Upload
ledat
View
214
Download
0
Embed Size (px)
Citation preview
The mechanical properties of various chemical vapor deposition diamondstructures compared to the ideal single crystalPeter Hess Citation: J. Appl. Phys. 111, 051101 (2012); doi: 10.1063/1.3683544 View online: http://dx.doi.org/10.1063/1.3683544 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v111/i5 Published by the American Institute of Physics. Related ArticlesBias-enhanced nucleation and growth processes for improving the electron field emission properties of diamondfilms J. Appl. Phys. 111, 053701 (2012) Direct visualization and characterization of chemical bonding and phase composition of grain boundaries inpolycrystalline diamond films by transmission electron microscopy and high resolution electron energy lossspectroscopy Appl. Phys. Lett. 99, 201907 (2011) Impurity impact ionization avalanche in p-type diamond Appl. Phys. Lett. 99, 202105 (2011) Ultrathin ultrananocrystalline diamond film synthesis by direct current plasma-assisted chemical vapor deposition J. Appl. Phys. 110, 084305 (2011) Diamond nanoparticles with more surface functional groups obtained using carbon nanotubes as sources J. Appl. Phys. 110, 054321 (2011) Additional information on J. Appl. Phys.Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors
APPLIED PHYSICS REVIEWS—FOCUSED REVIEW
The mechanical properties of various chemical vapor deposition diamondstructures compared to the ideal single crystal
Peter Hessa)
Institute of Physical Chemistry, University of Heidelberg, Im Neuenheimer Feld 253,Heidelberg D-69120, Germany
(Received 13 October 2011; accepted 13 January 2012; published online 2 March 2012)
The structural and electronic properties of the diamond lattice, leading to its outstanding
mechanical properties, are discussed. These include the highest elastic moduli and fracture strength
of any known material. Its extreme hardness is strongly connected with the extreme shear modulus,
which even exceeds the large bulk modulus, revealing that diamond is more resistant to shear
deformation than to volume changes. These unique features protect the ideal diamond lattice also
against mechanical failure and fracture. Besides fast heat conduction, the fast vibrational
movement of carbon atoms results in an extreme speed of sound and propagation of crack tips with
comparable velocity. The ideal mechanical properties are compared with those of real diamond
films, plates, and crystals, such as ultrananocrystalline (UNC), nanocrystalline, microcrystalline,
and homo- and heteroepitaxial single-crystal chemical vapor deposition (CVD) diamond, produced
by metastable synthesis using CVD.
Ultrasonic methods have played and continue to play a dominant role in the determination of the
linear elastic properties, such as elastic moduli of crystals or the Young’s modulus of thin films
with substantially varying impurity levels and morphologies. A surprising result of these extensive
measurements is that even UNC diamond may approach the extreme Young’s modulus of single-
crystal diamond under optimized deposition conditions. The physical reasons for why the stiffness
often deviates by no more than a factor of two from the ideal value are discussed, keeping in mind
the large variety of diamond materials grown by various deposition conditions.
Diamond is also known for its extreme hardness and fracture strength, despite its brittle nature.
However, even for the best natural and synthetic diamond crystals, the measured critical fracture
stress is one to two orders of magnitude smaller than the ideal value obtained by ab initiocalculations for the ideal cubic lattice. Currently, fracture is studied mainly by indentation or
mechanical breaking of freestanding films, e.g., by bending or bursting. It is very difficult to study
the fracture mechanism, discriminating between tensile, shear, and tearing stress components
(mode I–III fracture) with these partly semiquantitative methods. A novel ultrasonic laser-based
technique using short nonlinear surface acoustic wave pulses, developing shock fronts during prop-
agation, has recently been employed to study mode-resolved fractures of single-crystal silicon.
This method allows the generation of finite cracks and the evaluation of the fracture strength for
well-defined crystallographic configurations. Laser ultrasonics reaches the critical stress at which
real diamond fails and therefore can be employed as a new tool for mechanistic studies of the frac-
ture behavior of CVD diamond in the future. VC 2012 American Institute of Physics.
[doi:10.1063/1.3683544]
TABLE OF CONTENTS
I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
A. Structure and mechanical properties of an
ideal diamond crystal . . . . . . . . . . . . . . . . . . . . 2
B. Natural and synthetic diamond . . . . . . . . . . . . 2
C. Purity and morphology of real crystals . . . . . 3
D. Elastic and inelastic behavior . . . . . . . . . . . . . 3
II. DIAGNOSTIC METHODS . . . . . . . . . . . . . . . . . . . 4
A. Methods to study linear elastic behavior . . . 4
1. Ultrasonics with bulk acoustic waves. . . . 4
2. Laser ultrasonics with surface acoustic
waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3. Resonant ultrasonic spectroscopy . . . . . . . 4
4. Brillouin scattering . . . . . . . . . . . . . . . . . . . 5
B. Methods for studying nonlinear mechanical
behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1. Indentation . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2. Bending and bursting tests . . . . . . . . . . . . . 5a)Author to whom correspondence should be addressed. Electronic mail:
0021-8979/2012/111(5)/051101/15/$30.00 VC 2012 American Institute of Physics111, 051101-1
JOURNAL OF APPLIED PHYSICS 111, 051101 (2012)
3. Laser-based impulsive fracture method . . 6
III. ELASTIC PROPERTIES OF DIAMOND . . . . . . 6
A. Ideal single-crystal diamond . . . . . . . . . . . . . . 6
B. Natural and synthetic single-crystal
diamond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
C. Homo- and heteroepitaxial CVD diamond . . 7
D. Microcrystalline diamond films . . . . . . . . . . . 7
E. Nanocrystalline diamond . . . . . . . . . . . . . . . . . 8
IV. NONLINEAR MECHANICAL
BEHAVIOR OF DIAMOND . . . . . . . . . . . . . . . . . 9
A. Fracture strength of ideal crystals . . . . . . . . . 11
B. Hardness and fracture of single-crystal
diamond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
C. Hardness and fracture of microcrystalline
diamond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
D. Hardness and fracture of nanocrystalline
diamond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
V. OUTLOOK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
I. INTRODUCTION
A. Structure and mechanical properties of an idealdiamond crystal
Diamond crystallizes in a body-centered cubic structure,
where the carbon atoms are tetrahedrally bonded by sp3
hybridization, and is considered as a prototype of covalent
crystals with cubic symmetry. To realize outstanding me-
chanical properties of a solid, strong equivalent chemical
bonds are needed in three dimensions. To achieve this, at
least four strong bonds per atom are needed with angles of
109.47� between them, generating the cubic symmetry. Car-
bon is the smallest atom that fulfills all these requirements.
The resulting small bond length leads to a high density of
bonding electrons and a tetrahedral structure that is stabilized
by resonance of the bonding electrons among adjacent
bonds.1 This yields the high dissociation energy of the cova-
lent single carbon–carbon bond (339 kJ/mol (Ref. 2), 347 kJ/
mol (Ref. 3)). The high density of carbon atoms, with the
highest covalent bond density of any material, contributes to
the exceptional mechanical properties. As an additional con-
sequence of the stability of such an arrangement of atoms, a
large amount of energy is needed to remove a single carbon
atom from the three-dimensional lattice. Another conse-
quence of the extreme stability of this framework is the
chemical inertness of diamond. Furthermore, the extremely
high surface energy is responsible for the highest cleavage
energy in fracture of any known material (5.5 J/m2 (Ref. 2),
5.3 J/m2 (Ref. 3)). Diamond’s weakest cleavage plane is the
octahedral f111g plane, but cleavage may occur also on
other planes, such as (110) and (221).3
Consequently, the exceptional mechanical (and other)
macroscopic properties are due to the high density of carbon
atoms, which are connected in three dimensions by strong
chemical bonds. In fact, diamond has the highest elastic
moduli and fracture strength of any known material. A
possible exception could be only the bulk modulus of
osmium.2
Especially, the shear modulus is exceptional, with the
consequence that plasticity is not observed at room tempera-
ture and moderate stresses. Plastic flow is associated with
dislocation motion, which is connected with an activation
energy equal to twice the large bandgap in covalent dia-
mond.2 Therefore, diamond is a nearly ideal brittle material,
with no contribution from plastic or viscous deformation
during fracture at low temperatures.
Based on its high specific surface energy, it is clear that
diamond is very abrasion or scratch resistant and extremely
hard. The hardness is also strongly connected with the
extreme shear modulus, which even exceeds the large bulk
modulus, revealing that diamond is more resistant to shear
deformation than to volume changes. These particular fea-
tures protect defect-free diamond also against mechanical
failure and fracture, which often starts from the surface in a
single- or multi-mode process.
The angular frequency of lattice vibrations is proportional
to the square root of the restoring force and inversely propor-
tional to the square root of the mass. The strong carbon bonds
and their low mass mean that the diamond lattice possesses
high-frequency vibrations of up to 4.0� 1013 Hz (1332.2
cm�1 (Ref. 4)). Besides fast heat conduction, the fast
vibrational movement of carbon atoms results in an extreme
speed of sound and propagation of crack tips with comparable
velocity. In fact, a speed of 7200 m/s has been measured for
surface cracks, which is about 0.7 of the speed of Rayleigh
waves.3
In summary, a consequence of all these unique features
is a strong resistance of diamond to all kinds of mechanical
distortions, such as tension, compression, torsion, and
bending, that usually lead to elastic deformation, plastic
flow, or destruction by fatigue or fracture processes. The dis-
cussion clearly indicates that it is not very probable that a
compound crystal or structural arrangement can be found
that combines all these unique features of diamond in one
single material.
B. Natural and synthetic diamond
Natural and synthetic diamonds originate from different
growth environments and conditions that controlled the dep-
osition process and, thus, their structure and composition.
Synthetic diamond can be grown under equilibrium condi-
tions or in a kinetically controlled process. Here, primarily,
the aspects important for the mechanical properties and
behavior of synthetic diamond grown by metastable chemi-
cal vapor deposition (CVD) are discussed.
The quality of natural diamonds is often characterized
according to their optical absorption behavior and appear-
ance. They are usually divided into type I and II crystals.
The mostly yellow-colored transparent type I crystals con-
tain molecular nitrogen defects with a high concentration of
200–3000 ppm (type Ia) or nitrogen atoms are substituted in
carbon positions (type Ib), but this is found relatively sel-
dom. Diamonds of type II contain less than 2 ppm nitrogen.
In this class, crystals with a specific resistance of�105 X cm
(type IIa) are distinguished from those with>106 X cm (type
IIb).
051101-2 Peter Hess J. Appl. Phys. 111, 051101 (2012)
Under conditions in which the diamond phase is thermo-
dynamically stable, diamonds are grown by an industrialized
process.5 In the presence of solvent metals, such as Fe, Co,
or Ni, diamonds can by synthesized at pressures of 5–6 GPa
(60 kbar) and temperatures of 1600–2000 K, the so-called
high pressure, high temperature (HPHT) method. Mostly
small grains of type Ib are produced this way, used for grind-
ing and abrasive applications. The size of the crystals
increases with well-controlled growth time. By adding a get-
ter mixture (e.g., Ti, Al, and Zr), which preferentially binds
nitrogen, it is also possible to obtain type IIa diamonds. It is
important to note that for the growth of pure diamonds from
an impurity-free carbon melt pressures >10 GPa and temper-
atures >4000 K may be required. The realization of diamond
growth under such extreme conditions without getter or sol-
vent material has not yet been achieved.
Diamond can also be grown in the metastable regime of
the phase diagram at low gas pressures of �10 kPa and sub-
strate temperatures of �500–1000 K. For more detailed
reviews published recently on low pressure diamond growth,
its properties, and applications see Refs. 5 and 6. Note that
the CVD process is driven by kinetics and not by thermody-
namics. In the low pressure deposition process, a source gas
containing, for example, methane (usually< 5%) in hydro-
gen is heated to about 2000–2300 K, generating atomic
hydrogen. To induce the relevant chemistry in the gas phase,
a simple hot filament or net of filaments is employed in hot
filament chemical vapor deposition (HFCVD). Alternatively,
plasma is used in plasma-assisted CVD (PACVD), e.g.,
microwave plasma CVD (MPCVD). The complex chemistry
in the gas phase and at the surface leads to the formation of
carbon atoms in sp3 hybridization at the surface, while gra-
phitic sp2-hybridized carbon atoms are etched away by
hydrogen atoms. A report of recent progress in the under-
standing of the chemical mechanism can be found in Ref. 7.
Since, in the CVD process, the purity of the source gases can
be controlled quite well, the CVD of diamond gives access
to high purity and high quality materials in the form of large
crystals, thick plates, or thin films and coatings that can be
deposited on shaped substrates.
C. Purity and morphology of real crystals
As already mentioned above, natural diamond crystals
contain varying amounts of impurities, especially nitrogen
molecules in interstitial positions or atomic nitrogen substi-
tuting for carbon atoms. In the CVD process, the nitrogen
content can be controlled and often nitrogen is introduced
intentionally to increase the growth rate. In addition, other
local or extended structural defects are usually present. The
same is true for diamond crystals prepared by the HPHT
method, where different elements are introduced to reduce
the effective pressure and temperature that are needed for
thermodynamic conversion of graphite into diamond in the
experimentally accessible region. The individual impurities
and structural defects must be determined for each particular
sample separately.
While, for metastable diamond synthesis, extremely
pure precursor gases can be employed, the morphology
depends strongly on the substrate surface and deposition con-
ditions. For homoepitaxial growth on bulk diamond,8 no
extra nucleation step is needed, as in the case of heteroepi-
taxial growth on a non-diamond substrate, such as iridium
buffer layers.9,10 The non-spontaneous nucleation process is
mainly responsible for the development of more-or-less
quasi-monocrystalline, textured, and polycrystalline dia-
mond. The average grain size at the nucleation surface
depends on the nucleation density and increases with the
thickness of the film during competitive columnar growth of
the nucleated crystallites. At nucleation rates of �1011 cm�2,
high quality microcrystalline diamond can be grown with
large crystallites, which, of course, strongly affect the me-
chanical properties. It is clear that, with decreasing grain
size, the increasing influence of grain boundaries, e.g., with
non-sp3 bonding, distorts the regular cubic lattice and,
consequently, deteriorates the outstanding mechanical
behavior. X ray diffraction (XRD) is the method of choice to
obtain detailed information on the crystalline phase, such as
grain size and preferential orientation of crystallites.
The steady growth of crystallites can be restricted by re-
nucleation occurring in an Ar/CH4 plasma or at higher CH4
concentrations, leading to small equiaxed crystallites of
< 100 nm (nanocrystalline) or even < 10 nm grain size
(ultrananocrystalline (UNC)).11 Reduction of the grain size
results in a larger surface-to-volume ratio with enhanced dis-
order by non-diamond carbon and hydrogen bonded at the
grain surfaces. As a consequence, stiffness and hardness of
the material is expected to be reduced in comparison with
single-crystal or microcrystalline diamond, where nearly the
ideal values of these properties have been measured by vari-
ous methods. On the other hand, small grain sizes reduce the
surface roughness, a property of immense practical
importance.
D. Elastic and inelastic behavior
The linear elastic coefficients (since the elastic moduli
are temperature dependent, the widespread term elastic
“constants” should be avoided) of single-crystal diamond are
directly connected with its outstanding mechanical properties,
such as high resistance to extension, compression, and shear,
and its low thermal expansion.5,6 They are responsible for the
fact that diamond exhibits the highest currently known bulk,
shear, and Young’s moduli. The elastic coefficients of the
synthetic forms of diamond deposited by HPHT and low-
pressure CVD can be compared with the elastic moduli of the
highest quality single crystals presently being investigated.12
Since diamond crystals are anisotropic, the stiffness tensor
has three independent components instead of only two for an
isotropic material. However, very often, isotropic behavior is
assumed and only the Young’s modulus is measured to char-
acterize the stiffness. Furthermore, the mechanical properties
can be compared with first principles calculations of an ideal
crystal to find out to what extent the particular specimen
approaches the ideal stiffness and strength of a defect-free
three-dimensional diamond lattice (see Ref. 13 and references
therein). It will turn out that the strongly nonlinear mechani-
cal failure processes may be extremely sensitive to lattice
051101-3 Peter Hess J. Appl. Phys. 111, 051101 (2012)
distortions and especially to extended defects, whereas linear
mechanical properties, such as stiffness and elasticity, are far
less sensitive to point and extended defects. In fact, it is diffi-
cult to measure the impact of grain boundaries on the elastic
behavior of high quality CVD diamond, since their influence
may be smaller than the experimental error involved. On the
other hand, a single defect or flaw that hardly affects the lin-
ear elastic properties can be responsible for fracture. The sen-
sitivity of hardness to irregularities and distortions of the
ideal network is expected to lie somewhere between these
two extremes of elastic stiffness and inelastic fracture. To
explain this difference in the defect sensitivity, we may
refer to the fact that elasticity is an atomistic property,
depending on the individual bonds, while fracture strength
depends on the probed volume or surface region and, thus,
is a size-dependent quantity, depending on flaw size and
concentration in an integral manner.
II. DIAGNOSTIC METHODS
A. Methods to study linear elastic behavior
The following methods have been used to obtain the
most accurate stiffness coefficients of diamond specimens.
Conventional transducer ultrasonics, using bulk acoustic
waves (BAWs), provided the first accurate measurements for
relatively small natural crystals. Laser ultrasonics (LU),
employing surface acoustic waves (SAWs), is mainly used
for the investigation of layered systems with CVD diamond
films, while resonant-ultrasonic spectroscopy (RUS) and
Brillouin scattering (BS) have been employed extensively to
investigate freestanding CVD diamond plates, but layered
systems are also accessible to them.
1. Ultrasonics with bulk acoustic waves
In the first accurate measurements of the elastic coeffi-
cients of natural type I diamonds, the velocities of ultrasonic
longitudinal and shear waves were measured with the pulse-
superposition method, using conventional quartz plates as
transducers and small diamond specimens with lateral
dimensions of a few millimeters (�1 carat).14 Since the
transducer needs to be in direct contact with the sample, con-
ventional transducer ultrasonics is not contact free. Refined
ultrasonic experiments employing the pulse-superposition
method and a 22-carat type II crystal yielded a set of elastic
moduli in excellent agreement with the former results.15
This ultrasonic method is based on an accurate measurement
of the sound velocity via the round-trip transit time. The
sound velocity is directly connected with the corresponding
elastic stiffness C via the mass density q (C¼qv2). Since di-
amond is an anisotropic cubic crystal, at least three inde-
pendent velocity measurements are required to determine the
complete set of three elastic coefficients.
2. Laser ultrasonics with surface acoustic waves
The investigation of CVD films and coatings has been
dominated by laser ultrasonics employing SAWs, which are
excited thermoelastically by focusing a nanosecond or pico-
second laser pulse onto the surface to launch a broadband
elastic pulse16 or by using an optically generated transient
laser grating to generate a narrowband wave train.17 The
propagating SAW pulse can be registered either optically
with a laser or, in the case of a rough diamond surface, with
a versatile piezoelectric foil transducer.18 Thick layers (e.g.,
100 lm) sustain a large number of acoustic modes, non-
dispersive and dispersive with normal or anomalous disper-
sion, which allow the extraction of a complete set of elastic
coefficients.19 For films with a thickness of a few micro-
meters and a spectral range of several hundred megahertz, in
principle, the Young’s modulus, density, and/or film thick-
ness can be extracted if the dispersion curve is strongly non-
linear.20 This is the case when the acoustic contrast between
film and substrate material is high enough, as in the system
diamond/silicon, and the film thickness becomes comparable
with the acoustic wavelength. For diamond films, anomalous
dispersion is observed, where the phase velocity increases
with increasing frequency of the wave, because, in a coher-
ent broadband pulse, the longer wavelengths, propagating
mainly in the substrate, possess a lower velocity than the
shorter wavelengths, which are located preferentially in the
stiffer diamond layer.21 In thinner nano- and microcrystalline
films with a thickness of several hundred nanometers or less,
the dispersion curve is usually linear and only one property,
e.g., the Young’s modulus, can be obtained from the slope of
the linear dispersion curve. The Poisson ratio, usually taken
from the literature, serves as the second elastic property if
isotropy is assumed.
3. Resonant ultrasonic spectroscopy
Originally, resonant ultrasound spectroscopy (RUS) was
developed to determine all the independent bulk elastic coef-
ficients of a homogeneous solid from the mechanical reso-
nance frequencies of the particular specimen.22 The free
vibrational resonance frequencies depend on the specimens
shape, crystallographic orientation, mass density, elastic coef-
ficients, and dissipation. They can be measured by sandwich-
ing the sample between two transducers, by a piezoelectric
tripod transducer to minimize the influence of contact on the
resonance frequencies, or by contact-free laser-Doppler inter-
ferometry, where the frequency of a reflected laser beam
slightly changes with the frequency of the surface vibration.23
This yields the particle velocity and out-of-plane displace-
ment of the surface. Careful comparison of the measured
with the calculated displacement distribution allows the
excited acoustic modes to be identified and the elastic coeffi-
cients to be extracted by solution of the inverse problem.
Recently, this method has been extended to layered sys-
tems by measuring the free vibrational resonance frequencies
with and without the diamond film. The analysis is based on
the shift of the resonant frequencies in a deposited layer and
requires highly accurate measurement of the resonance fre-
quencies and careful mode identification of the individual
resonances. Microcrystalline diamond films often show trans-
verse isotropy or hexagonal symmetry with five independent
elastic coefficients owing to their columnar structure, texture,
residual stress, and local incohesive bonds or microcracks.23
These coefficients can be determined by RUS.
051101-4 Peter Hess J. Appl. Phys. 111, 051101 (2012)
4. Brillouin scattering
Brillouin scattering is based on the spectral analysis of
the inelastic interaction between laser photons and the long
wavelength acoustic surface and bulk phonons obeying energy
and wave-vector conservation. From the measured frequency
shifts of the Brillouin components, the sound velocities and
finally the elastic coefficients can be extracted. Measurements
have been performed with a piezoelectrically scanned, multi-
pass, high-resolution Fabry-Perot interferometer. Contrary to
the situation encountered normally in Brillouin scattering, in
the special situation of diamond, Brillouin experiments are
more accurate than conventional ultrasonics, owing to the
wide transparency of diamond, the large frequency shifts
resulting from the high sound velocities, and the possibility of
using specimens with very small size. In fact, one of the early
highly accurate sets of elastic coefficients was measured using
a 1/3 carat diamond sample.24,25
It is also possible to study thick diamond films and to
reveal the acoustic modes confined within the film material.
For example, surface and bulk modes have been analyzed by
Brillouin light scattering to extract the three independent
single-crystal elastic coefficients of a 1-mm-thick homoepi-
taxial diamond film.26 Polycrystalline and fine-grained dia-
mond films have also been investigated by this method.27
B. Methods for studying nonlinear mechanicalbehavior
Diamond is the stiffest and hardest material; however,
because of its brittle nature, mechanical failure is, neverthe-
less, a very important issue in practical applications. Its brit-
tle behavior is due to the very low mobility of dislocations at
room temperature, which can be explained by the localized
nature of dislocation motion in a covalent crystal owing to
large barriers.
Usually, it is the surface that is exposed to harsh impulsive
shocks or prolonged mechanical pressure. Therefore, the sur-
face is the location where, normally, the highest stresses are
exerted, and for these reasons, it can be expected that nuclea-
tion and growth of microcracks quite often start from the sur-
face. Thus, the mechanical surface properties, such as its
roughness and morphology, play a dominant role in the irre-
versible destruction of materials, e.g., by impulsive load or fa-
tigue failure. This particular fracture process often leads to
partially closed surface-breaking cracks. Conventional
methods currently applied to study fracture processes are
mainly (nano)indentation, bending of plates, and bursting
of membranes. In the case of extensive grain boundaries
and extended defects, such as microcracks in the bulk, the
situation may change and failure may start in the bulk
material.
1. Indentation
A versatile widespread tool quite often used for hard-
ness, fracture toughness, and strength analysis is the scratch
tester, where a diamond stylus is drawn across the surface
under increasing load to determine the critical load, or the
stylus is used as an indenter. These methods are versatile,
but besides being strongly influenced by the properties of the
system itself, they also depend on several test parameters,
such as scratching velocity and stylus properties, which
affect the critical load. Because of the mechanistic complex-
ity arising from the fact that other failure processes in addi-
tion to plastic deformation are involved, in the presence of
the strongly inhomogeneous deformation fields, it is gener-
ally very difficult to extract quantitative values of the cohe-
sion or fracture strength.28 Therefore, in most cases, only the
hardness and Young’s modulus are extracted from the inden-
tation curves. These values depend on the indenter type, the
applied load, and dynamics of the indentation process, as
well as the surface quality and configuration of the sample.
Especially in the case of single-crystal diamond, these
quasi-static methods are only capable of delivering semi-
quantitative results, since the diamond probe tip and the sam-
ple possess a similar mechanical strength, and thus, the tip is
susceptible to brittle damage. The diamond indenter can defi-
nitely not be considered as rigid and, therefore, its deforma-
tion must be taken into account. Its integrity must be
checked continually throughout the experiments, owing to its
intrinsic brittleness. It seems that, at the hardness of diamond
(>60 GPa), indentation is no longer controlled by plastic de-
formation. Despite these drawbacks, indentation is one of the
methods used to study fracture strength as well as hardness
and stiffness also.29 Various forms of sharp and blunt inden-
ters, such as cube-corner Berkovich, pyramidical Vickers, or
sphere Rockwell indenters, have been applied in these inden-
tation experiments.
With nanoindentation using a Berkovich diamond in-
denter, extremely small volumes can be studied and, by care-
fully recording the loading and unloading responses, it is
possible to improve the determination of the hardness and
elastic modulus in comparison to Vickers microhardness
measurements.30 Data analysis and processing of the load–-
displacement curves can be performed with well-established
evaluation procedures.31 It seems that the nanosized region is
less affected by crack formation and dislocation activities.
In the recently developed nanoindentation methods, low
loads can be used and an imprint image of the residual
indent is not needed anymore, because the penetration
depth into the material is measured on the nanometer scale.
Hardness measurements can be performed with loads
for which the formation of cracks at the indent site may
be avoided, since the normal force is controlled on the
50–300 mN scale.30
2. Bending and bursting tests
Three-point and four-point bend tests are applied to
study the fracture strength and toughness if suitable diamond
specimens are available. The latter method stresses a larger
volume and, therefore, delivers more reliable results. The
CVD process allows the deposition of free-standing discs and
plates of millimeter thickness suitable for bending studies.
The method has well-known drawbacks, such as the rapid
drop of the stress from its maximum value below the central
support and the fact that the sample edges are exposed to the
same stress as the tested surface. For these and sample
051101-5 Peter Hess J. Appl. Phys. 111, 051101 (2012)
availability reasons, fracture strengths of diamond based on
this approach are limited; however, they clearly demonstrate
the applicability of this destructive method.32 Advantages are
the possibility of studying separately the material strength of
the nucleation side and growth side of a diamond plate. Fur-
thermore, a larger volume is stressed than in an indentation
test, sampling a more extended part of the intrinsic flaw dis-
tribution. Besides these bending methods, bursting-disk
tests are frequently applied to determine the strength of dia-
mond.33 Here, the maximum stress is at the center of the
disk. A novel method also not susceptible to edge effects is
the “ball on three balls test” procedure.34 This technique
has been successfully employed to measure the bending
strength of 50–130-lm-thick self-supporting nanocrystal-
line diamond (NCD) foils with a diameter of 135 mm.34
Four-point bend tests have been used to determine not only
the strength, but also the fracture toughness of free-
standing CVD diamond plates.35
The Young’s modulus and fracture strength of sub-mi-
crometer-thick specimens have been obtained by bending
tests on free-standing microcantilevers and by a membrane
deflection technique.36 By the latter method, free-standing
sub-micrometer films with a special dog-bone geometry
were investigated, minimizing boundary-bending effects; the
extended area in the membrane center allowed the applica-
tion of line loads. The membranes were loaded by a nano-
indenter, and an interferometer was used to record the
membrane deflection in these tensile test experiments of
UNC films.36,37 These techniques replace the so-called blis-
ter or bulge tests, where a circular area of the free specimen
that is supported around its periphery is pressurized,3 and the
bursting disk tests.33
It is important to note that the strength is not a constant
material property but a statistical parameter that depends on
the specimen shape and size. According to the “weakest-
link” model, the material fails as soon as the strength is ex-
hausted at one location of the probed structure. While the
strength distribution of brittle materials cannot be described
by a Gaussian distribution, Weibull statistics provides a
power law Pk¼ (r(xk)/r0)m, which allows the determination
of strength values, i.e., the failure probabilities at defined
stress levels. Here, r0 is the stress scale parameter
(“characteristic strength”), m the Weibull modulus (shape
distribution parameter), and r(x) the maximum principal
stress at point x. The Weibull modulus m, which describes
the shape of the failure distribution, can be extracted from
the slope of a plot of ln[–ln(1–Pk)] versus ln(strength). For 3
� m � 5, the distribution is wide, indicating unreliable mate-
rials; for m � 10, the narrow distribution characterizes high
strength materials.37 To characterize the intrinsic strength, it
is necessary to take into account the size of the surface or
volume subjected to tensile stress.38
3. Laser-based impulsive fracture method
Laser-excited strongly nonlinear SAW pulses, develop-
ing steep shock fronts during propagation in a nonlinear me-
dium, have been introduced as a new tool to study the
fracture strength of materials.39,40 If the stress reaches the
material’s strength, a surface-breaking microcrack is gener-
ated with length and depth essentially controlled by the dura-
tion of the elastic stress pulse and the speed of crack
propagation, which is on the order of the Rayleigh velocity.
For such a pump–probe experiment, a pulsed laser is needed
to launch the nonlinear SAW pulse and a continuous-wave
laser is employed to monitor the transient surface displace-
ment or surface velocity of the propagating SAW pulse at a
defined distance from the source. A disadvantage of the
method is that relatively large samples are needed with a
size of square centimeters. The method has been applied to
determine the fracture strength of single-crystal silicon,
where high-quality wafers and thick plates of this size with
well-defined crystallographic orientation are easily available.
The technique allows the realization of stresses of up to
approximately 10 GPa in the shocked SAW pulse. Since the
fracture strengths measured up to now for all kinds of natural
and synthetic diamond crystals are in this experimentally ac-
cessible range, in principle, the impulsive SAW-fracture
method can also be applied for detailed mechanistic studies
of configuration-resolved diamond fracture studies.
III. ELASTIC PROPERTIES OF DIAMOND
A. Ideal single-crystal diamond
For anisotropic single-crystal diamond, the linear stiff-
ness tensor has the three independent components C11, C12,
and C44, instead of only two for an isotropic material. While
C11 and C44 describe the longitudinal and shear stiffness,
respectively, C12 has no such direct physical interpretation.
The most accurate theoretical second-order elastic
coefficients of diamond have been calculated by ab initio all-
electron density-functional theory. The corresponding
values of C11¼ 1043 6 5 GPa, C12¼ 128 6 5 GPa, and
C44¼ 534 6 17 GPa deviate in the percent region from the
best currently accepted experimental values (see Table I).41
In addition, calculated values of the bulk modulus
B¼ 433 6 5 GPa, shear modulus G¼ 502 6 10 GPa, and
Poisson ratio l¼ 0.082 6 0.005 have been reported. For
comparison, calculations on the effect of normal stress on
the ideal shear strength delivered the equilibrium bulk modu-
lus B¼ 426 GPa and the shear modulus G¼ 528 GPa for the
slip system f111gh112i.42
The exact values of the Young’s modulus and Poisson
ratio depend on the crystalline configuration, stress direction,
and anisotropy and can be calculated from the well-known
set of elastic coefficients.43 While the Young’s modulus
TABLE I. Second-order elastic coefficients of diamond obtained by ab ini-
tio theory, bulk ultrasonics, and Brillouin scattering for natural, homoepitax-
ial CVD, and heteroepitaxial CVD crystals.
C11 (GPa) C12 (GPa) C44 (GPa)
Ab initio calculation 1043 6 5 128 6 5 534 6 17 [41]
Ultrasound, natural type I,II 1079 6 5 124 6 5 578 6 2 [15]
Brillouin, natural type IIa,b 1080.4 6 0.5 127.0 6 1 576.6 6 0.5 [25]
Homoepitaxial, CVD 1155 6 20 267 6 50 600 6 20 [26]
Heteroepitaxial, CVD 1146 6 4.8 178 6 46 562 6 3.7 [19]
051101-6 Peter Hess J. Appl. Phys. 111, 051101 (2012)
shows relatively little anisotropy with a minimum of 1050
GPa in the [100] direction and a maximum of 1210 GPa in
the [111] direction, the Poisson ratio varies strongly between
0.00786 and 0.115.43
B. Natural and synthetic single-crystal diamond
As discussed in Sec. II, elastic moduli of natural type I14
and type II15 single-crystal diamonds have been measured at
298 K by using bulk ultrasonic waves. Abundant measure-
ments of the longitudinal and shear sound velocity in differ-
ent crystallographic directions provided the first accurate set
of elastic coefficients of C11¼ 1079 6 5 GPa, C12¼ 124 6 5
GPa, and C44¼ 578 6 2 GPa (see Table I).15 A further
improvement of the accuracy has been achieved by Brillouin
scattering experiments using natural type IIa and IIb dia-
monds,24 but also synthetic crystals of type IIa quality with
varying isotopic composition grown by HFCVD and
HPHT.25 These experiments delivered the generally accepted
set of elastic coefficients of C11¼ 1080.4 6 0.5 GPa,
C12¼ 127.0 6 1.0 GPa, and C44¼ 576.6 6 0.5 GPa with sub-
stantially smaller error bars.25 It is important to note that,
within the larger errors involved in transducer-based ultra-
sonics, the previously obtained set of elastic stiffness coeffi-
cients agrees with the most accurate one obtained by
Brillouin scattering, as can be seen in Table I. Note that all
moduli are for the natural isotope composition of 98.9% 12C
and 1.1% 13C. The elastic moduli of 13C are approximately
0.5% higher. Brillouin scattering yielded a bulk modulus of
B¼ 444.8 6 0.8 GPa. In the following, the elasticity of all
kinds of synthetic forms of diamond will be judged by com-
paring their elastic moduli with the Brillouin data given
above. It is important to note that no difference has been
found in the elastic coefficients between high quality natural
and synthetic diamond.
For type IIa single-crystal diamond plates, the six inde-
pendent third-order elastic coefficients have been obtained
by combining shock wave compression experiments with
hydrostatic compression data.44 These coefficients can serve
as a first step to quantify the nonlinear elastic response of di-
amond, e.g., to model the lattice anharmonicity or to develop
empirical interatomic potentials. This first complete set of
third-order elastic coefficients improves our understanding
of the anharmonic, but also the anisotropic response of dia-
mond crystals under high pressure conditions.
C. Homo- and heteroepitaxial CVD diamond
A complete elastic characterization of�1-mm-thick
homoepitaxial diamond layers by Brillouin scattering has
been performed on free-standing CVD diamond plates after
removal of the HPHT diamond substrate used for prepara-
tion.26 Controlled amounts of nitrogen (2–50 ppm) were
added during MPCVD to study the film quality for different
impurity levels and growth rates. The following elastic mod-
uli have been determined for the lowest growth rate of
7.9 lm/h (4% CH4, 2 ppm N2): C11¼ 1155 6 20 GPa,
C12¼ 267 6 50 GPa, and C44¼ 600 6 20 GPa. A compari-
son with the accepted set of moduli shows reasonable agree-
ment of C11 and C44, however, a very large deviation for C12
(calculated from the reported value of (C11 – C12)/2). This
much larger deviation indicates a much lower sensitivity for
the determination of C12. The situation does not really
improve with increasing growth rate, e.g., for 27 lm/h (4%
CH4, 50 ppm N2), where the agreement of C12 improves, but
the deviation for C44 increases. A clear interpretation of
these changes in the elastic coefficients in terms of the mor-
phology seems to be difficult.
A complete set of moduli has also been determined for a
110-lm-thick anisotropic heteroepitaxial diamond layer.19 In
the multimode photoacoustic technique employed for analy-
sis, acoustic surface and interface modes and SAW propaga-
tion in different crystallographic directions have been
included in the fitting procedure. An advantage of this photo-
acoustic method would be that the diamond film does not
necessarily have to be removed from the Ir/YSZ (yttria-
stabilized zirconia) Si(001) substrate if the properties of the
nucleation layer and substrate can be taken into considera-
tion. The analysis yielded the following stiffness coeffi-
cients: C11¼ 1146 6 4.8 GPa, C12¼ 178 6 46 GPa, and
C44¼ 562 6 3.7 GPa.19 Similar to the homoepitaxial case,
C12 shows the largest deviation, but C11 and C44 also deviate,
within experimental error, from the most accurate set of
elastic moduli. Possible systematic errors involved in the
evaluation of the effects of the nucleation layer cannot be
excluded.
Table I exhibits a comparison of the linear elastic coeffi-
cients of natural and synthetic diamond, obtained by homoe-
pitaxial and heteroepitaxial growth, which were determined
by ab initio calculations, bulk and surface ultrasonic waves,
and Brillouin scattering experiments.
D. Microcrystalline diamond films
One of the first determinations of the set of three elastic
coefficients performed on freestanding microcrystalline dia-
mond films grown by HFCVD with a thickness of up to
400 lm and columnar crystallites of 20–40 lm lateral exten-
sion at the film surface has been achieved by Brillouin scat-
tering.45 From the Rayleigh velocity, the authors concluded
that the film had a (110) texture. However, such columnar
structured films with transverse isotropy (or hexagonal sym-
metry), in reality, have five independent elastic coefficients,
namely C11, C33, C12, C13, and C44. Thin HFCVD films
(3.8 lm, 7.1 lm, and 17.3 lm) were investigated by RUS
and laser-Doppler interferometry together with their silicon
substrate, while two thicker PACVD films (289 lm and
525 lm) were isolated from their substrate.23 The detailed
results for the elastic coefficients show elastic anisotropy
between in-plane and out-of-plane directions and reduced
stiffness. Three reasons for the compliance and anisotropy of
the films were considered by the researchers, namely residual
stress, texture, and local incohesive bonds or microcracks.
The authors conclude from x ray diffraction experiments that
stress effects were small and, thus, negligible, and from
micromechanics modeling, they infer the dominance of inco-
hesive bonds at the grain boundaries.
Many reports are available on the elastic properties of
thin, microcrystalline diamond films deposited on various
051101-7 Peter Hess J. Appl. Phys. 111, 051101 (2012)
substrates. In most studies, the assumption is made that the
material is elastically isotropic, reducing the number of inde-
pendent elastic properties to only two, i.e., the Young’s mod-
ulus and the Poisson ratio. Of course, the CVD process
allows the deposition of highly oriented structures with tex-
ture axes along the [100], [110], and [111] directions. The
corresponding values of the Young’s moduli and Poisson
ratios of fiber-textured CVD diamond deposits have been
estimated from the set of elastic constants.43 For randomly
oriented and densely packed aggregates of diamond grains
(neglecting contributions of the grain boundaries!), an aver-
age Young’s modulus of 1143 GPa and an average Poisson
ratio of l¼ 0.0691 have been calculated.43 Similarly, a sur-
prisingly small value, namely l¼ 0.082, has been found for
the Poisson ratio of the ideal diamond lattice by ab initio cal-
culations.41 Experimental values of the Poisson ratio are
scarce for CVD diamond. An accurate value of l¼ 0.075
has been obtained by measuring two different natural fre-
quencies of a freely vibrating plate and is based on the rela-
tions between the mechanical resonance frequencies and the
dynamic elastic coefficients.46 All these values are outside
the range 0.2�l� 0.5 obtained by classical elasticity theory
for the Poisson ratio of isotropic materials.47 Thus, extremely
hard materials, such as diamond, are outside the range that
can be described by classical elasticity. Very accurate exper-
imental data that are able to clearly resolve the anisotropy of
the Young’s modulus and Poisson ratio are still missing.
A suitable tool for the measurement of the Young’s
modulus in layered systems is the dispersion of SAWs.16 As
described above, up to three film properties, such as the
Young’s modulus, film thickness, and density, may be
extracted if the thickness is in the micrometer range and the
bandwidth of the broadband SAW pulse approaches several
hundred megahertz in a system with large acoustic contrast.
On the other hand, an essentially linear dispersion effect is
observed for thinner films or a lower frequency bandwidth,
providing only one property, e.g., the Young’s modulus.
Since the dispersion method is not very sensitive to the Pois-
son ratio, values obtained by this method may not be very
accurate and it is not critical to take its value from the
literature.
The laser-based SAW technique has been extensively
used to study the dependence of the Young’s modulus of
CVD films on the deposition conditions. For example, by
varying the methane concentration in the source gas between
0.5% and 2.0%, it was shown that the Young’s modulus, ex-
trapolated to zero methane pressure, approaches the value of
E¼ 1143 GPa calculated for random aggregates in polycrys-
talline diamond. For the lowest measured content of 0.5%
CH4, the stiffness was E¼ 1080 GPa.20 However, we have to
bear in mind that the lower the methane content, the lower
the growth rate and, of course, no growth occurs at a meth-
ane pressure of zero.
Another example is the observation of SAW dispersion
generated by gradients in the mechanical and elastic proper-
ties of millimeter-thick microcrystalline diamond plates.48
These measurements of the Rayleigh velocity clearly demon-
strated the variation of the mechanical properties between
the growth and nucleation side. Another application is the
investigation of the positive role played by oxygen on the
growth rate and on the stiffness as a function of the oxygen
content in the source gas.49
The Young’s modulus is often measured as part of the
characterization of the mechanical properties of diamond
using, besides laser-based SAW dispersion, diverse methods,
such as the pressure–displacement relationship of freestand-
ing films, when the bursting strength is measured,50,51
measurement of the velocity of ultrasonic waves with the
pulse-echo method,52 and, most frequently, nanoindentation
in connection with the evaluation of hardness.53–56
E. Nanocrystalline diamond
By increasing secondary nucleation, e.g., by adding nitro-
gen to the source gas, the grain size may drastically decrease
from several micrometers to several nanometers, yielding
NCD or ultrananocrystalline diamond (UNCD). A simple esti-
mate indicates that the fraction of atoms connected with grain
boundaries may increase to about 10% for crystal sizes of a
few nanometers.57 RUS and laser-Doppler interferometry
have been used to determine sets of elastic coefficients,
assuming cubic symmetry.58 It turned out that the diagonal
stiffness coefficients C11 and C44 decreased as the grain size
decreased, while, simultaneously, the off-diagonal coefficient
C12 increased significantly in comparison with microcrystal-
line diamond. A similar behavior has been found for hexago-
nal symmetry.59 In other words, with increasing nitrogen
concentration in the source gas, C11, C33, and C66 decrease,
while C12 and C13 increase. Several explanations have been
suggested to explain this unusual behavior, such as thin gra-
phitic phases at the grain boundaries.58
Ab initio calculations based on the assumption of an iso-
tropic material and the presence of stacking faults inside the
grains, which behave as graphitic bonds, can explain only
qualitatively the decrease of the stiffness, i.e., the Young’s
and shear moduli (experiment:< 20%–30%) and the increase
of the Poisson ratio (experiment:>200%). In principle, the
deterioration of the stiffness is not surprising for an increase
of the average interplane distance, while the Poisson ratio
increases with an increase in the volume fraction of defects,
but the simple model cannot reproduce these changes quanti-
tatively.60 Owing to the very large variety of possible mor-
phologies and nanostructures, the relationship between
microscopic structure and mechanical properties is still an
open issue. This is true, especially, for NCD and UNCD dia-
mond with strongly extended grain boundaries.
NCD diamond with a crystallite size< 100–200 nm can
reach a Young’s modulus of 1100 GPa for low methane con-
centration (0.5%) and high power densities in the range of
25 W/cm3, even when the film thickness is only 140 nm.61
Bulge test measurements on freestanding membranes
showed that, with an increase of the methane concentration
to 20% and a resulting decrease of the grain size below 10
nm, the stiffness is reduced to about 700 GPa. The impor-
tance of the nucleation density on the mechanical quality has
been demonstrated for columnar-structured diamond films
with column diameters of less than 100 nm. The Young’s
modulus obtained for low nucleation density (� 1010 cm�2)
051101-8 Peter Hess J. Appl. Phys. 111, 051101 (2012)
of 517 GPa was roughly half that of a film and plate grown
with much higher nucleation density (>1011 cm�2) of 1120
GPa.62 The finding of a measurable reduction of the stiffness
at a grain size below approximately 100 nm is in good agree-
ment with SAW dispersion measurements on NCD samples
with a grain size between 60 nm and 9 nm, where the Young’s
modulus ranged from 1050 to 700 GPa.57 However, quite dif-
ferent stiffnesses of 609 GPa63 and 1160 GPa64 have been
reported for comparable grain sizes of approximately 150 nm.
While the content of hydrogen impurities (sp3-CH)
increases, the concentration of sp2-carbon in the grain boun-
daries decreases with decreasing grain size. For micrometer-
thick columnar-structured films with a diameter of the cylin-
drical microstructure below 60 nm, it was shown that the lat-
eral grain size of the h100i-oriented columns essentially did
not increase with film thickness and the (100)-faceted films
had a root-mean-square (rms) roughness of only�15 nm.65
Typically, the roughness is 50–100 nm rms for NCD films.
UNCD diamond with low surface roughness can also be
deposited with the argon-rich CH4/Ar plasma chemistry,
using very small quantities of hydrogen.66 This process
yields sp3-bonded grains of 3–5 nm size and atomic grain
boundaries of 0.2–0.4 nm thickness with substantial sp2
coordination.67 For this material, surprisingly high Young’s
moduli between 930 GPa and 970 GPa have been measured
by membrane deflection experiments, which did not change
very much for micro- and nano-seeding.36,37 Obviously, the
further dramatic extension of the surface of grain boundaries
by the extremely small grain size does not necessarily lead
to a dramatic reduction in the elastic stiffness in comparison
with microcrystalline diamond. This finding points more to
direct strong bonding, e.g., by double bonds between grains,
than to grains embedded in an amorphous matrix, as
expressed by the term atomic grain boundaries. It should be
pointed out, however, that also a much lower Young’s modu-
lus of�550 GPa was found by the SAW method for the stiff-
ness of 2- and 6-lm-thick UNCD film.68 These results show
that, for both NCD and UNCD, a variation of the elastic stiff-
ness by a factor of about two has been observed. As
expected, the Young’s modulus may decrease with the extent
of amorphous grain boundaries.69,70
An important aspect with respect to practical applica-
tions is the surface roughness, which sensitively depends on
the nucleation density, initial diamond growth, and final film
thickness. For the nucleation or bottom surface of a free-
standing 1-lm-thick UNCD film, a rms roughness of 3.4 nm
has been observed by atomic force microscopy (AFM),
whereas the growth or top surface had a substantially higher
rms roughness of 20.3 nm.71 The growth of very smooth
UNCD of 4–6 nm rms roughness could be achieved by add-
ing a 10-nm-thick tungsten layer on the silicon substrate as a
template layer that increased the nucleation density.72
In Fig. 1, selected data for the Young’s modulus,
obtained by various methods, are collected and presented as
a function of the average grain size. Whenever a range of
values was given, the mean value of Young’s modulus has
been taken. Similarly, a mean grain size was used when the
grain size of the nucleation and growth sides was specified
or a series of experiments was performed. For columnar
structures, the diameter of the columns has been taken as
grain size. It is clear that a certain ambiguity could not be
avoided, especially in the selection of a mean grain size.
The plot documents a surprisingly small change of the
highest measured stiffness values with the drastic decrease
of the mean grain size by many orders of magnitude from
single crystal to high quality UNCD material. Some smaller
values are included as well to demonstrate the stiffness
reduction if optimum conditions for the substrate preparation
(e.g., nucleation density), the source gas (e.g., methane con-
centration), and the deposition conditions (e.g., density of
plasma processing) were not employed.
Table II shows a comparison of relevant linear mechani-
cal properties of single-crystal diamond and the best values
of mostly microcrystalline CVD diamond, such as bulk and
shear moduli, Poisson ratio, longitudinal sound velocity,
Rayleigh velocity, vibrational frequency, and expansion
coefficient.
IV. NONLINEAR MECHANICAL BEHAVIOR OFDIAMOND
Nonlinear mechanical properties used to characterize di-
amond are its hardness, normally connected with plastic
FIG. 1. (Color online) Young’s modulus as a function of grain size for natu-
ral diamond and CVD diamond. The line is a guide to the eye.
TABLE II. Outstanding linear mechanical properties of single-crystal dia-
mond (calculated or measured) and CVD diamond (measured).
Single-crystal diamond CVD diamond
Bulk modulus 433 GPa [41] 443 GPa [2]
Shear modulus 502 GPa [41] 507 GPa [2]
Young’s modulus, anisotropy 1050–1210 GPa
(random) crystallites 1143 GPa [43] � 500–1200 GPa
Poisson ratio, anisotropy 0.00786–0.115
(random) crystallites 0.0691 [43] 0.075 [46]
Sound velocity, long. (111) 19039 m=s [52] 18784 m=s [52]
Sound velocity, long. (100) 18038 m=s [52]
Sound velocity, long. (110) 18182 m=s [52]
Rayleigh velocity, (110) texture 10753 m=s [45] 10326 m=s [45]
Rayleigh velocity, polycrystal. 10930 m=s [48] 10850 m=s [48]
Vibrational frequency (Raman) 1332.2 cm–1 [4] 1332 cm–1 [6]
Expansion coefficient 0.8� 10–6 K–1 [6]
0.9� 10–6 K–1 [5]
051101-9 Peter Hess J. Appl. Phys. 111, 051101 (2012)
deformation, the fracture toughness, describing the propaga-
tion of the tip of an already existing microcrack, and the
fracture strength or critical failure stress. In principle, all
three properties are connected with irreversible failure and
formation of cracks, since, for a brittle material such as dia-
mond, it is very difficult to clearly separate deformation and
fracture.
Hardness can be defined as the intrinsic resistance of
materials to deformation by an applied force. It is a rather
complex quantity that cannot be quantified easily on an abso-
lute scale.73 The most rigorous approach is first principles
calculations of the ideal strength, providing the stress–strain
profile for large deformations, taking into account the
changes in the electronic structure connected with such
extensive strains. The ideal strength can be calculated for
different crystallographic directions and is given by the max-
imum stress in the stress–strain curve found in the weakest
tensile-stretch or shear-slip direction. After determination of
the weakest tensile direction, the critical shear stress is calcu-
lated by studying the shear deformation in the easy slip plane
perpendicular to this direction. As discussed before, the ideal
shear strength of diamond exceeds, to some extent, the ideal
tensile strength, and thus, tensile or brittle failure and not
plastic deformation is expected at normal temperatures.
In a much simpler approach, hardness may be connected
with the resistance to volume changes, as described by the
bulk modulus. Of course, diamond has the highest bulk mod-
ulus of all covalent solids; however, we have to bear in mind
several drawbacks of such a simple definition. The force
must be applied isotropically and not confined to a specific
location, as is the case in the usually employed indentation
experiments. Furthermore, the bulk modulus is an equilib-
rium property, which is not defined for large deformations.
Definitely, it is not sufficient to consider hydrostatic com-
pression alone; both tensile and shear load must be taken
into account separately for localized forces. Consequently,
hardness measurements with an indenter provide only semi-
quantitative information on nonlinear mechanical properties,
because hardness is not only affected by the stiffness, but
also by the strength of the material, and it is very difficult to
separate these two properties, owing to the non-uniform dis-
tribution of stresses under a sharp indenter tip.
Since diamond is a brittle material, it exhibits, at room
temperature, an essentially linear elastic behavior of the
stress–strain relationship, up to the point where it breaks. Ir-
reversible deformation, therefore, leads to fracture, e.g., by
cleavage into two parts. Therefore, the process of indentation
may involve nucleation and propagation of cracks. In fact, in
natural crystals, usually simultaneous fracture is observed,
whereas, in CVD diamond, cracks may be absent, owing to
internal stresses and grain boundaries. While the initiation of
fracture is not well understood, widely accepted models are
available that describe the growth of a pre-existing crack.
Cracks generated during indentation can be used to
obtain the fracture toughness, which characterizes the resist-
ance to fracture on the macroscale. This information leads to
a better understanding of the size of defects or so-called
flaws in the material and their distribution. The fracture
toughness is considered to be a true, or nearly true, materials
constant characterizing flaw-containing real materials. In the
simplest form of the Griffith relation, the flaw size a0 is
given by the relation between the strength or critical failure
stress rcr and the fracture toughness or critical stress inten-
sity factor KIc, which can be written as rcr ¼ KIc=wa01=2.74
Here, w is a constant depending on flaw geometry. If the
fracture stress and toughness are known, the flaw size can be
estimated. The subscript I in KIc refers to mode I cracking,
where opening failure occurs perpendicularly to the applied
tensile load. Similar expressions can be derived for the criti-
cal failure stress when the crack is subject to in-plane shear
(sliding or mode II) and out-of-plane shear (tearing or mode
III) load.
The fracture strength of a brittle material depends on the
size and distribution of defects on the surface and flaws in
the bulk and, thus, on its fracture toughness. The real fracture
strength of a flaw-containing crystal may be orders of magni-
tude lower than the ideal strength and is extremely dependent
on flaw size and distribution, e.g., on the surface area and its
roughness or the probed defective volume. In the case of
brittleness and a random distribution of flaws, we expect that
a smaller stressed surface region or volume will tend to ex-
hibit a higher fracture strength, owing to fewer flaws causing
failure in the probed region. Such size effects gain impor-
tance in components of microstructural dimensions, such as
thin films and advanced coatings in micromechanical sys-
tems (MEMS) and nanomechanical systems (NEMS). A
defect-free lattice, of course, possesses the highest strength,
the so-called ideal or theoretical strength.
A theoretical analysis of the intrinsic cleavage process
indicates that the opening stress at the crack tip must reach
the theoretical cohesive strength, which is on the order of 1/
10 of the elastic modulus for cubic materials or �E/10,
where E is the Young’s modulus.75 Accordingly, we esti-
mate, with this rule of thumb, an ideal strength of roughly
110 GPa for diamond. Note that, here, a nonlinear mechani-
cal property is estimated from a linear one.
Surprisingly, the highest real fracture strength values of
diamond, measured by several independent techniques, such
as indentation, bending, and bursting, are in the range of sev-
eral gigapascals only and, therefore, not higher than the
measured fracture strength of silicon, for example. If the
crystallographic orientation of the diamond specimen and
the exact geometry of the destructive impact are not well
defined, it is difficult to make an accurate comparison of the
experimental results with each other or with the theoretical
strength. Unfortunately, this is the current situation for most
available measurements. From a large number of independ-
ent experiments on high-quality natural and synthetic dia-
mond samples, which yield a fracture strength between 1 and
7 GPa, we come to the conclusion that the strength of the
best diamond crystals is one to two orders of magnitude
below the theoretical strength of an ideal defect-free dia-
mond lattice. This conclusion is confirmed by general theo-
retical considerations.76 It is important to note that such a
large difference between ideal and real properties provides
an enormous potential for improving the mechanical quality
and fracture strength of CVD diamonds. Since elastic proper-
ties are less sensitive to defects, inelastic properties should
051101-10 Peter Hess J. Appl. Phys. 111, 051101 (2012)
be used to control the success of purity and structural
improvements.
A. Fracture strength of ideal crystals
The theoretical strength of a defect-free diamond lattice
has been calculated for several crystallographic configura-
tions using a first-principles approach. For tension parallel to
the h100i, h110i, and h111i directions, a maximum critical
stress of 225, 130, and 90 GPa was found, respectively.77
These values are consistent with dominant cleavage along
the f111g plane, as observed experimentally. Independent
calculations of the ideal tensile strength for a stress oriented
in the h111i direction yielded 95 GPa, with a critical strain
of 0.13, and an ideal shear strength for shear along the f111geasy cleavage plane in the h112i direction of 93 GPa, with a
critical strain of 0.3.78 These upper limits for the critical
stress are in good agreement with a study of the microscopic
bond-breaking processes, which gave 92.9 GPa for the ten-
sile strength, with a critical strain of 0.15, and 96.3 GPa for
the shear strength under f111gh112i shear, with a critical
strain of 0.35.79 A somewhat higher ideal shear strength of
96.6 GPa (critical strain 0.31) has recently been confirmed
by ab initio density functional theory.80
These calculations reveal that, for diamond, the ideal
tensile and shear strengths are nearly identical, with a shear
strength that is even somewhat higher. The critical shear
stress plays an important role in dislocation nucleation in a
pristine crystal and the phenomenon of plasticity. It is impor-
tant to note that the C–C bonds remain strong up to the
bond-breaking point, explaining the comparable tensile and
shear strength. Microscopically, at the tensile strain of 0.16,
the layers in the (111) plane become essentially flat, as
expected for sp2 bonding. At the shear strain of 0.355, a
transformation into the thermodynamically stable graphite
structure takes place, which is connected with a large volume
expansion of 53%.78 This transformation of the diamond into
the graphite structure upon shear instability has been associ-
ated with the ability of carbon to form p bonds.78
This unique feature of quite similar ideal tensile and
ideal shear strengths definitely contributes to the outstanding
mechanical properties of diamond, especially its hardness.
According to ab initio calculations, the ideal strength of dia-
mond could be identified with the tensile failure of the f111gcleavage plane. Thus, the tensile fracture strength of 93
GPa79 could be considered as the theoretical hardness of dia-
mond if we accept the rigorous theoretical hardness defini-
tion mentioned above.
B. Hardness and fracture of single-crystal diamond
In recent years, gem-sized single crystals of diamond
have been grown homoepitaxially with very high growth rate
(�100 lm/h) by MPCVD. Colorless crystals of 2 carats can
be synthesized routinely, and up to 10 carats have been fabri-
cated by this method.81 The normal hardness of about
60–110 GPa of diamond crystals was surpassed by low pres-
sure/high temperature (LPHT) and high pressure/high tem-
perature (HPHT) annealing. For annealed crystals,
exceptionally high Vickers hardness values of 125 GPa
(LPHT) and 170 GPa (HPHT) have been reported.82 Such
ultrahard diamond crystals clearly reach the limit of the in-
denter technique. Many of these ultrahard diamonds dam-
aged the softer indenter without leaving an imprint on the
surface that could be analyzed, and therefore, it is not clear
whether the measured property should be identified with
hardness.83 These diamond crystals, grown by nitrogen dop-
ing, show also high toughness values of 6–18 MPa m1/2.83
For annealed specimens, even an extreme fracture toughness
of�30 MPa m1/2 has been reported.81 These exceptional val-
ues have not been included in the collection of toughness
results in Fig. 3 and the nonlinear mechanical properties in
Table III. Unfortunately, no measurements of the critical
failure stress are currently available for this kind of ultrahard
and ultratough crystals to allow a comparison with the theo-
retical strength of diamond.
C. Hardness and fracture of microcrystalline diamond
Unlike an ideal crystal, real crystals contain various
amounts of atomic and extended defects, such as vacancies
or voids, impurities at interstitial positions or replacing regu-
lar lattice atoms, and a variety of larger structural defects,
such as grain boundaries or even microcracks and flaws.
These structural deficiencies have a pronounced effect, espe-
cially on the nonlinear mechanical properties. In fact, just
FIG. 3. (Color online) Fracture toughness vs grain size for natural diamond
and CVD diamond.
FIG. 2. (Color online) Hardness as a function of grain size for natural dia-
mond and CVD diamond. The line is a guide to the eye.
051101-11 Peter Hess J. Appl. Phys. 111, 051101 (2012)
one single defect may be responsible for failure of a natural
or synthetic diamond crystal, according to the weakest-link
model.
Microcrystalline diamonds typically have a deposition-
based columnar grain morphology with crystallites in the mi-
crometer range, increasing from the nucleation to the growth
side, and a large surface roughness in the micrometer range.
Hardness values extracted from the size of the indents usu-
ally vary in the wide range of about 60–110 GPa, owing to
the elastic recovery of the indent and the initiation of numer-
ous cracks. Specific values obtained, for example, with a
Vickers microhardness tester were around 75 GPa.84 The
mechanical behavior of freestanding CVD diamond plates
has been studied by Vickers indentation and tensile testing
of pre-notched samples.85 A relatively high average hardness
value of 96 GPa has been measured for these 150–200-lm-
thick plates. The hardness values of microcrystalline dia-
mond cover typically the range of 60–105 GPa,53–56 similar
to natural diamond. For an overview, see the collection of
hardness data shown in Fig. 2.
Up to now, mainly indenters have been used to study—
besides hardness—the fracture toughness of natural and
CVD diamond crystals, where, at the indent corners, radial
cracks emanate, which are used to estimate the toughness.
The values obtained for natural diamond from the length of
the corner cracks depend strongly on the orientation of the
crystal with respect to the easy cleavage direction and vary
between approximately 7 and 14 MPa m1/2.84 The fracture
toughness values of 5–8.4 MPa m1/2 obtained by indentation
for microcrystalline CVD diamond are in good agreement
with those for natural diamond,85–88 as can be seen in Fig. 3.
Very similar toughness values have been reported for
pre-cracked freestanding films using three-point bending
tests, which provided, in addition, the fracture strength.89
For strength measurements with the nucleation side in ten-
sion, a critical stress of �0.75 GPa has been obtained, which
decreased to �0.5 GPa for the growth side in tension for
films with a thickness of about 700 lm. The fracture tough-
ness was KIC¼ 6.8 MPa m1/2 for pre-notched freestanding
samples and KIC¼ 9.2 MPa m1/2 for a laser-notched diamond
sample. Subsequent independent measurements of the tensile
strength confirmed the values of the nucleation side, but
reported only 0.28 GPa for the growth side for 1.89-mm-
thick CVD diamond specimens.90 For the fracture toughness,
these authors obtained a value of KIC¼ 8.5 MPa m1/2.35 In
this paper, previous toughness measurements are reviewed.
The fracture strength of polycrystalline diamond has
been studied by pressure burst tests,91,92 three-point bend-
ing,90,93 and four-point bending90,94 experiments. The frac-
ture strength of the fine-grained nucleation side was
consistently higher than that of the coarse-grained growth
side,90,91,93 with values typically below 1 GPa, as discussed
above. On the other hand, tensile strengths of up to 5.2 GPa
have been reported in the literature for CVD diamond92 and
7.5 GPa for natural IIa diamond.95
Such high tensile strengths have also been obtained by
microindentation tests performed with a Rockwell sphere in-
denter for microcrystalline diamond films deposited on a SiC
substrate. A finite element method (FEM) analysis, based on
the critical indentation force, indentation depth, and diameter
of the observed Hertzian ring cracks, was employed to deter-
mine the fracture strength of 7.1 GPa for a 35-lm-thick
film.96 This seems to be one of the highest strength values
reported to date for CVD diamond. In a more recent paper,
the authors report similar strengths, namely 6.4 GPa for a
21-lm and 4.0 GPa for a 70-lm-thick film, as can be seen in
Fig. 4.97
D. Hardness and fracture of nanocrystalline diamond
The hardness values exhibited in Fig. 2 for NCD and
UNCD clearly show that a comparable hardness can be
achieved for these materials, despite their small grain size,69
but also much lower values have been observed in the case
of extended amorphous grain boundaries.70 The hardness of
such films with low roughness has been investigated in stud-
ies on protective coatings with a low roughness
of< 10 nm.98,99 The influence of total gas pressure, substrate
temperature, and methane concentration on grain size, sur-
face roughness, and hardness of NCD has been studied in
Ref. 100. A comparison of hardness values of microcrystal-
line diamond and UNCD has been performed.101 For 14%
CH4 in the source gas, a grain size of 13.5 nm, a surface
roughness of 13.4 nm, and hardness of 65 GPa has been
measured. The hardness of UNCD, deposited from CH4/H2
using a combination of microwave and radio frequency (rf)
capacitive discharge deposition, was 70 GPa.102 A higher
hardness of 91 GPa has been reported for a grain size of 10
nm using 10% methane in the CH4/H2 source gas,103 and an
even higher one of 98 GPa has been observed for UNCD.104FIG. 4. (Color) Fracture strength vs grain size for natural diamond and
CVD diamond.
TABLE III. Highest values reported for diamond hardness, fracture tough-
ness, and fracture strength.
Hardness
(GPa)
Fracture toughness
(MPa m1=2)
Fracture strength
(GPa)
Ideal crystal �110 �93 [79]
Single crystal 50–110 [83] 6–18 [83] < 1 [93]
Microcrystalline 118 [53] 13 [82] 7.1 [96]
Nanocrystalline 86 [57] 3.8 [106]
Ultrananocrystalline 98 [104] 8.7 [104] 5.1 [104]
051101-12 Peter Hess J. Appl. Phys. 111, 051101 (2012)
Figure 2 presents an overview of selected hardness val-
ues as a function of grain size. Whenever possible, mean val-
ues of a series of samples or experiments have been
included. Quite similar to the situation encountered for the
Young’s modulus, the hardness decreases only from 110
GPa to about 90 GPa from single-crystal diamond to UNCD
samples. This striking correlation of hardness with the linear
elastic modulus is quite surprising. It can be understood by
assuming that different inelastic processes and, thus, distinct
quantities are measured in the case of ultrasmall crystallites
as compared to a single crystal.
Information on the fracture toughness is relatively
scarce for both NCD and UNCD. First values for UNCD
have been obtained by the membrane deflection technique
using sharp cracks and blunt notches.104 The corresponding
stress-intensity factors were 4.5 MPa m1/2 and 8.7 MPa m1/2,
respectively. A similar value of 5.3 MPa m1/2 has been
reported in Ref. 105.
Figure 3 shows a collection of toughness values versus
grain size of CVD diamond extending from 4 to 9 MPa m1/2,
with very few exceptions. This range of fracture stress-
intensity factors agrees quite well with that of natural single-
crystal diamond. Toughness values for a grain size �100 lm
are all displayed at 100 lm grain size. Obviously, high qual-
ity grain boundaries, e.g., in UNCD, do not deteriorate the
toughness of the material drastically, pointing to the conclu-
sion that intergrain failure should be able to compete with
intragrain failure.
As shown in Refs. 106 and 107, low nitrogen pressure in
the deposition gas may lead to columnar structures with a
diameter<200 nm, whereas, at higher nitrogen content,
renucleation occurs and the crystal size does not exceed sev-
eral tens of nanometers in all three dimensions. After nuclea-
tion by bias pretreatment, the concentration of atomic
hydrogen was reduced during growth by addition of nitrogen
or by increasing the total pressure. The fracture strengths
have been measured from the bending of diamond cantile-
vers under defined force, which delivers the stress–strain de-
pendence. The strength of�3 GPa obtained for the columnar
structure (NCD) was somewhat lower than the�3.9 GPa for
equiaxed grains (UNCD).106
A review on the science and technology of diamond
UNCD films has been published recently.108 The fracture
strength of freestanding UNCD films has been studied by the
membrane deflection method. According to the weakest-link
model, it may be expected that failure strength decreases,
owing to the much smaller grain size and extended grain
boundaries. A complete Weibull analysis has been per-
formed for the large data set published in Ref. 37. For a 63%
probability of cumulative failure, a nominal strength of 4.1
GPa with a Weibull modulus of 10.8 has been found.37 This
strength depends on the surface or volume subjected to ten-
sile stress and therefore does not describe the intrinsic
strength of the material. Both the measured strength and the
nominal strength derived from a Weibull plot do not take
into consideration the stressed surface area. This is taken
into account by the characteristic strength of 1.87 GPa,
which refers to the effective strength of a 1 cm2 stressed
sample and a Weibull modulus of 12.8.38 These stress values
should be compared with the measured strength of 3.97
GPa.37
The size effect or scaling parameter in fracture of
UNCD films has been analyzed by Weibull statistics.72 For
one particular sample, e.g., a characteristic strength of 4.2
GPa and a Weibull modulus of 12.2 has been obtained. The
analysis pointed to a failure mechanism controlled by a dis-
tribution of volume defects and not surface defects for the
investigated samples. This conclusion that, in UNCD, failure
most likely initiates from interior (volume) defects has been
confirmed.104 The characteristic strengths for two specimen
sizes were 4.0 and 5.1 GPa with a Weibull modulus of 11.6.
In Fig. 4, strength values are presented as a function of
grain size independent of film thickness and probed surface
area or volume. Most reported measurements resulted in a
fracture strength below 1 GPa, with a higher value if the
nucleation side of the sample was stretched. The highest
reported strengths of several gigapascals are comparable for
nano-, micro-, and single-crystal diamond and, thus, 1–2
orders of magnitude below the theoretical strength.
V. OUTLOOK
Despite its extreme mechanical and tribological proper-
ties, the scientific and engineering applications of diamond
have been limited by scarcity; availability of suitable sizes,
shapes, and forms; as well as expense. With the continuous
improvement of the CVD processes, this situation is chang-
ing, and coatings, films, plates, etc. are becoming available
with the quality tailored to the specific application. For
example, its high stiffness and excellent resistance to thermal
shocks, pressure loading, scratches, and erosion makes dia-
mond an ideal material for diverse applications where a hard
and wear-resistant material is needed. Even for applications
where the material does not have to be of optimal quality in
any respect, the control of its mechanical properties may be
a necessity.
The Young’s modulus provides a suitable control of the
stiffness. Several methods are available for its measurement,
with or without substrate, such as dispersion of SAWs, in-
dentation, beam bending, and bulging of freestanding mem-
branes. If nanodiamond films are used to reduce the surface
roughness, the stiffness may decrease from about 1100 GPa
to about 900 GPa only. Since the actual stiffness reduction
cannot be recognized from the grain size, it is necessary to
determine the Young’s modulus experimentally for each
specimen. A small reduction occurs at a grain size below
100–200 nm, sometimes used to define NCD. A reduction by
a factor of two already indicates a clear deterioration of the
mechanical quality, since the elastic modulus is not very sen-
sitive to high quality grain boundaries, as the nearly ideal
stiffness obtained for microcrystalline diamond proves.
Nevertheless, a stiffness of 500 GPa may still qualify smooth
nanocrystalline films for coatings and SAW applications.
Besides elastic properties, such as stiffness, nonlinear
properties, such as deformation, and irreversible failure play
an important role in many applications. Quite often hardness
is used for characterization of destruction despite the rather
complex meaning of this quantity at the hardness level of
051101-13 Peter Hess J. Appl. Phys. 111, 051101 (2012)
diamond (60–110 GPa). As long as a theoretical standard for
comparison is lacking, results can only be compared with the
hardness of other materials. However, measurements of high
quality diamond with a diamond indenter have to be consid-
ered with great care, due to brittle fracture of the sample to-
gether with deformation or irreversible destruction of the
indenter. It seems to be an open question whether it is mean-
ingful to extend the comparative hardness scale beyond dia-
mond into the region of the so-called ultrahard materials.109
It looks like ultrahardness cannot be described by a single
number before the fundamental aspects of hardness are clari-
fied, and a meaningful claim must present not only a detailed
description of the materials involved, but also a rigorous
description of the measurement procedure.
A much more interesting, but difficult-to-measure non-
linear quantity is the critical fracture strength. For “the ideal”
property, we have have accurate ab initio calculations, which
allow an absolute judgment of the strength of any real dia-
mond crystal. Presently available strength values clearly
show a huge difference between the ideal tensile strength of
about 93 GPa and the highest measured critical fracture stress
of about 7 GPa (see Table III). The discrepancy of 1-2 orders
of magnitude between the best real and ideal diamond docu-
ments the sensitivity of the fracture strength to defects that
are registered in an integral manner in the probed region.
This indicates that the macroscopic stiffness, which is charac-
teristic of harmonic extensions near the equilibrium position,
is far less sensitive to lattice distortions than strong displace-
ments, leading to bond dissociation. It is important to realize
that this situation offers unique possibilities for industrial
quality control, but also for controlling the step-by-step
improvement of the material. Of course, breaking freestand-
ing diamond plates or ruining a diamond indenter is an ex-
pensive way to achieve strength information. The nonlinear
SAW technique using ultrasonic shock pulses offers new pos-
sibilities here, when large diamond samples with well-
defined crystallographic configuration become available.110
The disparity between ideal values and real bulk proper-
ties can be reduced only by gradually improving the impurity
level and defect structure of the whole material. Important
new insights into the role played by different impurities,
defects, and morphologies will be gained by the development
of novel precise techniques for the analysis of microscopic
and nanoscopic samples with smaller and smaller probed
regions.
1J. J. Gilman, Mater. Res. Soc. Symp. Proc. 383, 281 (1995).2J. J. Gilman, Mater. Res. Innovations 6, 112 (2002).3J. E. Field, EMIS Datareviews Series, INSPEC 9, 36 (1994).4J. B. Cui, K. Amtmann, J. Ristein, and L. Ley, J. Appl. Phys. 83, 7929
(1998).5R. S. Balmer, J. R. Brandon, S. L. Clewes, H. K. Dhillon, J. M. Dodson,
I. Friel, P. N. Inglis, T. D. Madgwick, M. L. Markham, T. P. Mollart, N.
Perkins, G. A. Scarsbrook, D. J. Twitchen, A. J. Whitehead, J. J. Wilman,
and S. M. Woollard, J. Phys.: Condens. Matter 21, 364221 (2009).6J. J. Gracio, Q. H. Fan, and J. C. Madaleno, J. Phys. D: Appl. Phys. 43,
374017 (2010).7J. E. Butler, Y. A. Mankelevich, A. Cheesman, J. Ma, and M. N. R. Ash-
fold, J. Phys.: Condens. Matter 21, 364201 (2009).8P. M. Martineau, M. P. Gaukroger, K. B. Guy, S. C. Lawson, D. J.
Twitchen, I. Friel, J. O. Hansen, G. C. Summerton, T. P. G. Addison, and
R. Burns, J. Phys.: Condens. Matter 21, 364205 (2009).
9M. Schreck, F. Hormann, H. Roll, J. K. N. Lindner, and B. Stritzker,
Appl. Phys. Lett. 78, 192 (2001).10M. Schreck, S. Gsell, R. Brescia, M. Fischer, P. Bernhard, P. Prunici, P.
Hess, and B. Stritzker, Diamond Relat. Mater. 18, 107 (2009).11O. A. Williams, Diamond Relat. Mater. 20, 621 (2011).12A. Migliori, H. Ledbetter, R. G. Leisure, C. Pantea, and J. B. Betts, J.
Appl. Phys. 104, 053512 (2008).13X. Luo, Z. Liu, B. Xu, D. Yu, Y. Tian, H.-T. Wang, and J. He, J. Phys.
Chem. 114, 17851 (2010).14H. J. McSkimin and W. L. Bond, Phys. Rev. 105, 116 (1957).15H. J. McSkimin and P. Andreatch, Jr., J. Appl. Phys. 43, 2944 (1972).16A. M. Lomonosov, A. P. Mayer, and P. Hess, in Modern Acoustical Tech-
niques for the Measurement of Mechanical Properties, edited by M.
Levy, H. E. Bass, R. Stern (Academic, Boston, 2001), Vol. 39, pp.
65–134.17J. A. Rogers and K. A. Nelson, J. Appl. Phys. 75, 1534 (1994).18H. Coufal, R. Grygier, P. Hess, and A. Neubrand, J. Acoust. Soc. Am. 92,
2980 (1992).19Z. H. Shen, A. M. Lomonosov, P. Hess, M. Fischer, S. Gsell, and M.
Schreck, J. Appl. Phys. 108, 083524 (2010).20R. Kuschnereit, P. Hess, D. Albert, and W. Kulisch, Thin Solid Films
312, 66 (1998).21P. Hess, Proc. SPIE 4703, 1 (2002).22R. G. Leisure and F. A. Willis, J. Phys.: Condens. Matter 9, 6001
(1997).23N. Nakamura, H. Ogi, and M. Hirao, Acta Mater. 52, 765 (2004).24M. Grimsditch and A. K. Ramdas, Phys. Rev. B 11, 3139 (1975).25R. Vogelgesang, A. K. Ramdas, S. Rodriguez, M. Grimsditch, and T.
Anthony, Phys. Rev. B 54, 3989 (1996).26P. Djemia, A. Tallaire, J. Achard, F. Silva, and A. Gicquel, Diamond
Relat. Mater. 16, 962 (2007).27P. Djemia, C. Dugautier, T. Chauveau, E. Dogheche, M. I. De Barros,
and L. Vandenbulcke, J. Appl. Phys. 90, 3771 (2001).28R. H. Lacombe, Adhesion Measurement Methods: Theory and Practice
(CRC, Boca Raton, FL, 2006).29J. E. Field and C. S. J. Pickles, Diamond Relat. Mater. 5, 625 (1996).30S. Chowdhury, E. de Barra, and M. T. Laugier, Surf. Coat. Technol. 193,
200 (2005).31W. C. Oliver and G. M. Pharr, J. Mater. Res. 7, 23 (1992).32C. S. J. Pickles, Diamond Relat. Mater. 11, 1913 (2002).33A. R. Davies, J. E. Field, and C. S. J. Pickles, Philos. Mag. 83, 4059
(2004).34R. Ramakrishnan, M. A. Lodes, S. M. Rosiwal, and R. F. Singer, Acta
Mater. 59, 3343 (2011).35A. R. Davies, J. E. Field, K. Takahashi, and K. Hade, J. Mater. Sci. 39,
1571 (2004).36H. D. Espinosa, B. C. Prorok, B. Peng, K. H. Kim, N. Moldovan, O.
Auciello, J. A. Carlisle, D. M. Gruen, and D. C. Mancini, Exp. Mech. 43,
256 (2003).37H. D. Espinosa, B. Peng, B. C. Prorok, N. Moldovan, O. Auciello, J. A.
Carlisle, D. M. Gruen, and D. C. Marcini, J. Appl. Phys. 94, 6076 (2003).38C. A. Klein, J. Appl. Phys. 97, 016105 (2005).39P. Hess, Phys. Today 55, 42 (2002).40A. M. Lomonosov and P. Hess, Phys. Rev. Lett. 89, 095501 (2002).41D. G. Clerc and H. Ledbetter, J. Phys. Chem. Solids 66, 1589 (2005).42Y. Umeno and M. Cerny, Phys. Rev. B 77, 10010 (2008).43C. A. Klein and G. F. Cardinale, Diamond Relat. Mater. 2, 918 (1993).44J. M. Lang and Y. M. Gupta, Phys. Rev. Lett. 106, 125502 (2011).45X. Jiang, J. V. Harzer, B. Hillebrands, Ch. Wild, and P. Koidl, Appl.
Phys. Lett. 59, 1055 (1991).46L. Bruno, L. Pagnotta, and A. Poggialini, J. Eur. Ceram. Soc. 26, 2419 (2006).47P. H. Mott and C. M. Roland, Phys. Rev. B 80, 132104 (2009).48G. Lehmann, M. Schreck, L. Hou, J. Lambers, and P. Hess, Diamond
Relat. Mater. 10, 686 (2001).49Z. H. Shen, P. Hess, J. P. Huang, Y. C. Lin, K. H. Chen, L. C. Chen, and
S. T. Lin, J. Appl. Phys. 99, 124303 (2006).50T. J. Valentine, A. J. Whitehead, R. S. Sussmann, C. J. H. Wort, and
G. A. Scarsbrook, Diamond Relat. Mater. 3, 1168 (1994).51D. K. Reinhard, T. A. Grotjohn, M. Becker, M. K. Yaran, T. Schuelke,
and J. Asmussen, J. Vac. Sci. Technol. B 22, 2811 (2004).52S.-F. Wang, Y.-F. Hsu, J.-C. Pu, J. C. Sung, and L. G. Hwa, Mater.
Chem. Phys. 85, 432 (2004).53M. A. Nitti, G. Cicala, R. Brescia, A. Romeo, J. B. Guion, G. Perna, and
V. Capozzi, Diamond Relat. Mater. 20, 221 (2011).
051101-14 Peter Hess J. Appl. Phys. 111, 051101 (2012)
54S. Chowdhury, E. de Barra, and M. T. Laugier, Diamond Relat. Mater.
13, 1625 (2004).55A. Kant, M. D. Drory, N. R. Moody, W. J. Moberlychan, J. W. Ager III,
and R. O. Ritchie, Mater. Res. Symp. Proc. 505, 611 (1998).56J. Hu, Y. K. Chou, R. G. Thompson, J. Burgess, and S. Street, Surf. Coat.
Technol. 202, 1113 (2007).57M. Wiora, K. Bruhne, A. Floter, P. Gluche, T. M. Willey, S. O.
Kucheyev, A. W. Van Buuren, A. V. Hamza, J. Biener, and H.-J. Fecht,
Diamond Relat. Mater. 18, 927 (2009).58R. Ikeda, H. Tanei, N. Nakamura, H. Ogi, M. Hirao, A. Sawabe, and M.
Takemoto, Diamond Relat. Mater. 15, 729 (2006).59H. Tanei, N. Nakamura, H. Ogi, and M. Hirao, Phys. Rev. Lett. 100,
016804 (2008).60H. Tanei, K. Tanigaki, K. Kusakabe, H. Ogi, N. Nakamura, and M. Hirao,
Appl. Phys. Lett. 94, 041914 (2009).61O. A. Williams, A. Kriele, J. Hees, M. Wolfer, W. Muller-Sebert, and
C. E. Nebel, Chem. Phys. Lett. 495, 84 (2010).62J. Philip, P. Hess, T. Feygelson, J. E. Butler, S. Chattopadhyay, K. H.
Chen, and L. C. Chen, J. Appl. Phys. 93, 2164 (2003).63S. O. Kucheyev, J. Biener, J. W. Tringe, Y. M. Wang, P. B. Mirkarimi, T.
van Buuren, S. L. Baker, and A. V. Hamza, Appl. Phys. Lett. 86, 221914
(2005).64A. Kriele, O. A. Williams, M. Wolfer, D. Brink, W. Muller-Sebert, and
C. E. Nebel, Appl. Phys. Lett. 95, 031905 (2009).65C. J. Tang, S. M. S. Pereira, A. J. S. Fernandes, A. J. Neves, J. Gracio,
I. K. Bdikin, M. R. Soares, L. S. Fu, L. P. Gu, A. L. Khokin, and M. C.
Carmo, J. Cryst. Growth 311, 2258 (2009).66D. M. Gruen, S. Liu, A. R. Krauss, J. Luo, and X. Pan, Appl. Phys. Lett.
64, 1502 (1994).67D. M. Gruen, S. Liu, A. R. Krauss, and X. Pan, J. Appl. Phys. 75, 1758
(1994).68Y. C. Lee, S. J. Lin, V. Buck, R. Kunze, H. Schmidt, C. Y. Lin, W. L.
Fang, and I. N. Lin, Diamond Relat. Mater. 17, 446 (2008).69S. A. Catledge and Y. K. Vohra, J. Appl. Phys. 84, 6469 (1998).70W. Kulisch, C. Popov, S. Boycheva, L. Buforn, G. Favaro, and N. Conte,
Diamond Relat. Mater. 13, 1997 (2004).71B. Peng, C. Li, N. Moldovan, H. D. Espinosa, X. Xiao, O. Auciello, and
J. A. Carlisle, J. Mater. Res. 22, 913 (2007).72N. N. Naguib, J. W. Elam, J. Birrell, J. Wang, D. S. Grierson, B. Kabius,
J. M. Hiller, A. V. Sumant, R. W. Carpick, O. Auciello, and J. A. Carlisle,
Chem. Phys. Lett. 430, 345 (2006).73J. S. Tse, J. Superhard Mater. 32, 177 (2010).74B. R. Lawn, J. Mater. Res. 19, 22 (2004).75E. van der Giessen and A. Needleman, Annu. Rev. Mater. Res. 32, 141
(2002).76M. Marder and J. Fineberg, Phys. Today 49, 24 (1996).77R. H. Telling, C. J. Pickard, M. C. Payne, and J. E. Field, Phys. Rev. Lett.
84, 5160 (2000).78D. Roundy and M. L. Cohen, Phys. Rev. B 64, 212103 (2001).79Y. Zhang, H. Sun, and C. Chen, Phys. Rev. Lett. 94, 145505 (2005).80Y. Umeno, Mater. Res. Soc. Symp. Proc. 1086, 1086-U07–03 (2008).81Q. Liang, C. Yan, Y. Meng, J. Lai, S. Krasnicki, H. Mao, and R. J. Hem-
ley, Diamond Relat. Mater. 18, 698 (2009).
82Q. Liang, C. Yan, Y. Meng, J. Lai, S. Krasnicki, H. Mao, and R. J. Hem-
ley, J. Phys.: Condens. Matter 21, 364215 (2009).83C. S. Yan, H. K. Mao, W. Li, J. Qian, Y. Zhao, and R. J. Hemley, Phys.
Status Solidi A 201, R25 (2004).84N. V. Novikov and S. N. Dub, Diamond Relat. Mater. 5, 1026 (1996).85M. D. Drory, R. H. Dauskardt, A. Kant, and R. O. Ritchie, J. Appl. Phys.
78, 3083 (1995).86R. S. Sussmann, J. R. Brandon, G. A. Scarsbrook, C. G. Sweeney, T. J.
Valentine, A. J. Whitehead, and C. J. H. Wort, Diamond Relat. Mater. 3,
303 (1994).87A. Kant, M. D. Drory, and R. O. Ritchie, Mater. Res. Soc. Symp. Proc.
383, 289 (1995).88Z. Xia, W. A. Curtin, and B. W. Sheldon, Acta Mater. 52, 3507 (2004).89F. X. Lu, Z. Jiang, W. Z. Tang, T. B. Huang, and J. M. Liu, Diamond
Relat. Mater. 10, 770 (2001).90A. R. Davies, J. E. Field, K. Takahashi, and K. Hada, Diamond Relat.
Mater. 14, 6 (2005).91G. F. Cardinale and C. J. Robinson, J. Mater. Res. 7, 1432 (1992).92D. S. Olson, G. J. Reynolds, G. F. Virshup, F. I. Friedlander, B. G. James,
and L. D. Partain, J. Appl. Phys. 78, 5177 (1995).93S. E. Coe and R. S. Sussmann, Diamond Relat. Mater. 9, 1726 (2000).94T. E. Steyer, K. T. Faber, and M. D. Drory, Appl. Phys. Lett. 66, 3105 (1995).95J. E. Field, in Properties of Diamond, edited by J. E. Field (Academic,
London, 1992), pp. 286–310.96R. Ikeda, H. Cho, M. Takamoto, and K. Ono, J. Acoust. Emission 22, 119
(2004).97R. Ikeda, M. Hayashi, A. Yonezu, H. Cho, T. Ogawa, and M. Takemoto,
Trans. Jpn. Soc. Mech. Eng., Ser. A 73, 57 (2007).98W. B. Yang, F. X. Lu, and Z. X. Cao, J. Appl. Phys. 91, 10068 (2002).99Y. Hayashi and T. Soga, Tribol. Int. 37, 965 (2004).
100M.-S. You, F. C.-N. Hong, Y.-R. Jeng, and S.-M. Huang, Diamond Relat.
Mater. 18, 155 (2009).101A. R. Kraus, O. Auciello, D. M. Gruen, A. Jayatissa, A. Sumant, J. Tucek,
D. C. Mancini, N. Moldovan, A. Erdemir, D. Ersoy, M. N. Gardos, H. G.
Busmann, E. M. Meyer, and M. Q. Ding, Diamond Relat. Mater. 10, 1952
(2001).102M. Karaskova, L. Zajıckova, V. Bursıkova, D. Franta, D. Necas, O. Bla-
hova, and J. �Sperka, Surf. Coat. Technol. 204, 1997 (2010).103H. Yoshikawa, C. Morel, and Y. Koga, Diamond Relat. Mater. 10, 1588
(2001).104H. D. Espinosa, B. Peng, N. Moldovan, T. A. Friedmann, X. Xiao, D. C.
Mancini, O. Auciello, J. Carlisle, C. A. Zorman, and M. Merhegany,
Appl. Phys. Lett. 89, 073111 (2006).105O. Auciello, J. Birrell, J. A. Carlisle, J. E. Gerbi, X. Xiao, B. Peng, and H.
D. Espinosa, J. Phys.: Condens. Matter 16, R539 (2004).106F. J. Hernandez Guillen, K. Janischowsky, W. Ebert, and E. Kohn, Phys.
Status Solidi A 201, 2553 (2004).107F. J. Hernandez Guillen, K. Janischowsky, J. Kusterer, W. Ebert, and E.
Kohn, Diamond Relat. Mater. 14, 411 (2005).108O. Auciello and A. V. Sumant, Diamond Relat. Mater. 19, 699 (2010).109V. Brazhkin, N. Dubrovinskaia, M. Nicol, N. Novikov, R. Riedel, V.
Solozhenko, and Y. Zhao, Nature Mater. 3, 576 (2004).110P. Hess, Diamond Relat. Mater. 18, 186 (2009).
051101-15 Peter Hess J. Appl. Phys. 111, 051101 (2012)