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Internal stresses in CVD diamond layers
Diamond and Related Materials, 3 (1994) 757 764
757
Internal stresses in CVD diamond layers D. Schwarzbach, R. Haubner and B. Lux Institute.for Chemical Technology of Inorganic Materials, Technical University of Vienna, Getreidemarkt 9/161, A- 1060 Vienna (Austria)
Abstract Chemically vapour deposited diamond layers always contain internal stresses. Two types of layer stressescan be distinguished: tensile intrinsic stresses and compressivethermal stresses. Intrinsic stresses were measured in situ during diamond deposition onto silicon substrates using a bending-plate method in a hot-filament reactor. The intrinsic tensile stresses increased with the total gas pressure and the methane content in the reaction gas. The measured stress curves became steeper with decreasing filament temperature. They also depended strongly on the layer morphology and grain size. The intrinsic stresses within a layer increased with decreasing grain size and the transition from well facetedcrystallitesto spherolitic ballas-typecrystals having an extremelyfine-grainradial polycrystalline structure. The thermal stresses, determined after cooling to room temperature, increase with higher deposition temperature and show acceptable agreement with calculated values using a simple model.
1. Introduction In the last decade, considerable effort and expense have been invested in research into the field of diamond CVD coating synthesis [1, 2]. D i a m o n d has numerous interesting properties, such as the highest known thermal conductivity, high electrical resistivity, hardness and chemical inertness [3]. There are now a number of emerging technological applications for diamond layers, ranging for instance from coatings for cutting tools [-4, 5], to heats sinks for electronic devices [-63 and fleestanding membranes [-7]. Deposition of diamond films using CVD techniques is always linked with the formation of internal stresses, which influence the main properties of the layer, such as adhesion [-8] or flatness and suitability for the manufacture of items such as diamond membranes [.73. The internal stresses of a coating on a substrate are composed of two types [-9]. (1) Intrinsic stresses are tensile stresses which always occur during layer growth. The reasons for their formation are yet not well understood, (2) Thermal stresses are compressive stresses due to the difference between the thermal expansion coefficients of the layer and the substrate. They are generated after deposition during cooling of the coating and the substrate to room temperature, In the case of low-pressure diamond synthesis, internal stresses are generally determined using X-ray diffraction [ 1 0 - 1 2 ] or R a m a n scattering [,13-15] methods, However, these methods give only the sum of the intrinsic and thermal layer stresses since the measurements are taken at r o o m temperature after deposition,
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In the investigation reported here, the intrinsic stresses in hot-filament CVD diamond layers deposited under various conditions were measured in situ, during deposition, and the thermal stresses were measured during cooling from the deposition temperature to r o o m temperature as done earlier by Sch~ifer et al. in a microwave plasma reactor [-16]. The influences of the deposition conditions on both intrinsic and thermal stresses are discussed.
2. Experimental details 2.1. A p p a r a t u s
The layers were produced in a hot-filament CVD reactor [-173 consisting of a stainless steel tube, approximately 700 m m long and with 150 m m diameter. The reaction gas was blown through a nozzle between the two filaments (constant flow rate 500 standard cm 3 m i n t ) . Two parallel coils of 0.8 m m tantalum wire (6 m m inner diameter, 48 m m length, 4 windings per centimetre) are fixed between two pairs of molybdenum electrodes. The distance between the filaments was approximately 10 m m (Fig. 1). The filament temperature was determined with a two-colour IR pyrometer. The substrates were placed 15 m m below the filaments on a heatable molybdenum box. The substrate temperature was measured using an encased NiCr/Ni thermocouple placed in a hole in the centre of the sample holder, 1 m m below the substrate [-18]. The gas flow rate was set with two mass flow controllers; the gas pressure, measured with a piezo-resistive sensor, was regulated with a needle valve. The filaments were precar-
© 1994
ElsevierSequoia. All rights reserved SSDI 0925-9635(93)05110-X
758
D. Schwarzbach et al. / Internal stresses in CVD diamond
Filament supensions
i - - Gas inlet
Mirror
.......... 'J =,., sam,>,e
Front view
S/de view
Screen
.I!i
'
Fig. 2. Principle of the bending-plate method. The incident laser beam comes from the left, is split and directed toward the substrate surface. After reflection the beams are projected onto a screen.
Fig. 1. Experimental set-up of the dual-hot-filament CVD equipment. TABLE 1. Experimental conditions
Gas flow rate (standard cm3 min -1 ) Total gas pressure (mbar) Substratc surface temperature (°C) Filament temperature (°C) Deposition time (h) Filament precarburization (h)
Methane 0.74-2.20 500 20-100 730-900 2150-2400 10-25 2-3
~
0 '. . . . . . . . . . . . . . . . . .
~
. s
10
lS
Time [hI
2.2. Determination of internal stress The internal stresses were measured with the bendingplate method assuming that the stressed layer (thickness d) strains the substrate (thickness D) elastically until equilibrium is reached, that the layer is well bonded to the substrate, and that the layer is much thinner than the substrate. Since the thickness D of the substrate can be neglected compared with the bending radius R, the internal stresses tr can be calculated from eqn. (1) 1-20, 21]: EsubD2 (1) 6(1 - VSub)Rd
~
(b)
E .1.0
.
o
,
s
,
.
.
Io ~m. 011
,
is
(c) _. !
I
0 . 0 ~ - - - - - - - - -. -. -. .-.-.-. -. .- .-.-.~. . . . . . . t ~
t
i 0.5 1.0
0
=
4
6
Average layer thickness [microns]
is the Young's modulus of the substrate, and
VSub its Poisson ratio.
The curvature radius was determined by deflection of parallel laser beams on the substrate surface (Fig. 2). After reflection the beams were projected o n t o a s c r e e n , The spot distance is registered using a c a m e r a with a time interval shutter and determined with a ruler, Figure 3 shows the steps for obtaining a typical stress curve. The resolution was 0.10-0.15 GPa. The depth scale was calculated using an average growth rate obtained from the weight gain during deposition, the substrate area, the density of natural diamond and the total deposition time.
two
I- - il
t
o.ol ~"E~m~Im4~ff~7.-~ ~.o.s
Esu b
i
i_.
burized for 2-3 h before the deposition experiment [ 19]. The experimental conditions are summarized in Table 1.
where
(e) icoo.ng J
Deposition
20
Fig. 3. In situ determination of the intrinsic stresses with the bendingplate method. The solid triangle at the end of each curve corresponds to the superimposed thermal stresses after the sample was cooled to room temperature: (a) raw data (spot distance vs. time) as collected during measurement; (b) slope of the inverse curvature radius with time as calculated from the laser deflection; (c) calculated stress curve obtained with eqn.(1).
2.3. Substrates andpretreatment Mirror-polished (100) silicon wafers were used as substrates. They were pretreated by rubbing two pieces against each other with 0.25 lxm diamond paste in between for about 10 rain to enhance the nucleation
D. Schwarzbach et al. / Internal stresses in CVD diamond
density. Then they were cut into pieces of approximately 5 × 35 mm 2, rinsed and cleaned ultrasonically with methanol and acetone,
3. Results 3.1. Influence of the deposition parameters and microstructure on the intrinsic stresses 3.1.1. Methane contentinthegasphase At 2300 °C filament temperature, 780 °C substrate temperature and 50 mbar total gas pressure there was a tendency for the layer stresses to increase with increasing methane content (Fig. 4(a)), while thelayer showed cubic (100) and octahedral (111) crystal habits and the grain size decreased (Fig. 4(b)-4(d)). 3.1.2. Total gas pressure Figure 5(a) shows the influence of the total gas pressure on the internal stresses at 2300 °C filament temperature, 900 °C substrate temperature and 1.00% methane in the reaction gas. With increasing pressure, the grain size (octahedral crystal habits) decreased (Fig. 5(b)-(d)). At lower substrate and lower filament temperatures, an increase in the gas pressure gave rise to better faceted
759
layers (ballas at 20 mbar, fine-grained octahedra at 50 mbar). However, the stress curves are nearly the same for both morphologies. 3.1.3. Filament temperature Figure 6(a) shows the dependence of the layer stresses at 780 °C substrate temperature, 20mbar and 1.00% C H 4 o n the filament temperature. The higher the filament temperature, the flatter is the slope of the stress curve. The morphology changed from fine-grained ballas-type (smooth surface) (Tvil. . . . t =2050°C) to spherolitic shaped ballas (TF, . . . . t=2150 °C) and finally to octahedra at Tvi~. . . . t--2300 °C (Fig. 6(b) 6(d)). Nevertheless, there seems to be no significant difference in the stress behaviour between the ballas layer at 2150 °C and the well-faceted layer at 2300 °C. TABLE 2. Thermal stresses
Substrate temperature (°c)
Average thermal stresses (GPa)
9oo _+20
0.61 _+ 0.045 - 0 . 4 9 _+ 0.057 -0.41
780 _+ 25 730 _+ 10
(b)
(a) 0.01
............................................
(c)
1.o
%/.40% (d)
1.80% ........ 0
I ......... 1
I ......... 2
I ......... 3
I ......... 4
I ......... 5
I ......... 6
I,.. 7
Layer thickness [microns]
Fig. 4. Internal stress curves at various m e t h a n e contents (a) and corresponding scanning electron micrographs: (b) 0.74%, (c) 1.40%, (d) 1.80%. Tvll. . . . t = 2300 °C, Tsub = 780 °C, p = 50 mbar.
760
D. Schwarzbach et al. / Internal stresses in CVD diamond
3.2. Thermal stresses During cooling of the composite layer/substrate to room temperature, a compressive stress component developed. Table 2 shows thermal stresses measured at room temperature: the higher the deposition temperature was, the higher the measured thermal stresses were. Figure 7 shows the development of the thermal stresses during cooling. First, the tensile layer stresses increased, Below 650 °C a compressive stress component developed and at room temperature the final overall thermal stress component was compressive,
This phenomenon is however too close to the detection limits of the bending-plate method to be measured accurately. Stage II. After the isolated crystals began to touch each other and a closed layer formed, the stresses increased linearly with time as well as with the layer thickness. Owing to the high particle density induced by surface pretreatment, the crystallites began growing randomly oriented; only those with an orientation having a high and optimal growth rate formed the layer texture [23, 24].
4. Discussion
Four stages can be distinguished with respect to the behaviour shown by the stress curves (Fig. 8). Stage I. During the early period of isolated crystal growth, before the layer had formed, the stresses remained constant at zero level or seemed to increase slightly. This observed increase might be an artefact due to the formation of a surface carbide layer on the silicon wafer causing swelling of the outmost substrate surface and thus curvature. A similar effect was seen by Schfifer et al. [ 16] on molybdenum substrates. Another reason might be tantalum evaporating from the filament [22].
Stage III. Once the steady state was reached and the columnar texture had formed, crystals were no longer overgrown. The layer grew without the formation of any more stress. In the newly deposited diamond, the density of grain boundaries remained constant and no more stress formation occurred [9]. Most of our experiments had to be stopped before this stage was reached, because the laser spots were diffused by the rough surface of the diamond layer and were therefore no longer detectable. Stage IV. After turning off the gas flow and the heating, the temperature decreased to room temperature. The changes in stress curve were due to the development
(b)
(a) i""
0.0q a.
(c) /
0.5
! 1.0 100 m l ~ r
k'~ 1.5
.........
| ......... I
I ..... 2
illnl
Layer thickness
.........
3
(d)
I ........
4
[microns]
Fig. 5. Internal stress curves at various total gas pressures (a) and corresponding scanning electron micrographs: (b) 20 mbar, (c) 50 mbar, (c) 100 mbar. TFilament=2300 °C, Tsub=900 °C, 1.00% CH 4.
761
D. Schwarzbach et al. / Internal stresses in CVD diamond
(b)
(a)
0.0~
...................................
Q..
0.5
2o5o°c z15o~
1.0
......... 0
I ..... 1
,,,,I
.........
I .........
2
! .........
3
4
Layerthickness[microns]
Fig. 6. Internal stress curves at various filament temperatures (a) and corresponding scanning electron micrographs: (b) 2300 'C, (c) 2150 "C, (d) 2050 °C. Tsub= 780 °C, p = 50 mbar, 1.00% CH4.
6 ~ [.-
,,,
~
Silicon .~ ......
:. 1.. 1.., ._. l.. ....
~i
I,..-..-" .".':..': • ," -_]
o
,
(a)
t
!
o
i
]Ii
0o
-0.5~1.,,,~ j~ -~
° /
Rnal thermal stress component
~,
--1.o
~= 8oo~ ,
o
(b)
Depositiont tRoom emperaturetSmperst . . . . . . .u . .re ...................
,
200
~ ,
,
4o0
,
.,
,
soo
, coo
, looo
Substrale temperature [~]
Fig. 7. Correlation between the observed thermal stresses and the thermal expansion coefficients at different substrate deposition temperatures: (a) linear thermal expansion coefficients of silicon and diamond at different temperatures [-32, 33]; (b) course of the total layer stresses during cooling after deposition as a function of the substrate temperature; the minima in the stress curves correspond to the intersection of the two expansion coefficient curves,
Fig. 8. Schematic representation of a stress curve.
of thermal stress, mainly between the silicon substrate and the diamond layer. The depth scales in the stress curves are only approximations. The growth delay and the different growth rates of isolated crystallites before layer closure as well as layer growth after closure are sources of error.
Furthermore, it must be mentioned that, at deposition temperatures above 800 °C, the substrate curvature and hence the measured stresses may be caused not only by the layer stresses, but also by a superimposition from a plastic deformation of the silicon wafer [ 16]. However, this effect was found to be negligible in our case; the stress data were not corrected.
762
D. Schwarzbach et al. / Internal stresses in CVD diamond
4.1. Intrinsic stresses
The obtained results are in good agreement with those published in the literature. Sch/ifer et al. [16] found increasing tensile stresses with decreasing grain size. Windischmann et al. [25] found decreasing grain sizes with increasing methane content. However, they expected an increasing compressive stress component with increasing sp2-carbon and hydrogen incorporation in the layer. But their assumption was verified only on very thin layers (0.75 jam) where texture formation might not yet be accomplished, The diamond layer morphology is controlled by the gas activation and thus by the deposition conditions [26-28]. The results discussed above show clearly that the intrinsic stresses in diamond films are linked with the layer morphology and its grain size. Even though the processes leading to tensile stress formation are not
yet fully understood, grain boundaries are reported to be a major reason for stress formation during diamond deposition [9, 16]. Owing to the high melting point of diamond and the low mobility of carbon atoms in the diamond lattice, there is no possibility for internal crystal growth (recrystallization) after the diamond is deposited. Recent transmission electron microscopy studies revealed that the different diamond layer morphologies contain different kinds of structural misfits [29]: octahedral crystals contain domains with dislocations and domains with twins, while cubic crystals contain mainly dislocations and the highly twinned areas are near the grain boundaries. Thus stress formation should depend mainly on the changes in grain boundary density with increasing layer thickness due to the formation of columnar texture. The higher intrinsic stresses in ballas-type diamond
(b)
(c) (a)
oO°'~ o.o
,
-!"~
~
..............................
'. ~
(110
~, ~'
(d)
o.5 \
":'~'...mo\,, b,,llms
"' "~ '4,,
(lOO) illllll,ll,,,,
1
,,llhl,,,,,,,lllll,
2
Jlllh/,,,,,,
9
4
Layer thickness [microns]
(e)
l O~m
Fig. 9. Layermorphologyand intrinsic stresses:(a) stresscurves;scanning electronmicrographs of(b) coarse octahedral crystals,(c) fine octahedral crystals, (d) cubiccrystal faces,and (e) ballas-typediamond.
763
D. Schwarzbach et al. / Internal stresses in CVD diamond
layers are supposedly caused by their much higher density of grain and subgrain boundaries as well as structural misfits. Figure 9 illustrates the dependence of the morphology on the intrinsic stresses: when the morphology changed from coarse-grained and fine-grained (111)octahedra/ cuboctahedral habits via cubic (100) to ballas, the layer stress increased. 4.2. Thermal stresses
The observed thermal stresses were independent of the diamond layer morphology and thickness and nearly the same at similar deposition temperatures (Table 2). This is due to the fact that diamond and silicon have different thermal expansion properties and the differences in linear shrinkage cause additional substrate bending, In Fig. 7, the layer stresses during cooling are correlated with the thermal expansion coefficients of silicon and diamond; the observed minimum of the thermal stress curve at Y s u b ~ 6 5 0 °C corresponds to the intersection of the expansion coefficient curves. Above 650 ~;C tensile stresses were formed, and below 650 °C compressive thermal stresses were formed. The final thermal stress component (at room temperature) remained cornpressive. The measured thermal stresses increased with deposition temperature, which is consistent with results reported the literature (for example refs. 15 and 30). 4.3. Calculation of thermal stresses
While in the stress curves the thermal stress component was determined experimentally from substrate bending, in the following section the thermal stresses are calculated directly from material constants using the well known formula [9, 30]: Ed O'th --
1 --
~d
(~d -- ~Si)(TSub - - TR)
(2)
where Eo is Young's modulus of diamond (Ed= 1210 GPa), vd is Poisson's ratio of diamond ( V d = 0 . 1 ) , (Zd and 0~Sub are thermal expansion coefficients of diamond and silicon (Fig. 7(a)), r s u b is the substrate temperature during deposition, and T R is room temperature. The thermal stresses were calculated by determining an average difference A~i of the thermal expansion coefficients within a temperature interval ATi = ( T i - Ti 1) (100 °C), eqn. (3), multiplying them by ATi and summing from the deposition temperature r s u b to room temperature TR, eqn. (4), similar to the method described by Windischmann et al. [25]: Ao~i ~ (~d,l'i
1 - - 0~Si, Ti 1) -~- (~d. Ti - - 0~Si,Ti)
(3)
2 Ed 0~th-
~'S~b 2
1 -
i'd rR
(a/ATi)
Figure 10 shows the results of these calculations in comparison with those of Windischmann et al. [25] and Baglio et al. [31] as well as our experimental results and those of Schtifer et el. [16]. The values for the linear thermal expansion coefficients of silicon and diamond were taken from refs. 32 and 33 respectively (Fig. 7(a)). There is a maximum between 600 ~C and 700 °C which corresponds to the intersection of the silicon and diamond thermal expansion curves. The comparison with experimental values shows an acceptable agreement with the theory. A certain tendency for the experimentally determined thermal stresses to rise with decreasing temperature compared with the calculated values can be seen however. This can be explained by assuming a temperature gradient across the silicon substrate during deposition, causing an additional tensile stress component as shown by Bichler et al. [8] on SiA1ON substrates. However, our calculated values showed a strong discrepancy with those calculated and published by Windischmann et al. [25]. There is no clear explanation for this fact, especially since we used the same values for the thermal expansion coefficients and the same method for calculation of the thermal stresses. The experimental results support our calculation. Thermal stress values calculated by Baglio et al. [31] using Timoshenko's bimetallic strip formula also gave a very different curve from ours since it was based on different initial assumptions [34]. It seems that in the temperature interval where diamond deposition is usually carried out, the thermal stresses are similar using both models. Quantitatively, both models gave theoretical values of approximately
(4)
0.0
~"
'..
-0.2
.O. N ~ -0.4 -~ ~ ,'-
k-
•~•
• "" "-
• • •".ll "" "'"-..
-0.6
-0.8
"" . . . . . .
." "'" Ik " ["" 'g" ° .~.l"~
!
. . . . . . . . . . . . . . . . . . . . . .
0
200
400
1000 Deposition temperature [°C] 6oo
aoo
'
1200
Fig. 10. Calculation of thermal stresses: . . results of the present calculations; --- calculations from Baglio et al, [311; calculations by Windischman et al. [25]; • experimental values obtained in the present investigation (Table 2); ± experimental values from Sch/ifer
et al. [16].
764
D. Schwarzbach et al. / lnternal stresses in CVD diamond
0.5 GPa for compressive thermal s t r e s s e s 700-900 °C and agreed with the experiments. -
at
5. Conclusion
The deposition conditions influence the intrinsic stresses in CVD diamond layers via control of the gas activation and hence the layer morphology. Well faceted, octahedral and coarse-grained layers showed lower stresses than cubic crystallite layers or fine-grained spherolitic ballas-type layers. This supports theories
which correlate stress generation with grain boundary phenomena. The thermal stresses can be controlled by the deposition temperature and can be calculated with acceptable accuracy using simple models. Acknowledgments The authors thank Dr. Naoji Fujimori and the Sumitomo Electric Ltd. in Itami, Japan, for helpful discussions and financial support. The authors also acknowledge Dr. C.-P. Klages a n d D r . L. Sch~fer from the Fraunhofer Institut in Hamburg for helpful discussions and practical hints, in particular concerning the construction of our equipment. References 1 R. Davis (ed.), Diamond Films and Coatings, Noyes, Park Ridge, NJ, 1992. 2 Y. Tzeng, M. Yoshikawa, M. Murakawa and A. Feldman (eds.), Applications of Diamond and Related Materials, Elsevier, Amsterdam, 1991. 3 J. E. Field, Properties of Diamond, Academic Press, London, 1979. 4 B. Lux, R. Haubner and P. Renard, Diamond Relat. Mater., 1 (1991) 1035. 5 H. Schachner, B. Lux, K. G. Stjernberg, A. G. Thelin and H. Tippmann, European Patent 0166708A3, June 27, 1984. 6 R. C. Eden, in Y. Tzeng, M. Yoshikawa, M. Murakawa and A. Feldman (eds.), Applications of Diamond and Related Materials, Elsevier, Amsterdam, 1991, p. 259. 7 L. Sch~ifer, A. Bluhm, C.-P. Klages, B. L6chel, L. M. Buchmann and H.-L. Huber, Diamond Relat. Mater., 2 (1993) 1191.
8 R. Bichler, R. Haubner and B. Lux, High Temp. High Pressure, 21
(1989) 575.
9 M. F. Doerner and W. D. Nix, CRC Crit. Rev. Solid State Mater. Sci., 14 (1988) 225. 10 P. R. Chalker, A. M. Jones, C. Johnston and I. M. Buckley-Golder, Surf. Coat. Technol., 47 (1991) 365. 11 H. Guo and M. Alam, in Y. Tzeng, M. Yoshikawa, M. Murakawa and A. Feldman (eds.), Applications of Diamond and Related Materials, Elsevier, Amsterdam, 1991, p. 149. 12 N. S. VanDamme, D. C. Nagle and S. R. Winzer, Appl. Phys. Lett.,
58 (25) (1991) 2919. 13 E. Gheeraert, A. Deneuville, A. M. Bonnot and L. Abello, Diamond Relat. Mater., 1 (1992) 525. 14 C. Johnston, A. Crossley, P. R. Chalker, I. M. Buckley-Golder and K. Kobashi, Diamond Relat. Mater., 1 (1992) 450. 15 M. Yoshikawa, G. Katagiri, H. Ishida, A. Ishitani, M. Ono and K. Matsumura, Appl. Phys. Lett., 55 (25) (1989) 2608. 16 L. Sch/ifer, X. Jiang and C.-P. Klages, in Y. Tzeng, M. Yoshikawa, M. Murakawa and A. Feldman (eds.), Applications of Diamond and Related Materials, Elsevier, Amsterdam, 1991, p. 121. 17 R. Haubner and B. Lux, Diamond Relat. Mater., 2 (1993) 1277. 18 R. Haubner, S. Okoli and B. Lux, Refract. Met. Hard Mater., 11 (1992) 259. 19 S. Okoli, R. Haubner and B. Lux, Surf. Coat. Technol., 47 (1991) 59. 20 G. G. Stoney, eroc. R. Soc. London, Ser. A, 32 (1909) 172. 21 J. D. Finnegan, AEC Tech. Rep., 15, Case Institute of Technology, Cleveland, OH, 1961. 22 M. Griesser, G. Stingeder, M. Grasserbauer, H. Baumann, F. Link, P. Wurzinger, H. Lux, R. Haubner and B. Lux, Diamond Relat. Mater., 3 (1994) 638. 23 C. Wild, P. Koidl, W. Mtiller-Sebert, H. Walcher, R. Kohl, N. Herres, R. Locher, R. Samlenski and R. Brenn, Diamond Relat. Mater., 2 (1993) 158. 24 H. Jehn, in H. Fischmeister and H. John (eds.), Hartstoffschichten zur Verschleiflminderung, DGM Informationsgesellschaft, Stuttgart, 1987, p. 45. 25 H. Windischmann, G. F. Epps, Y. Chong and R. W. Collins, J. Appl. Phys., 69 (4) (1991) 2231. 26 R. Haubner and B. Lux, J. Refract. Hard Met., 6 (4) (1987) 210. 27 A. Lindlbauer, R. Haubner and B. Lux, Refract. Met. Hard Mater., 11 (1992) 247. 28 R. Brunsteiner, R. Haubner and B. Lux, Diamond Relat. Mater., 2 (1993) 1263. 29 M. Joksch, P. Wurzinger, P. Pongratz, R. Haubner and B. Lux, Diamond Relat. Mater., 3 (1994) 681. 30 R. W. Hoffman, in C. H. S. Dupuy and A. Cachard (eds.), Physics of Nonmetallic Thin Films, Plenum, New York, 1974, p. 306. 31 J. A. Baglio, B. C. Farnsworth, S. Hankin and D. O'Neil, Thin Solid Films, 212 (1992) 180. 32 A. Schulze, Z. Tech. Phys., 11 (1930) 442. 33 G. A. Slack and S. F. Bartram, J. Appl. Phys., 46 (1) (1975) 89. 34 S. P. Timoshenko, Mech. Eng., 259 (1923) 233.
757
Internal stresses in CVD diamond layers D. Schwarzbach, R. Haubner and B. Lux Institute.for Chemical Technology of Inorganic Materials, Technical University of Vienna, Getreidemarkt 9/161, A- 1060 Vienna (Austria)
Abstract Chemically vapour deposited diamond layers always contain internal stresses. Two types of layer stressescan be distinguished: tensile intrinsic stresses and compressivethermal stresses. Intrinsic stresses were measured in situ during diamond deposition onto silicon substrates using a bending-plate method in a hot-filament reactor. The intrinsic tensile stresses increased with the total gas pressure and the methane content in the reaction gas. The measured stress curves became steeper with decreasing filament temperature. They also depended strongly on the layer morphology and grain size. The intrinsic stresses within a layer increased with decreasing grain size and the transition from well facetedcrystallitesto spherolitic ballas-typecrystals having an extremelyfine-grainradial polycrystalline structure. The thermal stresses, determined after cooling to room temperature, increase with higher deposition temperature and show acceptable agreement with calculated values using a simple model.
1. Introduction In the last decade, considerable effort and expense have been invested in research into the field of diamond CVD coating synthesis [1, 2]. D i a m o n d has numerous interesting properties, such as the highest known thermal conductivity, high electrical resistivity, hardness and chemical inertness [3]. There are now a number of emerging technological applications for diamond layers, ranging for instance from coatings for cutting tools [-4, 5], to heats sinks for electronic devices [-63 and fleestanding membranes [-7]. Deposition of diamond films using CVD techniques is always linked with the formation of internal stresses, which influence the main properties of the layer, such as adhesion [-8] or flatness and suitability for the manufacture of items such as diamond membranes [.73. The internal stresses of a coating on a substrate are composed of two types [-9]. (1) Intrinsic stresses are tensile stresses which always occur during layer growth. The reasons for their formation are yet not well understood, (2) Thermal stresses are compressive stresses due to the difference between the thermal expansion coefficients of the layer and the substrate. They are generated after deposition during cooling of the coating and the substrate to room temperature, In the case of low-pressure diamond synthesis, internal stresses are generally determined using X-ray diffraction [ 1 0 - 1 2 ] or R a m a n scattering [,13-15] methods, However, these methods give only the sum of the intrinsic and thermal layer stresses since the measurements are taken at r o o m temperature after deposition,
0925--9635/94,/$7.00
In the investigation reported here, the intrinsic stresses in hot-filament CVD diamond layers deposited under various conditions were measured in situ, during deposition, and the thermal stresses were measured during cooling from the deposition temperature to r o o m temperature as done earlier by Sch~ifer et al. in a microwave plasma reactor [-16]. The influences of the deposition conditions on both intrinsic and thermal stresses are discussed.
2. Experimental details 2.1. A p p a r a t u s
The layers were produced in a hot-filament CVD reactor [-173 consisting of a stainless steel tube, approximately 700 m m long and with 150 m m diameter. The reaction gas was blown through a nozzle between the two filaments (constant flow rate 500 standard cm 3 m i n t ) . Two parallel coils of 0.8 m m tantalum wire (6 m m inner diameter, 48 m m length, 4 windings per centimetre) are fixed between two pairs of molybdenum electrodes. The distance between the filaments was approximately 10 m m (Fig. 1). The filament temperature was determined with a two-colour IR pyrometer. The substrates were placed 15 m m below the filaments on a heatable molybdenum box. The substrate temperature was measured using an encased NiCr/Ni thermocouple placed in a hole in the centre of the sample holder, 1 m m below the substrate [-18]. The gas flow rate was set with two mass flow controllers; the gas pressure, measured with a piezo-resistive sensor, was regulated with a needle valve. The filaments were precar-
© 1994
ElsevierSequoia. All rights reserved SSDI 0925-9635(93)05110-X
758
D. Schwarzbach et al. / Internal stresses in CVD diamond
Filament supensions
i - - Gas inlet
Mirror
.......... 'J =,., sam,>,e
Front view
S/de view
Screen
.I!i
'
Fig. 2. Principle of the bending-plate method. The incident laser beam comes from the left, is split and directed toward the substrate surface. After reflection the beams are projected onto a screen.
Fig. 1. Experimental set-up of the dual-hot-filament CVD equipment. TABLE 1. Experimental conditions
Gas flow rate (standard cm3 min -1 ) Total gas pressure (mbar) Substratc surface temperature (°C) Filament temperature (°C) Deposition time (h) Filament precarburization (h)
Methane 0.74-2.20 500 20-100 730-900 2150-2400 10-25 2-3
~
0 '. . . . . . . . . . . . . . . . . .
~
. s
10
lS
Time [hI
2.2. Determination of internal stress The internal stresses were measured with the bendingplate method assuming that the stressed layer (thickness d) strains the substrate (thickness D) elastically until equilibrium is reached, that the layer is well bonded to the substrate, and that the layer is much thinner than the substrate. Since the thickness D of the substrate can be neglected compared with the bending radius R, the internal stresses tr can be calculated from eqn. (1) 1-20, 21]: EsubD2 (1) 6(1 - VSub)Rd
~
(b)
E .1.0
.
o
,
s
,
.
.
Io ~m. 011
,
is
(c) _. !
I
0 . 0 ~ - - - - - - - - -. -. -. .-.-.-. -. .- .-.-.~. . . . . . . t ~
t
i 0.5 1.0
0
=
4
6
Average layer thickness [microns]
is the Young's modulus of the substrate, and
VSub its Poisson ratio.
The curvature radius was determined by deflection of parallel laser beams on the substrate surface (Fig. 2). After reflection the beams were projected o n t o a s c r e e n , The spot distance is registered using a c a m e r a with a time interval shutter and determined with a ruler, Figure 3 shows the steps for obtaining a typical stress curve. The resolution was 0.10-0.15 GPa. The depth scale was calculated using an average growth rate obtained from the weight gain during deposition, the substrate area, the density of natural diamond and the total deposition time.
two
I- - il
t
o.ol ~"E~m~Im4~ff~7.-~ ~.o.s
Esu b
i
i_.
burized for 2-3 h before the deposition experiment [ 19]. The experimental conditions are summarized in Table 1.
where
(e) icoo.ng J
Deposition
20
Fig. 3. In situ determination of the intrinsic stresses with the bendingplate method. The solid triangle at the end of each curve corresponds to the superimposed thermal stresses after the sample was cooled to room temperature: (a) raw data (spot distance vs. time) as collected during measurement; (b) slope of the inverse curvature radius with time as calculated from the laser deflection; (c) calculated stress curve obtained with eqn.(1).
2.3. Substrates andpretreatment Mirror-polished (100) silicon wafers were used as substrates. They were pretreated by rubbing two pieces against each other with 0.25 lxm diamond paste in between for about 10 rain to enhance the nucleation
D. Schwarzbach et al. / Internal stresses in CVD diamond
density. Then they were cut into pieces of approximately 5 × 35 mm 2, rinsed and cleaned ultrasonically with methanol and acetone,
3. Results 3.1. Influence of the deposition parameters and microstructure on the intrinsic stresses 3.1.1. Methane contentinthegasphase At 2300 °C filament temperature, 780 °C substrate temperature and 50 mbar total gas pressure there was a tendency for the layer stresses to increase with increasing methane content (Fig. 4(a)), while thelayer showed cubic (100) and octahedral (111) crystal habits and the grain size decreased (Fig. 4(b)-4(d)). 3.1.2. Total gas pressure Figure 5(a) shows the influence of the total gas pressure on the internal stresses at 2300 °C filament temperature, 900 °C substrate temperature and 1.00% methane in the reaction gas. With increasing pressure, the grain size (octahedral crystal habits) decreased (Fig. 5(b)-(d)). At lower substrate and lower filament temperatures, an increase in the gas pressure gave rise to better faceted
759
layers (ballas at 20 mbar, fine-grained octahedra at 50 mbar). However, the stress curves are nearly the same for both morphologies. 3.1.3. Filament temperature Figure 6(a) shows the dependence of the layer stresses at 780 °C substrate temperature, 20mbar and 1.00% C H 4 o n the filament temperature. The higher the filament temperature, the flatter is the slope of the stress curve. The morphology changed from fine-grained ballas-type (smooth surface) (Tvil. . . . t =2050°C) to spherolitic shaped ballas (TF, . . . . t=2150 °C) and finally to octahedra at Tvi~. . . . t--2300 °C (Fig. 6(b) 6(d)). Nevertheless, there seems to be no significant difference in the stress behaviour between the ballas layer at 2150 °C and the well-faceted layer at 2300 °C. TABLE 2. Thermal stresses
Substrate temperature (°c)
Average thermal stresses (GPa)
9oo _+20
0.61 _+ 0.045 - 0 . 4 9 _+ 0.057 -0.41
780 _+ 25 730 _+ 10
(b)
(a) 0.01
............................................
(c)
1.o
%/.40% (d)
1.80% ........ 0
I ......... 1
I ......... 2
I ......... 3
I ......... 4
I ......... 5
I ......... 6
I,.. 7
Layer thickness [microns]
Fig. 4. Internal stress curves at various m e t h a n e contents (a) and corresponding scanning electron micrographs: (b) 0.74%, (c) 1.40%, (d) 1.80%. Tvll. . . . t = 2300 °C, Tsub = 780 °C, p = 50 mbar.
760
D. Schwarzbach et al. / Internal stresses in CVD diamond
3.2. Thermal stresses During cooling of the composite layer/substrate to room temperature, a compressive stress component developed. Table 2 shows thermal stresses measured at room temperature: the higher the deposition temperature was, the higher the measured thermal stresses were. Figure 7 shows the development of the thermal stresses during cooling. First, the tensile layer stresses increased, Below 650 °C a compressive stress component developed and at room temperature the final overall thermal stress component was compressive,
This phenomenon is however too close to the detection limits of the bending-plate method to be measured accurately. Stage II. After the isolated crystals began to touch each other and a closed layer formed, the stresses increased linearly with time as well as with the layer thickness. Owing to the high particle density induced by surface pretreatment, the crystallites began growing randomly oriented; only those with an orientation having a high and optimal growth rate formed the layer texture [23, 24].
4. Discussion
Four stages can be distinguished with respect to the behaviour shown by the stress curves (Fig. 8). Stage I. During the early period of isolated crystal growth, before the layer had formed, the stresses remained constant at zero level or seemed to increase slightly. This observed increase might be an artefact due to the formation of a surface carbide layer on the silicon wafer causing swelling of the outmost substrate surface and thus curvature. A similar effect was seen by Schfifer et al. [ 16] on molybdenum substrates. Another reason might be tantalum evaporating from the filament [22].
Stage III. Once the steady state was reached and the columnar texture had formed, crystals were no longer overgrown. The layer grew without the formation of any more stress. In the newly deposited diamond, the density of grain boundaries remained constant and no more stress formation occurred [9]. Most of our experiments had to be stopped before this stage was reached, because the laser spots were diffused by the rough surface of the diamond layer and were therefore no longer detectable. Stage IV. After turning off the gas flow and the heating, the temperature decreased to room temperature. The changes in stress curve were due to the development
(b)
(a) i""
0.0q a.
(c) /
0.5
! 1.0 100 m l ~ r
k'~ 1.5
.........
| ......... I
I ..... 2
illnl
Layer thickness
.........
3
(d)
I ........
4
[microns]
Fig. 5. Internal stress curves at various total gas pressures (a) and corresponding scanning electron micrographs: (b) 20 mbar, (c) 50 mbar, (c) 100 mbar. TFilament=2300 °C, Tsub=900 °C, 1.00% CH 4.
761
D. Schwarzbach et al. / Internal stresses in CVD diamond
(b)
(a)
0.0~
...................................
Q..
0.5
2o5o°c z15o~
1.0
......... 0
I ..... 1
,,,,I
.........
I .........
2
! .........
3
4
Layerthickness[microns]
Fig. 6. Internal stress curves at various filament temperatures (a) and corresponding scanning electron micrographs: (b) 2300 'C, (c) 2150 "C, (d) 2050 °C. Tsub= 780 °C, p = 50 mbar, 1.00% CH4.
6 ~ [.-
,,,
~
Silicon .~ ......
:. 1.. 1.., ._. l.. ....
~i
I,..-..-" .".':..': • ," -_]
o
,
(a)
t
!
o
i
]Ii
0o
-0.5~1.,,,~ j~ -~
° /
Rnal thermal stress component
~,
--1.o
~= 8oo~ ,
o
(b)
Depositiont tRoom emperaturetSmperst . . . . . . .u . .re ...................
,
200
~ ,
,
4o0
,
.,
,
soo
, coo
, looo
Substrale temperature [~]
Fig. 7. Correlation between the observed thermal stresses and the thermal expansion coefficients at different substrate deposition temperatures: (a) linear thermal expansion coefficients of silicon and diamond at different temperatures [-32, 33]; (b) course of the total layer stresses during cooling after deposition as a function of the substrate temperature; the minima in the stress curves correspond to the intersection of the two expansion coefficient curves,
Fig. 8. Schematic representation of a stress curve.
of thermal stress, mainly between the silicon substrate and the diamond layer. The depth scales in the stress curves are only approximations. The growth delay and the different growth rates of isolated crystallites before layer closure as well as layer growth after closure are sources of error.
Furthermore, it must be mentioned that, at deposition temperatures above 800 °C, the substrate curvature and hence the measured stresses may be caused not only by the layer stresses, but also by a superimposition from a plastic deformation of the silicon wafer [ 16]. However, this effect was found to be negligible in our case; the stress data were not corrected.
762
D. Schwarzbach et al. / Internal stresses in CVD diamond
4.1. Intrinsic stresses
The obtained results are in good agreement with those published in the literature. Sch/ifer et al. [16] found increasing tensile stresses with decreasing grain size. Windischmann et al. [25] found decreasing grain sizes with increasing methane content. However, they expected an increasing compressive stress component with increasing sp2-carbon and hydrogen incorporation in the layer. But their assumption was verified only on very thin layers (0.75 jam) where texture formation might not yet be accomplished, The diamond layer morphology is controlled by the gas activation and thus by the deposition conditions [26-28]. The results discussed above show clearly that the intrinsic stresses in diamond films are linked with the layer morphology and its grain size. Even though the processes leading to tensile stress formation are not
yet fully understood, grain boundaries are reported to be a major reason for stress formation during diamond deposition [9, 16]. Owing to the high melting point of diamond and the low mobility of carbon atoms in the diamond lattice, there is no possibility for internal crystal growth (recrystallization) after the diamond is deposited. Recent transmission electron microscopy studies revealed that the different diamond layer morphologies contain different kinds of structural misfits [29]: octahedral crystals contain domains with dislocations and domains with twins, while cubic crystals contain mainly dislocations and the highly twinned areas are near the grain boundaries. Thus stress formation should depend mainly on the changes in grain boundary density with increasing layer thickness due to the formation of columnar texture. The higher intrinsic stresses in ballas-type diamond
(b)
(c) (a)
oO°'~ o.o
,
-!"~
~
..............................
'. ~
(110
~, ~'
(d)
o.5 \
":'~'...mo\,, b,,llms
"' "~ '4,,
(lOO) illllll,ll,,,,
1
,,llhl,,,,,,,lllll,
2
Jlllh/,,,,,,
9
4
Layer thickness [microns]
(e)
l O~m
Fig. 9. Layermorphologyand intrinsic stresses:(a) stresscurves;scanning electronmicrographs of(b) coarse octahedral crystals,(c) fine octahedral crystals, (d) cubiccrystal faces,and (e) ballas-typediamond.
763
D. Schwarzbach et al. / Internal stresses in CVD diamond
layers are supposedly caused by their much higher density of grain and subgrain boundaries as well as structural misfits. Figure 9 illustrates the dependence of the morphology on the intrinsic stresses: when the morphology changed from coarse-grained and fine-grained (111)octahedra/ cuboctahedral habits via cubic (100) to ballas, the layer stress increased. 4.2. Thermal stresses
The observed thermal stresses were independent of the diamond layer morphology and thickness and nearly the same at similar deposition temperatures (Table 2). This is due to the fact that diamond and silicon have different thermal expansion properties and the differences in linear shrinkage cause additional substrate bending, In Fig. 7, the layer stresses during cooling are correlated with the thermal expansion coefficients of silicon and diamond; the observed minimum of the thermal stress curve at Y s u b ~ 6 5 0 °C corresponds to the intersection of the expansion coefficient curves. Above 650 ~;C tensile stresses were formed, and below 650 °C compressive thermal stresses were formed. The final thermal stress component (at room temperature) remained cornpressive. The measured thermal stresses increased with deposition temperature, which is consistent with results reported the literature (for example refs. 15 and 30). 4.3. Calculation of thermal stresses
While in the stress curves the thermal stress component was determined experimentally from substrate bending, in the following section the thermal stresses are calculated directly from material constants using the well known formula [9, 30]: Ed O'th --
1 --
~d
(~d -- ~Si)(TSub - - TR)
(2)
where Eo is Young's modulus of diamond (Ed= 1210 GPa), vd is Poisson's ratio of diamond ( V d = 0 . 1 ) , (Zd and 0~Sub are thermal expansion coefficients of diamond and silicon (Fig. 7(a)), r s u b is the substrate temperature during deposition, and T R is room temperature. The thermal stresses were calculated by determining an average difference A~i of the thermal expansion coefficients within a temperature interval ATi = ( T i - Ti 1) (100 °C), eqn. (3), multiplying them by ATi and summing from the deposition temperature r s u b to room temperature TR, eqn. (4), similar to the method described by Windischmann et al. [25]: Ao~i ~ (~d,l'i
1 - - 0~Si, Ti 1) -~- (~d. Ti - - 0~Si,Ti)
(3)
2 Ed 0~th-
~'S~b 2
1 -
i'd rR
(a/ATi)
Figure 10 shows the results of these calculations in comparison with those of Windischmann et al. [25] and Baglio et al. [31] as well as our experimental results and those of Schtifer et el. [16]. The values for the linear thermal expansion coefficients of silicon and diamond were taken from refs. 32 and 33 respectively (Fig. 7(a)). There is a maximum between 600 ~C and 700 °C which corresponds to the intersection of the silicon and diamond thermal expansion curves. The comparison with experimental values shows an acceptable agreement with the theory. A certain tendency for the experimentally determined thermal stresses to rise with decreasing temperature compared with the calculated values can be seen however. This can be explained by assuming a temperature gradient across the silicon substrate during deposition, causing an additional tensile stress component as shown by Bichler et al. [8] on SiA1ON substrates. However, our calculated values showed a strong discrepancy with those calculated and published by Windischmann et al. [25]. There is no clear explanation for this fact, especially since we used the same values for the thermal expansion coefficients and the same method for calculation of the thermal stresses. The experimental results support our calculation. Thermal stress values calculated by Baglio et al. [31] using Timoshenko's bimetallic strip formula also gave a very different curve from ours since it was based on different initial assumptions [34]. It seems that in the temperature interval where diamond deposition is usually carried out, the thermal stresses are similar using both models. Quantitatively, both models gave theoretical values of approximately
(4)
0.0
~"
'..
-0.2
.O. N ~ -0.4 -~ ~ ,'-
k-
•~•
• "" "-
• • •".ll "" "'"-..
-0.6
-0.8
"" . . . . . .
." "'" Ik " ["" 'g" ° .~.l"~
!
. . . . . . . . . . . . . . . . . . . . . .
0
200
400
1000 Deposition temperature [°C] 6oo
aoo
'
1200
Fig. 10. Calculation of thermal stresses: . . results of the present calculations; --- calculations from Baglio et al, [311; calculations by Windischman et al. [25]; • experimental values obtained in the present investigation (Table 2); ± experimental values from Sch/ifer
et al. [16].
764
D. Schwarzbach et al. / lnternal stresses in CVD diamond
0.5 GPa for compressive thermal s t r e s s e s 700-900 °C and agreed with the experiments. -
at
5. Conclusion
The deposition conditions influence the intrinsic stresses in CVD diamond layers via control of the gas activation and hence the layer morphology. Well faceted, octahedral and coarse-grained layers showed lower stresses than cubic crystallite layers or fine-grained spherolitic ballas-type layers. This supports theories
which correlate stress generation with grain boundary phenomena. The thermal stresses can be controlled by the deposition temperature and can be calculated with acceptable accuracy using simple models. Acknowledgments The authors thank Dr. Naoji Fujimori and the Sumitomo Electric Ltd. in Itami, Japan, for helpful discussions and financial support. The authors also acknowledge Dr. C.-P. Klages a n d D r . L. Sch~fer from the Fraunhofer Institut in Hamburg for helpful discussions and practical hints, in particular concerning the construction of our equipment. References 1 R. Davis (ed.), Diamond Films and Coatings, Noyes, Park Ridge, NJ, 1992. 2 Y. Tzeng, M. Yoshikawa, M. Murakawa and A. Feldman (eds.), Applications of Diamond and Related Materials, Elsevier, Amsterdam, 1991. 3 J. E. Field, Properties of Diamond, Academic Press, London, 1979. 4 B. Lux, R. Haubner and P. Renard, Diamond Relat. Mater., 1 (1991) 1035. 5 H. Schachner, B. Lux, K. G. Stjernberg, A. G. Thelin and H. Tippmann, European Patent 0166708A3, June 27, 1984. 6 R. C. Eden, in Y. Tzeng, M. Yoshikawa, M. Murakawa and A. Feldman (eds.), Applications of Diamond and Related Materials, Elsevier, Amsterdam, 1991, p. 259. 7 L. Sch~ifer, A. Bluhm, C.-P. Klages, B. L6chel, L. M. Buchmann and H.-L. Huber, Diamond Relat. Mater., 2 (1993) 1191.
8 R. Bichler, R. Haubner and B. Lux, High Temp. High Pressure, 21
(1989) 575.
9 M. F. Doerner and W. D. Nix, CRC Crit. Rev. Solid State Mater. Sci., 14 (1988) 225. 10 P. R. Chalker, A. M. Jones, C. Johnston and I. M. Buckley-Golder, Surf. Coat. Technol., 47 (1991) 365. 11 H. Guo and M. Alam, in Y. Tzeng, M. Yoshikawa, M. Murakawa and A. Feldman (eds.), Applications of Diamond and Related Materials, Elsevier, Amsterdam, 1991, p. 149. 12 N. S. VanDamme, D. C. Nagle and S. R. Winzer, Appl. Phys. Lett.,
58 (25) (1991) 2919. 13 E. Gheeraert, A. Deneuville, A. M. Bonnot and L. Abello, Diamond Relat. Mater., 1 (1992) 525. 14 C. Johnston, A. Crossley, P. R. Chalker, I. M. Buckley-Golder and K. Kobashi, Diamond Relat. Mater., 1 (1992) 450. 15 M. Yoshikawa, G. Katagiri, H. Ishida, A. Ishitani, M. Ono and K. Matsumura, Appl. Phys. Lett., 55 (25) (1989) 2608. 16 L. Sch/ifer, X. Jiang and C.-P. Klages, in Y. Tzeng, M. Yoshikawa, M. Murakawa and A. Feldman (eds.), Applications of Diamond and Related Materials, Elsevier, Amsterdam, 1991, p. 121. 17 R. Haubner and B. Lux, Diamond Relat. Mater., 2 (1993) 1277. 18 R. Haubner, S. Okoli and B. Lux, Refract. Met. Hard Mater., 11 (1992) 259. 19 S. Okoli, R. Haubner and B. Lux, Surf. Coat. Technol., 47 (1991) 59. 20 G. G. Stoney, eroc. R. Soc. London, Ser. A, 32 (1909) 172. 21 J. D. Finnegan, AEC Tech. Rep., 15, Case Institute of Technology, Cleveland, OH, 1961. 22 M. Griesser, G. Stingeder, M. Grasserbauer, H. Baumann, F. Link, P. Wurzinger, H. Lux, R. Haubner and B. Lux, Diamond Relat. Mater., 3 (1994) 638. 23 C. Wild, P. Koidl, W. Mtiller-Sebert, H. Walcher, R. Kohl, N. Herres, R. Locher, R. Samlenski and R. Brenn, Diamond Relat. Mater., 2 (1993) 158. 24 H. Jehn, in H. Fischmeister and H. John (eds.), Hartstoffschichten zur Verschleiflminderung, DGM Informationsgesellschaft, Stuttgart, 1987, p. 45. 25 H. Windischmann, G. F. Epps, Y. Chong and R. W. Collins, J. Appl. Phys., 69 (4) (1991) 2231. 26 R. Haubner and B. Lux, J. Refract. Hard Met., 6 (4) (1987) 210. 27 A. Lindlbauer, R. Haubner and B. Lux, Refract. Met. Hard Mater., 11 (1992) 247. 28 R. Brunsteiner, R. Haubner and B. Lux, Diamond Relat. Mater., 2 (1993) 1263. 29 M. Joksch, P. Wurzinger, P. Pongratz, R. Haubner and B. Lux, Diamond Relat. Mater., 3 (1994) 681. 30 R. W. Hoffman, in C. H. S. Dupuy and A. Cachard (eds.), Physics of Nonmetallic Thin Films, Plenum, New York, 1974, p. 306. 31 J. A. Baglio, B. C. Farnsworth, S. Hankin and D. O'Neil, Thin Solid Films, 212 (1992) 180. 32 A. Schulze, Z. Tech. Phys., 11 (1930) 442. 33 G. A. Slack and S. F. Bartram, J. Appl. Phys., 46 (1) (1975) 89. 34 S. P. Timoshenko, Mech. Eng., 259 (1923) 233.