Origin of characteristic subgap optical absorption in CVD diamond films

Article abstract of DOI:10.1103/PhysRevB.54.5552

Photothermal deflection spectroscopy (PDS) is used to study subgap optical absorption in chemical vapor deposited (CVD) diamond films. The high PDS sensitivity over a broad investigated spectral range enabled the investigation of the main defect structure in CVD diamond upon changing the deposition conditions. All films exhibit a very characteristic shape of the subgap optical absorption in the infrared and visible spectral regions, which scales with the content of amorphous carbon phase in grain boundaries. In best-quality films the character of the fundamental absorption edge is attained and the subgap absorption reduced. For nanocrystalline approaching films, an absorption spectrum resembling closely the one for amorphous carbon (a-C:H) films is observed. Based on these results a mathematical model for the description of the optical absorption coefficient in the subgap region is developed. A numerical deconvolution procedure is applied to fit the experimental data.

Full text of DOI:10.1103/PhysRevB.54.5552

PHYSICAL REVIEW B
VOLUME 54, NUMBER 8
15 AUGUST 1996-II
Origin of characteristic subgap optical absorption in CVD diamond films
´
M. Nesladek, K. Meykens, and L. M. Stals
Materials Physics Division, Institute for Materials Research, Limburgs Universitair Centrum,
Wetenschapspark, B-3590 Diepenbeek, Belgium
ˇˇ
M. Vanecek and J. Rosa
Institute of Physics, Academy of Sciences of the Czech Republic, Cukrovarnicka 10, CZ-162 00 Prague, Czech Republic
͑Received 25 September 1995; revised manuscript received 16 April 1996͒
Photothermal deflection spectroscopy ͑PDS͒ is used to study subgap optical absorption in chemical vapor
deposited ͑CVD͒ diamond films. The high PDS sensitivity over a broad investigated spectral range enabled the
investigation of the main defect structure in CVD diamond upon changing the deposition conditions. All films
exhibit a very characteristic shape of the subgap optical absorption in the infrared and visible spectral regions,
which scales with the content of amorphous carbon phase in grain boundaries. In best-quality films the
character of the fundamental absorption edge is attained and the subgap absorption reduced. For nanocrystal-
line approaching films, an absorption spectrum resembling closely the one for amorphous carbon (a-C:H͒ films
is observed. Based on these results a mathematical model for the description of the optical absorption coeffi-
cient in the subgap region is developed. A numerical deconvolution procedure is applied to fit the experimental
data. ͓S0163-1829͑96͒08331-2͔
I. INTRODUCTION
play an essential role in amorphous carbon (a-C:H͒ or
higher-ordered tetrahedrally bonded amorphous carbon
(ta-C͒ films. It is widely accepted that in a-C:H ␲-bonded
Diamond as a covalently bonded wide-band-gap semicon-
ductor has been widely studied, but only recent develop-
ments in the chemical vapor deposited ͑CVD͒ technology
opened up new possibilities for the preparation of thin-film
devices and for new extensive research activities in this
field.1–3 Recently, it has been shown that in some homoepi-
taxially grown CVD diamond films the charge carrier mo-
bilities start to approach values found for natural
diamond.3–5 Nevertheless, tested devices perform rather
modestly4 in comparison with the existing ones, based on Si,
Ge or III-V, II-VI materials. One of the key issues in the
preparation of high-quality CVD diamond films is the de-
fects and a deeper understanding of their localized defect
states in the band gap is needed for further progress to be
made in the CVD diamond-electronics field.
*
carbon atoms cause the formation of a ␲-␲ pseudogap,
*
emerging about midgap ͑in between the ␴-␴ bands͒ and
dominating the optical absorption.15–17 Similarly to a-C:H
and ta-C, the D ͑disorder͒ and G ͑graphite͒ Raman scatter-
ing bands are present18,19 in lower-quality diamond films.
Optical absorption methods have proven to be very pow-
erful for studying the electronic structure in tetrahedrally
bonded semiconductors.20–22 When looking for defect-
induced subgap absorption in the ppm ͑parts per million͒
range and lower, as is required for an electronic-quality ma-
terial, very sensitive techniques must be used. In commercial
spectrophotometers absorbance values down only to the
10Ϫ2 level can be measured. Moreover, light scattering in
CVD diamond films ͑due to bulk inclusions and to grain
boundaries in polycrystalline films͒ make standard transmis-
sion measurements difficult.23
Defects, classified generally as N, GR, R, TR, H,6 have
been thoroughly investigated in the past in natural and high-
pressure high-temperature ͑HPHT͒ synthetic diamond. Many
of these defects are related to the presence of nitrogen, which
is the main extrinsic impurity in diamond. The defects and
their electronic models are discussed in detail in the
literature.6–10 In this paper we concentrate on defects in CVD
films. Recently, we showed that CVD films exhibit a char-
acteristic subgap continuum absorption,11 not observed for
natural or HPHT synthetic diamond.
Specific of the low-pressure CVD diamond is the prepa-
ration in the pressure-temperature region, in which graphite
is thermodynamically stable over diamond. This fact may
account for the observed differences. One possibility to con-
sider is the presence of ␲-bonded carbon. Thermodynamic
growth models,12–14 based on the preferential etching of the
nondiamond phase and/or diamond surface stabilization by
atomic hydrogen, always include a certain probability for the
incorporation of sp2 carbon in the diamond films. sp2 sites
To study the defect-induced optical absorption in CVD
diamond we recently applied photothermal deflection spec-
troscopy ͑PDS͒.11,24–26 This method made it possible to mea-
sure absorbances down to a 10Ϫ5 level, and thus to investi-
gate deep defects on a ppm scale and below, even in 10-
␮m-thick diamond films.11 The main aim of this paper is to
report in detail on the characteristic subgap continuum ab-
sorption in CVD diamond films deposited under various con-
ditions and to compare their absorption with the known ab-
sorption spectra of type IIa ͑natural͒ and type Ib ͑HPHT͒
diamond.
In Sec. IV of this paper we develop a theoretical model
for the description of the defect induced optical absorption in
CVD diamond. We show that the optical absorption coeffi-
cient can be fitted to the experimental data when considering
optical transitions due to the presence of ␲-bonded carbon in
the grain boundaries.
0163-1829/96/54͑8͒/5552͑10͒/$10.00
54
5552
© 1996 The American Physical Society

54
ORIGIN OF CHARACTERISTIC SUBGAP OPTICAL . . .
5553
TABLE I. The deposition conditions for samples of sets I ͑constant temperature͒ and II ͑constant methane
concentration͒. Summarized are also the characteristics of samples and the ratio of intensities I(D)/I(G) of
D ͑disorder͒ and G ͑graphite͒ Raman peaks at full width at half maximum ͑FWHM͒.
Sample Pressure Temp. CH4 MW Depos. Appearance
power time
Morphology
Grain I(D)/I(G)
size at FWHM
(␮m͒ Raman peaks
͑Pa͒
͑K͒
͑%͒
͑W͒
͑min͒
d2.1
d2.2
d2.3
d2.4
d2.6
d2.7
d2.8
d2.9
D11
6600
6600
6600
6600
6600
6600
6600
6600
13 200
1173 0.65
800
800
800
800
800
800
800
800
4000
720
720
720
720
720
720
720
720
1440
White
Gray
Gray
Random
Random
Random
5
3
3
No G peaka
Ϸ 5
Ϸ 1
Ϸ0.2
Only G
Ϸ 0.2
Ϸ 0.5
1173
1173
1173
1173
1093
1033
1173
1033
1
1.5
2
2.5
3
3
3
3b
Brownish Nanocrystalline Ͻ1
Brownish Nanocrystalline Ͻ0.1
Gray
Gray
Dark brown Nanocrystalline Ͻ0.1
White Random 15
Nanocrystalline Ͻ1
͕100͖ Ϸ2
Only G
No G peaka
a1332 cmϪ1 peak only.
bAlso 1 sccm ͑cubic centimeter per minute at standard temperature and pressure͒ O2.
II. EXPERIMENTAL DETAILS
A. Diamond samples
Bulk type IIa ͑natural͒ and Ib ͑HPHT͒ DeBeers monoc-
rystalline diamond plates 250 ␮m thick and also a ͕100͖-
oriented homoepitaxial diamond ͑sample D17͒ deposited on
IIA DeBeers plate were used for comparison.26
Diamond films were grown by means of plasma-enhanced
chemical vapor deposition in an ASTEX PDS-17 microwave
plasma reactor. Undoped Si wafers were used as substrate
material. Care was taken to maintain a low background pres-
sure in the vacuum vessel before starting the deposition and
about the cleanness of the reactor walls, to reduce the incor-
poration of extrinsic impurities in the deposited films. The
deposition processes consisted of two consecutive steps: ͑1͒
the bias-enhanced nucleation step27,28 for creating a suffi-
ciently high diamond nucleation density ͑1010 cmϪ2) on a
mirror polished Si substrate; ͑2͒ the growth step. The depo-
sition parameters of sets I ͑constant temperature͒ and II ͑con-
stant methane concentration͒ are summarized in Table I.
The main experimental interest was to find out how
␲-bonded carbon can be incorporated into CVD diamond
films. Because one possible site for amorphous carbon are
grain boundaries18,29–32 we attempted to change their area by
changing the film morphology. By keeping the nucleation
step constant and changing the growth-step deposition con-
ditions various film morphologies were prepared, ranging
from those containing well-defined crystals to those showing
a finely grained morphology. The characteristics of the de-
posited films are summarized in Table I. We attempted to
compare the optical absorption of these films with our best
film ͑D11͒, prepared at the optimized conditions.11 The
samples D11 and d2.1 only exhibited the Raman diamond
peak at 1332 cmϪ1. All other films exhibited a typical broad
feature at 1550 cmϪ1 in the Raman spectra (G peak͒ varying
in intensity ͑see Table I͒.
B. Optical characterization
Self-supporting CVD diamond films of dimensions typi-
cally 4ϫ4 mm2, 2ϫ2 mm2 type IIa, and Ib monocrystalline
diamond plates and homoepitaxial diamond films deposited
on 2ϫ2 mm2 type IIa diamond plates were used for the
optical absorption measurement by use of PDS.
The principle of PDS is the following: A high-intensity
light beam from a monochromator ͑pumping beam͒ is modu-
lated by a mechanical chopper and focused on the diamond
film, which is immersed in spectroscopically pure CCl4.
Photons absorbed in the film cause a temperature elevation in
the illuminated area. The thermal wave generated in the
CVD diamond film propagates into CCl4 and modulates the
index of refraction of the CCl4 liquid.33–35 A He-Ne laser
beam, directed parallel and as closely as possible to the
sample surface ͑probe beam͒ is periodically deflected, the
deflection ⌬␸ being proportional to the absorptance A of the
measured thin film and the pumping beam intensity. The
deflection ⌬␸ is detected by a silicon position sensitive four
quadrant diode33–35 as a small ac voltage signal ͑PDS signal
S), proportional to ⌬␸. The PDS signal S is measured by a
lock-in amplifier. By scanning the wavelength of the mono-
chromatic pump beam from the region of complete light ab-
sorption in the film ͑e.g., the saturation of PDS signal
SϭSsat is reached͒ to the region of low absorption, a full
PDS spectrum is obtained.
The growth rate was typically 1.5 ␮m/h, and films with a
thickness between 15 and 20 ␮m were deposited ͑with the
exemption of the film D11, which was prepared under dif-
ferent conditions and which was 100 ␮m thick͒. After the
deposition the diamond films had to be removed from the
substrates to allow the PDS measurements. Therefore, the Si
substrate was etched away in a HNO3/HF/H2O mixture and
traces of surface graphitic carbon were removed in a
H2SO4/CrO3 solution.
Full details of the experimental setup11,25,33–35 and the
evaluation procedure11,25,33–35 for the calculation of the opti-
cal absorption coefficient ␣ can be found elsewhere. Here,
only the details,11,25,36 important for the measurement of
CVD diamond thin films are given. It is important to note
that the effect of light scattering in the bulk or at the rough
surface of CVD diamond films is much less significant for
PDS than for conventional transmission measurements, be-

5554
NESLADEK, MEYKENS, STALS, VANECEK, AND ROSA
54
cause just the light scattered ͑and absorbed͒ in the direction
of the probe beam gives some contribution to the PDS signal.
Under typical experimental conditions ͑chopping fre-
quency 14 Hz, sample thickness 20–200 ␮m, spectral reso-
lution of the monochromator 8 nm͒ the PDS signal S, nor-
malized to a constant pumping beam intensity, can be written
in the incoherent limit34,35 as
SϭSsat 1Ϫexp ␣d͒ ,
͑1͒
͓
͑
͔
where Ssat is the saturated signal in the spectral region where
a full absorption occurs, d is the sample thickness, and ␣ is
the optical absorption coefficient. The reflectance is assumed
to be approximately spectrally independent in the investi-
gated spectral range.37 Some films ͑d2.6, d2.9 see Table I͒
showed a saturation of the PDS signal Ssat in the investigated
spectral range. For these films the optical absorption was
directly calculated using Eq. ͑1͒. For the samples of which
the saturation signal was not reached in the investigated
spectral range ͑d2.1, d2.4, d2.7, d2.8͒, the following proce-
dure was used: all parameters of the experimental setup were
kept the same, hence, the proportionality coefficient34,35 Ssat
between the probe beam deflection ⌬␸ and the PDS signal
S in Eq. ͑1͒ remains the same. In the high-quality diamond
sample D11 ͑see Table I͒ the optical transmission and reflec-
tion were measured using a Perkin-Elmer Spectrophotom-
eter, equipped with a light-integrating sphere. These data
were used as a cross check for the proportionality coefficient
value in a higher photon energy (E) part of the spectrum. In
a low absorption region the optical absorption coefficient ␣
was determined from the PDS signal S, using the above de-
scribed procedure.
To reduce the experimental noise, a 1-mW He-Ne laser
with a high pointing stability and an optimized beam geom-
etry was used, focused as closely as possible to the smooth
side of the diamond film ͑the side of the Si/diamond inter-
face after the chemical etch͒. Care was taken to reduce vi-
brations of the experimental setup and to avoid scattering of
the laser beam optical path in air and in the transparent de-
flection liquid ͑CCl4) due to particulates. Additionally, a
long black tube was placed in front of the position sensor to
stop ambient scattered light modulated at the chopper fre-
quency and to define a narrow field of view, just letting pass
the probing He-Ne laser beam.The whole apparatus was
computer controlled and typically 300 readings of the lock-in
amplifier were averaged at each photon energy.
FIG. 1. The optical absorption coefficient ␣ for 100-␮m-thick
polycrystalline sample D11, for 373-␮m-thick ͕100͖-oriented
sample D17, and for type IIa ͑natural͒ diamond and type Ib ͑high-
pressure high-temperature synthetic͒ diamond plates, calculated
from the optical transmission measurements. The inset shows the
indirect optical transitions at the fundamental absorption edge of
diamond.
ingly, reduction of the substrate temperature again led to the
deposition of films with defined grain sizes, and approaching
the ͕100͖ morphology.
Twinning38 must be at least partially responsible for the
development of the observed finely grained film morphology
͑samples d2.4, d2.6, d2.7, d2.9͒. If only a drift from the
diamond deposition domain39 causes the development of the
finely grained morphology, a reduction of the deposition
temperature would not restore regular crystal shapes ͑sample
d2.8͒. Similar conclusions about the influence of the twin-
ning on successively smaller areas were recently drawn by
Tamor and Everson.40
Stacking or multiple stacking faults are very often found
in diamond films.38,40 This is not surprising because the en-
ergy difference for the correct and incorrect hexagonal ring
stacking ͑staggered to eclipsed͒ is very small. On the atomic
scale, twinning means in-plane incorporation of the boat type
six-membered rings instead of an incorporation of the correct
chair ring type. When the film grows these two cases differ
only in the next-nearest-neighbor environment.38 In case of a
very small misorientation between crystallites a bond-angle
disorder can be induced to bond adjacent crystals. It appears
likely that for crystallites with higher misorientations, amor-
phous carbon regions on grain boundaries can be produced to
reduce the total energy of the system. The amorphous carbon
on grain boundaries was experimentally confirmed by vari-
ous techniques such as Raman spectroscopy and electron-
energy-loss spectroscopy.30–32
III. RESULTS AND DISCUSSION
A. Twinning and development of texture
As the main objective was to trace the development of
characteristic defects in CVD diamond while changing the
deposition conditions, there was particular interest in inves-
tigating films containing various amounts of sp2 carbon ͑see
Table I͒.
The surface morphology of investigated films is shown in
our previous work.25 To summarize, at low methane concen-
trations ͑samples d2.1, d2.2, d2.3͒, films were white and/or
transluctant ͑light scattering͒ and nontextured. As the meth-
ane concentration increased the films became very finely
grained and they were brown and optically smooth. Surpris-
B. Optical measurements
1. Transmission measurements
In Fig. 1 the optical absorption coefficient ␣, calculated

54
ORIGIN OF CHARACTERISTIC SUBGAP OPTICAL . . .
5555
FIG. 3. The optical transmission spectra ͑without correction for
the elastic light scattering͒ of samples prepared at a methane con-
centration of 3% and at various substrate temperatures ͑set II͒.
FIG. 2. The optical transmission spectra ͑without correction for
the elastic light scattering͒ of samples prepared at various methane
concentrations and at a constant substrate temperature of 900 C°
͑set I͒. Transmission of IIa diamond plate is shown for comparison.
above the detection limit of the PDS apparatus used. The
polycrystalline diamond shows a distinct optical absorption
in the subgap region.
The optical absorption of the samples from the sets I and
II is shown in Fig. 5. It can be seen that all these samples
also show the characteristic spectral dependence of the opti-
cal absorption coefficient ␣.25 The absolute value of the op-
tical absorption coefficient increases with the nondiamond
carbon content ͑the closer to nanocrystalline films the higher
from the optical transmission measurements, for the poly-
crystalline sample D11, homoepitaxial layer D17, and type
IIa and type Ib diamond plates is plotted. IIa diamond shows
a typical indirect absorption edge at 5.45 eV. This absorption
can be described by the ␣Ϸ(EϪEgϮEph)2 dependence for
parabolic bands22 ͑see the inset of Fig. 1͒. Similarly, it was
found by Sussmann and et al.41 that the UV absorption spec-
tra for polished high-quality CVD films approach those for
IIa diamond. The homoepitaxially grown diamond ͑D17͒ and
the best polycrystalline material ͑D11͒ attain the character of
the fundamental absorption edge, but a broadening, espe-
cially for polycrystalline material, can be clearly observed in
Fig. 1. It should be noted that, because the absorption coef-
ficient falls off sharply in this region, light scattering at grain
boundaries cannot cause the broadening, as proved by mod-
eling.
The absorption in the high-quality CVD diamond film
D11 attains the character of the fundamental absorption
edge. This statement is not true for all CVD films. Figures 2
and 3 show the optical transmission data between two limit-
ing cases: first ͑samples d2.1, d2.2͒, following the optical
transmission of IIa diamond ͑measured without the integra-
tion sphere͒, but with much lower detected transmission be-
cause of the scattering losses and, second, a slowly decaying
transmission from the energy corresponding to the expected
band edge for diamond. Similar tailing of the optical absorp-
tion coefficient into the band gap was also observed for
a-C:H films.42,43
2. PDS measurements
To study the weak subgap absorption the PDS measure-
ments were performed, shown in Figs. 4 and 5. In Fig. 4 the
optical absorption coefficient ␣ of our best polycrystalline
material ͑D11͒ is plotted and compared with the optical ab-
sorption coefficient for IIa diamond.25 The absorption in IIa
diamond ͑investigated only in the 1–3-eV region͒ is just
FIG. 4. The spectral dependence of the optical absorption coef-
ficient ␣ obtained from PDS measurements for the film D11 and for
the bulk IIa diamond sample ͑investigated only in the range 1–3
*
*
eV͒. Plotted are also numerical fits to Eq. ͑2͒ (␲-␲ and ␲-␴
transitions͒ and to Eq. ͑8͒ ͑indirect transitions and exponential
broadening͒, discussed in Sec. IV.

5556
NESLADEK, MEYKENS, STALS, VANECEK, AND ROSA
3. Light scattering
54
In this section, we would like to show that the typical
shape of the optical absorption in the films, measured in this
work, cannot be due to elastic light scattering. The spectral
dependence of the optical scattering coefficient should then
indeed be dramatically changed in going from large grain
sized films to optically smooth nanocrystalline films ͑e.g.,
from Mie-type scattering on large defects as grain boundaries
to Raleigh-type scattering due to defects on a microscopical
scale͒. These two scattering types have quite different spec-
tral dependencies.45
Further, it follows from its operation principle that PDS is
much less sensitive to optical scattering than transmission
measurements33 because the contribution of scattered light to
the PDS signal comes only from the scattered pump-beam
light in the direction along the He-Ne probe beam. If the
sample is strongly light scattering and weakly absorbing in
the infrared and visible parts of the spectrum ͑as the high-
quality sample D11 in Fig. 4͒ and if we use the usual way of
setting the PDS signal to the absolute scale with the help of
the saturated PDS signal value33–35 then the absorption coef-
ficient in a low absorption region can be overestimated. This
has been observed for amorphous silicon thin films deposited
on rough glass substrates. The reason is that a relative con-
tribution of the scattered light is higher in the low absorption
region but negligible in the saturation region ͑strongly ab-
sorbed light does not penetrate through the film towards the
light-scattering surface͒. For large-grain sized films we partly
overcame this problem by setting the PDS spectrum on the
absolute scale with the help of the transmittance-reflectance
measurement using the light-integration sphere. For optically
smooth homoepitaxial and finely grained films ͑d2.9, etc.͒
the effect is negligible.
To summarize, the main support that the typical spectral
dependence of subgap absorption is not substantially influ-
enced by the light scattering comes from the fact that all
investigated films, ranging from rough to very smooth, show
the similar spectral shape of the optical absorption coeffi-
cient ␣ in the 0.9–3-eV region. This absorption is shifted
just for various films on the absolute scale. We observe a
higher subgap absorption in the optically smooth, nonscatter-
ing films.
FIG. 5. The spectral dependence of the absorption coefficient
␣ obtained from PDS measurement for the films of set I ͑samples
differ in CH4 concentration in the H2 feed gas during deposition͒
and for the films of set II ͑samples are deposited at different sub-
strate temperatures͒. Plotted are also numerical fits to Eq. ͑2͒.
the absorption͒ and the saturation of the PDS signal S occurs
at lower photon energy, for the same sample thickness. For
the better quality samples ͑d2.1, d2.8͒ with the regular crys-
tal morphologies the subgap absorption is reduced.
Photocurrent measurements in polycrystalline CVD dia-
mond films44 showed a similar type of spectral dependence
as the optical absorption measurements presented in this
work. It has been suggested that the optical transitions in-
volved are due to the presence of an electronic level 1 eV
distant from the valence-band edge.
According to the development of the absorption spectra
when continuously changing the deposition conditions ͑and
correspondingly developing the finely grained morphology͒
and according to a close similarity of the measured absorp-
tion spectra with the absorption spectra of a-C:H films17,42,43
*
we propose that the ␲-␲ band-type transitions are respon-
*
sible for the observed absorption. The model of the ␲-␲
pseudogap was originally developed by Roberston and
IV. MODEL OF THE OPTICAL ABSORPTION
IN CVD DIAMOND
O’Reilly15 and Bredas and Street16 for amorphous carbon.
Recently, photoluminescence studies in polycrystalline CVD
diamond accompanied with micro-Raman measurements at-
tributed the observed broadband luminescence, centered
about 2 eV, to amorphous carbon in grain boundaries.30,31
Nitrogen related absorption in CVD diamond films was
investigated recently.25,26 A typical absorption spectrum due
to substitutional nitrogen in type Ib HPHT diamond is shown
in Fig. 1. Investigated films show a clearly different spectral
dependence of the optical absorption coefficient compared to
Ib diamond.
Based on the above presented experimental results it can
be concluded that there is enough evidence to suggest that
the main defect structure ͑dominating the defect-induced op-
tical absorption in polycrystalline diamond films͒ is associ-
ated with ␲-bonded carbon.
In-plane bonded six-membered hexagonal rings with
120° bonding angle between each C atom exist in pure
graphite. Their incorporation into the diamond structure with
109.5° tetragonally bonded C would lead to an enormous
distortion. A much more probable explanation was presented
in Sec. III. This is that amorphous carbon is located mainly
in grain boundaries and at other defects. It is interesting to
note that already some time ago ␲-bonded carbon clusters
were suggested as an explanation of the optical absorption
induced by radiation damage in IIa diamond.46
Secondary-ion-mass spectroscopy ͑SIMS͒ analysis25
shows typically 30 ppm N, 500 ppm H, and 100 ppm O in
the undoped films investigated in this study. The subgap ab-
sorption varies for the measured samples by 4 orders of mag-
nitude. So it is difficult to explain the observed spectra by
impurity absorption.

54
ORIGIN OF CHARACTERISTIC SUBGAP OPTICAL . . .
5557
of these optical transitions to include the fact that the value
of the optical matrix elements associated with each particular
optical transition can be different.
*
A. ␲ - ␲ transitions
The origin of the pseudogap in a-C:H and highly-ordered
materials, such as tetrahedrally bonded amorphous carbon
(ta-C͒, is still subject to discussion.49–54 The molecular-
dynamics study of Wang, Ho, and Chan51 suggested that
graphitic clusters are arranged in planar sheets, containing
mostly 6- and some 5- or 7-membered rings. It has been
proposed that in highly-ordered structures the clustering of
aromatic rings will not be energetically favorable. Recently
Stephan et al.50 suggested that with increasing atomic den-
sity the number of fused rings decreases. For dense films, the
existence of single rings, or pairs, or single defects is fa-
vored. This idea was supported by Drabold, Fedders, and
Strum,52 who introduced ␲-bonded defect pairs as the main
defect. Recently, it was shown54 in the framework of the
tight-binding Huckel Hamiltonian that a single, distorted aro-
matic ring, forming a boat or chair configuration and conse-
quently causing ␲-␴ mixing, can produce states lying close
to the midgap in ta-C.
FIG. 6. The schematic model of the density of states in the gap
*
of CVD diamond, discussed in Sec. IV. The ␲ and ␲ bands ap-
pear due to the presence of amorphous carbon regions. Shown are
*
also two parabolic bands ⌺,⌺ for diamond and the exponentially
*
broadened tails ␴, ␴ ͑starting at the energy Em). Optical transi-
tions are marked by arrows.
In our model, the localized ␲ band is approximated by a
statistical Gaussian distribution of the density of localized
states, centered at Ei (iϭ1,2) and with a half-width 2w.
Various models of optical absorption, such as those based
on theories of Tauc,20 Abe and Toyoazawa,47 and others
were developed to describe the optical absorption in materi-
als retaining only short-range order such as amorphous sili-
con (a-Si͒ and amorphous hydrogenated silicon (a-Si:H͒.
For amorphous carbon the situation is more complex because
of the mixed ␲-␴ bonding environment. Theoretical consid-
erations for possible optical transitions in a-C:H were dis-
cussed by Robertson,17 based on the Tauc model of a virtual
crystal. In Tauc’s model20 only the conservation of energy
and no conservation of momentum is assumed. The momen-
tum matrix elements T are then replaced by their average
value T0.22 Upon assuming the Velicky summation rule22 the
optical absorption coefficient ␣ can be written as
*
Both ␲ and ␲ bands are taken as symmetrical around the
midgap, as expected from recent molecular-dynamics
simulations.49 The ␲(iϭ1) and ␲ (iϭ2) band density of
*
states are
Ϫ1/2
gi͑iϭ1,2͒ ⑀͒ϭA 2␲w2͒
͑
͑
ϫexp Ϫ ⑀ϪE ͒2/2w2
.
͔
͑3͒
͓
͑
i
The convolution integral Eq. ͑2͒ can then be solved ana-
lytically, leading to the expression for the optical absorption
coefficient ␣
:
␲␲
*
Ϫ1
␣
E͒ϭA2 2E␲w2͒
exp Ϫ ⑀ϪE ͒2/2w2 exp Ϫ ⑀
͑
͑
͓
͑
͔
͓
͑
͵
␲␲
1
*
␣ E͒ϭCEϪ1 g ⑀͒f ⑀͒g ⑀ϩE͒ 1Ϫf ⑀ϩE͒ d⑀,
͑
͑
͑
͑
f
͓
͑
͔
͵
i
ϪE2͒2/2w2 d⑀
͑4͒
͔
͑2͒
and after integration (A and B1 are constants͒,
where gi and gf are the initial and final density of states,
respectively, f(⑀) and f(Eϩ⑀) are the Fermi-Dirac occupa-
tion functions for the initial and final states and the constant
C is proportional to the momentum matrix elements T20. The
approximate spectral independence of the matrix elements
was experimentally confirmed for a-Si:H by Jackson et al.48
Our model of the subgap optical transitions is based on
the currently accepted electronic picture of amorphous
carbon17 and on recent molecular-dynamics simulations,
E͒ϭA2
Ϫ1exp Ϫ E
ϪE͒2/4w2
͔
2Eͱ␲͒
ϭB1EϪ1exp Ϫ E
␣
͑
͑
͓
͑
␲␲
␲␲
*
*
ϪE͒2/4w2
.
͔
͑5͒
͓
͑
␲␲
*
Equation ͑5͒ has a clear interpretation. The convolution
integral reaches its maximum for a photon energy equal to
the energy distance between two symmetrical ␲-␲ density
of states distributions. The final joint density of states is
again a Gaussian distribution but with a half-width of 4w.
The optical absorption coefficient, calculated according to
*
*
showing the presence of ␲,␲ states in the computed elec-
tronic density of states.49,50 Our model is shown in Fig. 6.
Eq. ͑5͒, is plotted for various values of
w
and
*
E0 (ϭE1ϪE2) in Fig. 7. The similarity between the spectra
in Fig. 7 and the experimental data, presented in Fig. 5, and
the data measured by Bounough et al.43 and by Veerasamy
et al.55 for a-C:H suggests the validity of our model for the
The following transitions are considered: ͑1͒ ␲-␲ ͑Gauss-
ian bands͒ transitions in amorphous carbon regions. ͑2͒
␲-␴ transitions in amorphous carbon regions. ͑3͒ Indirect
transitions between the valence and conduction bands of dia-
*
*
mond.
␲-␲ transitions.
As shown in the following sections, an independent value
for the preintegration factor in Eq. ͑2͒ is considered for each
The fit of the experimental subgap absorption spectra ͑see
Sec. IV D and Figs. 4 and 5͒ using the model based on the

5558
NESLADEK, MEYKENS, STALS, VANECEK, AND ROSA
54
FIG. 8. The plot of the value of the optical absorption coeffi-
cient ␣ at 3.5 eV for films from Figs. 4 and 5 as a function of the
preintegration constant B1 from Eq. ͑5͒, obtained by numerical fit-
ting procedure.
Eq. ͑5͒ and supporting our model.
It should be noted that this approximation is different
from the approach used frequently in the literature for
a-C:H, where the Tauc plot, approximating the optical tran-
sitions between two parabolic bands in amorphous semicon-
ductors, is used to fit experimental data.
*
B. ␲ - ␴ transitions
It has been suggested that localized-localized state
*
␲-␲ transitions are allowed, if they occur with the initial
and final states localized on the same cluster.17 The ␲-␴
*
transitions would be possible if ␲-␴ bond mixing in amor-
phous carbon regions occurs. Bond-angle distortions due the
immersing the sp2 atoms in the sp3 environment were sug-
gested in recent molecular-dynamics work.49,50 Because, the
sp2 sites are neighboring the sp3 sites, a sufficient bonding
FIG. 7. Numerical calculation of the optical absorption coeffi-
*
cient ␣ according to Eq. ͑5͒ due to ␲-␲ transitions, described by
the parameters w,E ,B1. Simulations are shown for different val-
␲␲
*
*
angle distortion could enhance the ␲-␴ optical transitions.
ues (w); E
is set to 3.5 eV in these calculations ͑a͒, and simu-
␲␲
*
lations for various values of E
eV ͑b͒.
͑see also Fig. 6͒; w is set to 0.5
This allows one to use the same approximation as in
Sec. IV A for the calculation of the optical absorption coef-
ficient ␣ ͓Eq. ͑2͔͒ but with a different preintegration factor.
This is in the agreement with Ref. 49, where it has been
suggested that the optical transitions in tetrahedrally bonded
amorphous carbon can be calculated by the convolution of
the initial ͑occupied͒ and final ͑empty͒ states.
␲␲
*
Eq. ͑5͒ yields a very good agreement for all films, ranging
from nanocrystalline to well-faceted diamond films. Practi-
cally, the main difference between fitting parameters is the
value of constant B1 in Eq. ͑5͒, which is proportional to the
product of the optical matrix elements and the concentration
To include potential fluctuations due to distorted bonds
*
prefactor of ␲ and ␲ distributions of states ͓Eq. ͑3͔͒. The fit
we consider the ␴ band density g as an exponential, declin-
␴
yields typically 3.5 eV for energy distance between the two
ing below the energy EϭEm from the parabolic g⌺ band,
which is expected for diamond ͑see Fig. 6͒. The ␲-band
densities are taken as previously in Eq. ͑3͒. So we have for
⑀ϾEm ,
*
maxima of the ␲-␲ distributions E
for investigated
␲␲
*
CVD diamond films. The molecular-dynamics simulations49
suggested a value between 4 and 5 eV for E in tetrahe-
␲␲
*
drally bonded amorphous carbon. In Fig. 8 we plot the value
of optical absorption coefficient ␣ at 3.5 eV as a function of
the preintegration constant B1, obtained from Eq. ͑5͒ by us-
ing the least-square fitting procedure. It can be seen that the
␣(B1) dependence has a linear character as expected from
1/2
g
⑀͒ϭD ⑀ϪE
͒
͑6͒
͑
͑
⌺
*
⌺
*
and for ⑀ϽEm :

54
ORIGIN OF CHARACTERISTIC SUBGAP OPTICAL . . .
5559
responds to the fundamental absorption in type IIa diamond.
As seen in Fig. 1 tailing off of the optical absorption coeffi-
cient into the band gap is present. Therefore it is expected
that the optical absorption coefficient ␣ is the sum of the
indirect transitions between the valence and conduction para-
bolic bands, and the transitions from the localized states in
the broadened exponential tails to states above the band
edges:
␣ E͒ϭG EϪE ϮE ͒2ϩH
g
⑀͒g
⌺
⑀ϩE͒d⑀,
͑
*
͑
͑
͑
͵
g
ph
␴
͑8͒
where G and H are constants, Eg is the fundamental indirect
gap, g is the initial density of states, which is approximated
␴
by Eqs. ͑6͒ and ͑7͒, assuming EsϭEg , and g⌺ is the den-
*
sity of final states above the band edge. It is interesting to
note that the calculated absorption spectra according to Eq.
͑8͒ for film D11 in Fig. 4 resemble the UV-continuum ab-
sorption, present in IIa diamond.7 The origin of the con-
tinuum absorption has not yet been satisfactorily explained
in the literature. It has been suggested that it is connected
with the presence of amorphous carbon clusters.46
FIG. 9. Numerical calculation of the optical absorption coeffi-
*
cient ␣ due to ␲-␴ transitions in amorphous carbon regions, de-
scribed by Eqs. ͑6͒ and ͑7͒. Simulations are shown for various
*
energy distances of the ␲-⌺ density of states distributions, de-
scribed by the parameter E
in these calculations.
, the value for E
is set to 3.5 eV
␲⌺
␲␲
*
*
D. Deconvolution of experimental data
The least-squares fitting procedure was used to fit Eq. ͑2͒,
*
*
assuming both ␲-␲ and ␲-␴ transitions, to the experi-
mental values of the optical absorption coefficient, obtained
from the PDS measurements. The calculated optical absorp-
tion coefficient is plotted together with the experimental data
in Figs. 4 and 5. A very good agreement with the experimen-
tal data can be established. The parameters of the fit ͑the
preintegration constants B1 and B2, the energy distance
g ϭ G E ϪE
͒
1/2/exp E ϪE ͒/E1
͓͑
͑
͔
͖
͕
␴
m
⌺
m
⌺
*
*
ϫexp ⑀ϪE ͒/E0
,
͑7͒
͓͑
͔
⌺
*
where D and G are constants, E0 is the slope of the expo-
*
nential tail, and E⌺ is the bottom of the ⌺ diamond con-
*
duction band. The densities described by Eqs. ͑6͒ and ͑7͒ are
(
)
equal at EϭEm . This gives the preexponential factor in Eq.
͑7͒. In this case the convolution integral Eq. ͑2͒ can only be
solved numerically. The numerical solution of Eq. ͑2͒ for
various values of Em and E1 is shown in Fig. 9. And param-
eters of the numerical fit are shown in Table II.
*
E
between two maxima of the ␲
state distributions
and their half width at half maximum 2w, the energy dis-
tance E between the maximum of the ␲ state distribution
␲␲
*
␲⌺
*
*
and the bottom edge of the ⌺ band and the slope E1 of the
exponential broadening of the parabolic ⌺ band, starting at
the energy Em) are summarized in Table II. In Fig. 8, the
relation between the numerically obtained value for the con-
stant B1 and the measured value of the optical absorption
coefficient ␣ at 3.5 eV is plotted. According to these results,
by increasing the film quality, the absolute number of ␲-
C. Indirect transitions in diamond
These transitions are described by a ␣Ϸ(EϪEgϮEph)2
dependence,22 where Eph is the phonon energy and Eg is the
width of the forbidden gap, assuming little change of the
matrix elements around the band edges. Such transition cor-
*
␲
transitions is decreasing ͑e.g., lower defect induced op-
tical absorption͒ and the distance between the two Gaussian
distributions ͑pseudogap͒ is widening. This is in agreement
with data obtained for a-C:H films.
To summarize, the proposed theoretical model describes
the character of the characteristic continuum subgap optical
absorption in CVD diamond films. The least-square fitting is
in a very good agreement with experimental data of the op-
tical absorption spectra for all investigated samples, starting
from nanocrystalline films to high-quality CVD diamond.
TABLE II. Parameters of the least-squares fitting procedure to
Eq. ͑2͒ (w, E
, E
, Em , B1, B2, E1), described in the Sec.
␲-⌺
*
␲-␲
*
IV and Fig. 6, obtained by a numerical deconvolution of the experi-
mental absorption data.
Sample w ͑eV͒
E
E
Em
B1
B2
E1
␲-␲
␲-⌺
*
*
d2.1
d2.3
d2.4
d2.6
d2.7
d2.8
d2.9
D11
0.65
0.54
0.57
0.54
0.52
0.47
0.50
0.55
3.50
3.07
3.06
3.00
3.08
3.24
2.90
3.40
3.89
3.80
3.70
3.80
3.70
3.80
3.62
4.45
4.00
95
0.8
0.10
0.10
0.10
0.15
0.10
3.89 301 0.8
3.80 1017 0.3
3.88 1270 0.4
3.78 1038 0.4
3.89 444 0.2 0.1036
3.63 1440 0.2
4.55 2.4 1.6
V. CONCLUSIONS
In this paper, the subgap absorption in CVD diamond
films was investigated and a mathematical model for its de-
scription developed. The PDS allowed the detailed study of
the subgap absorption and the measurement of the optical
absorption coefficient in a very broad dynamic range from
0.20
0.1

5560
NESLADEK, MEYKENS, STALS, VANECEK, AND ROSA
54
␣ Ͻ10Ϫ2 cmϪ1 to ␣Ͼ103 cmϪ1. Based on these data we
followed up the development of the main defect structure in
CVD diamond from sub-ppm concentration upon changing
the deposition conditions.25
Experimental data, presented in Sec. IIIB showed that all
measured CVD films exhibited a very similar feature in the
subgap absorption, starting from an onset at about 1 eV. The
observed feature resembles the optical absorption spectra of
amorphous carbon (a-C:H͒ films, which are believed due to
trogen or by other extrinsic impurities.
A numerical deconvolution procedure was developed in
Sec. IV D to fit the experimental data, according to the theo-
retical model. The fit yielded a very good agreement with
experimental data and the fitted parameters are very close to
those expected from current knowledge of the electronic
structure of a-C:H.
ACKNOWLEDGMENTS
*
optical transitions in the ␲-␲ pseudogap.
This work was supported by a NATO International Sci-
entific Exchange Programmes Linkage Grant, Project No.
HTECH.LG 940890, and NFWO ͑Nationaal Fonds voor
Wetenschappelijk Onderzoek, Brussels͒ Research Pro-
gramme, Project No. G.0014.96 and by the Grant Agency of
the Czech Republic, Project No. 202/96/0446. The authors
would like to thank E. Sleeks and RUCA in Antwerp for
help with optical transmission measurements. The authors
appreciate the fruitful discussions with Dr. C. Quaeyhaegens,
Professor P. Nagels and Dr. S. Kadlec and provision of IIa
and Ib diamond plates and homoepitaxial diamond sample
by Dr. J. Schermer and Dr. G. Janssen from the University of
Nijmegen in The Netherlands.
The morphology study proved that the films with highest
subgap absorption were very finely grained with an increased
surface of grain boundaries. Raman spectroscopy detected an
increased amount of sp2 carbon in these films. These films
were optically smooth ͑low scattering losses͒ and optical
transmission measurements showed that the characteristic in-
direct absorption edge of the diamond structure was lost. On
the other hand the best-quality CVD diamond attained the
character of the fundamental absorption edge, although opti-
cal absorption in this region exhibited a broadening. In com-
parison with absorption data for type Ib ͑HPHT͒ diamond
and the SIMS analysis25 it has been shown that the charac-
teristic absorption cannot be explained by substitutional ni-
1 J.L. Davidson, in Synthetic Diamond, Emerging CVD Science and
Technology, edited by K.E. Spear and J.P. Dismukes ͑Wiley,
New York, 1994͒, p. 355.
Tauc ͑Plenum, New York, 1974͒.
21 M. Vanecek, J. Kocka, J. Stuchlik, Z. Kozisek, O. Stika, and A.
Triska, Solar Energy Mater. 8, 411 ͑1983͒.
2 N.R. Parikh, J.D. Hunn, E. McGucken, M.L. Swanson, C.W.
White, R.A. Rudder, D.M. Malta, J.B. Posthill, and R.J. Marku-
nas, Appl. Phys. Lett. 61, 3124 ͑1992͒.
22 G.D. Cody, in Semicondutors and Semimetals, edited by J.J.
Pankove ͑Academic, New York, 1984͒, Vol. 21B, p. 11.
23 Xiang Xi Bi, P.C. Eklud, J.G. Zhang, A.M. Rao, T.A. Perry, and
C.P. Beetz, Jr., J. Mater. Res. 5, 811 ͑1990͒.
3 B.A. Fox, M.L. Hartsell, D.M. Malta, H.A. Wynands, C.-T. Kao,
L.S. Plano,G.J. Tessmer, R.B. Henard, J.S.Holmes, A.J. Tess-
mer, and D.L. Dreifus, Diamond Relat. Mater. 4, 622 ͑1995͒.
4 S.A. Grot, C.W. Hatfield, G. Sh. Gildenblat, A.R. Badzian, and T.
Badzian, Appl. Phys. Lett. 58, 1542 ͑1991͒.
24 M. Vanecek, J. Rosa, M. Nesladek, and L. M. Stals, in Proceed-
ings of the Diamond ’95 Conference, Barcelona, Spain, 11-15
September 1995 ͓Diamond Relat. Mater.͔ ͑to be published͒.
25 M. Nesladek, M. Vanecek, and L.M. Stals, Phys. Status Solidi A
154, 283 ͑1996͒, special issue on diamond films.
26 K. Meykens, M. Nesladek, J.J. Schermer, G. Janssen, M.
Vanecek, J. Rosa, L.M. Stals, and C. Quaeyhaegens, in pro-
cee dings of the Diamond ’95 Conference ͑Ref. 24͒.
27 B.A. Fox, B.R. Stoner, D.M. Malta, P.J. Ellis, R.C. Glass, and
F.R. Sivazlian, Diamond Relat. Mater. 3, 382 ͑1994͒.
28 C. Wild, R. Kohl, N. Herres, W. Muller-Sebert, and P. Koidl,
Diamond Relat. Mater. 3, 373 ͑1994͒.
5 J.A. VonWindheim, V. Venkatesan, D.M. Malta, and K. Das, J.
Electron. Mater. 22, 391 ͑1993͒.
6 J. Walker, Rep. Prog. Phys. 42, 1605 ͑1979͒.
7 C.D. Clark, E.W.J. Mitchel, and B.J. Parsons, in Properties of
Diamond, edited by J. E. Field ͑Academic, New York, 1979͒, p.
22.
8 A. Mainwood, Phys. Rev. B 49, 7934 ͑1994͒.
9 P.R. Briddon and R. Jones, Physica B 185, 179 ͑1993͒.
10 G. Davies and A.T. Collins, Diamond Relat. Mater. 2, 80 ͑1993͒.
11 M. Nesladek, M. Vanecek, J. Rosa, C. Quaeyhaegens, and L.M.
Stals, Diamond Relat. Mater. 4, 697 ͑1995͒.
29 R.G. Buckley, T.D. Moustakas, Ling Ye, and J. Varon, J. Appl.
Phys. 66, 3595 ͑1989͒.
30 L. Bergmann, B.R. Stoner, K.F. Turner, J.T. Glass, and R.J. Ne-
manich, J. Appl. Phys. 73, 3951 ͑1993͒.
12 B.V. Deryagin and D.V. Fedoseev, Growth of Diamond and
Graphite from the Gas Phase ͑Nauka, Moscow, 1977͒.
13 K.E. Spear and M. Frenklach, in Synthetic Diamond, Emerging
CVD Science and Technology ͑Ref. 1͒, p. 243.
14 W. Piekarczyk, J. Cryst. Growth 119, 345 ͑1992͒.
15 J. Robertson and E.P. O’Reilly, Phys. Rev. B 35, 2946 ͑1987͒.
16 J. L. Bredas and G.B. Street, J. Phys. C 18, L651 ͑1985͒.
17 J. Robertson, Philos. Mag. B 66, 199 ͑1992͒.
18 R.E. Schroder, R.J. Nemanich, and J.T. Glass, Phys. Rev. B 41,
3738 ͑1990͒.
19 D.S. Knight and W.B. White, J. Mater. Res. 4, 385 ͑1989͒.
20 J. Tauc, in Amorphous and Liquid Semiconductors, edited by J.
31 L. Bergmann, M.T. McClure, J.T. Glass, and R.J. Nemanich, J.
Appl. Phys. 76, 3020 ͑1994͒.
32 P.J. Fallon and L.M. Brown, Diamond Relat. Mater. 2, 1004
͑1993͒.
33 W. Jackson, N.M. Amer, A.C. Boccara and D. Fournier, Appl.
Opt. 20, 1333 ͑1981͒.
34 N.M. Amer and W.B. Jackson, in Semiconductors and Semimet-
als, ͑Ref. 22͒, p. 83.
35 H. Curtins and M. Favre, in Amorphous Silicon and Related Ma-
terials, edited by H. Fritzsche ͑World Scientific, Singapore,
1989͒, p. 328.

54
ORIGIN OF CHARACTERISTIC SUBGAP OPTICAL . . .
5561
36 E. Gheeraert, A. Deneuville, E. Bustarret, and F. Fontaine, Dia-
mond Relat. Mater. 4, 684 ͑1995͒.
46 C.D. Clark, R.W. Ditchburn, and H.B. Dyer, Proc. R. Soc. Lon-
don A 237, 75 ͑1956͒.
37 H.R. Philipp and E.A. Taft, Phys. Rev. 136, A1445 ͑1964͒.
38 C. Agnus, M. Sunkara, S.R. Sahaida, and J.T. Glass, J. Mater.
Res. 7, 3001 ͑1992͒.
47 S. Abe and Y. Toyazawa, J. Phys. Soc. Jpn. 50, 2185 ͑1981͒.
48 W.B. Jackson, S.M. Kelso, C.C. Tsai, J.W. Allen, and S.-J. Oh,
Phys. Rev. B 31, 5187 ͑1985͒.
39 P.A. Bachmann, D. Leers, and H. Lydtin, Diamond Relat. Mater.
1, 1 ͑1991͒.
40 M.A. Tamor and M.P. Everson, J. Mater. Res. 9, 1839 ͑1994͒.
41 R.S. Sussmann, J.R. Brandon, G.A. Scarsbrook, C.G. Sweeney,
T.J. Valentine, A.J. Whithead, and C.J. Wort, Diamond Relat.
Mater. 3, 303 ͑1994͒.
49 N. A. Marks, D. R. McKenzie, B. A. Pailthorpe, M. Bernasconi,
and M. Parrinello, Phys. Rev. Lett. 76, 768 ͑1996͒.
50 U. Stephan, Th. Frauenheim, P. Blaudeck, and G. Jungnickel,
Phys. Rev. B 50, 1489 ͑1994͒.
51 C.Z. Wang, K.M. Ho, and C.T. Chan, Phys. Rev. Lett. 70, 611
͑1993͒.
52 D.A. Drabold, P.A. Fedders, and P. Stumm, Phys. Rev. B 49,
16 415 ͑1994͒.
53 Ch.H. Lee, W.R.L. Lambrecht, B. Segall, P. C. Kelires, T.
Frauenheim, and U. Stephan, Phys. Rev. B 49, 11 448 ͑1994͒.
54 J. Robertson, Diamond Relat. Mater. 4, 297 ͑1995͒.
55 V.S. Veerasamy, G.A.J. Armatunga, W.I. Milne, P. Hewitt, P.J.
Fallon, D.R. McKenzie, and C.A. Davis, Diamond Relat. Mater.
2, 782 ͑1993͒.
42 N. Savvides, J. Appl. Phys. 58, 518 ͑1985͒.
43 Y. Bounouh, M.L. Theye, A. Dehbi-Alaoui, A. Matthews, J. Cer-
nogora, and J.L. Fave, C. Colliex, A. Gheorgiu, and C. Sen-
emaud, Diamond Relat. Mater. 2, 259 ͑1993͒.
44 K. Okumura, J. Mort, and M. Machonkin, Philos. Mag. Lett. 65,
105 ͑1992͒.
45 H.C. Van De Hulst, Light Scattering by Small Particles ͑Wiley,
New York, 1957͒.