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Cylindrically symmetric diamond parts by hot-filament CVD
be avoided
ensional
ting, wire dies for drawing wire and diamond test tubes, that is, diamond tubes with can be deposited by a nu processes including ame and hot-filament deposition. Plasma processes are most useful for planar parts. Three-dimensional parts in plasma processes cause problems because electric field concentrations at edges and other high surface curvature regions cause uneven heating and uneven deposition of Diamond. Flame and hot-filament deposition can deposit cylindrically symmetric diamond parts if the part is rotated about its axis of symmetry during deposition [ 11. Such a situation is similar to that of a rotisserie where a piece of meat is evenly browned and cooked as it is slowly rotated over a heat source. Rotation of the substrate to achieve an even deposition of CVD diamond is not without problems. First, the rotation of the substrate requires special vacuum feedthrough, motors and substrate mounts that are 092%9635/9’7/$17.00 0 1997 Elsevier Science S.A. All rights reserved. Pld SO925-9635(97)00130-l
by using
few filaments
as
symmetry;
on its deposition
tem
and compressive stresses and stress gradients. substrate is removed, these stresses in the fre from its desired shape or may even cause the part to spontaneously crack and fail. Consequently, it is advantageous to be able to deposit diamond parts without substrate rotation. we will examine how the hot-filament C recess can be used to form iamond parts without [2,3]. Although a high number of sy filaments can always generate an ev riumber of filaments may be net substrate is not overheated. It wih be shown that an even deposition can be accomplished with a surprisingly
low number of straight filaments if the para~~~etersof the process are carefully selected [2]. In fact, even a single straight filament will suffice to make a threedimensional diamond part that is cylindrically symmetrical to within a few percent under certain situations. 2.2. Filament radiation heating oJ‘the s~~st~at~
2.1. Constraints on hot-filament diamond deposition The cumber of filaments, their diameter, temperature
and relative displacement from the substrate will determine the substrate temperature in the absence of forced substrate cooling. Stresses in the CVD diamond deposition process limit the practical range of substrate temperatures [3-IO]. Because the diamond part should be relatively stress free when it is removed from the substrate, diamond deposition should be carried out at substrate temperatures where the diamond is free from intrinsic stresses. At substrate temperatures > 740 “C, intrinsic tensile stresses are produced in a CVD diamond deposit. Similarly, at substrate temperatures < 740 “C, intrinsic compressive stresses are generated in a CVD diamond deposit. Both types of stresses increase approximately linearly with the deviation of the substrate temperature from 740 “C [4] although there is some disagreement in the literature about the exact temperature of zero intrinsic stresses [4,5]. Deposition stresses generate thre- dimensional strains that, in general, are not relieved when the substrate is removed from the part. As a result, large stresses will remain in the part making it susceptible to catastrophic failure. Consequently, to avoid problems with excessive stresses, the diamond deposit should be made at substrate temperatures between 700 and 800 “C where intrinsic stresses are minimized. The temperature of the hot filament is also constrained to a relatively narrow range. Filament temperatures < 2000 “C do not generate enough atomic hydrogen to insure that good quality diamond is deposited during the CVD process. On the other hand, filament temperatures > 2100 “C lead to contamination of the diamond deposit with an excess amount of metal from the filament which can ruin diamond properties such as its thermal conductivity. The size of the hot filament is also determined by other practical considerations. If the filament diameter is too large, it will require large currents at low voltages which are not compatible with ordinary laboratory Power supplies or standard power feedthroughs into the CVD deposition chamber. If the filament diameter is too small, filament breakage becomes a problem. Thus, filament diameters, in practice, usually range between 0.254 mm (0.010”) and 0.762 mm (0.030”). Finally, substrate cooling for cylindrically symmetric
Let us then calculate t substrate temperatasre i sidering only radiation
sivity of the filame e filament, d is the radiation consta nd Tr is the filam Consider the case where N long fil of Rf are aligned parallel tion Ifs impinging symmetry of the substrate. on the substrate from N filaments equally space distance d from the substra angle Q = 2R,/2n subtended he substrate as seen from each filament times th filament times the number N of filaments.
be heated to a temperature it will also radiate heat f9 away according to the Stefan-Boltzman Law: I, =cSA,aT:
(3)
where I, is the radiation emitted by the hot substrate, E, is the emissivity of the substrate, A, is the substrate area and 0 is the Stefan-Boltzman radiation constant. If radiation is the only means of substrate heating and cooling, the substrate will reach an equilibrium temperature when the incoming radiation, that is absorbed by the substrate, equals the outgoing radiation emitted by the substrate: ~Jfs = I,
(4)
where the radiation absorbed by the substrate has been reduced from the impinging radiation by the emissivity E, of the substrate because the substrate is not a perfectly absorbing black body. For a filament-substrate separation of 8.4 mm, a filament radius of 0.25 mm and a substrate radius of 2.54 mm and a substrate emissivity of 0.8, a substrate temperature of 750 “C (1023 K) and the Stefan-Boltzman constant of 5.67 x lo-l2
ration mat~c~~atica 2.3. Substmte heating bb)gas conhctiorr Another means of transferring energy from the hot filament to the substrate is the normal gas conduction mechanism where hot molecules collide with cold molecules and pass their kinetic down a temperature gradient. It can be shown that the heat conduction Qc from N parallel hot filaments with temperatures Tf to a substrate with a temperature TS in an infinite medium with a thermal conductivity K is given by [ 111:
where n is the filament-substrate separation and R, and Rr are the radii of the substrate and filament, respectively. Because of the linearity of the Laplace Equation from which Eq. (6) is derived, the solution to the heat flow problem from a single filament can
not only the flow o chemical species in a the amount of heat p hydrogen is similar to
. W [ll]:
(7)
where D is the diffusivity of atomic hydrogen in the gas mixture, AH is the bond en;thalpy of molecular hydrogen and C,r is the concentration of atomic hydrogen at the hot filament. Other symbols in Eq. (7) have the same meaning as in Eq. (6). again, because of the linearity of the Laplace Equation from which Eq. (7) is derived, a single the solution to the heat flow problem fro filament can be superimposed to arrive at the solution for N filaments [ Eq. (7)] just by multiplying the solution for a single filament by N. Eq. (7) assumes that all atomic hydrogen which reaches the substrate recombines
;md that no atomic hydr en is lost as it diffuses fro the filament to the subs e. That is, we assume tllat there are no nearby solid objects that can act as a recombination surface and that three-body collisions in the gas are rare in the time period required for atomic hydrogen to diffuse from the filament to the substrate. Under these conditions, the heat carried between the filament and substrate by atomic hydrogen is directly proportional to its concentration at the filament surface. The diffusion coefficient of atomic hydrogen in hydrogen at 2050 “C is 45 cm2 s-l at atmospheric pressure [ 121. Since CVD diamond is typically deposited at pressures of ca l/100 of an atmosphere, the diffusion constant of atomic hydrogen at this lower pressure would increase to about 4500 cm2 s-r. The bond enthalpy AH of molecular hydrogen is 4.5 eV or 4.35 x 10s J mol -l. The concentration CH of atomic hydrogen at a filament temperature of 2050 ‘C and a gas pressure of l/100 atm is ca 4.4 x lo-’ mol cm3. Putting all of these variable into Eq. (7), for a filament-substrate separation of 8.4 mm, a filament radius of 0.25 mm and a substrate radius of 2.54 mm, the rate of heat transfer between one filament and the substrate is 59 Wcm-‘. Our calculation overestimates the actual amount of atomic-hydrogen heat transfer as Eq. (7) assumes that all of the atomic hydrogen created at the filament is recombined on the substrate. In a real chamber, some of the atomic hydrogen will recombine on the chamber walls, filament holders or in the gas itself in three-body collisions with hydrogen or two-body collisions with the 1% methane component. Neverth (2)-(4) probably do give the t each phenomena to the transfe the substrate and show that atomic hydrogen sport is the most important means of moving energy from the filament to the substrate.
2.5. Substrateheatingfrom the convectivegas flow generated by atomic and molecular hydrogen reactions at thejiiament and substrate Because atomic hydrogen is created and destroyed, respectively, at the filament and substrate with the absorption and release of its high heat of formation, the transport of atomic hydrogen in the gas will also transport energy from the filament to the substrate. The transport of a species in a gas occurs both by diffusion and convective flow. Consequently, the heat flux & produced by the transport of atomic hydrogen from the filament to the substrate consists of two terms: ( 1) a diffusion term - QDX/ax where Q is the heat of formation of atomic hydrogen, D is the diffusion constant of atomic hydrogen in the gas and aC/& is the concentration gradient of atomic hydrogen in the gas;
(2) a c~llve~tive
ter
ofa tive JH = Q[ - DX?/c?x+ CLJ]. The generation an hydrogen at the filame cause a convective flow 2 moles of atomic are of molecular hydrogen that is d
hydrogen that is created: The net creation of 1
te a corresponding CQMgus mixture from the
ecause the energy content of atomic hydrogen in the gas far exceeds the usual heat capacity of the gas, the convective transfer of heat can be assigned to the atomic hydrogen in the gas. The increase in energy transfer from the hot filament to the substrate because of this convective gas flow is 5% ( 10% atomic hydrogen x 50% gas flow) of the diffusion heat flow given in Eq. (7) or 3.0 W cm- ‘. In summary, radiation heat transfer, heat conduction and atomic hydrogen diffusion and atomic hydrogen convection from each hot filament contribute ca 7.9, 13.1 and 59.2 and 3.0 W cm-‘, respectively, to the substrate with typical CVD diamond deposition conditions. The majority (74.8%) of the heat transfer is a result of atomic hydrogen generation and recombination and transport. The absolute and fractional contributions of each mechanism are listed in Table 1.
3. Anisotropyof the diamo
US
As discussed above, practical considerations restrict to a filament-substrate separation of < 1 cm and to
tally symmetric substrate with long straight hot filaments parallel to the ax& of symmetry of the substrate, our problem further reduces to the solution of Equation in two dimensions. Since we can only use one or two filaments because of the practical restrictions noted above, let us consider each case in turn.
If C(R) is the concentration of active diamond depositing species in our reaction chamber and R is the radial distance vector from the symmetry axis of the substrate and 6 is the angle around this symmetry axis [see Fig. l(a)]. then the solution to Laplace’s Equation is [ 141: -di)+Co
ln(R+di)-2C,
In(R)
(11)
where di is the radical distance vector from the symmetry axis of the substrate to the center of the two filaments, respectively, and COis the concentration of active species at the hot filament.
) ~~~pi~igi~lgon the surface of Fick’s Law:
The diffusion flux the substrate is given -=dR
3.1. Two-jlamen t case
C(R) = CO ln(
Fig. I. (a) Two opposing filaments with the substrate in between. The filumetat substrate spa&q is ti and the angle around the substrate center is II. R, IS the substrate radius. (b) The anisotropy of the deposit m Lhc tcv+lilamcnt LX&C.The ratto A,B is the ratio ot’ tbc deposit thickness along the center line and perpendicular to the center line 011 the substrate.
1
D C
+
1
2
1 02)
where D is the diffusion constant of the active species in the gas mixture. The growth rate I/(&, 0) of CV diamond on the substrate is given by:
JW,, 0)
V(R,, 0)= -
Ncl
where ibi is the moles per unit volume in diamond. Note that in Eq. (13), we have changed to a scalar notation from the vector notation in Eqs. ( 11) and ( 12).
The
aspect ratio A/
is given by the growth rate C’(R,,0) at 0 ==o divided by the growth rate 1/(& x/2) at 8 = n/2: deposit
1 A -= B
---
2
1
RS-d+R,+d 2
R, 2
( 14)
-
l/m--R,
In Table 2, the aspect ratio A/B of the CVD diamond is given versus the ratio of the substrate radius R, divided by the filament-substrate separation d. Table 2 shows that the aspect ratio A/B of the diamond deposit approaches 1 (complete symmetry) as the substrate radius R, becomes a small fraction of the filament-substrate separation d. The relative growth rate also increases as the substrate radius decreases because the depositing flux per unit area also increases. Consequently, with a given filament-substrate separation, symmetric CVD diamond parts can be quickly grown if their diameters are small relative to the filament-substrate spacing. Conversely, parts with large diameters will grow slowly and asymmetrically.
the hot filament. The diffusion flux the substrate is given
&posit
where D is the in the gas mixture.
where Nd is t moles Note that in . (171, notation from the vector
3.2. One-jlament case In some cases, even two filaments may overheat the substrate. Intuition would say that a symmetric deposit is not possible with a single filament. However, as we shall see below, a symmetric deposit is possible with a single filament under certain conditions that are more stringent than those for the two-filament case. Again, let us consider the case where the filament and substrate are long parallel rods. Our problem reduces to the twodimensional diffusion problem with a source (the filament) and a sink (the substrate) for the active CVD diamond depositing species, If C(R) is the concentration of active diamond depositing species in our reaction chamber and R is the radial distance vector from the symmetry axis of the substrate and 0 is the angle around this symmetry axis [see
(;I)
Table 2 The aspect ratio A/B of the diamond deposit parallel and perpendicular to the center he through the two hot filaments on opposite sides of the substrate versus the ratio of the substrate radius R, to the fi~~enmdmm spacing d. The relative diamond growth rates on the substrate are also given &Id
A/B
Relative diamond growth rate
0.5
2.4 1.29 1.12 1.05 1.02 1.01
1 2 4 10 20 40
0.2 0.1 0.05 0.02 0.01
Fig. 2. (a) A single filament and a substrate. The filament-substrate spacing is d and the angle around the substrate center is 0. R, is the substrate radius. (b) A single filament and a substrate. The ratio A/B is the ratio of the deposit thickness along the center line and to the deposit thickness perpendicular to the center line. The ratio of the thickness of the deposit on the side facing the filament to the thickness on the other side is A/A’.
Fig. 3. An abrasive water-jet for cutting hard materials. The water jet issues from a free jet nozzle. Abrasive particles are injected at the top of the mixing tube and become entrained in the water jet. The jet and entrained particles impinge on the workpiece material being cut.
,
A A’=
d+R, d-R,
I I
(19)
.
Table 3 The aspect ratio A/B of the diamond deposit parallel and perpendicular to the center line through the one hot filament and the substrate versus the ratio of the substrate radius R, to the filament-substrate spacing GCThe relative diamond growth rates on the substrate are also given
&Jd 0.1 0.05 0.02 0.01
AJB 1.22 1.10 1.04 1.02
Relative diamond growth rate 2 4 10 20
that
the filament-substrate separation d. A short physical explanation of why th the deposit in both the two- and onedepends on the ratio of the substrate diameter to the filament-substrate spacing is in order. If one sketches on a piece of paper a large substrate near a filament, it is apparent that the concentration of any active species diEusing from the filament to the substrate is going to Table 4 The ratio A/A’ of the thickness of the diamond deposit on the side of the substrate facing the filament to the thickness of the diamond deposit on the side away from the filament versus the ratio of the substrate radius R, to the filament-substrate spacing d
&Id
AJA’
0.1 0.05 0.02 0.01
1.22 1.10
1.04 1.02
Fig. 4. (a) An SEM of a water-jet nozzle produced by a one-filament CVD diamond process. (b) An SEM of a diamond tube produced by a ouefiliment CVD diamond deposkion.
be very different on opposing sides of the substrate because of the very steep concentration the substrate. As a result, the deposit will be much thicker on the substrate side facing the filament than the side facing away from the filament. However, if the substrate is pulled far away from the filament to drastically decrease the concentration gradients, the entire surface of the substrate is now in a region where the concentration gradients are very small SO that the active species concentration is almost equal on all sides of the substrate. In the latter case, the deposit will grow with a uniform thickness on all sides of the substrate.
4. Experimental 4.I. One-jilamen t case Fig. 3 shows a schematic diagram of a typical abrasive water-jet cutter [ 15,161. A high-pressure reservoir of water is connected to a free-jet nozzle from which a water jet shoots out. A mixing tube with attached funnel
is positioned so that the water jet shoots down the central axis of the mixing tube. Abrasive grain is fed into the funnel and is drawn into the water jet by the lower pressure around the water jet due to Bernoulli’s Principle. The abrasive grain is accelerated by the jet to the jet velocity and exits the mixing tube at the bottom onto a piece of material to be cut. Both the water-jet nozzle and the mixing tube have been made with hotfilament CVD diamond deposition using either one or two filaments in our laboratory. Fig. 4a shows a scanning electron micrograph (SEM ) of a water-jet nozzle of CVD diamond that was grown on a substrate with a radius of 8.89 x low3 cm (0.0035”) and a length of 28 cm from a single tungsten filament. The filament temperature was 2050 “C and the substrate temperature was 780 “C. The gas composition was 1% methane in hydrogen. The filament-substrate separation was 0.8 cm so that R,/d=O.Ol. The nozzle as shown was cut by a Q-switched Nd-YAG laser from the original diamond tube that was 28 cm long. The diamond nozzle shows no detectable asymmetry (< 1%) about its cylindrical axis. The symmetry is somewhat better than is predicted by our idealized theory (see Table 1).
a twoent d tube joins a es are Injected.
uropean Patent 0492160 (1 July .E. Kuhman. J. Vat. Sci. TeclInol.
ig. 5 shows a ~~br~~sive-jet a
amend tube with a diamo
was made with a two-filament process. conditions were the same as the one-filament case except that the filament-substrate spacing was 1 cm and substrate radius was 0.05 cm. The substrate included a funnel on one end with a radius of 0. i 5 cm at the funnel mouth. The higher R,/ct ratios (0.05 and 0.15) with the mixing tube deposition required two filaments tlo make a reasonab!y symmetric piece. The funnel mouth had an experimental A/B ratio of 1.05 at a R,/LI=O.15 while the main tube had an experimental A/B ratio of 1.02 at a R,/d=0.05. These ratios are the right order of magnitude but also somewhat better than our idealized theory predicts (see Table 2). For abrasive-jet mixing tubes, only the internal dimensions are critical so that these somewhat larger aspect ratios are adequate for this application.
5. Summary Without substrate cooling, substrate temperatures limit the number of hot-filaments that can be used in
[4] L. Schafer, X. Jian g, C.-P. Mlages. In: Y. Tzeng, M. Yos . Murakawa, A. Feldman, A. (Eds.) Proceedings of the 1st elated MateriInternational Conference of als Elsevier, New York, August, 1991, p. 121. [S] SK. Choi, D.V. Yung, H.M. Choi, J. Vat. Sci. Technol. ib2I4 ( 1996) 165. Windischmann, G.F. Epps, Y. Gong, R.W. Collins. J. Appl. tys. 69 ( 1991)2231. . Freund. J. Cryst. Growth 132 ( lW3 1 314. [Xl ED. Specht, R.E. Clausing, L. I-iedtheriy, J. Mater. Rcs. 5 (1990) 2351. [9] D. Schwarzbach, R. Haubner, 13.Lux, Dimond (1994) 757. [IO] P.R. Chalker, A.M. Jones, C. Johnstouc, B.M. BuckleySurf. Coatings Technol. 47 (1991) 365. [ 1l] J. Sucec, Heat Transfer. Simon and Schuster, New York, 1975. p. 75. [ 121 I. Langmuir, Science 35 (1912) 428. [ 131 T.R. Anthony, in: R.E. Clausing, L.L. Horton, J.C. Angus, P. Koidl (Eds.) Diamond and Diamond-like Films and Coatings. Plenum Press, New York, 1991, p. 560. [ 141 R.P. Feynman, R.B. Leighton, M. Sands, The Feynman Lectures on Physics, Vol. II, 7-1. Addison Wesley, Reading, MA, 1964. [ 151 F. Yeaple, Design News 46 ( 14) ( 1990) 59. [16] F. Yeaple. Design News 46 (14) (1990) 71.
ensional
ting, wire dies for drawing wire and diamond test tubes, that is, diamond tubes with can be deposited by a nu processes including ame and hot-filament deposition. Plasma processes are most useful for planar parts. Three-dimensional parts in plasma processes cause problems because electric field concentrations at edges and other high surface curvature regions cause uneven heating and uneven deposition of Diamond. Flame and hot-filament deposition can deposit cylindrically symmetric diamond parts if the part is rotated about its axis of symmetry during deposition [ 11. Such a situation is similar to that of a rotisserie where a piece of meat is evenly browned and cooked as it is slowly rotated over a heat source. Rotation of the substrate to achieve an even deposition of CVD diamond is not without problems. First, the rotation of the substrate requires special vacuum feedthrough, motors and substrate mounts that are 092%9635/9’7/$17.00 0 1997 Elsevier Science S.A. All rights reserved. Pld SO925-9635(97)00130-l
by using
few filaments
as
symmetry;
on its deposition
tem
and compressive stresses and stress gradients. substrate is removed, these stresses in the fre from its desired shape or may even cause the part to spontaneously crack and fail. Consequently, it is advantageous to be able to deposit diamond parts without substrate rotation. we will examine how the hot-filament C recess can be used to form iamond parts without [2,3]. Although a high number of sy filaments can always generate an ev riumber of filaments may be net substrate is not overheated. It wih be shown that an even deposition can be accomplished with a surprisingly
low number of straight filaments if the para~~~etersof the process are carefully selected [2]. In fact, even a single straight filament will suffice to make a threedimensional diamond part that is cylindrically symmetrical to within a few percent under certain situations. 2.2. Filament radiation heating oJ‘the s~~st~at~
2.1. Constraints on hot-filament diamond deposition The cumber of filaments, their diameter, temperature
and relative displacement from the substrate will determine the substrate temperature in the absence of forced substrate cooling. Stresses in the CVD diamond deposition process limit the practical range of substrate temperatures [3-IO]. Because the diamond part should be relatively stress free when it is removed from the substrate, diamond deposition should be carried out at substrate temperatures where the diamond is free from intrinsic stresses. At substrate temperatures > 740 “C, intrinsic tensile stresses are produced in a CVD diamond deposit. Similarly, at substrate temperatures < 740 “C, intrinsic compressive stresses are generated in a CVD diamond deposit. Both types of stresses increase approximately linearly with the deviation of the substrate temperature from 740 “C [4] although there is some disagreement in the literature about the exact temperature of zero intrinsic stresses [4,5]. Deposition stresses generate thre- dimensional strains that, in general, are not relieved when the substrate is removed from the part. As a result, large stresses will remain in the part making it susceptible to catastrophic failure. Consequently, to avoid problems with excessive stresses, the diamond deposit should be made at substrate temperatures between 700 and 800 “C where intrinsic stresses are minimized. The temperature of the hot filament is also constrained to a relatively narrow range. Filament temperatures < 2000 “C do not generate enough atomic hydrogen to insure that good quality diamond is deposited during the CVD process. On the other hand, filament temperatures > 2100 “C lead to contamination of the diamond deposit with an excess amount of metal from the filament which can ruin diamond properties such as its thermal conductivity. The size of the hot filament is also determined by other practical considerations. If the filament diameter is too large, it will require large currents at low voltages which are not compatible with ordinary laboratory Power supplies or standard power feedthroughs into the CVD deposition chamber. If the filament diameter is too small, filament breakage becomes a problem. Thus, filament diameters, in practice, usually range between 0.254 mm (0.010”) and 0.762 mm (0.030”). Finally, substrate cooling for cylindrically symmetric
Let us then calculate t substrate temperatasre i sidering only radiation
sivity of the filame e filament, d is the radiation consta nd Tr is the filam Consider the case where N long fil of Rf are aligned parallel tion Ifs impinging symmetry of the substrate. on the substrate from N filaments equally space distance d from the substra angle Q = 2R,/2n subtended he substrate as seen from each filament times th filament times the number N of filaments.
be heated to a temperature it will also radiate heat f9 away according to the Stefan-Boltzman Law: I, =cSA,aT:
(3)
where I, is the radiation emitted by the hot substrate, E, is the emissivity of the substrate, A, is the substrate area and 0 is the Stefan-Boltzman radiation constant. If radiation is the only means of substrate heating and cooling, the substrate will reach an equilibrium temperature when the incoming radiation, that is absorbed by the substrate, equals the outgoing radiation emitted by the substrate: ~Jfs = I,
(4)
where the radiation absorbed by the substrate has been reduced from the impinging radiation by the emissivity E, of the substrate because the substrate is not a perfectly absorbing black body. For a filament-substrate separation of 8.4 mm, a filament radius of 0.25 mm and a substrate radius of 2.54 mm and a substrate emissivity of 0.8, a substrate temperature of 750 “C (1023 K) and the Stefan-Boltzman constant of 5.67 x lo-l2
ration mat~c~~atica 2.3. Substmte heating bb)gas conhctiorr Another means of transferring energy from the hot filament to the substrate is the normal gas conduction mechanism where hot molecules collide with cold molecules and pass their kinetic down a temperature gradient. It can be shown that the heat conduction Qc from N parallel hot filaments with temperatures Tf to a substrate with a temperature TS in an infinite medium with a thermal conductivity K is given by [ 111:
where n is the filament-substrate separation and R, and Rr are the radii of the substrate and filament, respectively. Because of the linearity of the Laplace Equation from which Eq. (6) is derived, the solution to the heat flow problem from a single filament can
not only the flow o chemical species in a the amount of heat p hydrogen is similar to
. W [ll]:
(7)
where D is the diffusivity of atomic hydrogen in the gas mixture, AH is the bond en;thalpy of molecular hydrogen and C,r is the concentration of atomic hydrogen at the hot filament. Other symbols in Eq. (7) have the same meaning as in Eq. (6). again, because of the linearity of the Laplace Equation from which Eq. (7) is derived, a single the solution to the heat flow problem fro filament can be superimposed to arrive at the solution for N filaments [ Eq. (7)] just by multiplying the solution for a single filament by N. Eq. (7) assumes that all atomic hydrogen which reaches the substrate recombines
;md that no atomic hydr en is lost as it diffuses fro the filament to the subs e. That is, we assume tllat there are no nearby solid objects that can act as a recombination surface and that three-body collisions in the gas are rare in the time period required for atomic hydrogen to diffuse from the filament to the substrate. Under these conditions, the heat carried between the filament and substrate by atomic hydrogen is directly proportional to its concentration at the filament surface. The diffusion coefficient of atomic hydrogen in hydrogen at 2050 “C is 45 cm2 s-l at atmospheric pressure [ 121. Since CVD diamond is typically deposited at pressures of ca l/100 of an atmosphere, the diffusion constant of atomic hydrogen at this lower pressure would increase to about 4500 cm2 s-r. The bond enthalpy AH of molecular hydrogen is 4.5 eV or 4.35 x 10s J mol -l. The concentration CH of atomic hydrogen at a filament temperature of 2050 ‘C and a gas pressure of l/100 atm is ca 4.4 x lo-’ mol cm3. Putting all of these variable into Eq. (7), for a filament-substrate separation of 8.4 mm, a filament radius of 0.25 mm and a substrate radius of 2.54 mm, the rate of heat transfer between one filament and the substrate is 59 Wcm-‘. Our calculation overestimates the actual amount of atomic-hydrogen heat transfer as Eq. (7) assumes that all of the atomic hydrogen created at the filament is recombined on the substrate. In a real chamber, some of the atomic hydrogen will recombine on the chamber walls, filament holders or in the gas itself in three-body collisions with hydrogen or two-body collisions with the 1% methane component. Neverth (2)-(4) probably do give the t each phenomena to the transfe the substrate and show that atomic hydrogen sport is the most important means of moving energy from the filament to the substrate.
2.5. Substrateheatingfrom the convectivegas flow generated by atomic and molecular hydrogen reactions at thejiiament and substrate Because atomic hydrogen is created and destroyed, respectively, at the filament and substrate with the absorption and release of its high heat of formation, the transport of atomic hydrogen in the gas will also transport energy from the filament to the substrate. The transport of a species in a gas occurs both by diffusion and convective flow. Consequently, the heat flux & produced by the transport of atomic hydrogen from the filament to the substrate consists of two terms: ( 1) a diffusion term - QDX/ax where Q is the heat of formation of atomic hydrogen, D is the diffusion constant of atomic hydrogen in the gas and aC/& is the concentration gradient of atomic hydrogen in the gas;
(2) a c~llve~tive
ter
ofa tive JH = Q[ - DX?/c?x+ CLJ]. The generation an hydrogen at the filame cause a convective flow 2 moles of atomic are of molecular hydrogen that is d
hydrogen that is created: The net creation of 1
te a corresponding CQMgus mixture from the
ecause the energy content of atomic hydrogen in the gas far exceeds the usual heat capacity of the gas, the convective transfer of heat can be assigned to the atomic hydrogen in the gas. The increase in energy transfer from the hot filament to the substrate because of this convective gas flow is 5% ( 10% atomic hydrogen x 50% gas flow) of the diffusion heat flow given in Eq. (7) or 3.0 W cm- ‘. In summary, radiation heat transfer, heat conduction and atomic hydrogen diffusion and atomic hydrogen convection from each hot filament contribute ca 7.9, 13.1 and 59.2 and 3.0 W cm-‘, respectively, to the substrate with typical CVD diamond deposition conditions. The majority (74.8%) of the heat transfer is a result of atomic hydrogen generation and recombination and transport. The absolute and fractional contributions of each mechanism are listed in Table 1.
3. Anisotropyof the diamo
US
As discussed above, practical considerations restrict to a filament-substrate separation of < 1 cm and to
tally symmetric substrate with long straight hot filaments parallel to the ax& of symmetry of the substrate, our problem further reduces to the solution of Equation in two dimensions. Since we can only use one or two filaments because of the practical restrictions noted above, let us consider each case in turn.
If C(R) is the concentration of active diamond depositing species in our reaction chamber and R is the radial distance vector from the symmetry axis of the substrate and 6 is the angle around this symmetry axis [see Fig. l(a)]. then the solution to Laplace’s Equation is [ 141: -di)+Co
ln(R+di)-2C,
In(R)
(11)
where di is the radical distance vector from the symmetry axis of the substrate to the center of the two filaments, respectively, and COis the concentration of active species at the hot filament.
) ~~~pi~igi~lgon the surface of Fick’s Law:
The diffusion flux the substrate is given -=dR
3.1. Two-jlamen t case
C(R) = CO ln(
Fig. I. (a) Two opposing filaments with the substrate in between. The filumetat substrate spa&q is ti and the angle around the substrate center is II. R, IS the substrate radius. (b) The anisotropy of the deposit m Lhc tcv+lilamcnt LX&C.The ratto A,B is the ratio ot’ tbc deposit thickness along the center line and perpendicular to the center line 011 the substrate.
1
D C
+
1
2
1 02)
where D is the diffusion constant of the active species in the gas mixture. The growth rate I/(&, 0) of CV diamond on the substrate is given by:
JW,, 0)
V(R,, 0)= -
Ncl
where ibi is the moles per unit volume in diamond. Note that in Eq. (13), we have changed to a scalar notation from the vector notation in Eqs. ( 11) and ( 12).
The
aspect ratio A/
is given by the growth rate C’(R,,0) at 0 ==o divided by the growth rate 1/(& x/2) at 8 = n/2: deposit
1 A -= B
---
2
1
RS-d+R,+d 2
R, 2
( 14)
-
l/m--R,
In Table 2, the aspect ratio A/B of the CVD diamond is given versus the ratio of the substrate radius R, divided by the filament-substrate separation d. Table 2 shows that the aspect ratio A/B of the diamond deposit approaches 1 (complete symmetry) as the substrate radius R, becomes a small fraction of the filament-substrate separation d. The relative growth rate also increases as the substrate radius decreases because the depositing flux per unit area also increases. Consequently, with a given filament-substrate separation, symmetric CVD diamond parts can be quickly grown if their diameters are small relative to the filament-substrate spacing. Conversely, parts with large diameters will grow slowly and asymmetrically.
the hot filament. The diffusion flux the substrate is given
&posit
where D is the in the gas mixture.
where Nd is t moles Note that in . (171, notation from the vector
3.2. One-jlament case In some cases, even two filaments may overheat the substrate. Intuition would say that a symmetric deposit is not possible with a single filament. However, as we shall see below, a symmetric deposit is possible with a single filament under certain conditions that are more stringent than those for the two-filament case. Again, let us consider the case where the filament and substrate are long parallel rods. Our problem reduces to the twodimensional diffusion problem with a source (the filament) and a sink (the substrate) for the active CVD diamond depositing species, If C(R) is the concentration of active diamond depositing species in our reaction chamber and R is the radial distance vector from the symmetry axis of the substrate and 0 is the angle around this symmetry axis [see
(;I)
Table 2 The aspect ratio A/B of the diamond deposit parallel and perpendicular to the center he through the two hot filaments on opposite sides of the substrate versus the ratio of the substrate radius R, to the fi~~enmdmm spacing d. The relative diamond growth rates on the substrate are also given &Id
A/B
Relative diamond growth rate
0.5
2.4 1.29 1.12 1.05 1.02 1.01
1 2 4 10 20 40
0.2 0.1 0.05 0.02 0.01
Fig. 2. (a) A single filament and a substrate. The filament-substrate spacing is d and the angle around the substrate center is 0. R, is the substrate radius. (b) A single filament and a substrate. The ratio A/B is the ratio of the deposit thickness along the center line and to the deposit thickness perpendicular to the center line. The ratio of the thickness of the deposit on the side facing the filament to the thickness on the other side is A/A’.
Fig. 3. An abrasive water-jet for cutting hard materials. The water jet issues from a free jet nozzle. Abrasive particles are injected at the top of the mixing tube and become entrained in the water jet. The jet and entrained particles impinge on the workpiece material being cut.
,
A A’=
d+R, d-R,
I I
(19)
.
Table 3 The aspect ratio A/B of the diamond deposit parallel and perpendicular to the center line through the one hot filament and the substrate versus the ratio of the substrate radius R, to the filament-substrate spacing GCThe relative diamond growth rates on the substrate are also given
&Jd 0.1 0.05 0.02 0.01
AJB 1.22 1.10 1.04 1.02
Relative diamond growth rate 2 4 10 20
that
the filament-substrate separation d. A short physical explanation of why th the deposit in both the two- and onedepends on the ratio of the substrate diameter to the filament-substrate spacing is in order. If one sketches on a piece of paper a large substrate near a filament, it is apparent that the concentration of any active species diEusing from the filament to the substrate is going to Table 4 The ratio A/A’ of the thickness of the diamond deposit on the side of the substrate facing the filament to the thickness of the diamond deposit on the side away from the filament versus the ratio of the substrate radius R, to the filament-substrate spacing d
&Id
AJA’
0.1 0.05 0.02 0.01
1.22 1.10
1.04 1.02
Fig. 4. (a) An SEM of a water-jet nozzle produced by a one-filament CVD diamond process. (b) An SEM of a diamond tube produced by a ouefiliment CVD diamond deposkion.
be very different on opposing sides of the substrate because of the very steep concentration the substrate. As a result, the deposit will be much thicker on the substrate side facing the filament than the side facing away from the filament. However, if the substrate is pulled far away from the filament to drastically decrease the concentration gradients, the entire surface of the substrate is now in a region where the concentration gradients are very small SO that the active species concentration is almost equal on all sides of the substrate. In the latter case, the deposit will grow with a uniform thickness on all sides of the substrate.
4. Experimental 4.I. One-jilamen t case Fig. 3 shows a schematic diagram of a typical abrasive water-jet cutter [ 15,161. A high-pressure reservoir of water is connected to a free-jet nozzle from which a water jet shoots out. A mixing tube with attached funnel
is positioned so that the water jet shoots down the central axis of the mixing tube. Abrasive grain is fed into the funnel and is drawn into the water jet by the lower pressure around the water jet due to Bernoulli’s Principle. The abrasive grain is accelerated by the jet to the jet velocity and exits the mixing tube at the bottom onto a piece of material to be cut. Both the water-jet nozzle and the mixing tube have been made with hotfilament CVD diamond deposition using either one or two filaments in our laboratory. Fig. 4a shows a scanning electron micrograph (SEM ) of a water-jet nozzle of CVD diamond that was grown on a substrate with a radius of 8.89 x low3 cm (0.0035”) and a length of 28 cm from a single tungsten filament. The filament temperature was 2050 “C and the substrate temperature was 780 “C. The gas composition was 1% methane in hydrogen. The filament-substrate separation was 0.8 cm so that R,/d=O.Ol. The nozzle as shown was cut by a Q-switched Nd-YAG laser from the original diamond tube that was 28 cm long. The diamond nozzle shows no detectable asymmetry (< 1%) about its cylindrical axis. The symmetry is somewhat better than is predicted by our idealized theory (see Table 1).
a twoent d tube joins a es are Injected.
uropean Patent 0492160 (1 July .E. Kuhman. J. Vat. Sci. TeclInol.
ig. 5 shows a ~~br~~sive-jet a
amend tube with a diamo
was made with a two-filament process. conditions were the same as the one-filament case except that the filament-substrate spacing was 1 cm and substrate radius was 0.05 cm. The substrate included a funnel on one end with a radius of 0. i 5 cm at the funnel mouth. The higher R,/ct ratios (0.05 and 0.15) with the mixing tube deposition required two filaments tlo make a reasonab!y symmetric piece. The funnel mouth had an experimental A/B ratio of 1.05 at a R,/LI=O.15 while the main tube had an experimental A/B ratio of 1.02 at a R,/d=0.05. These ratios are the right order of magnitude but also somewhat better than our idealized theory predicts (see Table 2). For abrasive-jet mixing tubes, only the internal dimensions are critical so that these somewhat larger aspect ratios are adequate for this application.
5. Summary Without substrate cooling, substrate temperatures limit the number of hot-filaments that can be used in
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