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:

peter.hess@urz.uni-heidelberg.de.

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.

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