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Young's modulus, fracture strength, and Poisson's ratio of nanocrystalline diamond
films
Markus Mohr, Arnaud Caron, Petra Herbeck-Engel, Roland Bennewitz, Peter Gluche, Kai Brühne, and Hans-
Jörg Fecht
Citation: Journal of Applied Physics 116, 124308 (2014); doi: 10.1063/1.4896729
View online: http://dx.doi.org/10.1063/1.4896729
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/116/12?ver=pdfcov
Published by the AIP Publishing
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Young’s modulus, fracture strength, and Poisson’s ratio of nanocrystalline
diamond films
Markus Mohr,
1,a)
Arnaud Caron,
2
Petra Herbeck-Engel,
2
Roland Bennewitz,
2
Peter Gluche,
3
Kai Br€
uhne,
1
and Hans-J€
org Fecht
1
1
Institute of Micro and Nanomaterials, University of Ulm, Albert-Einstein-Allee 47, 89081 Ulm, Germany
2
INM Leibniz-Institute for New Materials, Campus D2.2, 66123 Saarbr€
ucken, Germany
3
GFD, Gesellschaft f€
ur Diamantprodukte mbH, Lise-Meitner-Straße 13, 89081 Ulm, Germany
(Received 5 August 2014; accepted 17 September 2014; published online 25 September 2014)
Young’s modulus, fracture stress, and Poisson’s ratio are important mechanical characteristics for
micromechanical devices. The Poisson’s ratio of a material is a good measure to elucidate its
mechanical behavior and generally is the negative ratio of transverse to axial strain. A
nanocrystalline (NCD) and an ultrananocrystalline (UNCD) diamond sample with grain boundaries
of different chemical and structural constitutions have been investigated by an ultrasonic resonance
method. For both samples, the elastic moduli are considerably reduced, compared with the elastic
modulus of single crystal diamond (sc-diamond). Depending on the chemical and structural
constitution of grain boundaries in nano- and ultrananocrystalline diamond different values for
Poisson’s ratio and for the fracture strength are observed. We found a Poisson’s ratio of
0.201 60.041 for the ultrananocrystalline sample and 0.034 60.017 for the nanocrystalline
sample. We discuss these results on the basis of a model for granular media. Higher disorder in the
grain boundary leads to lower shear stiffness between the single grains and ultimately results in a
decrease of Young’s and shear modulus and possibly of the fracture strength of the material.
V
C2014 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4896729]
I. INTRODUCTION
The Young’s modulus of metallic and non-metallic
materials, such as diamond, is known to decrease in general
with decreasing grain size.
1,2
However, the reported range of
values for the elastic moduli of diamond with grain sizes in
the range of 4–100 nm is quite substantial.
1
This large spread
is often attributed to the high variability of morphology and
structure of the investigated nanocrystalline and ultrananoc-
rystalline diamond films.
1
Simulations by Keblinski et al.
3
have shown that three- and four-coordinated atoms coexist in
the grain boundaries of diamond. In contrast, most of the
atoms in silicon grain boundaries are four-coordinated.
3
Monte Carlo simulations of the elastic properties of
ultrananocrystalline diamond
4
predict a decrease of
Poisson’s ratio for decreasing grain size, with a further
increase of Poisson’s ratio below a critical grain size of about
3 nm. These results are in agreement with simulation results
obtained by Fyta et al.,
5
where for an increasing amount of
sp
2
bonded carbon in the grain boundaries of ultrananocrys-
talline diamond a reduction of the Young’s and bulk modu-
lus was shown. However, simulations can only provide a
qualitative description of the elastic properties of nanocrys-
talline and ultrananocrystalline diamond, since the exact
atomic structure and chemical nature of the grain boundaries
are not known.
3–5
Poisson’s ratio is a fundamental measure to compare
materials properties and reflects the resistance of a material
against volume change per shape change.
6–8
The ratio of
shear modulus Gto bulk modulus Bdescribes the shear sta-
bility of a material
G
B¼3
2
12v
ðÞ
1þv:(1)
The G/B ratio for single crystalline diamond is 1.2011 (cal-
culated from values in Ref. 9) and therefore larger than for
other ceramics, glasses and semiconductors.
6
The Poisson’s ratio for single crystalline diamond is de-
pendent on the crystal direction and varies accordingly
between 0.00786 and 0.115,
1
having an average value
(assuming randomly oriented grains and neglecting grain
boundary effects), of 0.0691.
9
For polycrystalline diamond
with (110) fiber texture for Poisson’s ratio, again neglecting
effects from grain boundaries, an averaged value of 0.0730
for longitudinal orthogonal elongation and 0.0592 for trans-
versal orthogonal elongation has been reported.
9
For synthetically grown diamond films only a few meas-
urements of Poisson’s ratio can be found in the literature
using a range of different methods. Brillouin light scattering
measurements by Jiang et al.
10
on polycrystalline diamond
films with (110) fiber textured grains of sizes between 20
and 40 lm perpendicular to growth axis revealed an average
Poisson’s ratio of 0.0779, (calculated from the measured
elastic coefficients). Bruno et al.
11
measured Poisson’s ratio
of free-standing CVD diamond plates of 0.075 60.003 by
dynamic measurements and 0.083 60.025 by static measure-
ments. Adiga et al. found Poisson’s ratio of ultra-
nanocrystalline diamond to be 0.057 60.038.
12
In contrast,
Tanei et al.
13,14
reported a Poisson’s ratio of 0.07 for nano-
crystalline diamond grown without addition of nitrogen to
a)
Author to whom correspondence should be addressed. Electronic mail:
markus.mohr@uni-ulm.de
0021-8979/2014/116(12)/124308/9/$30.00 V
C2014 AIP Publishing LLC116, 124308-1
JOURNAL OF APPLIED PHYSICS 116, 124308 (2014)
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the gas phase of the hot filament CVD, while Poisson’s ratio
of nanocrystalline diamond was found to increase up to 0.18
after addition of 0.5% N
2
to the gas phase.
13,14
In a simple approach the Poisson’s ratio can be meas-
ured by the change of the width of a beam during elonga-
tion.
15
This approach is relatively inaccurate; especially in
the case where Poisson’s ratio is small, such as for diamond.
In this work, we applied a resonance method to measure the
Young’s and shear modulus of microfabricated nanocrystal-
line diamond beams using a transversal acoustical trans-
ducer, thus allowing to calculate Poisson’s ratio. We relate
the elastic and fracture behaviors of nanocrystalline diamond
to the structure of the non-diamond content within the grain
boundaries, which is investigated by means of Raman spec-
troscopy. We discuss our results on the basis of a simple
model for granular media.
16
II. MATERIAL
Nanocrystalline diamond films were grown on double
side polished Silicon wafers with a diameter of 3 inches and
an h100isurface by means of hot filament chemical vapor
deposition under two different process conditions. For both
films, the silicon wafers were pretreated (seeded) by immers-
ing the substrates into a solution containing nanodiamonds,
followed by ultrasonication, resulting in a seeding density in
the range of 10
11
cm
2
.
17
An ultrananocrystalline diamond
film (UNCD) was grown using a CH
4
/NH
3
/H
2
gas mixture.
The filament temperature was around 2000 C and the sub-
strate temperature was around 700 C.
Alternatively, a nanocrystalline diamond sample (NCD)
was grown using a CH
4
/Ar/H
2
gas mixture. The filament
temperature was around 1900 C and the substrate tempera-
ture was around 600 C.
Films of both types were grown with a thickness of typi-
cally 17 lm. Scanning electron microscopy (SEM) images of
the films surfaces are shown in Figures 1and 2.
In order to estimate the compressive residual stress in
the films, we produced Euler test structures:
18
beams with
different lengths (0.8 mm–2 mm), fixed on both sides. A
microscope image of these structures is shown in Figure 3.
The critical stress r
c
for which buckling occurs for a
beam of length Lcan be calculated as
rc¼p2Et2
31
L2;
where Eis the Young’s modulus of the beam material and t
the thickness of the beam. Using the later determined
Young’s moduli for both UNCD and NCD, the critical com-
pressive stress for the longest beam (L¼2 mm) was esti-
mated. As one can see in Figures 4(a) and 4(b), no buckling
was observable for any length of beam. Therefore, we con-
clude that the compressive stress is smaller than the critical
compressive stress for the longest beams, which is
120 MPa for the UNCD sample and 208 MPa for the
NCD sample.
FIG. 1. SEM image of the UNCD film surface.
FIG. 2. SEM image of the NCD film surface.
FIG. 3. Microscope picture of the free standing, double side anchored
beams.
FIG. 4. (a) SEM image under tilt, showing no buckling of the double side
clamped beams for the UNCD films. (b) SEM image under tilt, showing also
no buckling of the double side clamped beams of the NCD films.
124308-2 Mohr et al. J. Appl. Phys. 116, 124308 (2014)
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The average grain size of these films was determined
using x-ray diffraction in a standard Bragg-Brentano config-
uration and by application of Scherrer’s formula. The peak
broadening of the diamond (111) diffraction peak was used
to estimate the average grain size, neglecting peak broaden-
ing from micro strains. The XRD measurements are pre-
sented in Figure 5. The UNCD sample shows no apparent
texture (see Figure 6(a)) and the average grain size was
found hDi¼6 nm (see Figure 1). The NCD sample exhibits a
weak (110) fiber texture, confirmed by measurement of pole
figures for the (111) diffraction peak (see Figure 2(b)). In
this case, the average grain size hDiwas found to be 20 nm
(see Figure 5).
As-processed diamond films were used to build free
standing micro-beams attached to the silicon substrates.
Potassium hydroxide was used to remove the silicon sub-
strate in those areas, where later diamond micro-beams were
etched into the diamond film (using masking, standard pho-
tolithography, and an Ar/O
2
plasma). In Figure 7, the SEM
image of a processed sample is shown.
III. EXPERIMENTAL METHODS
A. Measurement of fracture strength
A nanoindenter was used as an instrument to bend
micro-beams with a length of 500 lm, a width of 200 lm,
and thickness of 17 lm. A special (wedge-shaped) stainless
steel tip was employed to apply a line load at the end of the
micro-beams (see Figure 8). The force Fand deflection d
were recorded simultaneously until fracture occurred. The
stress at fracture r
f
was then calculated according to
20
rf¼t
2
Ldmax
I
@F
@d;
where tis the thickness of the beam, d
max
the deflection of
the micro-beam at fracture, and Lis the beam length. The
moment of inertia Ifor rectangular beam cross sections of
width wand thickness tis given as
I¼wt3
12 :(2)
The aspect ratio (L/w) was only 2.5 and Euler-Bernoulli
beam theory is valid for aspect ratios larger than 10. The
Poisson’s ratio leads to a curvature of the beam also in trans-
verse direction (anticlastic curvature), when it is bend. This
curvature is suppressed by the beam fixture and stresses
build up that stiffens the beam close to the fixture. The stiff-
ening effect is also responsible for the increased bending
FIG. 5. X-ray diffraction peaks (111), for samples UNCD and NCD. An
additional small peak can be seen which is attributed to stacking faults.
19
FIG. 6. (a) Pole figure of the (111) peak of sample UNCD, showing no appa-
rent texture. (b) Pole figure of (111) peak of sample NCD, showing maxi-
mum intensities at tilt angle w¼35, indicating a weak monoaxial (110)
fiber texture.
FIG. 7. SEM image of free-standing micro-beams processed into the nano-
crystalline diamond film. The beams have the lengths 500 lm, 1000 lm, and
1500 lm. The shortest beam (500 lm in length) was broken in the fracture
strength experiment.
124308-3 Mohr et al. J. Appl. Phys. 116, 124308 (2014)
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resonance frequency for beams of small aspect ratio
21
and
the bending profile of cantilever beams with small aspect ra-
tio and surface stress, which is not following Stoney’s equa-
tion.
22
Obviously this effect is more pronounced for small
aspect ratios and large Poisson’s ratios.
23
In Ref. 24,
Meijaard presents a refinement of the classical beam theory
and an expression for the stiffness of a cantilever beam
clamped at one end and subjected to a general loading at the
other end for arbitrary aspect ratios. For an applied bending
force, the displacement is for a Poisson’s ratio of ¼0.3
(Ref. 24)
dcorr l
ðÞ¼F
EI
l3
30:0368l2wþ0:0144lw2þ0:0093w3
:
(3)
Comparing the bending stiffness kcorr ¼F
dcorr with the stiff-
ness as expected from simple beam theory k¼3EI
l3, the result
for the classical theory is only 4% smaller than after
Meijaards refined theory (using l¼500 lm, w¼200 lm).
Similar result is obtained using tabulated values for the cor-
rection factor in Ref. 25.
Since the Poisson’s ratios are expected to be smaller
than 0.3 (compare with our dynamic measurements), and
since the main contribution to the measurement uncertainty
results from measurement accuracy of the beam thickness
and the length (accuracy of positioning the nanoindenter tip),
we neglect the error made by application of the classical
beam theory.
The probability distribution of stress at fracture was
then fitted with a Weibull distribution function,
26
assuming a
weakest link model for failure. The probability of failure
p(r) for given stress ris given by
26
pr
ðÞ
¼1exp r
r0
m
!
;(4)
where the so called Weibull modulus mis a measure for the
scattering of the fracture stress. The stress scaling parameter
r
0
is the stress at which 63.2% of micro-beams failed.
Usually, r
0
is also referred to as the fracture strength.
1,27
Espinosa et al. showed in Refs. 28 and 29 that the Weibull
statistic can be applied to describe the fracture of (ultra-)
nanocrystalline diamond.
B. Static measurement of Young’s modulus
As a result of the manufacturing process, the beam cross
sections of the micro-beams were not perfectly rectangular
(the real cross sectional area deviated usually around 10%
from the perfect rectangular cross section). To minimize the
inaccuracies in measuring Young’s modulus, we precisely
determined the beam thickness tand the beams cross section
area Aby measuring the beam by a laser scanning micro-
scope (Keyence VK). The area moment of inertia Iwas cal-
culated from an effective width w
eff
¼A/t and the thickness,
following Eq. (2).
We statically bent two (1500 lm long) micro-beams
(not until fracture) at 10 different distances l(1320 lm–1500
lm) from the beam fixture and recorded the force Fand dis-
placement d. From the measured bending stiffness kof the
micro-beams, given as
k¼@F
@d¼3EI
l3;
we calculated the Young’s modulus of the micro-beams,
considering the elasticity of the support by replacing the
beam length lby corrected length lþl
0
.
30,31
The main con-
tribution to the uncertainty in the calculation of Young’s
modulus is the error in the determination of the area moment
of inertia.
Since the aspect ratio of the effective beam varied, with
the used length at which bending was done, from 6.6 to 7.5,
the classical beam theory
32
comes to its limits and the influ-
ence of the beam stiffening on the deflection has to be con-
sidered. We again compare the stiffness calculated by
classical theory with the one from Eq. (3) and for an aspect
ratio of 6.6 and an assumed Poisson’s ratio of 0.3 we only
find a deviation of 1.6%. Also, using tabulated values in Ref.
25, the error can be estimated to be below 3%, for the con-
servative assumption of a Poisson’s ratio of 0.3. Therefore,
we neglect the errors introduced by usage of classical theory.
C. Measurement of Poisson’s ratio
Bending and torsion oscillations of nanocrystalline and
ultrananocrystalline diamond micro beams (w¼200 lm,
L¼1500 lm) were excited using a transversal acoustical
transducer and were recorded by monitoring the dynamic lat-
eral and vertical beam deflection by means of the optical
deflection system of an AFM (Nanowizard II, JPK
Instruments, Germany) (see Figure 9). The resonance fre-
quencies of the nth bending and torsion modes were
extracted from amplitude spectra by fitting with a Lorentzian
function. The elasticity moduli Eand Gwere calculated
from the bending and torsion resonance frequencies of the
nth modes according to
24
fbending;n¼p2n1
ðÞ
2
2
1
2pL2ffiffiffiffiffiffiffiffi
t2E
12q
s;(5)
ftorsion;n¼2n1
2
1
Lt
wef f
ffiffiffiffi
G
q
s;(6)
FIG. 8. Schematic of the experimental setup for the fracture experiment.
124308-4 Mohr et al. J. Appl. Phys. 116, 124308 (2014)
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where f
bending,n
and f
torsion,n
are the resonance frequencies of
the nth bending and torsion modes, Lis the length, tis the
thickness, w
eff
is the effective width, qis the mass density,
and Eand Gare the Young’s modulus and the shear
modulus.
The length was measured by SEM and the mass density
was measured by the Archimedean method. The density of
sample UNCD was found to be 2.92 60.02 g/cm
3
and
3.27 60.02 g/cm
3
for sample NCD, respectively. The thick-
ness and effective width w
eff
were determined, using the pro-
cedure mentioned in Sec. III B.
In order to determine Young’s and shear modulus, the
resonance frequencies of bending and torsional resonances
were plotted against a function of their mode number nto be
able to do linear fitting (according to Eqs. (5) and (6)). In
Figures 10 and 11, representative bending and torsional fre-
quencies are plotted, together with a linear fit. From the
slope of the linear fit, together with the determined densities,
effective beam widths, lengths and thicknesses, and the
Young’s and shear modulus were calculated.
Assuming that the UNCD and NCD films are isotropic,
its corresponding Poisson’s ratio was calculated from E
and Gaccording to
¼E
2G1:
Note, that the value of Poisson’s ratio is independent of the
densities and its uncertainties.
The aspect ratio (L/w) of the used beams is 7.5 and since
the classical theory loses is accuracy for small aspect ratios,
we estimate the introduced error due to usage of the classical
model (Eqs. (5) and (6)). According to Looker et al.,
21
the
bending resonance frequencies have to be corrected by a fac-
tor C, dependent on Poisson’s ratio and aspect ratio
C¼fbending;corr
fbending
¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þZ
ðÞ
w=L
ðÞ
1þ12
ðÞ
Z
ðÞ
w=L
ðÞ
s;
where Zis a function depending only on Poisson’s ratio .
For an assumed Poisson’s ratio of ¼0.3 and the aspect ra-
tio of 7.5, the deviation of the bending resonance frequency
is calculated to be 0.89% and for a small Poisson’s ratio of
0.07, the error is only 0.05%. In the conservative estimation
for ¼0.3, the slope of the linear fit (see Figure 10) has to
be corrected by the same correction factor C¼1.0089. Since
the Poisson’s ratio is not known a priori, we added this
FIG. 9. Experimental set-up for the
investigation of the free resonant
behavior of nanocrystalline and ultra-
nanocrystalline diamond micro-beams.
FIG. 10. Bending resonance frequencies of sample NCD, beam 1. FIG. 11. Torsion resonance frequencies of sample NCD, beam 1.
124308-5 Mohr et al. J. Appl. Phys. 116, 124308 (2014)
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conservative correction for simplicity as an additional uncer-
tainty of the slope of the linear fit into the error calculations.
For higher bending modes, we find a reduced Young’s
modulus (max. 7% of relative deviation) which we attribute
to the finite stiffness of the cantilevers fixture.
IV. RESULTS AND DISCUSSION
By a simple geometric approximation, the relative
amount of atoms in the grain boundary N
gb
can be calcu-
lated. We assume spherical grains with grain size Dand one
atomic layer per grain which is associated with the grain
boundary. The number of atoms per volume n
atom
for dia-
mond is 1.764 10
29
m
3
.
33
With the number of atoms per
grain ngr ¼pD3
6natom and the number of atoms adjacent to
the grain ngb ¼pD2natom2=3, the relative amount of atoms
in the grain boundaries can be calculated by
Ngb ¼ngb
ngr
¼6
Dnatom1=3:
Using the measured grain sizes, the percentage of atoms
residing in the grain boundaries can be calculated to be 18%
for sample UNCD and 5% for sample NCD, which is how-
ever (due to the simple approximations) an underestimation.
In Figure 12, Raman spectra for both samples are
shown. The excitation wavelength was 532 nm. In the
Raman spectrum of the UNCD sample, we observe features
related to sp
2
bonded carbon: the G-band and the D-band at
1557 cm
1
and 1326 cm
1
.
34–36
The features at 1158 cm
1
and 1488 cm
1
were assigned to C-C and C-H vibration
modes.
34,36
In this spectrum, we do not observe the diamond
peak. Its absence is due to the smaller Raman cross-section
of diamond (about 200 times smaller) compared to sp
2
bonded carbon.
34,37
At the same time, the grain size of sample UNCD is
smaller (6 nm) than for sample NCD (20 nm) and therefore
the grain boundary volume is higher (around 18% of the
atoms reside in the grain boundaries), giving a stronger sig-
nal of sp
2
bonded carbons in the grain boundary.
Sample NCD exhibits the sharp diamond peak at
1334 cm
1
combined with a vibration at 1229 cm
1
caused
by small diamond clusters.
34
The further assignment of the
peaks follows the suggestions in Refs. 35 and 36. The peaks
at 1157 cm
1
and 1478 cm
1
are again C-C and C-H vibra-
tion modes.
34
A broad peak at 1566 cm
1
is visible, which is
the G-band. The diamond peak is enveloped in a broad band
at around 1329 cm
1
, which is the so called D-band.
To further analyze the different structures of the grain
boundaries of the UNCD and the NCD sample, the peaks
were fitted, as shown in Figure 6. We used Voigtian curves,
accounting for a Gaussian line broadening of the spectrome-
ter overlaid by an intrinsic Lorentzian line broadening. The
intensity ratio of the D-band to G-band for the UNCD sam-
ple is I(D)/I(G)¼1.22. For the NCD sample the D-band to
G-band ratio is I(D)/I(G)¼1.38. Together with the shift of
the G-band peak, we conclude, that following the argumenta-
tion in Ref. 35 the grain boundaries in the UNCD sample are
more disordered than in the NCD sample.
Figures 13 shows the Weibull plots of the UNCD and
NCD diamond film, respectively. For the UNCD samples we
found r
0
of 1.8 60.6 GPa and a Weibull modulus of
5.2 60.2, while for the NCD samples we found r
0
of
5.0 60.2 GPa and a Weibull modulus of 6.1 60.4.
Dynamic measurements of the Young’s and shear modu-
lus were performed on four micro-beams (1500 lm long) of
sample UNCD and NCD. For each sample, the beams were
chosen from a small area of the wafer to reduce variations
due to inhomogeneities. The static and dynamic measure-
ments of Young’s modulus are in good agreement. The
results of the static measurements are shown in Figure 8(a)
and are E¼501 649 GPa for sample UNCD and
E¼873 6164 GPa for sample NCD.
For the calculation of Poisson’s ratio, we assumed both
materials to be isotropic, since the fiber texture in the NCD
sample is monoaxial (the grains are randomly rotated around
the h110iaxis) and the expected difference between
Poisson’s ratio of an isotropic diamond film and a (110) tex-
tured film is small (neglecting effects of grain boundaries).
6
For the UNCD sample, the shear modulus was found to
be reduced by 64% and the Young’s modulus by 60% com-
pared to sc-diamond. Consequently, we find a Poisson’s ratio
of ¼0.201 60.041 for sample UNCD. The G/B ratio for
the UNCD sample is 0.77.
FIG. 12. Raman spectra of the ultrananocrystalline diamond sample
(UNCD) and the nanocrystalline diamond sample (NCD), together with the
fitted Voigtian curves. FIG. 13. Weibull plots for samples UNCD and NCD.
124308-6 Mohr et al. J. Appl. Phys. 116, 124308 (2014)
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The reduction of Young’s and shear modulus of the
NCD sample compared to sc-diamond is smaller than for the
UNCD sample. Sample NCD shows a decrease of Young’s
modulus of about 34% compared to sc-diamond, while the
shear modulus is reduced by 31% compared to sc-diamond.
This leads to a lower Poisson’s ratio of ¼0.034 60.017
and a higher G/B ratio of 1.35. The large error bars mainly
arise from measurements uncertainties for the geometry of
the beams.
In Table I, an overview over the measured values is
given. The measured values of Eand Gfor the eight micro-
beams are shown in Figure 14 and the resulting Poisson’s
ratios are illustrated in Figure 15.
A comparison of our measurement results with measure-
ments in literature are found in Figure 16. Poisson’s ratio of
sample NCD is smaller than the measured values for poly-
crystalline samples
10,11
and even smaller than measurements
for other ultrananocrystalline diamond.
12
This might be due
to systematic measurement errors and the limited amount of
beams measured in our investigation. In particular, the value
for sc-diamond lies within the measurement uncertainty for
two of the four measured beams.
The measured density and elastic moduli of sample
UNCD is much smaller than for UNCD films with similar
grain size, reported in literature.
12,38
We therefore expect the
grain boundaries of our UNCD films to possess a larger grain
boundary volume with disordered, weakly bonded atoms.
This is underpinned by the considerably reduced density of
the film, compared to sc-diamond. Also impurity atoms like
nitrogen and hydrogen are expected to reside in the grain
boundary and soften the material.
Therefore, the Young’s and shear moduli are even
considerably smaller than for hydrogen free tetrahedral
amorphous diamond-like carbon (ta-C), which shows
Young’s modulus of 759 622 GPa and Poisson’s ratio of
0.17 60.03.
39
Further characterization techniques like TEM, EELS, or
NEXAFS must be employed to completely understand the
(grain boundary) structure of our UNCD film.
However, our results seem to be in general agreement
with measurements of Tanei et al.,
13,14
who found an
increase of Poisson’s ratio for decreasing grain size and an
increasing sp
2
bonded carbon content.
The ratio of the number of atoms near a boundary to the
number of atoms inside a grain becomes larger when the av-
erage grain size of a material is decreased. The atoms in a
boundary usually form weaker bonds than within the grains.
Consequently, a decrease of the elastic moduli is usually
observed in nanocrystalline materials compared to the elastic
moduli of the same coarse-grained material.
4,40
The reduc-
tion of density in both of our materials, compared to
sc-diamond can be explained by the non-diamond content in
TABLE I. Measured properties of the investigated UNCD and NCD film,
the values for Young’s, shear modulus, and Poisson’s ratio are weighted
averages with their standard deviation.
UNCD NCD
Fracture stress (GPa) 1.8 60.6 5.0 60.2
Young’s modulus (GPa) 463 611 752 622
Shear modulus (GPa) 195 63 367 68
Poisson’s ratio 0.201 60.041 0.034 60.017
Density (g/cm
3
) 2.9260.02 3.27 60.02
FIG. 14. Measurements of Young’s and shear modulus for eight micro-
beams, also literature values for sc-diamond
6
are shown (dot-dashed line)
for comparison. (a) Bar graphs showing the measured Young’s moduli for
the eight micro-beams as determined by the resonance method. The error
bars were calculated using error propagation considering the measurement
uncertainties of density, beam length, width, and thickness. The two single
measurement points (sample UNCD, beam 3, and sample NCD, beam 4)
show the results of the static measurements for comparison. (b) Measured
shear moduli for the eight micro-beams, together with their error bars.
FIG. 15. The squared points show the derived values for Poisson’s ratio for
each micro-beam of samples UNCD and NCD. The error bars of Poisson’s
ratio values for each beam were derived using error propagation. The solid
lines show the weighted average value for each sample and the dashed lines
show the standard deviation from the weighted average. Also Poisson’s ratio
for sc-diamond
6
is shown for comparison (dot-dashed line).
124308-7 Mohr et al. J. Appl. Phys. 116, 124308 (2014)
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the grain boundaries (which has a lower density than
diamond).
Following the weakest link model, failure of a specimen
can occur as soon as a single defect in the stressed volume
triggers crack nucleation and growth that ends up in fracture
of the complete specimen. Considering the grain boundary
volume as a highly defective region, cracks are likely to initi-
ate preferentially within a grain boundary. Ovid’ko
41,42
has
suggested a theoretical model for the nucleation and growth
of nanocracks at triple junctions in nanocrystalline materials
due to grain boundary sliding. Simulations of Sha et al.
43
have shown that in ultrananocrystalline diamond, fracture is
initiated by slip of grains along grain boundaries, with subse-
quent crack initiation at a neighboring triple junction. In this
scheme, we might expect that a high resistance against grain
boundary sliding can increase the fracture resistance of nano-
crystalline diamond. We believe that the smaller shear stiff-
ness of sample UNCD, compared to sample NCD can favor
its smaller fracture stress.
We will now discuss the material characteristics, which
lead to the reduced elastic moduli of our investigated sam-
ples, compared to sc-diamond. Three properties of the grain
boundaries that might each influence the overall mechanical
properties must be noted, namely, their degree of disorder,
their hydrogen content, and their volume contribution to the
material.
The amount of hydrogen residing in the grain bounda-
ries, other impurities (e.g., nitrogen) in the grain boundaries,
the structure of the not sp
3
bonded carbons in the grain boun-
daries, as well as the grain boundary volume might influence
the mechanical behavior of the nano- and ultrananocrystal-
line diamond films. Molecular dynamic simulations by Mo
et al.
44
have shown that the shear resistance of the grain
boundary decreases for an increasing amount of hydrogen in
the grain boundaries. A smaller intensity of C-H vibration
modes, as seen in sample UNCD compared to sample NCD
is an effect of a smaller grain size.
45
The absence of well-
defined facets on the grain surfaces for smaller grain sizes
inhibits the observation of the C-H vibrations.
45–47
In
contrast, an increasing hydrogen concentration is usually
observed in diamond films for decreasing grain size due to
higher volume fraction of grain boundaries.
2,45,48,49
A
decrease in the elastic modulus has been reported for increas-
ing amount of hydrogen present in (ultra-) nanocrystalline
diamond films.
2,48
We thus expect that the more disordered
grain boundary (as we see from the measured Raman spec-
tra), together with a presumably higher hydrogen content, as
well as the higher grain boundary volume are responsible for
the stronger reduced elastic moduli of sample UNCD com-
pared to sample NCD.
A model for granular media
16,50
can be adopted to
describe the behavior of the nanocrystalline diamond films
investigated here. In a first order approximation, we assume
that the diamond grains are spherical. The interaction of two
neighboring spheres can then be described by a normal stiff-
ness constant k
n
and a tangential stiffness constant k
t
. For
simplification, we introduce the parameter k¼kt
kn. If a ran-
dom distribution of spheres is considered, it has been shown
in Ref. 16 that Poisson’s ratio relates to kas
v¼1k
4þk:(7)
Inserting the measured Poisson’s ratios in Eq. (7), we find
for the UNCD sample
kUNCD ¼0:16360:147
and for the NCD sample
kNCD ¼0:83660:081:
These results are consistent with the smaller shear modulus
of the UNCD sample compared to the NCD sample. Within
our description the higher grain boundary volume of the
UNCD sample compared to the NCD sample and the weaker
resistance of the grain boundary region to shearing might
contribute to a lower fracture stress.
V. CONCLUSION
Ultrananocrystalline and nanocrystalline diamond films
with grain boundaries of different chemical and structural
constitutions exhibit different Poisson’s ratios and different
fracture strengths. These differences can be rationalized on
the basis of a model for granular media. We conclude that
structurally more disordered grain boundaries (as in the
UNCD sample) lead to a lower shear stiffness between the
single grains, result in a smaller G/B ratio and might be a
reason for a smaller fracture stress.
ACKNOWLEDGMENTS
A.C., P.H.-E., and R.B. would like to thank Professor E.
Arzt for his support.
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