Process optimization in gas phase dry etching

A method of designing a reactor 10. The present reactor design method includes steps of providing a first plasma etching apparatus 10 having a substrate 21 therein. The substrate includes a top surface and a film overlying the top surface, and the film having a top film surface. The present reactor design method also includes chemical etching the top film surface to define a profile 27 on the film, and defining etch rate data from the profile region. A step of extracting a reaction rate constant from the etch rate data, and a step of using the reaction rate constant in designing a second plasma etching apparatus is also included.

BACKGROUND OF THE INVENTION 
The present invention relates to integrated circuits and their manufacture. 
The present invention is illustrated in an example with regard to plasma 
etching, and more particularly to plasma etching in resist strippers in 
semiconductor processing. But it will be recognized that the invention has 
a wider range of applicability in other technologies such as flat panel 
displays, large area substrate processing, and the like. Merely by way of 
example, the invention may be applied in plasma etching of materials such 
as silicon, silicon dioxide, silicon nitride, polysilicon, photoresist, 
polyimide, tungsten, among others. 
Industry utilizes or has proposed several techniques for plasma etching. 
One such method is conventional chemical gas phase dry etching. 
Conventional chemical gas phase dry etching relies upon a reaction between 
a neutral gas phase species and a surface material layer, typically for 
removal. The reaction generally forms volatile products with the surface 
material layer for its removal. In such method, the neutral gas phase 
species may be formed by way of a plasma discharge. 
A limitation with the conventional plasma etching technique is obtaining 
and maintaining etching uniformity within selected predetermined limits. 
In fact, the conventional technique for obtaining and maintaining uniform 
etching relies upon a "trial and error" process. The trial and error 
process often cannot anticipate the effects of parameter changes for 
actual wafer production. Accordingly, the conventional technique for 
obtaining and maintaining etching uniformity is often costly, laborious, 
and difficult to achieve. 
Another limitation with the conventional plasma etching technique is 
reaction rates between the etching species and the etched material are 
often not available. Accordingly, it is often impossible to anticipate 
actual etch rates from reaction rate constants since no accurate reaction 
rate constants are available. In fact, conventional techniques require the 
actual construction and operation of an etching apparatus to obtain 
accurate etch rates. Full scale prototype equipment and the use of actual 
semiconductor wafers are often required, thereby being an expensive and 
time consuming process. 
From the above it is seen that a method and apparatus of etching 
semiconductor wafers that is easy, reliable, faster, predictable, and cost 
effective is often desired. 
SUMMARY OF THE INVENTION 
According to the present invention, a plasma etching method that includes 
determining a reaction rate coefficient based upon etch profile data is 
provided. The present plasma etching method provides for an easy and cost 
effective way to select appropriate etching parameters such as reactor 
dimensions, temperature, pressure, radio frequency (rf) power, flow rate 
and the like by way of the etch profile data. 
In a specific embodiment, the present invention provides an integrated 
circuit fabrication method. The present method includes steps of providing 
a plasma etching apparatus having a substrate therein. The substrate 
includes a top surface and a film overlying the top surface. The film 
includes a top film surface. The present method also includes chemically 
etching the top film surface to define an etching profile on the film, and 
defining etch rate data which includes an etch rate and a spatial 
coordinate from the etching profile. A step of extracting a reaction rate 
constant from the etch rate data, and using the reaction rate constant in 
adjusting a plasma etching apparatus is also included. 
In an alternative specific embodiment, the present invention also provides 
a method of designing a reactor. The present method includes providing a 
first plasma etching apparatus having a substrate therein. The substrate 
has a top surface and a film overlying the top surface. The film has a top 
film surface. The present method also includes chemically etching the top 
film surface to define an etching profile on the film, and defining etch 
rate data which has an etch rate and a spatial coordinate from the etching 
profile. A step of extracting a reaction rate constant from the etch rate 
data, and using the reaction rate constant in designing a second plasma 
etching apparatus is also included. 
A further alternative embodiment provides another method of fabricating an 
integrated circuit device. The present method includes steps of providing 
a uniformity value for an etching reaction. The etching reaction includes 
a substrate and etchant species. The present method also includes defining 
etching parameter ranges providing the uniformity value. A step of 
adjusting at least one of the etching parameters to produce a selected 
etching rate is also included. The etching rate provides an etching 
condition for fabrication of an integrated circuit device. 
The present invention achieves these benefits in the context of known 
process technology. However, a further understanding of the nature and 
advantages of the present invention may be realized by reference to the 
latter portions of the specification and attached drawings.

DESCRIPTION OF THE SPECIFIC EMBODIMENT 
Plasma Etching Apparatus 
FIG. 1 is a simplified diagram of a plasma etching apparatus 10 according 
to the present invention. The plasma etching apparatus also known as a 
co-axial reactor includes at least three processing zones. The three 
processing zones are defined as a plasma generating zone (PG) 13, a 
transport zone (TZ) 15, a plate stack zone (PS) 17, and others. Also shown 
are a chemical feed F and exhaust E. The plasma generating zone provides 
for reactant species in plasma form and others. Excitation is often 
derived from a 13.56 MHz rf discharge 8 and may use either capacitor 
plates or a wrapped coil, but can also be derived from other sources. The 
co-axial reactor 10 also includes a chemical controller 14 and a 
temperature and pressure control 12, among other features. 
Chemical effects are often enhanced over ion induced effects and other 
effects by way of perforated metal shields 18 to confine the discharge to 
a region between an outer wall 16 and shields 18. The co-axial reactor 
relies substantially upon diffusion to obtain the desired etching 
uniformity. The co-axial reactor also relies upon a chemical etch rate 
which is diffusion limited. In particular, the chemical etch rate is 
generally defined as a chemical reaction rate of etchant species plus at 
least a diffusion rate of etchant species. When the diffusion rate of 
etchant species is much greater than the chemical reaction rate, the 
chemical etch rate is often determined by the diffusion rate. A more 
detailed analysis of such chemical etch rate will be described by way of 
the subsequent embodiments. 
Etchant species from the plasma generating zone diffuse through the 
transport zone 15 of the reaction chamber, and enter the plate stack zone 
space over surfaces of substrates 21. A concentration of etchant in the 
transport zone, which is often annular, between the plasma generating zone 
and the plate stack zone is defined as n.sub.oO. As etchant diffuses 
radially from the transport zone into the plate stack zone and over 
surfaces of the substrates, it is consumed by an etching reaction. A 
reactant concentration above the substrate can be defined as n.sub.o 
(r,z), where r is the distance from the center of the substrate and z is 
the distance above the substrate. A diffusive velocity v.sub.o of etchant 
species in the plate stack zone is characterized by Fick's law. 
##EQU1## 
In a specific embodiment, a gap d.sub.gap above the substrate is much less 
than the lateral extent d.sub.gap &lt;&lt;r and gas phase mass transfer 
resistance across the small axial distance is negligible so that the axial 
(z-direction) term of the concentration gradient can be ignored. The 
embodiment can be applied without this restriction; however, numerical 
mesh computer solutions are then required to evaluate the reaction rate 
constant and uniformity. In the embodiment, the surface etching reaction 
bears a first order form: 
EQU O+S.fwdarw.SO 
where 
S is a substrate atom (e.g., resist unit "mer"); and 
O is the gas-phase etchant (for example oxygen atoms) with certain etching 
kinetics. The first order etching reaction can be defined as follows: 
##EQU2## 
where 
R.sub.os defines a reaction rate; 
n.sub.o defines a concentration; 
A defines a reaction rate constant; 
T defines a temperature; 
E.sub.ACT defines an activation energy; and 
R defines a gas constant. An example of the first reaction is described in 
D. L. Flamm and D. M. Manos "Plasma Etching," (1989), which is hereby 
incorporated by reference for all purposes. Of course, other order 
reactions, reaction relations, and models may be applied depending upon 
the particular application. 
An example of an etched substrate 21 from the plate stack zone is 
illustrated by FIG. 1A. The substrate 21 is defined in spatial coordinates 
such as z and r. The substrate includes a bottom surface 23, sides 25, and 
a top surface film 27. As can be seen, the top surface film includes a 
convex region, or etching profile. The etching profile occurs by way of 
different etch rates along the r-direction of the substrate corresponding 
to different etchant species concentrations. A concentration profile 
n.sub.o (r,z) is also shown where the greatest concentration of reactant 
species exists at the outer periphery of the top surface film. In the 
present invention, an etch rate constant may be obtained by correlation to 
the etching profile. Byway of the etch rate constant, other etching 
parameters such as certain reactor dimensions including a distance between 
substrates, pressure, temperature, and the like are easily calculated. 
FIG. 2 illustrates an alternative example of an etching apparatus 50 
according to the present invention. The etching apparatus 50 is a single 
wafer etching apparatus with elements such as a chamber 53, a top 
electrode 55, a bottom electrode 57, a power source 59, a platen 64, and 
others. The bottom electrode 57 is at a ground potential, and the top 
electrode is operably coupled to the power source 59 at a high voltage 
potential. A plasma exists in a region 54 between the top electrode 55 and 
the bottom electrode 57, which is often a grid configuration or the like. 
Reactant species are directed via power source from a plasma source to a 
wafer substrate 61 by diffusion. A temperature and pressure controller 67 
and a flow controller 69 are also shown. The etching apparatus also 
includes a chemical source feed F and a exhaust E. Of course, other 
elements may also be available based upon the particular application. 
By way of a plate 63 interposed between the wafer substrate 61 and the 
bottom electrode 57, the reactant species do not directly bombard the 
wafer substrate. The plate is preferably made of an inert material 
appropriate for the particular etching such as pyrex or glass for resist 
ashing, alumina for fluorine atom etching of silicon, silicon nitride, or 
silicon dioxide, and the like. In an ashing reaction, the plate is placed 
at a distance ranging from about 5 mm to 50 mm and less from the wafer 
substrate 61. Of course, other dimensions may be used depending upon the 
particular application. The reactant species are transported via diffusion 
from the plasma source to the wafer substrate around the periphery of the 
plate 63. Accordingly, the reaction rate at the wafer substrate is 
controlled by a balance between chemical reaction and diffusion effects, 
rather than directional bombardment. 
By way of the diffusion effects, an etching rate constant may be obtained 
for the etching apparatus 50 of FIG. 2. In particular, the etching rate 
constant derives from a etching profile 65, which can be measured by 
conventional techniques. The present invention uses the etching rate 
constant to select other etching rate parameters such as reactor 
dimensions, spacing between the substrate and its adjacent surface, 
temperatures, pressures, and the like. But the present invention can be 
used with other reactor types where etching may not be controlled by 
diffusion. For example, the present invention provides a reaction rate 
which can be used in the design of reactors where diffusion does not 
control such as a directional etcher and the like. The reaction rate 
constant may also be used in the directional etcher to predict an extent 
of, for example, undercutting of unprotected sidewalls while ion 
bombardment drives reaction in a vertical direction. Of course, the 
invention may be applied to other reactors such as large batch, high 
pressure, chemical, single wafer, and others. The invention can also be 
applied to different substrate materials, and the like. 
Plasma Etching Method 
FIGS. 3-5 illustrate simplified flow diagrams of plasma etching methods 
according to the present invention. The present methods provide for 
improved etching conditions by way of a reaction rate constant derived 
from, for example, an etching profile. It should be noted that the present 
methods as illustrated should not be construed as limiting the invention 
as defined in the claims. One of ordinary skill in the art would easily 
recognize other applications of the inventions described here. 
In a specific embodiment, a method of extracting a rate constant 100 for a 
plasma etching step according to the present invention is illustrated by 
the flow diagram of FIG. 3. A substrate with an overlying film is placed 
into a plasma etching apparatus or the like. The overlying film is defined 
as an etching film. In the present embodiment, the overlying film is a 
photoresist film, but can also be other films such as a silicon film, a 
polysilicon film, silicon nitride, silicon oxide, polyimide, and the like. 
A step of plasma etching the film is performed by step 101. The plasma 
etching step occurs at constant pressure and preferably constant plasma 
source characteristics. More preferably, the plasma etching step occurs 
isothermally at temperature T.sub.1, but can also be performed with 
changing temperatures where temperature and time histories can be 
monitored. Plasma etching of the film stops before the endpoint (or etch 
stop). Alternatively, plasma etching stops at a first sign of the endpoint 
(or etch stop). The plasma etching step preferably stops before etching 
into an etch stop layer underlying the film to define a "clean" etching 
profile. 
The substrate including etched film is removed from the chamber of the 
plasma etching apparatus. The etched film includes an etching profile 
(step 103) made by way of plasma etching (step 101). The etching profile 
converts into a relative etch rate, relative concentration ratio, a 
relative etch depth, and the like at selected spatial coordinates. The 
relative etch rate is defined as an etch rate at a selected spatial 
coordinate over an etch rate at the substrate edge. The relative 
concentration ratio is defined as a concentration of etchant species at a 
selected spatial coordinate over a concentration of etchant at the 
substrate edge. 
In x-y-z coordinates, the relative etch rate in the z-direction, and the 
spatial coordinates are defined in the x-y coordinates. The etching 
profile is thereby characterized as a relative etch rate u, a x-location, 
and a y-location u, (x, y). In cylindrical coordinates, the relative etch 
rate is also in the z-direction, and the spatial coordinates are defined 
in the r and .theta. coordinates. The etching profile is characterized as 
a relative etch rate u, a r-location, and a .theta.-location (u, r, 
.theta.). An array of data points in either the x-y coordinates or 
r-.theta. coordinates define the etching profile. The array of data points 
can be defined as an n.times.3 array, where n represents the number of 
points sampled and 3 represents the etch rate and two spatial dimensions. 
Of course, the choice of coordinates depends upon the particular 
application. 
Optionally, in a non-isothermal condition, an average etch rate is 
measured. By approximate integration of a time dependent etch rate, 
suitable starting point approximations for an etching rate constant 
pre-exponential and activation energy can be selected. The etch rate is 
integrated over time (and temperature) using measured temperature-time 
data (or history). An etched depth profile and the etching rate from the 
integration can then be compared with actual data. A rate constant is 
appropriately readjusted and the aforementioned method is repeated as 
necessary. 
An etch constant (or a reaction rate constant) over diffusivity (k.sub.vo 
/D) and an etch rate at an edge is calculated at step 105. The etch 
constant over diffusivity correlates with data points representing the 
etch rate profile. In x-y coordinates, the relationship between k.sub.vo 
/D and the relative etch rate u(x,y) is often defined as follows: 
where 
a and b define substrate lengths in, respectively, an x-direction and a 
y-direction. 
##EQU3## 
In cylindrical coordinates, the relationship between the etch constant 
over diffusivity k.sub.vo /D and the relative etch rate u(r) is defined as 
follows: 
##EQU4## 
where 
a is an outer radius (or edge) of the substrate and I.sub.o is modified 
Bessel function of the first kind. 
In step 106, a diffusivity is calculated for the particular etchants. The 
binary diffusivity D.sub.AB may be calculated based upon the well known 
Chapman-Enskog kinetic theory equation: 
##EQU5## 
where 
T is a temperature; 
c is a total molar concentration; 
M.sub.A and M.sub.B are molecular weights; 
D.sub.AB is a binary diffusivity; 
.sigma..sub.AB is a collision diameter; and 
.OMEGA..sub.D,AB is a collision integral. 
The Chapman-Enskog kinetic theory equation is described in detail in part 
III of R. B. Bird, W. E. Stewart, and E. N. Lightfoot, "Transport 
Phenomena," Wiley (1960) which is hereby incorporated by reference for all 
purposes. Of course, other techniques for calculating a diffusivity may 
also be used. The equivalent volumetric reaction rate constant k.sub.vo is 
derived from the diffusivity as follows. 
##EQU6## 
Once the reaction rate constant k.sub.vo is extracted, the surface 
reaction rate constant k.sub.s may be isolated from the previous equation 
as follows. 
EQU K.sub.s =(k.sub.vo)d.sub.gap 
Repeat steps 101-106 at different temperatures T.sub.2, T.sub.3. . . 
T.sub.n to calculate additional reaction rate constants k(T.sub.2), 
k(T.sub.3) . . . k(T.sub.n). The steps are repeated at least two times and 
more, and preferably at least three times and more. Each temperature is at 
least 5.degree. C. greater than the previous temperature. Of course, the 
selection of temperatures and trial numbers depend upon the particular 
application. 
Extract an activation energy E.sub.act for a first order reaction from the 
data k(T.sub.2), k(T.sub.3) . . . k(T.sub.n) at T.sub.2, T.sub.3 . . . 
T.sub.n collected via step 109 by way of the following equation: 
##EQU7## 
The activation energy is preferably calculated by a least square fit of 
data collected at step 109 or any other suitable statistical technique. By 
way of the same equation, the present method calculates surface reaction 
rate constant k.sub.s at any temperature. 
In step 111, a concentration n.sub.o at the substrate edge is calculated. 
The concentration n.sub.o deduces from the following relationship: 
EQU n.sub.o =R.sub.15 /K.sub.s 
where 
R.sub.os is an etch rate. 
From the concentration and the surface reaction rate, the particular 
etching step can be improved by way of adjusting selected etching 
parameters. 
In an alternative specific embodiment, a method to "tune" a plasma source 
using a loading effect relationship (or equation) is illustrated by the 
simplified flow diagram 200 of FIG. 4. The method includes a step 201 of 
measuring an etch rate against an effective etchable area A.sub.w. The 
effective etchable area changes by varying the number m of wafers in the 
reactor, varying the size of the wafer, or the like. The effective area 
can be changed 209 by altering a gap between a wafer and its above surface 
211, changing wafer quantity in the reactor 213, and varying substrate 
support member dimensions 215. The method preferably occurs at constant 
temperature and pressure. However, the effective etchable area may also be 
varied by way of changing a temperature and/or a pressure. 
The method calculates a uniformity value (step 217) from the measured 
values of etch rate vs. effective area in steps 211, 213, and 215. The 
uniformity is calculated by, for example: 
##EQU8## 
where 
R.sub.MAX is a maximum etch rate; 
R.sub.MIN is a minimum etch rate; 
m is a sample number; 
R.sub.i is a general etch rate for an ith sample; uniformity is a planarity 
measurement in percentage. 
In a specific embodiment, a uniformity of about 90% and greater or 
preferably 95% and greater indicates that the effective area of the 
substrate is substantially equal to the actual substrate area (step 221) 
via branch 216. Of course, other methods of calculating a uniformity from 
etch rates and effective areas may also be used depending upon the 
particular application. Alternatively, an etching profile is measured and 
the effective area A.sub.eff is calculated (step 219) by way of, e.g., the 
loading effect relationship. 
At least two and more different effective etchable areas (step 223) are 
measured, or preferably at least three and more different etchable areas 
are measured. Alternatively, the flow diagram returns via branch 224 to 
step 209, and takes another etch rate measurement at a different effective 
area. The flow diagram then turns to step 203. 
In step 203, a supply of etchant S.sup.T in the reactor is calculated. 
Based upon the different etchable areas a slope mA.sub.eff deduces from 
the loading effect relationship as follows. 
##EQU9## 
where R.sub.os (m) is the etching rate at the boundary between the plate 
stack zone and transport zone when m substrates are present in the 
reactor. The first term includes a recombination term proportional to the 
total effective area A.sub.r which acts to catalyze loss of etchant on 
reactor surfaces in the reactor plus a convection term F. The second term 
is the loading effect relation, where the reciprocal etch rate is 
proportional to the amount of effective etchable substrate area A.sub.eff 
times the number of substrates m. When the etching across a substrate is 
uniform, A.sub.eff is the geometrical substrate area A.sub.w. When etching 
is nonuniform, on the other hand, A.sub.eff is a function of k.sub.vo /D 
and geometrical reactor dimensions. The supply of etchant S.sup.T may be 
calculated for a different plasma source or plasma source parameters such 
as temperature, pressure, or the like by repetition 207 of steps 201 and 
203. By way of the supply of etchant to the reactor, other plasma source 
parameters may be varied to obtain desired etching rates and uniformity 
for the particular reactor. 
Step 205 provides for the modification of chamber materials and the like to 
reduce slope numerator (k.sub.r, A.sub.r +F) in selecting the desired 
etching conditions. The chamber materials can be modified to reduce, for 
example, the recombination rate in the reactor. The recombination rate is 
directly related to the effective reactor recombination area A.sub.r. In 
step 205, the recombination rate can be adjusted by changing A.sub.r via 
changing chamber material, coating chamber surfaces with, for example, a 
product sold under the trademark TEFLON.TM. or KALREZ.TM. and the like, 
among others. Alternatively, the slope numerator flow term F is reduced 
when F contributes as a substantial loss term. Of course, the particular 
materials used depend upon the application. 
In step 207, the method changes plasma source parameters such as rf power, 
flow rate, and the like to select desired etching conditions. Once one of 
the aforementioned parameters is adjusted, the method returns to step 201 
via branch 208. At step 201, an etch rate vs. effective etchable area is 
measured and the method continues through the steps until desired etching 
condition are achieved. Of course, other sequences of the aforementioned 
step for tuning the plasma source may also exist depending upon the 
particular application. 
FIG. 5 is a simplified flow diagram for a method of selecting a desired 
uniformity and desired etching parameters within selected ranges to 
provide a desired etch rate for a particular etching process. The etching 
parameters include process variables such as reactor dimensions, a 
pressure, a temperature, and the like for a particular substrate and 
reactants. Other etching parameters may also be used depending upon the 
particular application. 
In step 301, select a uniformity for the selected substrate and the 
reactants. The selected uniformity becomes an upper operating limit for 
the reaction according to the present method. The upper operating limit 
ensures a "worst case" uniformity value for an etched substrate according 
this method. Uniformity can be defined by, for example: 
##EQU10## 
where 
R.sub.MAX is a maximum etch rate; 
R.sub.MIN is a minimum etch rate; 
m is a sample number; 
R.sub.i is a general etch rate for an ith sample; uniformity is a planarity 
measurement in percentage. 
In certain embodiments, the selected uniformity ranges from about 90% and 
greater or more preferably 95% and greater. Of course, other uniformity 
values may be selected based upon the particular application. 
Based upon the selected uniformity, use the selected uniformity as a 
stating point to extract a plurality of reaction rate constants k.sub.s. 
The reaction rate constants may be also be obtained by an input activation 
energy for the etching process, among other techniques (step 303). 
Alternatively, calculate k.sub.s at one or more temperatures, and 
preferably two or more temperatures (step 303) from a plurality of 
uniformity values. The uniformity values can be within the selected 
uniformity or outside the selected uniformity. 
In step 307, prepare an array of etching parameters including a temperature 
T, a pressure P, a characteristic reactor dimension, and a uniformity 
value. In an embodiment, the characteristic reactor dimension can be a gap 
d.sub.gap between the substrate and its adjacent surface. The array of 
etching parameters can be illustrated by way of a three dimensional plot. 
An example of such array is illustrated by way of a three dimensional plot 
500 in FIG. 5A. It should be noted that the illustration is merely an 
example of one application of the specific embodiment, and other examples 
can readily be determined by one of ordinary skill in the art. The plot 
includes a temperature axis, a pressure axis, and a gap axis. Each square 
region 501 represents a point defined by a specific temperature, pressure, 
and gap. Each square region 501 also includes a gray scale. Each different 
gray scale corresponds to a different uniformity value. In this example, 
the darker gray scale values 505 represent lower uniformity values than 
the lighter gray scale values 507. 
Based upon the array, compute locus of highest T, versus P and d.sub.gap, 
and of highest P, versus. T and d.sub.gap 511 where uniformity meets the 
specification, e.g., the selected uniformity from step 301. All points 
bounded within the highest T, versus P and d.sub.gap, and the highest P, 
versus T and d.sub.gap fall within the uniformity specification. Points 
outside the highest T. versus P and d.sub.gap and the highest P, versus T 
and d.sub.gap fall outside the uniformity specification. The points that 
fall within the Uniformity specification defines the calculated uniformity 
limit manifold having outer boundaries at P.sub.o and T.sub.o. 
In the calculated uniformity limit manifold, select a gap d.sub.gap, and 
adjust a locus of P and T below the calculated uniformity limit manifold 
by a predetermined amount to allow for statistical and experimental error 
and process drift. This step defines a new uniformity limit manifold, and 
ensures that points defined by a temperature, a pressure, and a gap, 
selected during subsequent steps fall within the selected uniformity (step 
301) despite any error or process drift from the calculation. The new 
uniformity limit manifold includes outer boundaries at P.sub.i and T.sub.i 
which are respectively less than P.sub.o and T.sub.o. 
In step 311, a maximum edge etch rate R.sub.os and supply of etchant from a 
plasma source (S) for a selected rf power, a reactant flow, a pressure, a 
temperature, and a gap within the new uniformity limit manifold is 
determined. The maximum edge etch rate can be used in defining a desired 
flow rate of source chemicals. Once the desired flow rate is determined, 
it should be held constant during subsequent steps in the embodiment. 
A step (step 313) of locating an intersection space of P&lt;Pi, T&lt;Ti, and a 
maximum etch rate (or an etchant supply) at selected rf power values is 
included. The intersection of space defines a maximum etch rate for the 
selected pressure P, temperature T, and gap d. Of course, other etching 
parameters may be adjusted depending upon the particular application. 
The method provides a resulting etch rate from the etching reaction using 
the aforementioned parameters which is compared with a desired etch rate. 
If the resulting etch rate is too low (or high), change power and/or 
reduce the effective etchable area, e.g., increase d.sub.gap, decrease 
number of substrates, use smaller substrates, and the like. Of course, 
other sequences of steps may be used in selecting a desired temperature, 
pressure, gap, and other parameters to provide the desired etch rate. The 
embodiment provides for a desired etch rate with a selected uniformity 
based upon a range of temperatures, pressures, and gap values, all within 
the selected uniformity specification. 
Theoretical Model of Apparatus 
1. Plasma Generating Zone 
In the specific embodiment, the plasma generating zone can be modeled as a 
"black box" where etchant flow of reactant species from the plasma 
generating source is determined from an etching rate at the plate stack 
zone. In particular, the etching rate is proportional to a product n.sub.o 
k.sub.s of etchant concentration n.sub.o above an etchable material film 
surface and an etching reaction rate constant k.sub.s. The etching 
reaction rate constant k.sub.s can be independently determined from 
uniformity data previously noted. Since the relative change in n.sub.o 
k.sub.s and the absolute value of k.sub.r (the effective surface 
recombination rate per unit reactor area) can be determined, n.sub.o is 
easily extracted and used to study the effects of discharge and surface 
parameters on production of etchant species in the plasma generating zone. 
Accordingly, the efficiency of radical production by the plasma generating 
zone (the source term in a mass the mass balance of n.sub.o) as a function 
of various parameters (pressure, power, temperature, etc.) can be 
extracted from indirect measurements. 
2. Transport Zone 
In the specific embodiment, etchant species concentrations in the transport 
space zone are approximated as "well-mixed". In the well-mixed embodiment, 
substantially all etchant species in the transport space zone are supplied 
by the plasma generating zone and are removed by at least: 1) etching 
reactions in the plate stack zone; 2) recombination; or 3) convection by 
flow out of the reactor. A supply S.sup.T of etchant from the plasma 
generating zone is equated to the three aforementioned loss terms as 
follows: 
EQU S.sup.T =k.sub.r A.sub.r n.sub.o +mA.sub.eff k.sub.s n.sub.o +Fn.sub.o 
where k.sub.r A.sub.r is an effective loss term with regard to 
recombination effects, k.sub.s is an etching reaction rate constant, 
A.sub.eff is an effective etchable area of a substrate, n.sub.o is the 
etchant concentration, m is the number of substrates and F is the gas flow 
rate out of the reactor. The equation may be rewritten in the form of a 
canonical loading effect relationship: 
##EQU11## 
Where R.sub.o (m) is the etching rate at the boundary between the plate 
stack zone and transport zone when m substrates are present in the 
reactor, and the first term includes a recombination term proportional to 
the total effective area A.sub.r which acts to catalyze loss of etchant on 
reactor surfaces plus convection F. The second term is the loading 
relation, wherein the reciprocal etch rate is proportional to the amount 
of effective etchable substrate area A.sub.eff times the number of 
substrates m. When the etching across a substrate is uniform, A.sub.eff is 
the geometrical substrate area A.sub.w. When etching is nonuniform, on the 
other hand, A.sub.eff is a function of k.sub.vo /D and geometrical reactor 
dimensions. Accordingly, A.sub.eff becomes a function of parameters such 
as temperature, pressure, reactor configuration, and the like. 
FIG. 6 shows etch rate data vs. the number of substrates in a reactor along 
with a line corresponding to the loading effect relationship in the form 
##EQU12## 
where 
C.sub.o =0.00030171936426 min/.ANG.; 
and C.sub.l A.sub.w =2.3003912550.times.10.sup.-5 are best fit constants 
for the conditions in FIG. 6. The equation gives an etching rate at the 
edge of the plate stack zone as a function of the number of substrates m 
and etchable exposed effective surface of a substrate A.sub.eff. Other 
variables such as the temperature, etchant generation rate, flow rate, and 
reactor size parameters were held constant. While R.sub.o (m) as written 
strictly applies to the etch rate at the edge of a substrate, when etching 
uniformity is high the etching rates at any other fixed relative position 
on the substrates are related to R.sub.o (m) by a constant factor of 
proportionality, and so they will also conform to the form of these 
relations. 
In the general case where etching is nonuniform across a substrate, the 
equivalent area A.sub.w is smaller than the geometrical substrate area by 
a constant factor as a function of k.sub.vo /D. It turns out that k.sub.s 
can be independently deduced from the profile of the etching rate in the 
stack zone, and in turn permits the absolute value of n.sub.o to be 
computed from the etching rate R.sub.o (m) at the edge of a substrate. If 
the slope of the isothermal loading effect curve 
##EQU13## 
is measured along with etching uniformity, the rate of etchant supplied by 
the source S.sup.T can be found by for substituting A.sub.eff (k.sub.vo 
/D) evaluated on the basis of etching uniformity measurements. 
3. Stack Zone 
For etchant mass transport from the transport zone into the plate stack 
zone, the distance between stacked wafers d.sub.gap is small compared to 
the lineal dimensions of a substrate in the embodiment. Consequently, it 
will be assumed that the concentration is substantially uniform in the 
axial z direction and there is equi-molal, isothermal, and isobaric 
counter-diffusion (e.g., no net flux, .SIGMA.n.sub.1 =0) x and y 
directions. Since the ashing reaction is proportional to n.sub.o, and 
O-atom consumption is proportional to the ashing rate, the continuity 
equation for O-atoms in two dimensions becomes: 
##EQU14## 
where k.sub.vo is the volume equivalent surface reaction rate constant, 
and v is the diffusive velocity of oxygen atoms. Inserting Fick's law 
EQU n.sub.o v=-D.gradient.n.sub.o 
the diffusion equation is obtained 
##EQU15## 
And at steady-state in two dimensions and where D is not a function of 
spatial coordinate(s), it is rewritten as 
##EQU16## 
Where u(x,y)=n.sub.o (x,y)/n.sub.oO and n.sub.oO is the etchant 
concentration at the outer edges of the substrates. The boundary 
conditions are therefore u=1. The equation is cast in dimensionless form 
as 
##EQU17## 
where L.sub.x and L.sub.y are characteristic independent lengths and 
widths of substrates. From this equation, it is clear that u(x/L.sub.x, 
y/L.sub.y) is a function solely of k.sub.vo /D and the boundary 
conditions. Consequently, if experimental values of u(x/L.sub.x, 
y/L.sub.y) are measured at two positions on the substrate (i.e., at the 
center and edge), two algebraic equations based on this measurement can be 
used to eliminate n.sub.o /n.sub.oO and solve for k.sub.vo /D. The 
diffusivity D can be calculated to good accuracy with the Hirshfelder 
equation; hence, k.sub.vo is measured with this procedure. 
For circular substrates, there is only one independent dimension (e.g., 
where r=a is the substrate radius). At steady state in one dimensional 
cylindrical coordinates the equation can be written: 
##EQU18## 
where u(r)=n.sub.o (r)/n.sub.oO and the boundary condition is u(a)=1 at 
the substrate (wafer) edge. 
In the subsequent sections, analytic solutions to these relationships are 
developed for rectangular and circular substrates (e.g., for flat panel 
display substrates and semiconductor wafers). The framework is used to 
derive uniformity relationships for flat panel resist stripping equipment. 
EXAMPLES 
1. Circular Substrate (Wafer) Stacked Etcher 
To prove the principles of the aforementioned embodiments, the present 
method and apparatus was applied to etching of circular substrates in a 
stacked etcher. Of course, the present method and apparatus can be applied 
to other geometries and etcher types. The present example is therefore not 
intended to be limiting in any way. The present method and apparatus is 
applied to the circular substrates as illustrated by way of FIG. 7. The 
present method relies upon etching of substrate material S by way of 
oxygen using a reaction which is substantially chemical etching. 
An illustration of a circular substrate according to the present invention 
is shown in FIG. 8. Assume that the distance d.sub.gap between stacked 
wafers is relatively small compared to the wafer radius a such that 
d.sub.gap &lt;&lt;a. Based upon the assumption, the oxygen concentration will be 
substantially uniform in the axial direction z. Accordingly, only radial 
diffusion in the r-direction needs consideration. Assuming an equi-molal 
counterdiffusion 
##EQU19## 
and an isobaric and isothermal stack zone, the problem reduces to two 
dimensions and becomes 
##EQU20## 
where u (r)=n.sub.o (r)/n.sub.oO. The boundary condition is u (a)=1 at a 
wafer edge, and the solution of the equation becomes 
##EQU21## 
where I.sub.0 and K.sub.0 are modified Bessel functions of the first and 
second kind, respectively, and c.sub.1 and c.sub.2 are constants. For a 
finite, normalized oxygen concentration u(O) at the center of the wafer, 
the equation requires c.sub.2 =0. The remaining boundary condition of 
u(a)=1, sets the solution: 
##EQU22## 
Note that the functional form u(r) describes both the relative etch rate 
profile R.sub.o (r)/R.sub.o (a) and the relative oxygen atom etchant 
concentration n.sub.o (r)/n.sub.o (a). The relative etch rate profile can 
easily be obtained by measuring an etching rate profile on a circular 
substrate made by way of the present method. 
FIG. 9 is a simplified plot of a normalized stripping rate vs. radial 
distance from a wafer center for the circular substrate example. The plot 
shows a profile of u(r) for k.sub.vo /D=0.1, and an a=150 mm. As can be 
seen, the normalized stripping rate is lower at a center region of the 
wafer, and increases to 1 at the wafer edge. Based upon a slope of the 
plot, a reaction rate coefficient can be extracted by way of a 
diffusivity. 
2. Rectangular Substrate Stack Asher 
To further provide the principle and operation of the present method and 
apparatus, the present method and apparatus is applied to a rectangular 
substrate configuration in a stack asher. Again, the present example 
should not be taken as limiting the scope of the claims described herein, 
but is merely an example. An analytical solution for etching profiles in 
the stack zone are derived for etching/ashing a stack of rectangular 
substrates as illustrated in FIG. 10. The rectangular substrate can be a 
flat panel display such as a liquid crystal display (LCD) plate and the 
like in the coordinate system of FIG. 10. To solve an equation for the 
present rectangular configuration where D is not dependent upon spatial 
coordinates, write the solution as: 
EQU u=u.sub.1 +u.sub.2 
where u.sub.1 is satisfied by the following equation 
##EQU23## 
where u.sub.1 =0 at y=.+-.b/2; 
and u.sub.2 is a solution that is 0 at x=.+-.a/2. The solution for u.sub.1 
(x,y)=X(x)Y(y) is obtained by a separation of variables as follows. 
##EQU24## 
The sign of the sum decomposing .lambda..sup.2 is chosen so that X(x) and 
Y(y) both have real values, as shown below. Since the boundary conditions 
on Y(y) are: 
EQU Y(-b/2)=Y(b/2)=0 
the solution is, 
EQU Y=c.sub.y cos.lambda..sub.y y. 
From the boundary conditions, c.sub.y =m/b where m=1,3,5, . . . Similarly, 
the solution for X is 
EQU x=c.sub.x cosh.lambda..sub.x x, 
where .lambda..sub.x is given by 
##EQU25## 
The general solution is the sum: 
##EQU26## 
where c.sub.m =0 for m=0, 2, 4, . . . to satisfy the boundary conditions. 
Setting u.sub.1 (a/2,y)=f(y), where f(y) is the even-function square wave 
of magnitude 1, the Fourier series is obtained, 
##EQU27## 
and after the integration 
##EQU28## 
which is zero when m is even, as required. Thus, the u.sub.1 part of the 
solution can be written 
##EQU29## 
Note that u.sub.1 (a/2,y)=1 for (-b/2&lt;y&lt;b/2). The solution for u.sub.2 can 
be obtained in a similar way. The solution is then 
##EQU30## 
where m is odd. As b.fwdarw..infin., this approaches the solution for 
1-dimensional diffusion (corresponding to an infinitely long strip): 
##EQU31## 
The previous two-dimensional equation is now applied to interpret ashing 
uniformity data and predict uniformity and the atomic oxygen concentration 
profile n.sub.o along the surface of a substrate for selected operating 
conditions. To use the relationship, values of k.sub.vo and D are 
required. For atomic oxygen diffusing through O.sub.2, diffusivity was 
computed as D(cm.sup.2 /s)=0.044T.sup.3/2 (T is in K) using relations in 
J. O. Hirschfelder, C. F. Curtiss, R. B. Byrd, "Molecular Theory of Gases 
and Liquids," pp. 538-541 and 578-582, John Wiley & Sons, 2nd Printing 
(1963), which is hereby incorporated by reference for all purposes. Of 
course, other techniques for calculating the diffusivity also exist. 
In general, k.sub.vo will be a function of at least gap, resist 
composition, temperature, and other parameters. In an example, k.sub.vo is 
unknown, although the activation energy for resist ashing is 
conventionally reported to be in the 11-12 kCal range from industry 
literature. However, the solutions for u(x,y) depend only on k.sub.vo /D 
and geometrical chamber dimensions such as gap (as incorporated into 
k.sub.vo), a, b, and the like. Accordingly, k.sub.vo /D is deduced from 
the etching rate profile, as previously described. 
In particular, k.sub.vo /D can be obtained from measurements of the amount 
of resist removed at two independent points (points where the 
theoretically predicted etch depth ratios u(X.sub.1,y.sub.1), 
u(X.sub.2,y.sub.2) are unequal by solving the appropriate equation for 
k.sub.vo /D and substituting for D(T,P). But the present example used a 
more robust procedure: determine k.sub.vo /D from a least squares fit to 
the entire experimental etch profile data set taken by a conventional 
stylus profilometer. 
FIG. 12 shows an experimental etching profile data taken on a 30.times.30 
cm resist-covered substrate spin-coated with 2.1 microns of MCPR 200 
resist (Mitsubishi Chemical Corp., equivalent to Tokyo Ohka Kogyo Co. OFPR 
800). A vertical axis 1201 defines an ashing rate R.sub.o with respect to 
an x-direction 1203 and a y-direction 1205. A grid pattern 1207 represents 
a "fitted" surface region via aforementioned equation representing ashing 
rates. Actual data points for each ashing rate are defined as the circular 
points 1211, and plots representing the fitted surface region are defined 
as cross points 1209. Ashing rate is greater around the periphery of the 
substrate, than substrate center regions. 
The reactor held 1.1 mm thick substrates with a 28.9 mm gap (d.sub.gap) 
above the wafer. A 4 kW rf power source sustained a plasma with pure 
oxygen gas flowing into the reactor at 3 liters/min. Thermocouple sensors 
and heaters kept the reactor chamber and substrates at T=220.degree. C. 
during the etch process, and a throttle valve maintained pressure at P=1.2 
Torr. Etching occurred for 5 min. Resist thickness was measured before and 
after etching using a Nanometrics Model 210 Nanospec Auto Film Thickness 
Monitor. The surface of FIG. 12 represents a least squares fit to the 
aforementioned equation for u(x,y) with k.sub.vo /D as the only adjustable 
parameter. The least squares fit gives k.sub.vo /D=0.047. At P=1.27 Torr 
and T=493K into D(cm.sup.2 /s)=0.044T.sup.3/2 yields D=400 Cm.sup.2 /s. By 
way of the relationship k.sub.vo /D=0.047, the etch rate constant is now 
k.sub.vo =19.5 sec..sup.-1. In the manner, e.g., by fitting profile data 
to the solution for given substrate geometry, k.sub.vo can be measured 
under various process conditions. By way of k.sub.vo, other parameters 
such as n.sub.o, k.sub.s, and the like may also be calculated. 
Once k.sub.vo is known as a function of temperature, ashing rate and 
uniformity can be calculated as a function of reactor size parameters 
(a,b) and process variables (p, T, and n.sub.o). While the etching rate is 
proportional to n.sub.oO, n.sub.oO does not affect the etch depth profile 
and need not be known to compute k.sub.vo. However, after k.sub.vo is 
obtained, n.sub.o0 can be computed from the experimentally measured 
etching rate per R.sub.os =k.sub.s n.sub.oO. The procedure applies up to 
endpoint (endpoint is the time at which resist has been "stripped" and is 
no longer covering the region of the substrate where etching was fastest). 
At endpoint, resist begins to be cleared from the substrate so that 
etchable area changes. Hence, n.sub.oO will start to change (increase) 
after endpoint. The magnitude of n.sub.oO during the steady-state period 
when resist is etching controlled by the plasma source, the number of 
substrates loaded into the reactor and (possibly) convective loss. 
Predicting Etch Rate 
The effect of profile uniformity on loading can be explicitly accounted for 
by defining a profile-average substrate area A.sub.eff 
##EQU32## 
so that n.sub.oO k.sub.vo A.sub.eff is the per substrate etchant 
consumption with nonuniformity resulting from effects of diffusion and 
reaction taken into account. Then for given plasma source (etchant 
supply), the etch rate/loading effect equation becomes: 
##EQU33## 
All of the terms can be computed explicitly from etch rate profile data, 
except for the rate of etchant production by the source. The etchant 
production rate can be computed from two measurements of etching rate when 
changing k.sub.vo A.sub.eff. A.sub.eff can be changed either by changing 
the number of substrates or changing the etch rate profile (with constant 
etchant supply). 
The present invention provides a method of selecting uniformity in chemical 
plasma etching as a function of processing parameters. The present 
invention also provides for a method of measuring absolute gas-surface 
reaction rates in commercial processing equipment without the benefit of 
sophisticated diagnostic equipment. 
Gas-surface radical reaction rates are often needed for the design of 
plasma processing equipment and for selection of desired reaction 
conditions. Unfortunately, few data are available on absolute reaction 
rates in systems of practical interest in the prior art. Most experimental 
data have been taken in difficult flow tube experiments, or by related 
techniques which require reactant concentrations to be quantified using 
sophisticated methods such as gas-phase titration, laser fluorescence or 
mass spectrometry. These measurements require great care and specialized 
instrumentation. In contrast, the present invention describes a technique 
for measuring etching rate constants. It can be carried out in commercial 
processing equipment and the like, and it does not require sophisticated 
instrumentation, direct radical measurements, or the like. An isothermal 
reaction rate constant may be derived from a single measurement of etching 
uniformity. From this information, the etching rate uniformity as a 
function of substrate spacing and pressure can be computed. If 
experimental data on uniformity are taken at several temperatures, an 
intrinsic activation energy can be derived and the effects of temperature 
can be expressed analytically. 
While the above is a full description of the specific embodiments, various 
modifications, alternative constructions and equivalents may be used. For 
example, while the description above is in terms of a plasma etching 
method, it would be possible to implement the present invention with other 
etching methods or the like. 
Therefore, the above description and illustrations should not be taken as 
limiting the scope of the present invention which is defined by the 
appended claims.