. 4.8 looks very similar with
the schematic variation of the proposed by Pirouz et all [27], shown in Fig. 4.10.
Figure. 4.10. Schematic variations of stress versus temperature in a crystal. (After Pirouz et all [27]).
The curve in Fig. 4.10 shows temperature dependence of the resolved shear stress, ,
needed for plastically deforming crystal at a constant strain rate. If a crystal loaded with
an increasing applied stress , then below transition temperature, reaches before
reaching and the crystal fracture, i.e. represents brittle regime. On the
other hand, above transition temperature, reaches first and the crystal yield
plastically, i.e., represents ductile regime. The brittle-to-ductile transition
temperature is defined from the intersection of the shear stress, and fracture stress,
71
curves. It can be seen that the curve in Fig. 4.10 is actually the same curve
as shown in Fig. 4.9.
CHAPTER 5
CONCLUSION
Experiments have been performed in this work to determine the basic characteristics of
brittle-to-ductile transition (BDT) temperature in single crystal InP using three
techniques; 4-point bend tests and indentation tests (static and dynamic). All three
techniques show that InP exhibits a sharp brittle-to-ductile transition temperature:
4-point bend tests ( ): between and .
Static indentation tests ( ): .
Dynamic indentation tests ( ): .
Of course the different techniques do not give absolutely identical values, since the
deformation mode is not identical for the different techniques. In particular, in the
indentation test, the presence of a hydrostatic pressure shifts the transition temperature to
a lower value, i.e. , in general.
At low temperatures, below , InP deformed (fractured) in a brittle manner at an
approximately constant stress. At the transition temperature, the deformation mode
changes and a higher stress is needed to break the crystal. As the temperatures increases
above , InP deformed plastically but the yield stress is lower and hence a lower
72
applied stress is required. This is because dislocation mobility increases with
temperature, and makes plastic shearing of the crystal easier.
TEM observations of InP sample indented by static indentation machine at
showed that plastic deformation of InP occurs by the operation of the slip
system.
73
APPENDIX
A series of tests were conducted to see how the transition temperature varies with the
strain rate. Such tests shed light on the correlation between dislocation mobility and the
transition temperature. This correlation has been previously reported for silicon [35]. As
described in section 3.1.1, in this series of tests, all the samples were prepared in one
batch to minimize variation between individual samples. Fig. A.1 shows the applied
stress versus temperature plot for tests performed at a strain rate of .
Figure. A.1. Applied stress versus temperature for samples tested at a strain rate of . The brittle-to-ductile transition temperature occurs between and .
74
It can be seen that at lower strain rate, the brittle-to-ductile transition temperature falls
between and . This result is unexpected; because the smaller the strain
rates the lower should be the brittle-to-ductile transition temperature. In other words, the
faster the crystal is loaded, the wider should be the temperature regime over which it
behaves in a brittle manner.
It was decided to re-determine the brittle-to-ductile transition temperature at a strain rate
of . The same precautions were taken in preparing the samples for these
tests. However, a small number of samples were used in this series of tests, because the
re-runs were simply a crosscheck to verify the validity of the previous results. Actually,
the strain rate used was close to, but not exactly . It was in fact
. The results are presented in the form of an applied stress versus
temperature plot in Fig. A.2.
This time the brittle-to-ductile transition temperature falls between and .
This result is about higher than the previous set of experiments, which could be due
the fact that these measurements were made at different times using samples taken from
different batches. Besides, the details of an experiment also affect the result: such
parameters include the flow rate of the argon gas, or the distance between the
thermocouple and the sample jig. Both parameters can affect the temperature reading.
Based on the re-run tests done above, we feel that only the first set of measurements, i.e.
those performed at a strain rate of are reliable.
75
Figure. A.2. Applied stress versus temperature for samples tested at a strain rate of . The brittle-to-ductile transition temperature occurs over a narrow temperature range of , and is characterized by a sudden increase of the applied stress.
The fact that the brittle-to-ductile transition temperature determined at a strain rate of
is higher than the brittle-to-ductile transition temperature at a strain rate
of , is not yet understood. However, in order to clarify this, more
experiments are required in particularly at low strain rates.
76
REFERENCES
[1] Heesuk Rho, “Indium Phosphide”, Department of Physics, University of Cincinnati, Cincinnati, Ohio, Nov 27, 1995.
[2] J. P. Hirth and J. Lothe, “ Theory of Dislocation”, John Wiley & Sons, 1982.
[3] A. George and J. Rabier, “Dislocations and plasticity in semiconductors. I. Dislocation structures and dynamics”, Rev. Phys. Appl., vol. 22, pp. 941-966, 1987.
[4] A. George and J. Rabier, “Dislocations and plasticity in semiconductors. II. The Relationship between dislocation dynamic and plastic deformation”, Rev. Phys. Appl., vol. 22, pp. 1327-1351, 1987.
[5] W. Shockley, “Dislocations and Edge States in the Diamond Crystal Structure”, Phys. Rev., vol. 91., 1953.
[6] A. H. Cottrell,“Dislocations and Plastic Flows in Crystals”, Oxford University Press, 1953.
[7] H. Alexander, “Dislocation in Covalent Crystal”, Elsevier Science Publisher B. V., 1986.
[8] K. Maeda and S. Takeuchi, “Enhancement of Dislocation Mobility in Semiconducting Crystals by Electronic Excitation”, In “Dislocations in Solids”, Edited by F. R. N. Nabarro and M. S. Duesbery, vol. 10, Chapter 54, pp. 443-504. Elsevier Science B. V., 1996.
[9] D. B. Holt, “Surface polarity and symmetry in semiconducting compounds”, J. Mater. Sci., vol. 23, pp. 1131-1136, 1988.
[10] C. R. Barrett, W. D. Nix, A. S. Tetelman, “The Principles of Engineering Materials”, Prentice Hall, Inc., New Jersey, 1973.
[11] B. R. Lawn and T. R. Wilshaw, “Fracture of Brittle Solids”, Cambridge University Press, Cambridge, 1975.
[12] C. E. Inglis, Trans Inst Naval Archit., vol. 55, p. 219, 1913.
[13] J. R. Rice and R. Thomson, “Ductile versus brittle behavior of crystals”, Phil.Mag., vol. 29, pp. 73-97, 1974.
77
[14] J. Schiөtz, L. M. Canel, and A. E. Carlsson, “Effects of crack tip geometry on dislocation emission and cleavage: A possible path to enhanced ductility”, Phys. Rev B., vol. 55, pp. 6211-6221, 1997.
[15] S. J. Noronha and D. Farkas, “Dislocation pinning effects on fracture behavior: Atomistic and dislocation dynamics simulation”, Phys. Rev B., vol. 66, pp. 132103-1 – 132103-4, 2002.
[16] P. B. Hirsch and S. G. Roberts, “Modeling plastic zones and brittle-ductile transition”, Philos. Trans. R. Soc. London, Ser. A 355. pp. 1992-2001, 1991 (1997).
[17] S. G. Roberts, P. B. Hirsch, A. S. Booth, M. Ellis, and F. C. Serbana, “Dislocation, crack, and brittleness in single crystals”, Phys. Scripta., T49, pp. 420-426, 1993.
[18] P. Pirouz, J. L. Demenet, and M. H. Hong, “On transition temperatures in the plasticity and fracture of semiconductors”, Phil. Mag. A, vol. 81, pp. 1207-1227, 2001.
[19] H. Nagai, “Dislocation velocities in indium phosphide”, Jpn. J. App. Phys., vol. 20, pp. 793-794, 1981.
[20] I. Yonenaga and K. Sumino, “Effects of dopants on dynamic behavior of dislocations and mechanical strength in indium phosphide”, J. App. Phys., vol. 74, pp. 917-924, 1993.
[21] P. Gall, J. P. Peyrade, R. Coquille`, F. Reynaud, S. Gabillet, and A. Albacete, “Thermal activation of glide in indium phosphide single crystals”, Acta Metall, vol. 35, pp. 143-148, 1987.
[22] T. Suzuki, T. Nishisako, T. Taru and T. Yasutomi, “Plastic deformation of indium phosphide at temperature between 77 and 500 K”, Phil. Mag. Lett., vol. 77, pp. 173-180, 1998.
[23] T. Suzuki, T. Yasutomi, T. Tokuoka, and I. Yonenaga, “Plasticity of III-V compounds at low temperature”, Phys. Stat. Sol., pp. 47-52, 1999.
[24] P. Boivin, J. Rabier, and H. Garem, ”Plastic deformation of GaAs single crystals as a function of electronic doping I: Medium temperatures (150-650˚C)”, Phil. Mag. A., vol. 61, No. 4, pp. 619-645, 1990.
[25] H. O. K. Kirchner, and T. Suzuki, ”Plastic Homology of Tetrabonded Crystals”, Acta mater., vol. 46, pp. 305-311, 1997.
[26] Chang-Hoon Kim and A. H. Heuer, “A high-temperature displacement-sensitive indenter for studying mechanical properties of thermal barrier coatings”, J. Mater. Res., Vol. 19, No. 1, pp. 351-355, 2004.
78
[27] P. Pirouz and M. Zhang, J. L. Demenet, and H. M. Hobgood, “Yield and fracture properties of the wide band-gap semiconductor 4H-SiC”, J. App. Phys, vol. 93, No. 6, pp. 3279-3290, 2001.
[28] K. Edagawa, H. Koizumi, Y. Kamimura, and T. Suzuki, “Temperature dependence of the flow stress of III-V compounds”, Phil. Mag. A, vol. 80, pp. 2591-2608, 2000.
[29] G. Patriarche and E. Le Bourhis, “In-depth deformation of InP under a Vickers indentor”, J. Mat. Sci., vol. 36, pp. 1343-1347, 2001.
[30] J. Samuels, “The Brittle to Ductile Transition in Silicon”, PhD Thesis, University of Oxford, 1987.
[31] I. –H. Lin and R. Thomson, “Cleavage, dislocation emission, and shielding for cracks under general loading”, Acta Metall, vol. 34, pp. 187-206, 1986.
[32] D. Farkas, M. Duranduru, W. A. Curtin, and C. Ribbens, “Multiple-dislocation emission from the crack tip in the ductile fracture of Al”, Phil. Mag. A., vol. 81, pp. 1241-1255, 2001.
[33] I. Yonenaga, “Mechanical properties and dislocation dynamics in III-V compounds”, J. Phys. III France, pp. 1435-1450, 1997.
[34] I. Yonenaga and T. Suzuki, “Cross-slip in GaAs and InP at elevated temperatures”, Phil. Mag. Lett., vol. 80, pp. 511-518, 2000.
[35] ST John, C., Phil. Mag., vol. 32, pp. 1193, 1975.
[36] E. Le Bourhis, J. P. Riviere, A. Zozime, “Material flow under an indentor in indium phosphide”, J. Mat. Sci., vol. 31, pp. 6571-6576, 1996.
[37] G. E. Beltz, D. M. Lipkin, L. L. Fischer, “Role of crack blunting in ductile versus brittle response of crystalline materials”, Phys. Rev. Lett., vol. 82., no. 22, pp. 4468-4471, 1999.
[38] T. L. Anderson, “Fracture Mechanics”, CRC Press, 1995.
[39] D. Hull, “Introduction to Dislocation”, Pergamon Press, 1975.
[40] A. Katz, “Indium Phosphide and Related Materials: Processing, Technology, and Devices”, Artec House, 1992.
[41] P. F. Thomason, “Ductile Fracture of Metals”, Pergamon Press, 1990.
79
[42] P. Pirouz, A. V. Samant, and M. H. Hong, “On temperature dependence of deformation mechanism and the brittle-ductile transition in semiconductors”, J. Mater. Res., vol. 14, No. 7, pp. 2783-2793, 1999.
[43] A. J. McEvily, “Metal Failures: Mechanism, Analysis, Prevention”, John Wiley & Sons, 2002.
[44] L. L. Fischer and G. E. Beltz, “Continuum mechanics of crack blunting on the atomic scale: elastic solution”, Mater. Sci. Eng., vol. 5, pp. 517-537, 1997.
[45] P. Pirouz, “Polytypic Transformation in SiC”, Solid State Phenomena., vol. 56, pp. 107-132, 1997.
[46] P. Pirouz, “Deformation Twinning in Bulk and Thin Film Semiconductor”. M. H. Yoo and M. Wuttig, eds. (The Minerals, Metals & Materials Society), pp. 275-295, 1994.
[47] A. A. Griffith, “The phenomena of rupture and flow in solids”, Phil. Trans. Royal Soc. London., vol. 221, pp. 163-198, 1921.
[48] G. T. Brown, B. Cockayne, W. R. Macewan. “Deformation behavior of single crystals of InP in uniaxial compression”, J. Mater. Sci., vol. 15, pp. 1469-1477, 1980.
[49] Seager, “Dislocations and Mechanical Properties of Crystals”, Wiley., 1957.
[50] R. Thomson, “Dislocation-crack interactions”, Pergamon. J, vol. 20, pp. 1473-1476, 1986.
[51] G. A. Wolff and J. D. Broder, “Microcleavage, Bonding Character and Surface Structure in Materials with Tetrahedral Coordination”, Acta Cryst., vol. 12, pp. 313-323, 1959.
[52] W. Y. Lum and A. R. Clawson, “Thermal degradation of InP and its control in LPE growth”, J. Appl. Phys., vol. 50, pp. 5296-5301, 1979.
[53] M. S. Abrahams and L. Ekstrom, “Dislocations and Brittle Fracture in Elemental and compound semiconductor”, Acta Metal., vol. 8, pp. 654-662, 1960.
[54] T. H. Courtney, “Mechanical Behavior of Materials”, McGraw-Hill, 1990.
[55] G. P. Cherepanov, “Mechanics of Brittle Fracture”, McGraw-Hill, 1979.
[56] Yu. S. Boyarskaya, D. Z. Grabko, M. I. Medinskaya, and N. A. Palistrant, “Mechanical properties of pure and doped InP single crystals determined under local loading”, Am. Ins. Phys., pp. 139-142, 1997.
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