Patent Publication Number: US-7901952-B2

Title: Plasma reactor control by translating desired values of M plasma parameters to values of N chamber parameters

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is a continuation-in-part of U.S. application Ser. No. 10/440,364, filed May 16, 2003 by Daniel Hoffman, entitled PLASMA DENSITY, ENERGY AND ETCH RATE MEASUREMENTS AT BIAS POWER INPUT AND REAL TIME FEEDBACK CONTROL OF PLASMA SOURCE AND BIAS POWER, issued as U.S. Pat. No. 7,247,218 on Jul. 24, 2007, and assigned to the present assignee. 
    
    
     BACKGROUND OF THE INVENTION 
     Plasma reactors employed in microelectronic circuit fabrication can etch or deposit thin film layers on a semiconductor substrate. In a plasma reactive ion etch process, the etch rate, ion density, wafer voltage and wafer current are critical in controlling etch selectivity, wafer heating, etch striations, ion bombardment damage, etch stopping, feature size and other effects. Such control becomes more critical as feature size decreases and device density increases. The main problem is that present techniques for measuring etch rate, ion density, wafer voltage and wafer current tend to be highly inaccurate (in the case of the wafer voltage) or must be performed by examining a test workpiece or wafer at the conclusion of processing (in the case of etch rate). There appears to be no accurate technique for measuring these parameters in “real time” (i.e., during wafer processing). As a result, the plasma reactor control parameters (source power, bias power, chamber pressure, gas flow rate and the like) must be selected before processing a current workpiece based upon prior results obtained by processing other workpieces in the chamber. Once target values for each of the reactor control parameters have been chosen to achieve a desired etch rate or a desired wafer voltage or a desired ion density, the target values must remain the same throughout the process step, and all efforts are dedicated to maintaining the chosen target values. If for example the chosen target value of one of the control parameters unexpectedly leads to a deviation from the desired processing parameter (e.g., etch rate), this error will not be discovered until after the current workpiece has been processed and then examined, and therefore the current workpiece or wafer cannot be saved from this error. As a result, the industry is typically plagued with significant losses in materiel and time. 
     A related problem is that plasma process evolution and design is slow and inefficient in that the discovery of optimal target values for the reactor control parameters of source power, bias power, chamber pressure and the like typically relies upon protracted trial and error methods. The selection of target values for the many reactor control parameters (e.g., source power, bias power, chamber pressure and the like) to achieve a particular etch rate at a particular wafer current (to control wafer heating) and at a particular wafer voltage (to control ion bombardment damage) and at a particular ion density (to control etch selectivity, for example) is a multi-dimensional problem. The mutual dependence or lack thereof among the various reactor control parameters (source power, bias power, chamber pressure, etc.) in reaching the desired target values of the process parameters (e.g., etch rate, wafer voltage, wafer current, ion density) is generally unknown, and the trial and error process to find the best target values for the reactor control parameters (bias and source power levels and chamber pressure) is necessarily complex and time consuming. Therefore, it is not possible to optimize or alter target values for the process parameters (e.g., etch rate, etc.) without a time-consuming trial and error process. Thus, real-time plasma process control or management has not seemed possible. 
     SUMMARY OF THE INVENTION 
     The invention concerns a method of processing a wafer in a plasma reactor chamber by controlling plural chamber parameters in accordance with desired values of plural plasma parameters. The method includes concurrently translating a set of M desired values for M plasma parameters to a set of N values for respective N chamber parameters. The M plasma parameters are selected from a group including wafer voltage, ion density, etch rate, wafer current, etch selectivity, ion energy and ion mass. The N chamber parameters are selected from a group including source power, bias power, chamber pressure, inner magnet coil current, outer magnet coil current, inner zone gas flow rate, outer zone gas flow rate, inner zone gas composition, outer zone gas composition. The method further includes setting the N chamber parameters to the set of N values. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  illustrates a plasma reactor and a measurement instrument therefor. 
         FIG. 2  illustrates an electrical model of the plasma reactor employed by the measurement instrument. 
         FIG. 3  illustrates the structure of the measurement instrument of  FIG. 1 . 
         FIG. 4  illustrates an input phase processor of the measurement instrument of  FIG. 3 . 
         FIG. 5  illustrates a transmission line transformation processor in the measurement instrument of  FIG. 3 . 
         FIG. 6  illustrates a grid-to-ground transformation processor in the measurement instrument of  FIG. 3 . 
         FIG. 7  illustrates a grid-to-wafer transformation processor in the measurement instrument of  FIG. 3 . 
         FIG. 8  illustrates a combined transformation processor in the measurement instrument of  FIG. 3 . 
         FIG. 9  illustrates a process feedback control system for a plasma reactor that includes the measurement instrument of  FIG. 3 . 
         FIG. 10  illustrates an alternative implementation of the process feedback control system. 
         FIG. 11  illustrates the measurement instrument of  FIG. 3 , a constant contour generator and a process set point controller connected in a system with a plasma reactor. 
         FIGS. 12 ,  13  and  14  illustrate different contours of constant performance parameter values produced by the system of  FIG. 11 . 
         FIG. 15  illustrates a method of finding an optimal operating point at the intersection of different contours of constant parameter values. 
         FIG. 16  illustrates the process set point controller in the system of  FIG. 11 . 
         FIGS. 17 ,  18  and  19  illustrate respective operations performed by the process set point controller of the contour generator in the system of  FIG. 11 . 
         FIG. 20  illustrates an overlay of contours of constant wafer voltage, contours of constant etch rate and contours of constant ion density at a chamber pressure of 100 mT. 
         FIG. 21  illustrates an overlay of contours of constant wafer voltage, contours of constant etch rate and contours of constant ion density at a chamber pressure of 30 mT. 
         FIG. 22  illustrates an overlay of contours of constant wafer voltage, contours of constant etch rate and contours of constant ion density at a chamber pressure of 70 mT. 
         FIG. 23  illustrates an overlay of contours of constant wafer voltage, contours of constant etch rate and contours of constant ion density at a chamber pressure of 150 mT. 
         FIG. 24  illustrates an overlay of contours of constant wafer voltage, contours of constant etch rate and contours of constant ion density at a chamber pressure of 200 mT. 
         FIG. 25  illustrates an overlay of contours of constant wafer voltage, contours of constant etch rate and contours of constant ion density at a chamber pressure of 250 mT. 
         FIG. 26  is a simplified block diagram of a plasma reactor in accordance with further embodiments of the invention. 
         FIGS. 27-32  depict a process for constructing single variable functions of different plasma parameters for the variables of source power, bias power and chamber pressure. 
         FIG. 27  depicts processes for constructing the single variable functions of the different plasma parameters in which the variable is the chamber parameter of plasma source power. 
         FIG. 28  depicts processes for constructing the single variable functions of the different plasma parameters in which the variable is the chamber parameter of plasma bias power. 
         FIG. 29  depicts processes for constructing the single variable functions of the different plasma parameters in which the variable is the chamber parameter of chamber pressure. 
         FIG. 30  depicts processes for constructing the single variable functions of the different plasma parameters in which the variable is the chamber parameter of inner magnet coil current. 
         FIG. 31  depicts processes for constructing the single variable functions of the different plasma parameters in which the variable is the chamber parameter of outer magnet coil current. 
         FIG. 32  depicts processes for constructing the single variable functions of the different plasma parameters in which the variable is the chamber parameter of gas flow rate or gas composition. 
         FIGS. 33-36  depict an example in which contours (i.e., surfaces) of constant value of four plasma parameters are produced from the single variable functions of  FIGS. 27-32  in a three dimensional control space with dimensions of source power, bias power and chamber pressure. 
         FIG. 33  depicts a process for producing contours of constant value for the plasma parameter of wafer voltage in the three dimensional control space. 
         FIG. 34  depicts a process for producing contours of constant value for the plasma parameter of etch rate in the three dimensional control space. 
         FIG. 35  depicts a process for producing contours of constant value for the plasma parameter of plasma ion density in the three dimensional control space. 
         FIG. 36  depicts a process for producing contours of constant value for the plasma parameter of wafer current in the three dimensional control space. 
         FIG. 37  depicts a process for controlling three plasma parameters using the contours of constant value of  FIGS. 33-36 . 
         FIG. 38  depicts the intersection of the contours of constant value in the three-dimensional control space in the process of  FIG. 37 . 
         FIG. 39  is a view of the three-dimensional control space corresponding to that of  FIG. 38  but for the underconstrained case in which only two contours of constant value are specified and therefore intersect along a curve,  FIG. 39  depicting a method of varying the chamber parameters along the curve of intersection. 
         FIGS. 40-43  depict an example in which contours (i.e., surfaces) of constant value of four plasma parameters are produced from the single variable functions of  FIGS. 27-32  in a four dimensional control space with dimensions of source power, bias power, gas flow rate (or composition) and magnet coil current. 
         FIG. 40  depicts a process for producing contours of constant value for the plasma parameter of wafer voltage in the four dimensional control space. 
         FIG. 41  depicts a process for producing contours of constant value for the plasma parameter of etch rate in the four dimensional control space. 
         FIG. 42  depicts a process for producing contours of constant value for the plasma parameter of plasma ion density in the four dimensional control space. 
         FIG. 43  depicts a process for producing contours of constant value for the plasma parameter of wafer current in the four dimensional control space. 
         FIG. 44  depicts a process for controlling four plasma parameters using the contours of constant value of  FIGS. 40-43 . 
         FIG. 45  depicts an under-constrained version of the process of  FIG. 44  in which only three plasma parameters are controlled in four dimensional control space by varying them along a trajectory or curve along which the three corresponding contours intersect in four-dimensional space. 
         FIG. 46  depicts a process for characterizing the reactor chamber and controlling M plasma parameters with N chamber parameters, 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     Introduction 
     The present description pertains to a plasma reactor having a plasma source power applicator (such as an overhead electrode or antenna) in which plasma bias power is applied to the wafer through the wafer support pedestal. I have discovered a measurement instrument (described below) that is the first one known to instantaneously and accurately measure wafer voltage, wafer current, ion density and etch rate. The measurement instrument uses only conventional electrical sensors at the bias power input that sense voltage, current and power at the output of an impedance match device coupled to the wafer support pedestal. The measurement instrument is therefore non-invasive of the plasma etch process occurring within the reactor chamber in addition to being accurate. The degree of accuracy is surprising, surpassing even the best known instruments and measurement techniques currently in use. 
     I have invented a plasma reactor having a feedback controller employing this same measurement instrument, in which plasma source power and plasma bias power are controlled in separate feedback control loops. In the bias power feedback control loop, plasma bias power is servoed or controlled to minimize the difference between a user-selected target value of the ion energy (or, equivalently, wafer voltage) and the actual ion energy sensed in real time by my measurement instrument. Simultaneously, in the source power feedback control loop, plasma source power is servoed or controlled to minimize the difference between a user-selected target value of the plasma ion density and the actual plasma ion density sensed in real time by my measurement instrument and a user-selected target value for the ion density. One surprising feature of my feedback controller is that a measurement at the bias power input is used to control the source power. 
     In addition, I have solved the problem of how to select the target values for ion density and ion energy. Because my measurement instrument provides instantaneous, accurate and simultaneous measurements of performance parameters such as wafer voltage (or, equivalently, ion energy), wafer current, ion density and etch rate, it has enabled me to observe accurately, for the first time, the real-time behavior of all these performance parameters simultaneously as a function of control parameters such as plasma source power, plasma bias power and others (e.g., chamber pressure, source power frequency, applied magnetic field, etc.). These observations have led to my discovery herein that the control parameters of plasma source power level and plasma bias power level affect the set of performance parameters (e.g., etch rate, ion energy, ion density) in the manner of a pair of independent variables. This discovery greatly simplifies the task of controlling plasma processing: by holding various other control parameters constant during processing (i.e., constant chamber pressure, constant gas flow rates, constant source power frequency and bias power frequency, etc.), the process is controlled entirely through the bias and source power levels. I have used this technique to parameterize all of the performance parameters (including etch rate, ion energy and others) as unique functions of two independent variables, namely source power level and bias power level. From this, I have generated curves in 2-dimensional source power-bias power space of constant etch rate, constant ion energy and constant ion density, for example. A process controller responds to user-selected ranges for the various performance parameters (etch rate, ion energy, ion density) using the curves of constant etch rate, constant ion density and constant ion energy to instantaneously find a target value for the source power level and the bias power level. This process controller provides the target values for the plasma source power level and plasma bias power level to the feedback controller referred to above. 
     As a result, a user need not have any knowledge of the control parameters (e.g., bias and source power levels) that may be required to realize a desired set of performance parameter values (e.g., etch rate) nor a corresponding understanding of the reactor&#39;s behavior in this regard. Instead, the user merely inputs to the control processor his set of desired performance parameter values or ranges, and the control processor instantly specifies target control parameter values (target source power and bias power values) to the feedback controller referred to above. Thereafter, control of the plasma process is entirely automatic, and can instantly accommodate any changes the user may introduce. For example, the user may specify different etch rates at different times during the same etch step, so that one etch rate prevails during the beginning of an etch process and another prevails toward the end of the process, for example. The user need not specify any control parameters, but only the results he desires (i.e., the performance parameters such as etch rate, etc.). 
     Instrument for Instantaneously Measuring Performance Parameters Including Etch Rate, Ion Density and Ion Energy 
     Referring to  FIG. 1 , a plasma reactor  100  has a chamber enclosure  105  enclosing a vacuum chamber  110  in which a wafer support pedestal  115  supports a semiconductor wafer  120  being processed. Plasma RF bias power from an RF bias power generator  125  is applied through an impedance match circuit  130  to the wafer support pedestal  115 . Conventional sensing circuits  132  within the impedance match circuit  130  have three output terminals  132   a ,  132   b ,  132   c  providing respective signals indicating the power (P bias ), voltage (V) and current (I) furnished at the output of the impedance match circuit  130  to the wafer support pedestal  115 . A measurement instrument  140 , which is the measurement instrument referred to above in this specification, uses the signals from the output terminals  132   a ,  132   b ,  132   c  to measure, simultaneously, etch rate on the wafer  120 , ion energy at the wafer surface (or equivalently, wafer voltage), ion density in the reactor chamber and electric current through the wafer  120 . The measurement instrument  140  employs processes based upon an electrical model of the reactor  100 . This model is illustrated in  FIG. 2 . 
       FIG. 2  depicts the plasma reactor of  FIG. 1  in greater detail, so that the individual elements of the wafer support pedestal  115  are visible, including an electrode  115 - 1 , a thin overlying dielectric (e.g., ceramic) layer  115 - 2 , an underlying dielectric (e.g., ceramic) layer  115 - 3 , and a conductive (e.g., aluminum) planar ground plate  115 - 4  at the bottom of the pedestal  115 . The electrode  115 - 1  takes the form of a conductive grid in the illustrated embodiment, and may be implemented in various forms such as a conductive solid plate or as a conductive mesh, for example. While the electrode  115   1  will hereinafter be referred to as a conductive grid, the term “grid” as employed in this specification refers to all forms that the electrode  115 - 1  may take, such as a conductive solid plate, or a conductive mesh, or a conductive screen, or a form combining aspects of any or all of the foregoing forms, for example. Also visible in  FIG. 2  is a coaxial cable  210  connecting the output of the impedance match circuit  130  to the grid  115 - 1 . The coaxial cable  210  has an inner conductor  212  and an outer conductor  214 . An electrical model with parameters depicted in  FIG. 2  characterizes the electrical properties of the plasma reactor  100 , which are readily determined using conventional techniques. Specifically, the coaxial transmission line or cable  210  is characterized by three quantities: (1) its length, (2) Z ch , its characteristic impedance, and (3) V ch , its complex phase velocity in the transmission line equation. The wafer support pedestal  115  is characterized by electrical properties of the overlying and underlying dielectric layers  115 - 2  and  115 - 3 . Specifically, the underlying dielectric layer  115 - 3  has a capacitance C D , which is a function of (1) the dielectric constant, ε D , of the dielectric layer  115 - 3 , and (2) the conductive loss component of the dielectric layer  115 - 3 , tan D , (3) the thickness, gap, of the dielectric layer  115 - 3  and (4) the radius of the wafer  120 . The overlying dielectric layer  115 - 2  has a capacitance C p  which is a function of (1) the thickness, gap p , of the dielectric layer  115 - 2 , (2) the dielectric constant, ε p , of the dielectric layer  115 - 2  and (3) the conductive loss component of the dielectric layer  115 - 2 , tan p . The plasma  220  is characterized by an admittance Y plasma  (to RF ground such as the interior chamber walls or ceiling) that consists of a real part (the conductance g) and an imaginary part (the susceptance b). Each of these electrical parameters has a role in the operation of the measurement instrument  140 . 
       FIG. 3  illustrates the structure of the measurement instrument  140  of  FIG. 1 . An input phase processor  310  receives the P bias , V and I signals from the impedance match sensing circuit  132  of  FIG. 1  and produces respective signals indicating a complex impedance Z, a complex input current I in  and a complex input voltage V in  at the near end of the coaxial cable  210  (i.e., the end nearest the impedance match circuit  130 ). A transmission line transformation processor  320  uses the characteristic impedance Z ch  and the complex loss coefficient V ch  (in the transmission line equation) from an electrical model  330  of the coaxial cable  210  to transform from Z, I in  and V in  at the near cable end to an admittance Y junction  at the far . cable end, i.e., at the junction between the coaxial cable  210  and the grid  115 - 1 . A grid-to-ground transformation processor  340  takes radius, gap, ε D  and tan D  from a model  345  of the grid-to-ground capacitance and produces a dielectric resistance R D  and dielectric capacitance C D . A grid-to-wafer transformation processor  350  takes radius, gap p , ε p  and tan p  from a model  355  of the grid-to-wafer capacitance and produces a plasma resistance R p  and a plasma capacitance C p . A combined transformation processor  360  accepts the outputs of all the other processors  320 ,  340 ,  350  and computes the admittance Y plasma  through the plasma from the wafer to RF ground and computes the wafer voltage V wafer  (or ion energy). From the plasma admittance and from the wafer voltage, the following quantities are computed: wafer current I wafer , the etch rate and the ion density. 
     In summary, electrical measurements are made at the output of the impedance match circuit  130 . The transmission line transformation processor  320  transforms these measurements at the near end of the cable  210  to an admittance at the far end. The grid to ground transformation processor  340  provides the transformation from the ground plane  115 - 4  near the far end of the cable to the conductive grid  115 - 1 . The grid-to-wafer transformation processor  350  provides the transformation from the conductive grid  115 - 2  to the wafer  120 . Using all of the foregoing transformations, the combined transformation processor  360  provides the transformation across the plasma in the form of the plasma admittance. From the plasma admittance, various performance parameters such as etch rate and plasma ion density are computed. 
     The transmission line model  330 , the model of the grid-to-ground capacitance  345  and the model  355  of the grid-to-wafer capacitance are not necessarily a part of the measurement instrument  140 . Or, they may be memories within the measurement instrument  140  that store, respectively, the coaxial cable parameters (V ch  and Z ch ), the grid-to-ground capacitance parameters (gap, ε D , tan D  and radius) and the grid-to-wafer capacitance parameters (gap p , ε p , tan p  and radius). 
       FIG. 4  illustrates the structure of the input phase processor  310  of  FIG. 3 . A delivered power arithmetic logic unit(ALU)  410  computes delivered power P from the outputs I and P bias  from the impedance match sensing circuit  132  as P bias  −(0.15)I 2 . A phase angle ALU  420  computes phase angle θ from the delivered power P and from V and I as cos −1 (P/VHI). An impedance ALU  430  computes the complex impedance Z as (V/I)e 10 , where i=(−1) 1/2 . An input current ALU  440  computes the input current I in  to the coaxial cable  210  as [P/Re(Z)] 1/2  An input voltage ALU  450  computes the input voltage V in  to the coaxial cable  210  as ZHI in . 
       FIG. 5  illustrates the structure of the transmission line transformation processor  320  of  FIG. 3 . The transmission line processor receives I in and V in  as inputs from the input phase processor  310  of  FIG. 4  and uses the transmission line model parameters V ch  and Z ch  (from the transmission line model or memory  330  of  FIG. 3 ) to compute the admittance Y junction  as follows: A junction current ALU  510  computes the current I junction  at the junction of the coaxial cable  210  and the grid  115 - 1  ( FIG. 1 ) as:
 
(I in )cosh[(V ch )(−length)]+(V in /Z ch )sinh[(V ch ) (−length)].
 
A junction voltage ALU  520  computes the voltage V junction  at the junction between the coaxial cable  210  and the grid  115 - 1  as:
 
(V in )cosh[(V ch )(−length)]+(I in Z Ch )sinh[(V ch )(−length)].
 
     A divider  530  receives I junction  and V junction  computes Y junction  as I junction /V junction . It should be noted that each of the electrical quantities in the foregoing computations (current, voltage, impedance, admittance, etc.) is a complex number having both a real part and an imaginary part. 
       FIG. 6  illustrates the structure of the grid-to-ground transformation processor  340  of  FIG. 3 . The grid-to-ground transformation processor  340  receives the parameters gap, ε D , tan D  and rad (the wafer radius) from the grid-to-ground model or memory  345  of  FIG. 3  computes the dielectric resistance R D  and the dielectric capacitance C D . The dielectric capacitance C D  is computed by a CD ALU  610  as follows:
 
(ε 0 )(ε D )π(rad) 2 /gap
 
where ε 0  is the electrical permittivity of free space. An RD ALU  620  uses the value of C D  from the CD ALU  610  and computes the dielectric resistance R D  as follows:
 
(tan D )/(ωC D gap 2 )
 
where ω is the angular frequency of the bias RF generator  125  of  FIG. 2 .
 
       FIG. 7  illustrates the structure of the grid-to-wafer transformation processor  350  of  FIG. 3 . The grid-to-wafer transformation processor  350  receives the parameters gap p , ε p , tan p  and rad from the grid-to-wafer model or memory  355  of  FIG. 3  and computes the plasma resistance R p  and the plasma capacitance C p . The plasma capacitance C p  is computed by a C P  ALU  710  as follows:
 
(ε 0 ) (ε p )π(rad) 2 /gap p  
 
where ε 0  is the electrical permittivity of free space. An RP ALU  720  uses the value of C p  from the CP ALU  710  and computes the plasma resistance R p  as follows:
 
(tan p )/(ωC p gap D   2 )
 
where ω is the angular frequency of the bias RF generator  125  of  FIG. 2 .
 
       FIG. 8  illustrates the structure of the combined transformation processor  360  of  FIG. 3 . The combined transformation processor  360  receives the parameters R D , C D  from the processor  340  of  FIG. 3 , receives the parameters R p , C p  from the processor  350  of  FIG. 3  and receives the parameter Y junction  from the processor  320  of  FIG. 3 . A grid impedance ALU  810  computes Z grid  (the impedance at the grid  115 - 1  of  FIG. 2 ) as follows:
 
[Y junction −1/(R D +(1/(iωC D )))] −1  
 
A wafer impedance ALU  820  uses the output of the grid impedance ALU  810  to compute Z wafer  (the impedance at the wafer  120  of  FIG. 2 ) as follows:
 
Z grid −1/(R p +(1/(iωC p )))
 
     A wafer voltage ALU  830  uses the outputs of both ALU=s  810  and  820  and V junction  from the divider  530  of  FIG. 5  to compute the voltage on the wafer  120  of  FIG. 2 , V wafer , as V junction  Z wafer /Z grid . A wafer current ALU  840  uses the outputs of the ALU=s  820  and  830  to compute the wafer current I wafer  as V wafer /Z wafer . An admittance ALU  850  uses the output of the ALU  820  to compute the admittance of the plasma, Y plasma , as 1/Z wafer . A susceptance ALU  860  uses the output of the ALU  850  to compute the plasma susceptance, b, as Im(Y plasma ). An etch rate ALU  870  uses the wafer voltage from the ALU  830  and the susceptance from the ALU  860  to compute the etch rate as b 2  V wafer   2 . An ion density ALU  880  uses the same outputs to compute the ion density as kb 2  V wafer   3/2 , where k is a constant given by:
 
(2 3/2 /3 2 )(1/[qε 0 A 2 π 2 f 2 T e   2 ])
 
where q is the electron charge, A is the area of the wafer  120  of  FIG. 2 , f is the frequency of the bias power generator  125  of  FIG. 2  and T e  is the electron temperature in volts. This relationship between ion density and the measured quantities b and V wafer  follows from an approximate formula for the plasma susceptance and a formula for the plasma sheath thickness. The plasma susceptance may be approximated as εAω/λ, where ε is the electrical permittivity within the plasma, A is the electrode area, co is the angular frequency of the bias power signal and λ is the plasma sheath thickness. The plasma sheath thickness may be approximated as [T e /(qη)] 1/2 [2V wafer /T e ] 3/4 , where T e  is electron temperature, q is the electron charge and λ is ion density. Substituting the expression for sheath thickness into the expression for the susceptance and solving for ion density yields an expression for ion density as a function of susceptance and wafer voltage.
 
Process Feedback Control System
 
       FIG. 9  illustrates a process feedback control system that uses the measurement instrument  140  of  FIG. 3 . A plasma reactor  900  includes all of the features of the plasma reactor  100  of  FIG. 1 , and in addition includes an overhead RF source power applicator  910  connected through an impedance match circuit  915  to an RF source power generator  920 . The RF source power applicator  910  may be, for example, a ceiling electrode that is insulated from the grounded chamber enclosure  105 . The power level of the RF plasma source power generator  920  generally controls the plasma ion density while the power level of the RF plasma bias power generator  125  generally controls the ion energy at the wafer surface. The measurement instrument  140  receives the power, voltage and current outputs from the sensor circuit  132  of the impedance match circuit  130 . From these quantities, the measurement instrument  140  computes the plasma susceptance b and computes the wafer voltage V wafer , which is output as a measurement signal. These computations are carried out in the manner described above with reference to  FIG. 5 . The measurement instrument  140  can then compute the ion density and/or the etch rate from b and V wafer , in the manner described above with reference to  FIG. 5 . At least two of the three measurement signals thus produced by the measurement instrument  140  can be used in a feedback control loop. 
     A feedback controller  950  uses the measurement signals from the measurement instrument  140  to create feedback signals to control the power level of the RF plasma bias power generator  125  and the power level of the RF plasma source power generator  920 . The ion energy at the wafer surface, which is equivalent to the wafer voltage V wafer , is directly controlled by the power level of the bias power generator  125 . Therefore, the wafer voltage measurement signal from the measurement instrument  140  (i.e., V wafer  from the ALU  830  of  FIG. 8 ) is used by the feedback controller  950  to control the bias power generator  125  in a bias power feedback control loop  957 . The source power generator  920 , on the other hand, directly controls plasma ion density. Therefore, plasma ion density measurement signal from the measurement instrument  140  (i.e., kb 2 V wafer   3/2  from the ALU  880  of  FIG. 8 ) is used by the feedback controller  950  to control the source power generator  920  in a source power feedback control loop  958 . 
     The bias power feedback control loop  957  includes a memory  960  that stores a selected or desired target value of the wafer voltage or ion energy, [V wafer ] TARGET . A subtractor  962  subtracts this target value from the sensed wafer voltage V wafer  to produce an error signal. The gain of the bias power feedback loop  957  is determined by a bias power feedback gain factor stored in a memory  964 . A multiplier  966  multiplies the error signal from the subtractor  962  by the gain factor in the memory  964  to produce a correction signal used to control the power level of the bias power generator  125 . The path of the bias power feedback control loop  957  is completed by the V, I and P bias  signals applied to the measurement instrument  140  to produce the measurement signal V wafer  representing the wafer voltage. 
     The source power feedback control loop receives from the measurement instrument  140  the sensed ion density value b 2 V wafer   3/2 . A memory  975  stores a selected or desired target value of the ion density, [b 2 V wafer   3/2 ]TARGET. A subtractor  980  computes the difference between the measured ion density and the ion density target value to produce an error signal. The gain of the source power feedback control loop  958  is determined by a source power feedback gain factor stored in a memory  985 . A multiplier  990  multiplies the error signal from the subtractor  980  by the gain factor from the memory  985  to produce a correction signal. This correction signal is used to control the power level of the RF source power generator  920 . The path of the source power feedback control loop  958  is completed by the V, I and P bias  signals applied to the measurement instrument  140  to produce the measurement signal b 2 V wafer   3/2  representing the ion density. 
     At the start of a plasma process step such as an etch process step, initial values for the power levels P s  and P B of the RF source power generator  920  and the RF bias power generator  125 , respectively, can be specified. If these initial values are sufficiently close to the optimum values, this feature can avoid unduly large initial corrections by the feedback controller  950 . For this purpose, the bias power feedback loop  957  includes a bias power command processor  992  coupled to receive the feedback correction signal from the multiplier  957  and to receive a target value for the bias power, [P bias ] TARGET . Before plasma processing begins, there is no feedback signal, and the bias power command processor  992  sets the power level of the bias power generator  125  to the initial target value [P bias ] TARGET . Once processing begins and a feedback signal is present, the bias power command processor  992  controls the bias power in accordance with the feedback correction signal from the multiplier  966  rather than the bias power target value. 
     Similarly, the source power feedback loop  958  includes a source power command processor  994  coupled to receive the feedback correction signal from the multiplier  990  and to receive a target value for the source power, [P source ] TARGET . Before plasma processing begins, there is no feedback signal, and the source power command processor  994  sets the power level of the source power generator  920  to the initial target value [P source ] TARGET . Once processing begins and a feedback signal is present, the source power command processor  994  controls the source power in accordance with the feedback correction signal from the multiplier  990  rather than the source power target value. 
     In accordance with another aspect, the source and bias power command processors  992 ,  994  can be instructed by the user to ignore their respective feedback control loops  957 ,  958  throughout much or all of the process step and instead maintain the source and bias power levels at the specified target values [P source ] TARGET  and [P bias ] TARGET . The user can change these values from time to time during processing. 
     Referring to  FIG. 10 , the feedback control processor  950  may employ the etch rate rather than the ion density as the measured parameter in the source power feedback control loop  958 . In the measurement instrument  140 , the etch rate measurement signal is taken from the ALU  870  of  FIG. 8  that computes b 2 V wafer   2 . In  FIG. 10 , a memory  975 ′ (in lieu of the memory  975  of  FIG. 9 ) stores a target value of the etch rate, [b 2 V wafer   2 ] TARGET . The subtractor  980  operates as described with reference to  FIG. 9  to produce an error signal. The remainder of the source power feedback control loop of  FIG. 10  generally is the same as in  FIG. 9 . 
     Process Set Point Controller 
     The feedback controller  950  requires a number of target values for various process control parameters. Specifically the feedback controller  950  of  FIG. 9  has a memory  975  storing the target value for the ion density, [b 2 V wafer   3/2 ] TARGET , and a memory  960  storing the target value for the ion energy (or, equivalently, wafer voltage), [V wafer ] TARGET , In the feedback controller of  FIG. 10 , the memory  975  is replaced by the memory  975 ′ storing the target value for the etch rate, [b 2 V wafer   2 ] TARGET . In addition, the feedback controller  950  can employ initial target values [P source ] TARGET  and [P bias ] TARGET  for the source and bias power levels respectively to initialize the feedback controller  950 , as discussed above. The selection or optimization of these target values can be left to the user=s efforts, which may involve an undue amount of trial and error and may be unreliable. Typically, a user who wishes to achieve certain process results (e.g., a certain etch rate, a certain ion energy, a reduction in etch processing artifacts such as striations, a reduction in heating due to wafer current, etc.) must conduct a time-consuming program of trial and error experiments to find the optimum process control parameters values to achieve the desired results. For this reason, the alteration of an existing process or the design of a new process must be undertaken over a very long development period. 
     In order to overcome this limitation, a process set point controller  1110  employed in the reactor of  FIG. 11  automatically and quickly (or instantaneously) finds the optimum target values of process control parameters based upon the user&#39;s selection of values for various performance parameters. For example, the process set point controller  1110  may determine the target values [P source ] TARGET  and [P bias ] TARGET  based upon a desired etch rate and/or a desired wafer voltage or other performance parameter specified by the user. Thus, a new process recipe can be designed nearly instantaneously. For present plasma reactors, this can take place in milliseconds, but could be made to be as fast as microseconds if needed. 
     There are many process control parameters (i.e., characteristics of the reactor under direct user control such as chamber pressure, source and bias power levels, etc.) and many process performance parameters (i.e., characteristics of the plasma and process not susceptible of direct control such as etch rate, ion density, ion energy, wafer current, etc.). A user can specify any one or more of these performance parameters as an objective for a given process. Any one or group of or all of the control parameters can be used to achieve the desired levels of the performance parameters chosen by the user. The question is whether or not the effects of some of the control parameters might be dependent upon others of the control parameters in controlling the performance parameters chosen by the user. Thus, the problem of selecting the right set of control parameters to achieve the desired results in the chosen performance parameters is complex and there appears to be no particularly optimum choice. 
     However, I have discovered that the source power and the bias power control the performance parameters of interest and do so in an independent manner. That is, source power P source  and bias power P bias  are independent variables and may be thought of as orthogonal entities forming a two-dimensional control space in which control of the performance parameters may be exercised with such versatility that no alteration of the other control parameters is required. This discovery greatly reduces the problem to only two variables. 
     Therefore, the following description will concern a control system in which the control parameters, with the exception of P source  and P bias , are held constant during processing. Thus, process control parameters including chamber pressure, gas composition, gas flow rate, source power frequency, bias power frequency, etc., are held constant. The source power and bias power levels (P source  and P bias ) are varied to achieve desired values in a specified set of performance parameters (e.g., etch rate and ion density). 
     The problem of finding target values for the various parameters given a set of user-defined values for a chosen set of performance parameters is solved by the process set point controller  1110  superimposing a set of constant parameter contours in the two-dimensional P source -P bias  space referred to above. Such constant parameter contours are obtained from a constant parameter contour generator  1120  in  FIG. 11 . For example, contours of constant ion density ( FIG. 12 ), contours of constant ion energy or wafer voltage ( FIG. 13 ), and contours of constant etch rate ( FIG. 14 ) are employed. How the constant parameter contour generator  1120  produces these contours using the measurement instrument  140  will be described later in this specification. The present description concerns their use by the process set point controller  1110 . 
     Referring to  FIG. 12 , a set of contours of constant plasma ion density in P source -P bias  space for a chamber pressure of 20 mT generally have a small negative slope and a small but positive first derivative d(P source )/d(P bias ). The top-most contour corresponds to a constant plasma density of 5H10 10  ions/cm 3  while the bottom contour corresponds to 1.5H10 10  ions/cm 3 . The vertical axis (P source ) ranges from 0 to 1500 Watts while the horizontal axis (P bias ) ranges from 2000 to 4500 Watts. Referring to  FIG. 13 , a set of contours of constant wafer voltage for the same chamber pressure (20 mT) have a positive slope and range from 600 volts (at the top) to 1800 Volts (at the bottom). Referring to  FIG. 14 , a set of contours of constant etch rate (in arbitrary units, e.g., where k=1) have a large negative slope. 
     The process set point controller  1110  determines how to simultaneously satisfy user-selected values of ion density, ion energy and etch rate. It does this by finding the intersection in P source -P bias  space of the corresponding contours of  FIGS. 12-14 . This intersection indicates the optimum target values for source and bias power, namely [P source ] TARGET  and [P bias ] TARGET . The problem is somewhat simpler if the user specifies values for only two performance parameters. For example, if the user specifies a wafer voltage of 1100 Volts and an ion density of 3.5H10 10  ions/cm 3 , then the correct point in P source -P bias  space is found by superimposing the constant wafer voltage contour for 1100 volts from  FIG. 12  and the constant density contour for 3.5H10 10  ions/cm 3  from  FIG. 13  and finding their intersection in P source -P bias  space. This procedure is performed by the process set point controller  1110  and is illustrated in  FIG. 15  in which the two curves intersect in P source -P bias  space at the point [850W, 3750W]. Therefore, in this example the user=s requirements are met by setting the source power level at 850 W and setting the bias power level at 3750 W. Thus, in this case the target values [P source ] TARGET  and [P bias ] TARGET  furnished to the source power command processor  994  and bias power command processor  992  of  FIG. 9  are 850 Watts and 3750 Watts, respectively. 
     It should be noted that this deduction of the target values of source and bias power levels may also result in the deduction of a target value for other parameters whose values have not been specified or limited by the user. As an illustration, in the foregoing example, the user has not specified a particular etch rate. However, a target value for the etch rate satisfying the user-selected values for ion density and energy can be found by superimposing the contours of  FIG. 14  onto  FIG. 15  (or vice versa). The point [850 W, 3750 W] lies on the contour of a constant etch rate of 2.101 (in arbitrary units) of  FIG. 14 , as indicated by the AX@ symbol in that drawing. Therefore, if the feedback controller of  FIG. 10  is employed, then the set point controller  1110  writes an etch rate target value of 2.101 in arbitrary units to the memory  975  of  FIG. 10 . 
     An advantage of this feature is that the contours of constant voltage, density, etch rate, etc., are characteristic of the reactor and generally do not change for given process conditions. They may therefore be determined by the constant parameter contour generator  1120  prior to processing and made available to the process set point controller  1110  constantly during use of the reactor, as indicated in  FIG. 11 . In this way, a target value for a particular parameter may be found instantly or whenever required in the manner illustrated in  FIG. 15 . 
     In operation, the bias power command processor  992  and the source power command processor  994  receive the target values [P source ] TARGET  and [P bias ] TARGET  from the process set point controller  1110  and receive feedback signals from the multipliers  958  and  957  respectively. During system initialization, the feedback signals are ignored, and the processors  992 ,  994  put the power levels of the RF generators  125 ,  920  to the target values [P source ] TARGET  and [P bias ] TARGET , respectively. After processing begins, the feedback signals are available and the processors  992 ,  994  can use the feedback control loops  957 ,  958  instead of the target values to control the source power and bias power levels. Alternatively, the power command processors  992 ,  994  may be programmed so that the target values [P source ] TARGET  and [P bias ] TARGET  determine the source and bias power levels not only at initialization but also during processing, while the feedback loops  957 ,  958  are ignored. 
       FIG. 11  shows that the user can apply to the process set point controller  1110  any one or a combination of user selected values for performance parameters, including etch rate, wafer voltage, ion density and wafer current. in response, the process set point controller  1110  uses the appropriate contours from the contour generator  1120  to produce not only source and bias power target values but, in some cases, target values for other parameters not limited or specified by the user, which may be a target value for the etch rate, the ion density, the ion energy or the wafer current. These target values are furnished to the feedback controller  950  for use in the manner described previously in this specification with reference to  FIG. 9 . 
       FIG. 16  illustrates the structure and operation of the process set point controller  1110  of  FIG. 11 . A first logic unit  1610  receives an etch rate command (if any) from the user and fetches from a memory  1615  the corresponding contour of constant etch rate in the set of contours of constant etch rates previously generated by the contour generator  1120 . A second logic unit  1620  receives an ion density command (if any) from the user and fetches from a memory  1625  the corresponding contour of constant ion density in the set of contours of constant ion density previously generated by the contour generator  1120 . A third logic unit  1630  receives a wafer voltage (ion energy) command (if any) from the user and fetches from a memory  1635  the corresponding contour of constant wafer voltage in the set of contours of constant wafer voltage previously generated by the contour generator  1120 . A fourth logic unit  1640  finds the intersection point in P source -P bias  space between any of the contours selected by the logic units  1610 ,  1620 ,  1630 . This intersection point is output to the feedback controller  950  of  FIG. 11  as [P source ] TARGET / [P bias ] TARGET . 
     Contour Generator  1120   
     Operation of the contour generator  1120  of  FIG. 11  is illustrated in  FIGS. 17 ,  18  and  19 .  FIG. 17  illustrates the operation of the contour generator  1120  in finding functions defining how certain performance parameters vary with bias power. These include functions for the performance parameters of wafer voltage, ion density and etch rate. As will be described below, the observations of changes in wafer voltage, ion density and etch rate with bias power are made for the contour generator  1120  by the measurement instrument  140  using the configuration of  FIG. 11 . In  FIG. 11 , the measurement instrument  140  transmits instantaneous measurements of wafer voltage, ion density and etch rate to the contour generator  1120 . The contour generator  1120  also receives the current source power and bias power commands, as indicated in  FIG. 11 , allowing it to correlate behavior of the performance parameters of wafer voltage, ion density and etch rate, with the control parameters of source power and bias power. 
       FIG. 18  illustrates the operation of the contour generator  1120  in finding functions defining how certain performance parameters vary with source power. As in  FIG. 17 , in  FIG. 18  these include functions for the performance parameters of wafer voltage, ion density and etch rate. Also as in  FIG. 17 , in the operation of  FIG. 18  is carried out using the configuration of  FIG. 11 . 
       FIG. 19  illustrates the operation of the contour generator  1120  in parameterizing the separate functions of source power and bias power discovered iii The operations of  FIGS. 17 and 18  into combined functions of both source power and bias power. Such combined functions represent the behavior of the performance parameters (wafer voltage, ion density, etch rate) in 2-dimensional P source -P bias  space. The contour generator  1120  then derives the contours of constant ion density, ion energy and etch rate from the respective combined functions. 
     The operation depicted in  FIG. 17  will now be described in detail with reference to both  FIGS. 11 and 17 . In the step of block  1710  of  FIG. 17 , the frequencies of the bias and source power generators  125 ,  920  of  FIG. 11  are set to constant values, the exhaust rate of a vacuum pump  1180  of the reactor of  FIG. 11  is controlled to achieve a constant chamber pressure, and mass flow rates from gas supplies  1182 ,  1184  are set through a mass flow controller  1186  of  FIG. 11  to constant values. In the step of block  1720  of  FIG. 17 , the power level of the source power generator  920  of  FIG. 11  is set to an initial set point, so that the entire process is at a steady state with the exception of the bias power level. In the step of block  1730  of  FIG. 17 , the power level of the bias power generator  125  of  FIG. 11  is set at the beginning of a predetermined range. The measurement instrument  140  then senses the voltage current and power at the impedance match  130  in order to measure wafer voltage, ion density and etch rate in the manner described previously with respect to  FIGS. 1-8  (block  1740  of  FIG. 17 ). These measurements are sent to the contour generator  1120  and stored in a memory  1120   a . In the next step (block  1750  of  FIG. 17 ), the power level of the bias power generator  125  of  FIG. 11  is incremented (by command of the controller  1110 ) to a slightly higher value and held at that value. A determination is then made in the step of block  1760  of  FIG. 17  as to whether or not the latest bias power level is at the end of the bias power range. If not (ANO@ branch of block  1760 ), the operation returns in a loop  1765  to the step of block  1740 . The steps within the loop  1765  are repeated in this manner until the end of the bias power range is reached (AYES@ branch of block  1760 ). The result is that three sets of data corresponding to functions of bias power defining the behaviors of wafer voltage, ion density and etch rate are stored in the memory  1120   a . Using conventional data fit algorithms, the contour generator uses the three sets of data to produce algebraic functions corresponding to the data, which are stored in the memory  1120   a  as follows: 
                     V   wafer     =       ⁢         f   a     ⁡     (     P   bias     )       i                 =       ⁢         f   b     ⁡     (     P   bias     )       i                 ER   =       ⁢         f   c     ⁡     (     P   bias     )       i                 
where η is plasma ion density, ER is etch rate and the index i refers to the current level of the source power generator  915  (block  1770 ). In the next step of  FIG. 17  (block  1780 ), the level of the source power generator  915  is incremented to a new value so that i6i+1. If the new source power level is not at the end of the source power range (ANO@ branch of block  1790 ), then the operation returns in a loop  1795  to the step of block  1730 , and the steps within the loop  1795  (i.e., blocks  1730  through  1790 ) are repeated until the source power level reaches the end of the source power range (AYES@ branch of block  1790 ). The result is that many sets of the functions
 
                     V   wafer     =       ⁢         f   a     ⁡     (     P   bias     )       i                 =       ⁢         f   b     ⁡     (     P   bias     )       i                 ER   =       ⁢         f   c     ⁡     (     P   bias     )       i                 
for all values of i within the source power range are stored in the memory  1120   a . This permits an analytical determination of whether or not the behavior of the three behavior parameters V wafer , η, ER with bias power changes with source power. I have discovered that it does not change to a great extent, so that bias power and source power are at least nearly independent variables. Thus, a single function of bias power for each of the parameters V wafer , η, ER generally suffices as a fairly accurate prediction of behavior over the entire range of the source power level, at least for the range chosen in the working examples given later in this specification. Thus, the loop  1795  of  FIG. 17  may not be strictly necessary. Instead, it may be acceptable to choose a single value for the source power level in the middle of the source power level range in step  1720  and perform the loop of  1765  to produce a single set of data for each of the three functions
 
                     V   wafer     =       ⁢       f   a     ⁡     (     P   bias     )                   =       ⁢       f   b     ⁡     (     P   bias     )                   ER   =       ⁢       f   c     ⁡     (     P   bias     )                   
These three functions of bias power are stored in the memory  1120   a.  
 
     The operation depicted in  FIG. 18  will now be described in detail with reference to both  FIGS. 11 and 18 . In the step of block  1810  of  FIG. 18 , the frequencies of the bias and source power generators  125 ,  920  of  FIG. 11  are set to constant values, the exhaust rate of a vacuum pump  1180  of the reactor of  FIG. 11  is controlled to achieve a constant chamber pressure, and mass flow rates from gas supplies  1182 ,  1184  are set through a mass flow controller  1186  of  FIG. 11  to constant values. In the step of block  1820  of  FIG. 18 , the power level of the bias power generator  125  of  FIG. 11  is set to an initial set point, so that the entire process is at a steady state with the exception of the source power level. In the step of block  1830  of  FIG. 18 , the power level of the source power generator  920  of  FIG. 11  is set at the beginning of a predetermined range. The measurement instrument  140  then senses the voltage current and power at the impedance match  130  in order to measure wafer voltage, ion density and etch rate in the manner described previously with respect to  FIGS. 1-8  (block  1840  of  FIG. 18 ). These measurements are sent to the contour generator  1120  and stored in the memory  1120   a . In the next step (block  1850  of  FIG. 18 ), the power level of the source power generator  920  of  FIG. 11  is incremented (by command of the controller  1110 ) to a slightly higher value and held at that value. A determination is then made in the step of block  1860  of  FIG. 18  as to whether or not the latest source power level is at the end of the source power range. If not (NO branch of block  1860 ), the operation returns in a loop  1865  to the step of block  1840 . The steps within the loop  1865  are repeated in this manner until the end of the source power range is reached (YES branch of block  1860 ). The result is that three sets of data corresponding to functions of source power defining the behaviors of wafer voltage, ion density and etch rate are stored in the memory  1120   a . Using conventional data fit algorithms, the contour generator  1120  uses the three sets of data to produce algebraic functions corresponding to the data, which are stored in the memory  1120   a  as follows: 
                     V   wafer     =       ⁢         f   a     ⁡     (     P   source     )       i                 =       ⁢         f   b     ⁡     (     P   source     )       i                 ER   =       ⁢         f   c     ⁡     (     P   source     )       i                 
where η is plasma ion density, ER is etch rate and the index i refers to the current level of the bias power generator  125  (block  1870 ). In the next step of  FIG. 18  (block  1880 ), the level of the bias power generator  125  is incremented to a new value so that i6i+1. if the new bias power level is not at the end of the bias power range (NO branch of block  1890 ), then the operation returns in a loop  1895  to the step of block  1830 , and the steps within the loop  1895  (i.e., blocks  1830  through  1890 ) are repeated until the bias power level reaches the end of the bias power range (YES branch of block  1890 ). The result is that many sets of the functions
 
                     V   wafer     =       ⁢         f   a     ⁡     (     P   source     )       i                 =       ⁢         f   b     ⁡     (     P   source     )       i                 ER   =       ⁢         f   c     ⁡     (     P   source     )       i                 
for all values of i within the bias power range are stored in the memory  1120   a . This permits an analytical determination of whether or not the behavior of the three behavior parameters V wafer , η, ER with source changes with bias power. I have discovered (as in the case of  FIG. 17 ) that it does not change to a great extent, so that bias power and source power are at least nearly independent variables, as discussed above. Thus, a single function of source power for each of the parameters V wafer , η, ER generally suffices as a fairly accurate prediction of behavior over the entire range of the bias power level, at least for the range chosen in the working examples given later in this specification, Thus, the loop  1895  of  FIG. 18  may not be strictly necessary. Instead, it may be acceptable to choose a single value for the bias power level in the middle of the bias power level range in step  1820  and perform the loop of  1865  to produce a single set of data for each of the three functions
 
                     V   wafer     =       ⁢       f   a     ⁡     (     P   source     )                   =       ⁢       f   b     ⁡     (     P   source     )                   ER   =       ⁢       f   c     ⁡     (     P   source     )                   
These three functions of source power are stored in the memory  1120   a . Thus, upon completion of the operations of  FIGS. 17 and 18 , the memory  1120   a  holds the following pair of functions for the wafer voltage:
 
 V   wafer   =f   a ( P   source )
 
 V   wafer   =f   a ( P   bias )
 
and following pair of functions for the ion density:
 
 =f   b ( P   source )
 
 =f   b ( P   bias )
 
and the following pair of functions for etch rate:
 
ER = f   c ( P   source )
 
ER = f   c ( P   bias )
 
     In the operation illustrated in  FIG. 19 , the contour generator  1120  combines each pair of functions having a single variable P source , or P bias , respectively, into a single combined function of the variable pair P source  and P bias . This produces the following three functions: 
     
       
         
           
             
               
                 
                   
                     V 
                     wafer 
                   
                   ⁡ 
                   
                     ( 
                     
                       
                         P 
                         source 
                       
                       , 
                       
                         P 
                         bias 
                       
                     
                     ) 
                   
                 
               
             
             
               
                 
                   η 
                   ⁡ 
                   
                     ( 
                     
                       
                         P 
                         source 
                       
                       , 
                       
                         P 
                         bias 
                       
                     
                     ) 
                   
                 
               
             
             
               
                 
                   
                     ER 
                     ⁡ 
                     
                       ( 
                       
                         
                           P 
                           source 
                         
                         , 
                         
                           P 
                           bias 
                         
                       
                       ) 
                     
                   
                   . 
                 
               
             
           
         
       
     
     Contours of constant parameter values (e.g., a contour of constant wafer voltage, a contour of constant etch rate, a contour of constant ion density) are found by setting the respective function to a constant value and then solving for P source  as a function of P bias . For example, in order to generate a contour of constant wafer voltage at 300 V, the function V wafer (P source , P bias ) is set equal to 300 V, and then solved for P source . 
     Operation of the contour generator  1120  of  FIG. 11  in carrying out the foregoing steps of generating the combined two-variable functions and then solving them for P source  as a function of P bias  at various constant values is illustrated in  FIG. 19 . Referring now to  FIG. 19 , the first step (block  1910 ) is to take the single variable functions of wafer voltage, i.e., V wafer (P source ) and V wafer (P bias ) and find their combined function. The next step (block  1920 ) is to take the single variable functions of ion density, i.e., η(P source ) and η(P bias ) and find their combined function η(P source , P bias ). The third step (block  1930 ) is to take the single variable functions of etch rate, i.e., ER(P source ) and ER(P bias ) and find their combined function ER(P source , P bias ). 
     Then, the contours of constant values are generated. To generate a contour of constant wafer voltage (block  1940  of  FIG. 19 ), the function V wafer (P source , P bias ) is set equal to a constant value of wafer voltage and the resulting expression is then solved for P source  as a function of P bias . This step is repeated for a range of constant wafer voltage values to generate a set of contours covering the range. These contours are stored in the memory  1120   a  of  FIG. 11  (block  1945  of  FIG. 19 ). 
     To generate a contour of constant ion density (block  1950  of  FIG. 19 ), the function η(P source , P bias ) is set equal to a constant value of ion density and the resulting expression is solved for P source  as a function of P bias . This step is repeated for a range of constant ion density values to generate a set of contours covering the range of ion density values. These contours are stored in the memory  1120   a  of  FIG. 11  (block  1955  of  FIG. 19 ). 
     To generate a contour of constant etch rate (block  1960  of  FIG. 19 ), the function ER(P source , P bias ) is set equal to a constant value of etch rate and the resulting expression solved for P source  as a function of P bias . This step is repeated for a range of constant etch rate values to generate a set of contours covering the range of etch rate values. These contours are stored in the memory  1120   a  of  FIG. 11  (block  1965  of  FIG. 19 ). 
     Generally, each combined two-variable function, e.g., V wafer (P source , P bias )) can be approximated by the product of the pair of individual functions, e.g., V wafer (P source ) and V wafer (P bias ). For example, ignoring all control parameters except RF power level and ignoring constants of proportionality:
 
 V   wafer   =f   a ( P   source )·[ P   source ] 1/2  
 
 V   wafer   =f   a ( P   bias )·[ P   bias ] 1/2  
 
so that the combined two-variable function is approximately: V wafer =F a (P source , P bias )=f a (P source ) f a (P bias ) [P source ] 1/2 [P bias ] 1/2  This expression, however is not exact. The exact function is best found by curve-fitting techniques involving all control parameters, namely P source  and P bias , as above, and in addition, source power frequency, bias power frequency, chamber pressure, and magnetic field (if any). I have found the following expression for V wafer  as a function of both P source  and P bias :
 
 V   wafer ( P   source   ,P   bias )= V   0 ( P   bias   /P   b0 ) 0.4 [( P   source   /P   s0 ) K   1 ( p/p   0 ) −1 +( p/p   0)   0.5 ] −0.5  
 
where P b0  is a maximum bias power value, P s0  is a maximum source power value, p 0  is a minimum chamber pressure, and p is the actual chamber pressure. In the reactor chamber described above, the maximum source power P s0  was 1500 Watts, the maximum bias power P b0  was 4500 Watts and the minimum pressure p 0  was 30 mT. These values may differ from the foregoing example depending upon chamber design and process design. V 0  is determined in accordance with the following procedure: the maximum bias power P b0  is applied to the wafer pedestal while the source power is held to zero and the chamber is held to the minimum pressure p 0 . The wafer voltage V wafer  is then measured and this measured value is stored as V 0 . K 1  is then determined by increasing the source power to its maximum value P s0  and then measuring the wafer voltage V wafer  again, and K 1  is adjusted until the foregoing equation yields the correct value for V wafer .
 
     The exponents in the foregoing equations were obtained by an extensive trial and error parameterization process for the reactor described in this specification. These exponents may be useful for other reactor designs, or the user may wish to try other exponents, depending upon the particular reactor design. 
     Ion density, η, and etch rate, ER, are both functions of V wafer  and b, the plasma susceptance or imaginary part of the plasma admittance, as described previously herein with reference to  FIG. 8 :
 
= b   2   V   wafer   2  
 
and
 
ER = kb   2   V   wafer   3/2  
 
     Therefore, only the plasma susceptance b need be specified in addition to V wafer  to define ER and η, for the sake of brevity. I have found the following expression for the plasma susceptance b as a function of both P source  and P bias :
 
 b ( P   source   ,P   bias )= b   0 ( P   bias   /P   b0 ) −0.25 [( P   source   /P   s0 )( p/p   0 ) −0.65   ][K   2  ( P   bias   /P   b0 ) −0.62 ( p/p   0 ) 3 +( p/p   0 ) 0.27 ]
 
where the definitions above apply and in addition b 0  is a reference susceptance value. The reference susceptance value b 0  is determined in accordance with the following procedure: the maximum bias power P b0  is applied to the wafer pedestal while the source power is held to zero and the chamber is held to the minimum pressure p 0 . The susceptance b is then measured at the wafer support pedestal (using a V/I meter, for example) and this measured value is stored as b 0 . K 2  is then determined by increasing the source power to its maximum value P s0  and then measuring the susceptance b again, and K 2  is adjusted until the foregoing equation yields the correct value for b.
 
     Ion density, η, and etch rate, ER, are then obtained by substituting the expressions for V wafer  and b into the foregoing expressions for η and ER . 
     The results of the contour generator operation of  FIG. 19  are illustrated for various chamber pressures in  FIGS. 20-26 .  FIG. 20  illustrates contours of constant wafer voltage, contours of constant ion density and contours of constant etch rate superimposed upon one another in P source -P bias  space. The chamber pressure for these contours was 100 mT. The contours of constant wafer voltage are depicted in solid lines. The contours of constant ion density are depicted in dashed lines. The contours of constant etch rate are depicted in dotted lines. The source power range (the vertical axis or ordinate) has a range from zero to 1200 Watts. The bias power range (the horizontal axis or abscissa) has a range from 200 Watts to 1200 Watts. The stated values of constant wafer voltage are RMS volts. The stated values of constant ion density are 10 10  ions/cm 3 . 
       FIGS. 20 ,  21 ,  22 ,  23 ,  24  and  25  correspond to  FIG. 20  for respective chamber pressures of 100 mT, 30 mT, 70 mT, 150 mT, 200 mT and 250 mT, respectively. 
     Once a complete set of contours of constant voltage, constant etch rate and constant ion density have been generated and permanently stored in the memory  120   a , the contour generator and even the measurement instrument may be discarded. In such an implementation, the process set point controller  1110  would control the entire process based upon the contours stored in the memory  120   a  in response to user inputs. In this case, the process set point controller  1110  could apply the bias and source power level commands directly to the bias and source power generators  125 ,  920 , respectively, so that the feedback controller  950  could also be eliminated in such an embodiment. 
     While the measurement instrument  140  has been described with reference to discrete processors  310 ,  320 ,  340 ,  350 ,  360  that carry out individual computations, these processors comprising the measurement instrument  140  can be implemented together in a programmed computer, such as a workstation or a personal computer rather than as separate hardware entities. The contour generator  1120  may also be implemented in a programmed computer or workstation. In addition, the feedback controller  950  of  FIG. 9  or  FIG. 10  may be implemented in a programmed computer. Moreover, the process set point controller may be implemented in a programmed computer. 
     The measurement instrument  140  has been described in certain applications, such as in a process control system. It is also useful as a tool for “fingerprinting” or characterizing a particular plasma reactor by observing the etch rate, ion density and wafer voltage measured by the instrument  140  at a selected process setting of source power, bias power, pressure and other parameters. 
     While the description of  FIG. 8  concerned an implementation in which etch rate is computed as ER =b 2  V wafer   2  and ion density as η=kb 2  V wafer   3/2 , other functions may be employed, such as, for example, [bV wafer ] 1 , or [bV wafer ] 2 , or gV wafer   3/2  (where g in this last expression is the conductance defined previously in this specification). 
     Reactor with Array of Chamber Parameters 
       FIG. 26  illustrates a plasma reactor similar to that of  FIG. 11  but having a greater number of chamber parameters capable of being controlled by the feedback controller  950 . Like elements in  FIGS. 11 and 26  have like reference numerals. In addition to the elements of  FIG. 11 , the reactor of  FIG. 26  also has inner and outer annular gas injection zones or showerheads  912 ,  914  within the overhead electrode  910 , plural gas supplies  1182   a  through  1182   f , each containing a different chemical species (or mixture) and coupled to the inner and outer gas injection zones  912 ,  914  through respective gas flow controllers  1186   a ,  1186   b . The gas flow controllers  1186   a ,  1186   b  control the gas flow rate and the composition or proportion of gas flow from each of the individual gas supplies  1182  to gas injection zones  912 ,  914 . Inner and outer magnet coils  1210 ,  1215  are connected to respective inner and outer DC coil current supplies  1220 ,  1225 . An optional DC chucking voltage supply  1230  is coupled to the bias feed center conductor  212 , in which case a DC isolation capacitor  1235  is connected in series between the center conductor  212  and the bias match  130 . 
     Chamber Characterization for Three Chamber Parameters 
     The reactor chamber of  FIG. 26  may be characterized by quantifying the behavior of, for example, four plasma parameters (such as wafer or sheath bias voltage, ion density, etch rate, wafer current) as functions of three chamber parameters (such as source power, bias power and chamber pressure). First, single value functions of the various plasma parameters are found with individual chamber parameters as the single variables, in the processes depicted in  FIGS. 27-32 . The first step is to initialize the chamber parameters (block  2001  of  FIG. 27 ). This step sets chamber parameters, such as source power P s , bias power P B , chamber pressure p ch , inner magnet current I inner , outer magnet current I outer , gas flow rate FR, to initial (e.g., mid-range) values. 
     The next major step is for the constant contour generator  1120  to find single variable functions of each plasma parameter in which bias power is the variable, which is depicted in  FIG. 27 . In the example of  FIG. 27 , functions are found for wafer voltage, V wafer (P B ), etch rate, ER(P B ), plasma ion density, η(P B ), and wafer current, I wafer (P B ). Referring to  FIG. 27 , the first step is to set P B to the beginning of its range (block  2003  of  FIG. 27 ). This range may be between zero and 1000 Watts at 13.56 MHz, as one possible example. The next step is to measure or sample plasma parameters of wafer voltage V wafer , etch rate ER, plasma ion density η and wafer current I Wafer  using the measurement instrument  140  of  FIG. 26  (block  2005 ). Then the contour generator  1120 , through the controller  950 , increments P B by a small predetermined amount or small fraction of the range (block  2007 ). A determination is then made as to whether the end of the bias power range has been reached (block  2009 ). If not (NO branch of block  2009 ), the process loops back to the step of block  2005 . If the end of range has been reached (YES branch of block  2009 ), then the process continues to the next step, namely block  2011 . The step of block  2011  consists of using the sampled data to construct functions V Wafer (P B ), ER(P B ), η(P B ) and I Wafer (P B ), which are stored in memory. These functions may be constructed by curve fitting techniques, for example. Then, in preparation for generation of functions depending upon other chamber parameters, the chamber parameter P B is returned to its initial value, preferably a mid-range value (block  2013 ). 
     The purpose of the next process, which is depicted in  FIG. 28 , is to find the single variable functions of which source power is the single variable, namely the functions V Wafer (P s ), ER(P s ), η(P s ) and I Wafer (P s ) Referring to  FIG. 28 , the first step is to set P s  to the beginning of source power range (block  2015  of  FIG. 28 ). The RF plasma source power range may be from zero to 3000 Watts at 162 MHz, as one possible example. The next step is to measure or sample the plasma parameters of wafer voltage V wafer , etch rate ER, plasma ion density η and wafer current I wafer  with the measurement instrument  140  (block  2017  of  FIG. 28 ). Then, the generator  1120 /controller  950  incremented P s  (block  2019 ). A determination is then made of whether the end of the source power range has been reached (block  2021 ). If not (NO branch of block  2021 ), the process returns to the step of block  2017 . Otherwise (YES branch of block  2021 ), the process continues to the next step of block  2023 . In the step of block  2023 , the sampled data (from block  2017 ) is used to construct the single variable functions V Wafer (P s ), ER(P s ), η(P s ) and I Wafer (P s ), and these functions are stored in memory. P S is then returned to its initial value (block  2025 ). 
     The purpose of the next process, which is depicted in  FIG. 29 , is to find the single variable functions of which chamber pressure is the single variable, namely the functions V Wafer (P ch ), ER (P ch ), η(p ch ) and I wafer (p ch ). The first step is to set p ch  to beginning of the chamber pressure range (block  2027 ). This range may lie between 0.5 mT and 200 mT, as one possible example. The next step is to measure or sample the plasma parameters of wafer voltage V Wafer , etch rate ER, plasma ion density η and wafer current I Wafer  with the measurement instrument  140  (block  2029 ). Then, the generator  1120 /controller  950  increments P ch  by a small fraction of the pressure range (block  2031 ). A determination is made at this point of whether the end of the chamber pressure range has been reached (block  2033 ). If not (NO branch of block  2033 ), the process loops back to the step of block  2029 . Otherwise (YES branch of block  2033 ), the process continues with the next step, namely block  2035 . In block  2035 , the sampled data from the step of block  2029  is used to construct the functions V Wafer (p ch ), ER(p ch ), η(p ch ) and I wafer (p ch ), which are then stored in memory. In block  2037 , p ch  is returned to its initial value. 
     The purpose of the next process, which is depicted in  FIG. 30 , is to find the single variable functions of which the current of the inner magnet coil  1210  of  FIG. 26 ) is the single variable, namely the functions V Wafer (I inner ), ER(I inner ), η(I inner ) and I Wafer (I inner ). In other embodiments, the current could be (instead) the AC current applied to MERIE magnets if these are present in the reactor. First, I inner  is set to the beginning of its range (block  2039 ). The next step is to measure or sample the plasma parameters of wafer voltage V Wafer , etch rate ER, plasma ion density η and wafer current I Wafer  with the measurement instrument  140  (block  2041 ). Then, the generator  1120 /controller  950  increments I inner  by a small predetermined fraction of its range (block  2043 ). As determination is made at that point of whether the end of the magnet coil current range has been reached (block  2045 ). If not (NO branch of block  2045 ), the process loops back to the step of block  2041 . Otherwise (YES branch of block  2045 ), the process proceeds to the next step, namely block  2047 . In the step of block  2047 , the sampled data from the step of block  2041  is used to construct the functions V Wafer (I inner ), ER(I inner ), η(I inner ) and I Wafer (I inner ), which are then stored in memory. The last step of this process is to return I inner  to its initial value (block  2049 ). 
     The purpose of the next process, which is depicted in  FIG. 31 , is to find the single variable functions of which the current supplied to the outer magnet coil  1215  of  FIG. 26  is the single variable, namely the functions V Wafer (I outer ), ER(I outer ) η(I outer ) and I Wafer (I outer ). The first step is to set I outer  to beginning of its range (block  2051  of  FIG. 31 ). The next step is to measure or sample the plasma parameters of wafer voltage V Wafer , etch rate ER, plasma ion density η and wafer current I Wafer  with the measurement instrument  140  (block  2053 ). The generator  1120 /controller  950  then increment I outer  by a predetermined small fraction of its range (block  2055 ). A determination is made as to whether the end of range has been reached (block  2057 ). If not, the process loops back to the step of block  2053 . Otherwise, the process proceeds with the next step. In the next step (block  2059 ), the sampled data from the step of block  2053  is used to construct the functions V Wafer (I outer ) , ER(I outer ), η(I outer ) and I Wafer (I outer ), which are then stored in memory. This process concludes by returning I outer  to its initial value (block  2061 ). 
     The purpose of the next process, which is depicted in  FIG. 32 , is to find the single variable functions of which the gas flow rate FR (or alternatively, the gas composition) is the single variable, namely the functions V Wafer (FR), ER(FR), η(FR) and I Wafer (FR). The gas composition may be the ratio between the carrier gas (e.g., argon) and the etchant species (fluorine or fluorocarbon or fluorhydrocarbon species), for example. The gas composition or gas flow rate may be separately defined as two different variables for each of the two (inner and outer) gas injection zones  912 ,  914 , for example. Thus, there are four possible variables or chamber parameters concerning gas flow that may be employed: inner zone gas flow rate, outer zone gas flow rate, inner zone gas composition, outer zone gas composition. The present example of  FIG. 32  concerns the use of a particular one of any of the foregoing gas flow-related chamber parameters, which will is labelled FR. 
     The first step of the process of  FIG. 32  is to set FR to beginning of range (block  2063 ). The next step is to measure or sample the plasma parameters of wafer voltage V Wafer , etch rate ER, plasma ion density η and wafer current I Wafer  with the measurement instrument  140  (block  2065 ). The next step is to increment FR (block  2067 ). A determination is then made of whether the end of range of the gas flow or gas composition parameter FR has been reached (block  2069 ). If not (NO branch of block  2069 ), the process loops back to the measurement step of block  2065 . Otherwise (YES branch of block  2069 ), the next step (block  2071 ) is performed. In the step of block  2071 , the sampled data is used to construct the functions V Wafer (FR), ER(FR), η(FR) and I Wafer (FR), which are then stores in memory. The construction of such functions may employ curve fitting techniques, for example. The final step of this process is to return FR to its initial value (block  2073 ). 
     Process Control in a 3-D Control Space—Translating Desired Plasma Parameter Values to Chamber Parameter Values 
     The single variable functions produced for the different plasma parameters in the processes of  FIGS. 27-32  may be employed in subsequent processes (depicted in  FIGS. 33-36 ) to construct a three-dimensional control space with two-dimensional surfaces of constant plasma parameter values for later use in controlling the reactor chamber during wafer processing. In the example of  FIGS. 33-36 , the three chamber parameters of source power, bias power and chamber pressure (i.e., P B , P S and p ch ) are selected. 
     The purpose of the process of  FIG. 33  (performed by the contour generator  1120 ) is to exploit the 3-D control space of P B , P s  and p ch to produce 2-D contours (surfaces) of constant V wafer . These will be accumulated in a collection of such surfaces for later use in controlling the reactor chamber during wafer processing. 
     The first step (block  2075 ) in the process of  FIG. 33  is to correlate or combine the single-variable functions V Wafer (P B ), V Wafer (P s ) and V Wafer (p ch ) into a single three-variable function V Wafer (P B , P S , p ch ). Curve fitting techniques of the type referred to earlier in this specification may be employed, for example, to accomplish this step. Next, an index “i” is initialized to one: by setting i=1 (block  2077 ). An equation is formed (block  2079 ) by setting the function V Wafer  (P B , P S , p ch ) equal to the i th  value in the range of values of V Wafer . This equation is solved to find the 2-D contour (surface) of constant V Wafer  for the i th  value of V Wafer . This contour is stored in the memory  1120   a  of the contour generator  1120 . The index i is then incremented by one by setting i=i+1 (block  2081 ), and the process loops back to block  2079  if the end of the range of values of V Wafer  has not been reached (NO branch of block  2183 ). Otherwise, if the end of range has been reached (YES branch of block  2183 ), the current process is complete and the next process is begun. 
     The purpose of the next process ( FIG. 34 ) (performed by the contour generator  1120 ) is to exploit the 3-D control space of P B , P S and p ch  to produce 2-D contours (surfaces) of constant etch rate (ER). These will be accumulated in a collection of such surfaces for later use in controlling the reactor chamber during wafer processing. 
     The first step (block  2085 ) in the process of  FIG. 34  is to correlate/combine the single variable functions ER(P B ), ER(P s ) and ER(p ch ) into a single three variable function ER(P B , P S , p ch ). An index i is initialized to one by setting i=1 (block  2087 ). Then an equation is formed by setting the function ER(P B , P S , p ch ) equal to the i th  value in the range of values of ER. This equation is solved to produce the 2-D contour (surface) of constant ER for the i th  value of ER. This contour is then stored in the memory  1120   a  (block  2089 ). The index i is incremented by one by setting i=i+1 (block  2091 ). A determination is then made of whether the end of the range of ER values has been reached (block  2093 ). If not (NO branch of block  2093 ), the process loops back to the step of block  2089 . Otherwise (YES branch of block  2093 ), the current process is complete and the next process is performed. 
     The purpose of the next process ( FIG. 35 ) (performed by the contour generator  1120 ) is to exploit the 3-D control space of P B , P S and p ch to produce 2-D contours (surfaces) of constant η. These will be accumulated in a collection of such surfaces for later use in controlling the reactor chamber during wafer processing. 
     The first step (block  2095 ) in the process of  FIG. 35  is to correlate or combine the single variable functions η(P B ), η(P s ) and η(p ch ) into a single three-variable function η(P B , P S , p ch ). The step may be carried out using curve fitting techniques, for example. Then, an index i is initialized to one by setting i=1 (block  2097 ). Then, an equation is created by setting the function η(P B , P s , p ch ) equal to the i th  value in the range of values of the plasma parameter η. This equation is solved for the 2-D contour (surface) of constant η for the i th  value of η. This contour is stored in the contour generator memory  1120   a  (block  2099 ). The index i is incremented by one by setting i=i+1 (block  2101 ). A determination is then made of whether the end of the range of ER values has been reached (block  2103 ). If not (NO branch of block  2103 ), the process loops back to the step of block  2099 . Otherwise (YES branch of block  2103 ), the current process is complete and the next process is performed. 
     The purpose of the next process ( FIG. 36 ) (performed by the contour generator  1120 ) is to exploit the 3-D control space of P B , P S and p ch to produce 2-D contours (surfaces) of constant I Wafer . These will be accumulated in a collection of such surfaces for later use in controlling the reactor chamber during wafer processing. 
     The first step (block  2105 ) in the process of  FIG. 36  is to correlate or combine the single variable functions I Wafer (P B ), I Wafer (P s ) and I Wafer (p ch ) to produce a single three-variable function I Wafer (P B , P S , p ch ). The next step is to initialize an index i to one by setting i=1 (block  2107 ). In the step of block  2109 , an equation is created by setting the function I wafer (P B , P S , p ch ) equal to the i th  value of I Wafer . This equation is solved for the 2-D contour (surface) of constant I Wafer  for the i th  value of I Wafer . This contour is then stored in the contour generator memory  1120   a . The index i is incremented by one by setting i=i+1 (block  2111 ). A determination is made of whether the end of range of the values of I Wafer  has been reached (block  2113 ). If not (NO branch of block  2113 ), the process loops back to the step of block  2109 . Otherwise (YES branch of block  2113 ), the current process is complete and the next process is performed. 
     The same processes may be performed for other chamber parameters. Such chamber parameters may include the gas flow rates of the inner and outer gas injection zones and the gas compositions of the different gas mixtures supplied to the inner and outer gas injection zones, for example. The foregoing procedures complete the characterization of the reactor chamber for the selected chamber parameters and plasma parameters. 
     The next process is to exploit the chamber characterization information obtained in the foregoing processes to provide a translation from desired values of selected plasma parameters to required values of selected chamber parameters. As one example of such a process,  FIG. 37  depicts a process for controlling three selected plasma parameters in the 3-D P B -P S -p ch control space in response to user-selected values for V Wafer , ER and η. This process is controlled by the process set point controller  1110  of  FIG. 26 . 
     The process of  FIG. 37  is controlled by the set point controller and begins with the set point controller  1110  receiving the user-selected values for V Wafer , ER and η (block  2115  of  FIG. 37 ). The next step is to fetch from memory the 2-D surface (contour) of constant wafer voltage corresponding to the unique user-selected value of V Wafer  and label the surface S v  (block  2117  of  FIG. 37 ). Then, the controller  1110  fetches from memory the 2-D surface (contour) of constant etch rate corresponding to the user-selected value of ER and label the surface S ER  (block  2119 ). Next, the controller  1110  fetches from memory the 2-D surface (contour) of constant plasma ion density corresponding to the user-selected value of η and labels the surface S η  (block  2121 ). The controller  1110  then locates the point of intersection (P B ′, P S ′, p ch ′) of the three surfaces S v , S ER  and S η  in 3-D P B -P S -p ch  space (block  2123 ). The set point controller  1110 , acting through the feedback controller  950 , then sets the RF bias power generator output level to the intersection value P B ′ (block  2125 ), sets the RF source power generator output level to the intersection value P S ′ (block  2127 ) and sets the chamber pressure to the intersection value p ch ′ (block  2129 ). This completes one control cycle of the process. 
     The intersection of three surfaces of constant plasma parameter values (of three different plasma parameters) in three dimensional chamber parameter space of the type exploited in the process of  FIG. 37  is depicted in  FIG. 38 . The surfaces are two-dimensional objects residing in three-dimensional space. The intersection of the three surfaces lies at a single point whose location is specified by the 3-vector (P B ′, P S ′, p ch ′). The three orthogonal axes of  FIG. 38  correspond to the three chamber parameters (P B , P S , p ch ). The three surfaces in  FIG. 38  are the surfaces of constant value for each of the plasma parameters V Wafer , ER and η, for which the values are the user-selected values. 
     Under-Constrained 3-D Control Space—Providing an Extra Degree of Freedom to Vary the Chamber Parameters 
     In the example of  FIG. 38 , the number of plasma parameters and the number of chamber parameters is the same. It is possible to obtain an additional degree of freedom by underconstraining the chamber parameters. This feature arises whenever the number of selected plasma parameters is less than the number of selected chamber parameters.  FIG. 39  illustrates such a case, in which the three chamber parameters are constrained by user selected values of only two plasma parameters. For example, in a three dimensional control space of P B , P S , p ch  the only constrained plasma parameters may be V Wafer  and η. In this case, there are only two surfaces intersecting the 3-D control space, and such an intersection occurs along a line or curve. The allows the user to vary the chamber parameters to any set of values (P B ′, P S ′, p ch ′) lying on that line while continuing to meet the user-selected values for V Wafer  and η. 
     Alternating Set Point Control in 3-D Control Space with Real Time Feedback Control 
     The foregoing chamber control process may be employed any time or all the time, but is particularly useful at the start of wafer processing, when no real time measurements of the plasma parameters are available. After plasma processing of the wafer is underway and measurements of plasma parameters become available through the measurement instrument  140 , control may be taken over by the feedback controller  950 . The feedback controller  950  compares actual real time measurements of selected plasma parameters (from the measurement instrument  140 ) with the user-selected values of those parameters. The feedback controller  950  minimizes those differences by correcting source power (for etch rate or ion density) and correcting bias power (for wafer voltage), as described earlier in this specification with reference to  FIGS. 9 and 10 . 
     If there is a significant change in one (or more) user selected values of plasma parameters, then the change can be immediately effected by reverting back to the control process of  FIG. 37 , in which the new chamber parameter settings are instantly ascertained by finding the intersection in 3-D control space of the contours corresponding to the (new) user-selected values of plasma parameters. This option enables the chamber to nearly instantaneously meet any changes in process recipe, a significant advantage. 
     Process Control in a 4-D Control Space—Translating Desired Plasma Parameter Values to Chamber Parameter Values 
       FIGS. 40-45  depict a control process example involving a 4-dimensional control space. This example involves generating four-variable functions from the single variable functions produced by the chamber characterization processes of  FIGS. 27-32 . Specifically,  FIGS. 40-45  illustrate the example of a 4-D control space of the chamber parameters P B , P s , FR and I inner . In the first process of this example, that of  FIG. 40 , the task is to produce 3-D contours (in 4-D space) of constant V Wafer . The first step (block  2131  of  FIG. 40 ) is to correlate or combine the four single variable functions V Wafer (P B ), V Wafer (P S ), V wafer (FR) and V wafer (I inner ) to produce a single four-variable function V wafer (P B , P s , FR, I inner ). Curve fitting techniques may be employed in carrying out this step. An index i is initialized to one by setting i=1 (block  2132 ). An equation is created by setting the i th  function V Wafer (P B , P s , FR, I inner ) equal to the i value in the range of values of V Wafer . This equation is solved for the 3-D contour (surface) of constant V Wafer  corresponding to the i th  value of V Wafer . This contour is stored in the contour generator memory  1120   a  (block  2133 ). The index i is then incremented by setting i=i+1 (block  2134 ). A determination is made of whether the end of the range of values for V Wafer  has been reached (block  2135 ). If not, the process returns to the step of block  2133  (NO branch of block  2135 ). Otherwise, the process is finished and the next process is begun (YES branch of block  2135 ). 
     In the next process of this example, that of  FIG. 41 , the task is to produce 3-D contours (in 4-D space) of constant ER. The first step (block  2137 ) is to correlate or combine the four single variable functions ER(P B ), ER(P s ), ER(FR) and ER(I inner ) into a single four-variable function ER(P B , P s , FR, (I inner ). This step may be carried out using curve fitting techniques. An index i is initialized to one by setting i=1 (block  2139 ). In the step of block  2141 , an equation is created by setting the function ER(P B , P s , FR, (I inner )) equal to the i th  value in the range of values of ER. This equation is solved for the 3-D contour (surface) of constant ER corresponding to the i th  value of ER. This contour is stored in the contour generator memory  1120   a . The index i is incremented (block  2143 ) and a determination is made of whether end of range of etch rate (ER) values has been reached (block  2145 ). If not (No branch of block  2145 ), the process returns to the step of block  2141 . Otherwise (YES branch of block  2145 ), the process is finished and the next process is begun. 
     In the next process of this example, that of  FIG. 42 , the task is to produce 3-D contours (in 4-D space) of constant η (etch rate). The first step (block  2147  of  FIG. 42 ) is to correlate or combine the four single-variable functions η(P B ), η(P s ) η(FR) and η(I inner ) into a single four-variable function η(P B , P S , FR, I inner ). This step may employ curve fitting techniques. An index i is initialized to one by setting i=1 (block  2149 ). In the step of block  2151 , an equation is created by setting the four-variable function η(P B , P s , FR, I inner ) equal to the i th  value of the range of ion density values η. This equation is solved for the 3-D contour (surface) of constant ηcorresponding to the i th  value of η. The contour is stored in the contour generator memory  1120   a . The index i is incremented by setting i=i+1 (block  2153 ) and a determination is made of whether the end of range of ion density (η) values has been reached (block  2155 ). If not (NO branch of block  2155 ), the process returns to the step of block  2151 . Otherwise (YES branch of block  2155 ), the process is finished and the next process is begun. 
     In the next process of this example, that of  FIG. 43 , the task is to produce 3-D contours (in 4-D space) of constant I Wafer . The first step (block  2157  of  FIG. 43 ) is to correlate or combine the four single-variable functions I Wafer (P B ), I Wafer (P s ), I Wafer (FR) and I Wafer ((I inner )) into a single four-variable function I Wafer  (P B , P s , FR, (I inner )). An index i is initialized to one by setting i=1 (block  2159 ). In the step of  2161 , an equation is created by setting the four-variable function I Wafer (P B , P S , FR, I inner ) equal to the i th  value of I Wafer . This equation is solved for the 3-D contour (surface) of constant I Wafer  corresponding to the i th  value of I Wafer . This contour is stored in the contour generator memory  1120   a . The index i is incremented by one by setting i=i+1 (block  2163 ). At this point, a determination is made whether the end of range of the values of I Wafer  has been reached (block  2165 ). If not (NO branch of block  2165 ), the process returns to the step of block  2161 . Otherwise (YES branch of block  2165 ), the process is finished. This completes the chamber characterization tasks required for the subsequent translation of user-selected values of four plasma parameters (e.g., V Wafer , ER, η and VI) to target values of four chamber parameters (e.g., P B , P s , FR, (I inner )). The processes for performing such a translation are now described with reference to  FIG. 44 . 
       FIG. 44  illustrates a process for controlling four selected plasma parameters (e.g., V wafer , ER, η and V Wafer ) in the 4-D P B -P S -FR-I inner  control space in response to user-selected values for the selected plasma parameters (V Wafer , ER, η and I Wafer ). This process translates the user-selected values for the plasma parameters V Wafer , ER, η and V Wafer  to required values for the chamber parameters P B , P S , FR and I inner . 
     The first step in the process of  FIG. 44  is for the process set point controller  1110  to receive the user-selected values for V Wafer , ER, η and I Wafer  (block  2167  of  FIG. 44 ). In the step of block  2169 , the process set point controller  1110  fetches from the contour generator memory  1120   a  the 3-D surface (contour) of constant wafer voltage corresponding to the unique user-selected value of V Wafer . This surface may be labelled S v . The controller  1110  also fetches from the memory  1120   a  the 3-D surface (contour) of constant etch rate corresponding to the user-selected value of ER, which may be labelled S ER  (block  2171 ). The controller  1110  fetches the 3-D surface (contour) of constant plasma ion density corresponding to the user-selected value of η and labels the surface S η  (bock  2173 ). Finally, the controller  1110  fetches the 3-D surface (contour) of constant wafer current corresponding to the unique user-selected value of I Wafer  and labels the surface S I  (block  2175 ). 
     The next step is for the set point controller  1110  to locate the point of intersection (P B ′, P S ′, FR′, I inner ′) of the four surfaces S v , S ER , S η and S I  in the 4-D P B -P S -FR-I inner  control space (block  2177 ). This four-dimensional step is analogous to the three dimensional case of three intersecting surfaces depicted in  FIG. 38 . The chamber parameters are then set to the respective values of P B ′, P S ′, FR′ and I inner ′ corresponding to the point of intersection. This is accomplished by the set point controller  1110  acting through the feedback controller  950  to effect the chamber parameters, as follows: setting the RF bias power generator output level to the intersection value P B ′ (block  2179 ), setting the RF source power generator output level to the intersection value P S ′ (block  2181 ), setting the gas flow rate to the intersection value FR′ (block  2183 ) and setting the inner magnet supply current to the intersection value (I inner )′ (block  2185 ). 
     Alternating Set Point Control in 4-D Control Space with Real Time Feedback Control: 
     The foregoing steps exploiting the 4-D control space bring the selected plasma parameters in line with their user-selected values. This fact can be verified by taking real time direct measurements of the plasma parameters from the measurement instrument  140 . As described earlier in this specification with reference to  FIGS. 9-11 , such real time measurements form the basis of a feedback control system in which chamber parameters (e.g., P s  and P B ) are changed to minimize the differences between the real time measurements and the user-selected or target values for the plasma parameters V Wafer , ER, η. For example, P s  is changed to bring either ER or η closer to the corresponding user-selected values, and P B  is changed to bring V wafer  closer to the corresponding user selected value. 
     Therefore, as one option the chamber control process of steps  2167 - 2185  of  FIG. 44  may be phased out and process control turned over to the feedback control loops of  FIGS. 9-11  based upon real time measurements of plasma parameters by the measurement instrument  140 . This option is depicted in the step of block  2187  of  FIG. 44 . The translation-based chamber control steps of blocks  2167 - 2185  may be employed at the beginning of a plasma process (when no real time measurements are available). Then, after the plasma process is sufficiently underway for real time measurements to become available, the step of block  2187  is performed to transition chamber control to the real time feedback control loops of  FIGS. 9-11 . Process control may be temporarily returned to the translation-based steps of blocks  2167 - 2185  whenever a significant change is made in the user selected values of one or more plasma parameters. This option enables the chamber to nearly instantaneously meet any changes in process recipe, a significant advantage. 
     Under-Constrained Case: Controlling Three Selected Plasma Parameters (e.g., V Wafer , ER, η) in the 4-D P B -P s -FR-I inner Control Space in Response to User-Selected Values for V Wafer , ER and η: 
       FIG. 45  depicts an example of providing an extra degree of freedom by operating in a chamber parameter space of dimensionality exceeding the number of user-controlled plasma parameters. This is a four-dimensional version of the under-constrained control case illustrated in  FIG. 39 , in which the chamber parameters are permitted to vary along a curve or trajectory determined by the intersecting surfaces. 
     The first step in the process of  FIG. 45  is for the set point controller  1110  to receive the user-selected values for the selected plasma parameters, e.g., V Wafer , ER and η (block  2189 ). The next step is for the set point controller  1110  to fetch from the contour generator memory  1120   a  the 3-D surface (contour) of constant wafer voltage corresponding to the unique user-selected value of V Wafer  and labels the surface S v  (block  2191 ). The set point controller  1110  also fetches the 3-D surface (contour) of constant etch rate corresponding to the user-selected value of ER and labels the surface S ER  (block  2193 ). And, the controller  1110  fetches the 3-D surface (contour) of constant plasma ion density corresponding to the user-selected value of η and labels the surface S η  (block  2195 ). The contour generator then locates the line or curve of intersection of the three surfaces S v , S ER  and S η in 4-D P B -P s -FR-I inner  space (block  2197 ). This curve lies along a set four dimensional location point (P B ′, P S ′, FR′, I inner ′) i  where the index i refers to a particular one of a theoretically infinite number of four-dimensional points along the line or curve of intersection. 
     The next step is to set the chamber parameters of P B , P s , FR, I inner  concurrently to any one of the four dimensional locations along the line/curve of intersection in 4-D space (block  2199 ). Thereafter, the chamber parameters P B , P s , FR, I inner  may be varied along the curve of intersection so that their concurrent values coincides with one of the four-dimensional point (P B ′, P S ′, FR′, I inner ′) i  along the line or curve of intersection (block  2201 ). 
     Alternating the Under-Constrained 4-D Control Space Method with Real Time Feedback Control: 
     A further option is to transition control over to the real time feedback control loops of  FIGS. 9-11  in the step of block  2187  of  FIG. 45 . Specifically, the chamber control process of steps  2189 - 2201  of  FIG. 45  may be phased out and process control turned over to the feedback control loops of  FIGS. 9-11  based upon real time measurements of plasma parameters by the measurement instrument  140 . This option is depicted in the step of block  2187  of  FIG. 45 . The translation-based chamber control steps of blocks  2167 - 2185  may be employed at the beginning of a plasma process (when real time measurements are available). Then, after the plasma process is sufficiently underway for real time measurements to become available, the step of block  2187  is performed to transition chamber control to the real time feedback control loops of  FIGS. 9-11 . Process control may be temporarily returned to the translation-based steps of blocks  2189 - 2201  whenever a significant change is made in the user selected values of one or more plasma parameters. This option enables the chamber to nearly instantaneously meet any changes in process recipe, a significant advantage. 
     Controlling M Plasma Parameters with N Chamber Parameters: 
     The processes described above in this specification concern two-dimensional, three-dimensional or four-dimensional control spaces. In fact, the invention may be carried out using any number of chamber parameters to simultaneously realized desired values of any number of plasma parameters. The plasma parameters subject to user-selected values may be selected from the group of plasma parameters that includes ion energy or wafer voltage, ion density, ion mass, etch rate, wafer current, etch selectivity, and so forth. The chamber parameters to that are controlled may be selected from the group that includes source power, bias power, chamber pressure, inner coil magnet current, outer coil magnet current, inner gas injection zone gas composition, outer gas injection zone gas composition, inner gas injection zone flow rate, outer gas injection zone flow rate, and so forth. Preferably, the number of selected plasma parameters is the same as the number of selected chamber parameters. However, the numbers may differ. For example, if the number of selected plasma parameters is less than the number of selected chamber parameters, then the system is under-constrained and at least one additional degree of freedom is present that permits the chamber parameters to be varied while continuing to meet the user-selected plasma parameter values. If the number of selected plasma parameters exceeds the number of selected chamber parameters, then the system is over constrained. In this case, the contours or surfaces of constant plasma parameter values may intersect at along several lines or points and control may require choosing between such points or interpolating between them. 
     The process of  FIG. 46  requires characterization of the chamber, which begins with the step of selecting a first one of N chamber parameters (block  2203 ) and ramping the selected chamber parameter while sampling M selected plasma parameters with the measurement instrument  140  (block  2205 ). A determination is made of whether all of the N chamber parameters have been selected (block  2207 ). If not (NO branch of block  2207 ), the next one of the N chamber parameters is selected (block  2209 ) and the process loops back from block  2209  to the step of block  2205 . Otherwise (YES branch of block  2207 ), the process continues with the next step, namely the step of block  2211 . In the step of block  2211 , the measured data from block  2205  is used to construct an N-variable function of each of the M plasma parameters. Each of these functions has all N chamber parameters as independent variables. In the next step (block  2213 ), for each possible value of each of the M plasma parameters, the contour generator  1120  constructs an N-1 dimensional contour of constant value in an N-dimensional space in which each of the N chamber parameters is a dimension. This completes the characterization of the chamber that will enable subsequent steps or process to translate the M plasma parameters to concurrent values of the N chamber parameters. 
     The next phase of the process of  FIG. 46  is to translate a set of M user-selected values for the M plasma parameters to concurrent set of N values for the N chamber parameters. This phase begins with the receipt of the user-selected values for the M plasma parameters (block  2215 ). 
     If the number of plasma and chamber parameters is the same (i.e., if M=N), then the next step is the step of block  2217 . In the step of block  2217 , the controller  1110  fetches the corresponding contour of constant value for each of the M plasma parameters and determines their point of intersection in N-dimensional space. Then, the feedback controller  950  sets the N chamber parameters to their respective values at the point of intersection (block  2219 ). 
     If the number of plasma parameters is less than the number of chamber parameters (e.g., if M=N−1), then the system is under-constrained so that there is at least one extra degree of freedom. For the case in which M is one less than M, the following steps may be performed: 
     Block  2221 : fetch the corresponding contour of constant value for each of the M plasma parameters and determine their line or curve of intersection in N-dimensional space; 
     Block  2223 : vary the N chamber parameters so that their respective values are restricted to lie along the line/curve of intersection. 
     The foregoing steps complete the configuration of the N chamber parameters to realize a set of user-selected values for the M plasma parameters, or (conversely) the translation of the user-selected values of the M plasma parameters to required concurrent values of the N chamber parameters. In the optional step of block  2187 , this process may be temporarily replaced by the real time feedback control process discussed above with reference to the feedback loops of  FIGS. 9-11  based upon real time measurements of plasma parameters by the measurement instrument  140 . 
     While the invention has been described in detail with reference to preferred embodiments, it is understood that variations and modifications thereof may be made without departing from the true spirit and scope of the invention.