The Effects of Geometry on Diffusion-Controlled Chemical Reaction in a Plasma James F. Battey Internationa~ Plasma Corporation, Hayward, CaliSornia 94544 Rates ABSTRACT The photoresist strip rate at the edge and center of an array of silicon wafers is calculated for an oxygen plasma. The etch rate at the edge is essen- tially independent of wafer diameter and spacing, whereas the etch rate at the center increases linearly with increasing wafer spacing and is inversely proportional to the square o~ wafer d~ameter. Thus, a plasma stripping or etch- ing system has a processing rate which can be specified in silicon area per unit time. The degree of inhomogeneity between edge and center also depends on the probability of reaction when a reactive specie hits the photoresist. Ex- perimental measurements confirm the predicted functional dependence, except with an oxygen plasma the strip rate at the center of the wafer varies in- versely as the cube of the wafer diameter rather than as the square. The im- plication is that two atomic oxygen atoms react simultaneously with the photoresist, perhaps as CH2 -t- 2 O --> CO -~ H~O, or more probably as O ~ O § wall-> O~ + wall. In most low pressure plasmas created by capacitively coupling rf energy into a cylindrical cross-section re- actor, the glow discharge is confined to a region near the walls of the reactor. The effect is analogous to the electrical skin effect in a metal conductor. The electri- cal current is carried in this. region and the active species are created in this region. In an oxygen plasma the active specie is atomic oxygen which has a suffi- ciently long life to diffuse through the interior of the reactor. If the flow rates are small, the concentration of atomic oxygen is substantially constant, apart from the region immediately adjacent to the walls. If a boatload of silicon wafers is introduced in an oxygen plasma, the atomic oxygen must diffuse be- tween the wafers in order to strip the photoresist. The strip rate is calculated for the edge of the wafer and the center of the wafer as a function of wafer spacing and diameter, background oxygen concentration, and probability that an oxygen atom reacts when it hits the photoresist. Experimental measurements are made to examine the predicted functional dependence. The mathematical model is also applicable to cal- culating the degree of inhomogeneity across the face of a wafer in an etching plasma or in a diffusion oper- ation. Experimental Arrangement The experimental work was done in an 8 in. diam- eter cylindrical quartz reactor 13 in. in length. A schematic of the apparatus is shown in Fig. 1. The back of the reactor is a spherical sectioned quartz dome, and the front is a flat quartz door. The gas is introduced into the reactor through a quartz mani- fold on the bottom of the reactor and parallel to its axis. The gas is exhausted by a forepump from the re- actor through a quartz manifold on the top of the re- actor and parallel to the reactor axis. The plasma is excited by a radio frequency generator operating at 13.56 MHz and coupled into the plasma through a 50~2 cable, a pi impedance matching network, and two half- cylinder metal electrodes placed around the outside walls of the cylindrical reactor. The top electrode is at electrical ground, and the bottom electrode oscillates between positive and negative voltage with the gen- erator. Key words: semiconductors, photoresist stripping, plasma etch- iag. CHAMBER i r..---1 r~-t r.,~t, FLOWRATOR REGULATOR GkDE VIEW NOz ]REGULATOR FLOWRATOR ~ IMPEDANCE MATCHING NETWORK 1 5o~ CABLE R.F. GENERATOR FRONT V|EW Fig. i. Diagram of the experimental apparatus The exhaust manifold is sufficiently large that the pressure drops in it are negligible, and the pressure in the reactor is measured with a Hastings VT4 vacuum gauge in a quartz tube leading out of the exhaust manifold. A metered flow of molecular oxygen is fed into the reactor from a cylinder of 99.6% minimum purity oxygen at high pressure through a regulator which reduces the pressure to approximately 1 atm, 437
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`438 J. Eleetrochem. Soc.: SOLID-STATE SCIENCE AND TECHNOLOGY March 1977 and a flowmeter, which is connected to the inlet mani- fold. The rf power in the reactor converts some of the molecular oxygen to atomic oxygen, and the flow of atomic oxygen in the exhaust can be measured by titrating it (2) with a flow of NOs which is introduced into the quartz tube which extends out of the exhaust manifold. In order to obtain reproducible stripping times, it was found necessary to put the single wafer on a large aluminum block which had a thermometer inserted in a hole in the block, and to place the block in the same position in the reactor. It was also found necessary to place two thermometers on the external walls of the reactor and require all three thermometers to assume the same temperatures as in a prior experiment. An additional requirement for reproducibility of strip times is that the internal walls of the quartz re- actor, the door, the inlet and exhaust manifolds, the aluminum block and its quartz support structure be cleaned using firepolishing with a glassblower's torch and acid etching. Quartz and aluminum oxide pre- sumably have a lower surface recombination rate for atomic oxygen when they are "clean." The presence os a second silicon wafer coaxial, parallel, and at a stipulated distance from the first wafer is simulated by a circular glass slide clamped to a heat sink which is thermally coupled to the heat sink on which the silicon wafer rests. The glass slide can be moved with respect to the silicon wafer by means r,, -" 2vo n=l coefficient of atomic oxygen in molecular oxygen, to the reaction rate at the surface which is assumed (and we have shown experimentally) to be linear in v. From kinetic theory the number of oxygen atoms striking the wafer surface per unit area and time is 1/4 v v, where is the average velocity. If p is the probability of react- ing per strike, then the boundary condition can be written DdV I 1_ = --vpv(z = O) dz z=o 4 or = hv(z = O) dz z=o where h = ~p/4D. Kinetic theory also says that D 1/3 Zg~ where 2~ is the mean free path so h ~ 3p/4L At 1 Torr ~ ~ 0.01 cm so h ~ 130, and an experimental determination of h will give an approximate value of p. The boundary condition that v ---- vo at r ---- a assumes that the photoresist stripping process causes a neg- ligible change in the atomic oxygen concentration out- side the region between the wafers. Since atomic oxygen generation or recombination between the wafers is assumed not to happen, con- servation of atomic oxygen requires that v be a solution to V2v = 0. The solution to the equation which con- forms to the boundary conditions is (3) Io(anr){aneOSanZ -}- hsinanz} sinanf + ~n (i -- c0sanf) of calibrated set screws. The experimental procedure was to scrape the photo- resist off a small area at the center and the edge of the wafer, measure the photoresist thickness with a Sloan and Dektak surface profile measuring system, strip the wafer for a length of time sufficient to remove a sig- nificant amount, but not all, of the photoresist from the center and edge of the wafer, and measure the thickness of the residual photoresist. Analysis The difference in strip rate between the edge and center of a wafer in a linear array can be calculated, if an assumption is made about the probability that an oxygen atom will chemically react with the photo- resist when it hits it. Assume two 2 in. wafers parallel to one another spaced 3/16 of an inch apart with their surfaces paral- lel to an axis through their centers, with photoresist covering the facing surfaces. The problem is to cal- dv culate the concentration of atomic oxygen as a function D dz of position between the wafers. If it is assumed that the primary method of atomic oxygen removal is by chemical reaction with the photoresist and volume re- combination mechanisms can be ignored, and if it is further assumed that the by-products of the reaction are present in concentrations that have a negligible and effect on the reaction rate, then the problem becomes mathematically the same as the conduction of heat in D a solid cylinder with a radiation boundary condition dz at the photoresist-covered wafer surfaces and a con- stant oxygen concentration or temperature at the side of the cylinder formed by drawing lines between the edges of the wafers. Let v represent the atomic oxygen concentration, z represent the distance along the axis between the wafers, one of which is at z ~ 0 and the other at z = .[, and r represent the radial distance from the center of the wafers to the edges which are at r = a. The radia- tion boundary condition equates the diffusion current , where D is the diffusion into the wafers, D "~z ~=o Io(ana) {(an ~ + h z) f + 2h} where the an are the positive roots of 2ah tan af ---- a 2 _ h z 6 Zo(~nr) = X.r -~=0 (m!) 2 is the zero order solution to the modified Bessel equa- tion. By the ratio test the series is absolutely con- vergent. Since the convergence is independent of z and r, it is uniformly convergent in any bound region. Thus v and its derivatives are continuous functions of z and r, and term by term differentiation of its series representation is permissible. The chemical reaction rate at any point on the wafer is equal to the diffusion current at that point into the wafer, or = 2Dvo z=O 'n=l r~a h anh ( sin,~nf + --(1 -- cos anf) } (~2n + h2)f + 2h z:O r=O 2Dvo f h anh ~ sin ~nf + M (1- cos,~nf)y ~n {(aen q- h2) f q- 2h}Io(ana) In numerically evaluating Io in the interval r ----- 0 to r : a, it should be noted that it is a power series, and the remainder value of the terms not calculated is equal to or less than the first term not calculated, evaluated at a. As ~nf gets large
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`Vol. 124, No. 3 CHEMICAL REACTION RATES 439 2anfhf tan an f -" (~j)2_ (kf)~ (..y)2 _ (n _ m)r~nf - 2Uy --- 0 if an ~ h, and m is the number of the first root to fall in the first quadrant, and 1 1 2h~" ~_ (n - m)~ + (n - m) if (n -- m)~ >> 8hr. Also h anh{ sin anf + -- (I -- cos anj') } ao R--i= ~" becomes (an 2 + h 2)f + 2h 2h2f n-m=r (n-- m)2n 2 for even n -- m, and goes to zero for odd n -- m. Now n-m=2 (n -- ~7~)2even 24 so the remainder, Re, for even n -- m is anh sin anf + -- (1 -- cos ~nf) Re - ~.r n-m=r (an 2 + h 2) f + 2h 2h2 f ~2 1 ~2 24 n-m=2 (n-- m) 2 The reaction rates at the edge of the wafer and at the center were calculated for values of h of 25, 2.5, 0.25, 0.025, and 0.0025. The results are shown in Table I. For the reaction rate at the edge of the wafer, 32 terms are calculated for h -- 25, 5 for h = 2.5 and 0.25, 3 for h = 0.025, and 2 for h = 0.0025 before cal- culating Re. A single term is adequate for the reaction rate at the center in all instances. The terms corre- sponding to values of anf lying close to odd n~'s are h quite uncertain since sin anf + -- (1 -- cos an f) is an the difference between two approximately equal num- bers. However, since these terms are small, it does not make much difference. The root equation can be written so that it is a func- tion of hf and af. Thus the roots listed here can be used for a different f spacing by making an appro- priate adjustment of the h values. Thus, the anf'S cal- culated here for 3/16 in. spacing can be used for 3/8 in. spacing by associating them with h values smaller by a factor of two than those listed here. Also, by symmetry, the case of one of the facing wafer surfaces coated and a 3/16 in. spacing corresponds to both faces coated at a 3/8 in. spacing. Similarly 1 d~ D Dvo dz z=0 r~a can be written so that it is equal to the product of h and a function that depends only on hf and a f, i.e. hi *, 2a,,f { sina.f + an--'~ (1- eos-,f)} ~=1 (any + hf) 2 Thus, the value of the function for a different f can be found by taking the value calculated here for 3/16 in. spacing but at an h value appropriate for the hf product physically desired. Thus, the functions calcu- lated here for 3/16 in. spacing can be used for 3/8 in. spacing by taking its value for an h that is twice the value physically desired. In Table I it can be observed that the calculated values of I dv ~D ......... Dvo dz z=O are equal to h within a few percent for h of 25 or 2.5, and more closely than that for smaller values of h. The implication of this is that the value of the function is substantially equal to one for any physically realizable values of h and values of j' small enough for the as- sumed boundary conditions to be applicable. I/D~,o dv/dz = h is the condition for a single plane surface. Thus, the strip rate at the edge of a wafer array is substantially the same as for a single wafer. If it is assumed that n CH2 groups are 'removed for every oxygen atom consumed, the chemical reaction rate for a single wafer is nhDvo. It is known that h -~ 0.158 cm -I at 160~ from the difference in etch rates at the edge and center in these experiments, vo has been measured by titrating the flow of atomic oxygen with NO2 and is about 7.4 X I014 at./cm 3. It is not clear what value of D should be used due primarily to the uncertainty in the gas temperature. However, a value of 700 cm/sec 2 does not seem unreasonable. For the power and pressure associated with vo, the strip time is about 94 sec. There are about 4.25 X 1021 CH2 groups per square centimeter, which leads to a value of n of about 550. Thus, although it has been shown that car- bon monoxide and water vapor can inhibit the reaction and thus are probably by-products of the reaction, it seems clear that most of the photoresist must come off in larger atomic groups. The roots and D d~/dz I z=0 do not depend on a, so the I r~a values of these quantities calculated here can be used for different diameter wafers. D dv/dz I z=o is inversely I r---~0 proportional to Io(ana). However, Io(ana) is propor- tional to a 2, and the calculated values of D d~/az I z=o [ r--O can be converted to a different diameter wafer by multiplying by the square of the ratio of the old diam- eter divided by the new diameter. For 4 in. wafers, the calculated values of D dv/dz [ r=oZ=~ for 2 in. wafers should be divided by four. If the values of h are small, 2ah/c~- 1r ~ 2h/a, tan %J' ~ aj' ~ 2h/~, and a ~ (2h/f) Vz. All the terms in the series summation for Io contain an an 2 and consequently are inversely proportional to f. Thus, the strip rate at the center is proportional to f. Thus, increasing wafer diameter by a factor, 5, leads to greater edge-to-center inhomogeneity in strip rates, which can be corrected by increasing wafer spacing by a factor f'~. Thus, a particular plasma stripping system has a capacity which can be specified in silicon area, independent of wafer diameter. Table I. Calculated relative strip rates at the edge and center of an array of wafers as a function of the relative probability that an oxygen atom reacts when it strikes the photoresist 1 dp I 1 d~, I --D-- D-- D~o dz ~:o D~o dz ~=o r=a r=o 25 24.19 1.416 (cid:141) 10 -~ 2.5 2.442 7.712 (cid:141) 10 -s 0.25 0.2494 0.04813 0.025 0.02500 0.02126 0.0025 0.002500 0.002458
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`440 J. EZectrochem. Soc.: SOLID-STATE SCIENCE AND TECHNOLOGY March 1977 9,000- 8 ,000- 7,000- 6,000- ._c S ,OOO" E o4 <C 4,000- cO 3,000- 2,000- 1,000- O O O O EDGE O~O O O O O 0 40DeC/finn 02 I 1.0 TORR I00 WATTS IEO~ o/ o o o~ ~ o/O / /~ / / I I 0.040 O.OSO I I I I 0.120 0.160 0,200 WAFER sEPARATION iNCHES Fig. 2. Strip rate at the wafer edge and center as a function of wafer separation under conditions of constant molecular oxygen flow rate, pressure, and power. The edge strip rate is independent of separation, whereas the center strip rate is linear in separation. 10,000' 9,000" 8,OOO. 7,000" 6,000" S ~000- c 4,000" I1: o_ 3,000- E,O00" (cid:12)9 400 CC/min 02 1,0 TORR ]00 WATTS o ?,r40 SEPARATION o o\ 0 0 CENTER 0 0 D Q 21 I ! I I I . 2.2 S 2.'t 2.5 2.6 {000 I TEMPERATURE ~ Fig. 3. Strip rate at the wafer edge and center as a function of temperature under conditions of constant molecular oxygen flow rate, pressure, and power. The edge strip rate varies exponentially as --E/RT, where E is approximately 6.5 kcal/mole. The center strip rate is approximately independent of temperature. Experimental Results The fact that D dz,/dz I z=o is proportional to f means I v=O that for small wafer spacing the center strip rate should be linear in f with a slope of 1 and, ignoring thermal effects and loading time, the wafer per hour capacity is independent of f. It is seen in Fig. 2 that the strip rate is linear in f. The reason for this is that larger wafer spacing allows more atomic oxygen be- tween the wafers. If the wafer spacing becomes too large, the boundary condition that v = ~o at r _-- a be- comes a poor approximation, and the analysis fails. Operation in the nonlinear region reduces the capacity of the equipment. The center strip rate experimentally approaches zero more rapidly at small f because f is approaching the mean free path which introduces an effect similar to a reduction in the diffusion coeffi- cient. In Fig. 3 the strip rate at the edge and the strip rate at the center are shown as a function of tem- perature. The strip rate at the edge varies exponen- tially as --E/RT where E is about 6.25 kcal/mole. To a first approximation the strip rate at the center is in- dependent of temperature in the range of 120~176 It should be emphasized that this is not true at lower temperatures where the center strip rates are much longer. Also, it will no longer be true once a ring around the edge of the wafer has cleared. Thus, an additional specification on a plasma stripping system is the time to heat a load of wafers from room tem- perature to 120~ The approximate temperature in- I0~ o ~ <[ uJ r a. Z.5" P o o_ CUBE OF WAFER DIAMETER INCHES S = A I I I I0 20 / SO SQUARE O WAFER DIAMETER IN HES 2 = [] Fig. 4. The ratio of the strip rate at the wafer edge and center as a function of wafer diameter under conditions of constant molecular oxygen flow rate, pressure, and power. The edge strip rate is independent of wafer diameter. The center strip rate is inversely proportional to the cube of the diameter.
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`Vol. I24, No. 3 CHEMICAL dependence of the center strip rate is explained by the fact that the product of the probability of the atomic oxygen atom reacting and the number that reach the center are approximately constant. In Fig. 4 the ratio of the strip rate at the center of the wafer to the strip rate at the edge is plotted as a function of the square and cube of the diameter of the wafer. The absolute strip rates were not very repro- ducible, presumably due to varying surface recom- bination rates at the jigging which held the wafer and the slide. However, the ratios were within a standard deviation of about 10%. Contrary to the predictions of the analysis, it appears that the ratio varies more nearly as the cube of the diameter. This can be qualitatively understood if it is assumed that two atomic oxygen atoms simultaneously react with a CH2 group to form CO and H20, or an appre- ciable amounf of the atomic oxygen recombines in a three-body reaction involving two oxygen atoms and the wafer. The latter explanation seems more probable since other experiments have shown the strip rate to be linear in oxygen concentration. Three bodies are needed to simultaneously conserve energy and mo- mentum. The correct boundary condition to V2~, ~- O would then involve equating the diffusion current of oxygen atoms into the wafer surface to a term pro- portional to the square of the oxygen atom concen- tration at the surface. Wafers of a given diameter have a certain ratio of oxygen concentration and strip rate at the edge and center. A larger wafer may be thought of as a wafer of the smaller diameter surrounded by an additional REACTION RATES 441 ring. The ratio of oxygen concentration and strip rate at the smaller diameter and center should be the same for the larger wafers as for the smaller wafers. How- ever, the ratio at the larger diameter to the smaller diameter would be lower for the square law than the linear boundary condition for the larger wafers be- cause the region is one of higher atomic oxygen con- centration than the region inside the smaller diameter. If the atomic oxygen concentration and strip rate are presumed to be a fixed quantity at the edge of the wafers, these quantities will be lower at the smaller diameter and center of the larger wafers than the linear boundary condition predicts. Thus, thesquare law boundary condition has the effect of making the ratio of the strip rate at the center to the edge have a larger than square law dependence on the diameter of the wafers. Manuscript received Sept. 2, 1976. Any discussion of this paper will appear in a Discus- sion Section to be published in the December 1977 JOURNAL. All discussions for the December 1977 Discus- sion Section should be submitted by Aug. 1, 1977. Publication costs of this article were assisted by the International Plasma Corporation. REFERENCES 1. J. F. Battey, IEEE Trans. Electron Devices, ed-24, 140 (1977). 2. A. T. Bell and K. Kwong, AIChE J., 18, 990 (1972). 3. H. S. Carslaw and J. C. Jaeger, "Conduction of Heat in Solids," p. 220, Clarendon Press, Oxford (1959). 4. J. F. Battey, This Journal, 124, 147 (1977). Liquid Phase Epitaxial Growth of Semi-insulating GaAs Crystals Mutsuyuld Otsubo and Hidejiro Miki Mitsubishi Electric Corporation, Central Research Laboratories, Itami, Hyogo 664 Japan ABSTRACT Semi-insulating GaAs crystals were grown by liquid phase epitaxy from a Ga solution doped with Cr metal. The room temperature resistivity of the layers was 10 -2 ,~ 107 D, (cid:12)9 cm, del:ending on the doping concentration of Cr, the background donor concenetration, arsenic vapor pressure, and the cooling rate. It was found that the grown layers became semi-insulating only when the background donor concentration of the undoped layer is lower than 1.0 X 1025 cm -~. Higher arsenic vapor pressure and the higher cooling rate give layers with higher resistivity. The threshold voltage, at which the I-V curve changes from linear to 9"3~4 dependence, varies with the square of the thick- ness with electron injecting contacts. The activation energy obtained from the temperature dependence of resistivity increases as the room temperature resistivity increases. The highest obtained was 0.71 eV. Semi-insulating GaAs crystals have been widely used as the substrates for microwave devices such as planar-type Gunn diodes, Schottky barrier gate field effect transistors (MESFET's), and integrated circuits. Semi-insulating GaAs bulk crystals can be grown from melt by O2 doping (1-3) and Cr doping (4). In O2-doped semi-insulating GaAs crystals, however, it is difficult to produce a homogeneous crystal over an ingot (3, 5-6) and the resistivity in the crystal is readily changed by heat-treatment (5, 7). From these points of view, bulk semi-insulating crystals are currently produced from melt by Cr doping. Cronin and Haisty (4) showed that the solubility of Cr in GaAs is on the order of 1017 cm -~ and an effective segregation coefficient is 6.4 X 10 -4. So, more than 1029 ,~ 1020 cm -3 of Cr should be added to the melt Key words: high resistivity materials, III-V compounds, field effect transistors, chromium metals, planar devices. in order to compensate the background donor con- centration of 1026 ,~ 1017 cm -3. Such a high Cr con- centration in the melt suggests the presence of a second phase. In applications for microwave devices such as planar-type Gunn diodes and MESFET's, the device performances at higher frequency are affected by the semi-insulating substrate properties (8-9). The most successful technique is use of an epitaxial buffer layer between a semi-insulating substrate and an epitaxial layer (10-11). The buffer layer must be high quality with very low net doping and, it is hoped, with lower total impurity content than the substrate materials. So epitaxial growth technology is required for the preparation of semi-insulating GaA s films with higher quality. The semi-insulating GaAs crystals with room tem- perature resistivities greater than 10 s ~2 (cid:12)9 cm can be grown from vapor phase by using chromyl chloride
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`LAM Exh 1002-pg 5