NUMERICAL INVESTIGATION OF FEEDBACK CONTROL IN PLASMA PROCESSING REACTORS INTRODUCTION

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1 NUMERICAL INVESTIGATION OF FEEDBACK CONTROL IN PLASMA PROCESSING REACTORS Shahid Rauf ad Mark J. Kusher Departmet of Electrical ad Computer Egieerig Uiversity of Illiois 1406 W. Gree Street, Urbaa, Illiois The cotiuously icreasig complexity of fabricatig microelectroics devices has made it ecessary to cosider utilizig feedback cotrol durig plasma processig steps. To ivestigate ad optimally select cotrol strategies, comprehesive plasma equipmet models are required. This paper describes a versatile simulatio tool, the Virtual Plasma Equipmet Model (VPEM), that has bee developed to computatioally ivestigate feedback cotrol i plasma processig equipmet. The VPEM is a extesio of a detailed plasma equipmet model, which has bee equipped with sesor, actuator ad cotroller modules. The VPEM was used to ivestigate feedback cotrol i iductively coupled plasmas usig cotrollers based o a respose surface methodology. The results from the VPEM suggest strategies whereby feedback cotrol ca be used to compesate for gas leaks, cotrol drift i process parameters such as power ad pressure, ad ullify the effect of log term chages i wall coditios. INTRODUCTION As microelectroics device dimesios cotiue to shrik ad wafers cotiue to icrease i size, it is becomig ecessary to have tighter toleraces durig the fabricatio process to maitai high yields. This is particularly true for plasma processig steps. Feedback cotrol has, therefore, become a importat issue i plasma processig equipmet desig. Theoretical ad experimetal research [1-3] has demostrated the utility of feedback cotrol i stabilizig plasma processes, cotrol of exteral disturbaces ad improvemet of importat etchig ad depositio characteristics. To ivestigate ad optimally select cotrol strategies, comprehesive equipmet models liked to cotrol algorithms are required. I this paper, we report o a geeral plasma equipmet simulatio tool, the Virtual Plasma Equipmet Model (VPEM), that has bee developed to theoretically ivestigate feedback cotrol i plasma processig equipmet. The VPEM makes use of the Hybrid Plasma Equipmet Model (HPEM) [4] to simulate the plasma. I the VPEM, the HPEM is coected to a exteral sesor module, actuator module ad a programmable cotroller module i a feedback cotrol loop.

2 We used the VPEM to study a umber of feedback cotrol related problems i iductively coupled plasmas (ICP) with Ar, Ar/N 2 ad Ar/Cl 2 gas mixtures. I these studies, the cotrollers were desiged usig a respose surface based methodology. Our results suggest meas whereby cotrollers ca be desiged to stably cotrol drifts i process parameters such as gas pressure ad iductive power, compesate for gas leaks ad ullify the effects of log term chages i wall coditios. DESCRIPTION OF THE PLASMA MODEL The geeral structure of the VPEM is illustrated i Fig. 1. The VPEM makes use of the HPEM for simulatig the plasma. Sice the HPEM has bee described i detail i several previous publicatios [4-6], it is oly briefly discussed here. The HPEM cosists of three coupled modules. The first module computes the iductive electromagetic fields. The secod module simulates electro eergy trasport usig either a Mote Carlo simulatio or by solvig the electro eergy equatio coupled with a solutio of the Boltzma equatio for rate coefficiets. The third module solves cotiuity ad mometum equatios for species desities, ad computes the electrostatic fields. The HPEM iterates the three coupled modules util quasi-steadystate coditios are obtaied. To make the HPEM suitable for ivestigatig feedback cotrol problems, three modules were added. I the actuator module, process parameters icludig gas pressure, iductive power depositio, electrode voltages, gas flow rate ad mole fractio of gases i the feed are adjusted. I the sesor module, the output of the HPEM is used to emulate quatities that are ideally measured by experimetal sesors. These iclude poit ad spatially averaged desities, flux o the wafer, flux at a give locatio ad io eergy flux at the wafer. The sesor ad actuator modules are liked through a programmable cotroller. I a typical VPEM simulatio, the user sets up the HPEM simulatio ad, i additio, specifies the sesors, actuators ad a desired set poit. The HPEM computes the steady-state plasma coditios. This iformatio is used by the sesor module to emulate sesor data, which is passed to the cotroller module. The cotroller module computes how much the actuators eed to be adjusted to meet the set poit, ad passes this iformatio to the actuator module. The actuator module adjusts process parameters ad rerus the HPEM simulatio. This process is repeated util the userspecified termiatio coditio has bee satisfied. CONTROLLER DESIGN PROCEDURE We used a respose surface based methodology to desig the cotrollers. The first step i the cotroller desig is to specify the sesors, actuators ad actuator parameter rage that have to be used. Usig this iformatio, simulatios are ru at selected poits withi the actuator parameter rage, ad respose surfaces of sesor

3 output as a fuctio of actuator settigs are costructed. We used desig of experimet techiques to miimize the umber of simulatios that must be performed. For cotroller desig, the useful iformatio that is extracted from these respose surfaces are the least mea square polyomial approximatios likig the sesors ad actuators. I our studies, we foud that quadratic polyomials were adequate for desigig stable cotrollers. We will, therefore, restrict our attetio to quadratic polyomials i the followig discussio. For a -actuator -sesor system, these polyomials have the form y j = c j + a jk ( xk xk0) + b jkl( xk xk0)( xl xl 0 ), (1) k= 1 k= 1 l = 1 where j=1,2,,. y j are the outputs (sesors), x k are the iputs (actuators), x k0 are the ceter poit withi the rage of x j, ad c j, a jk ad b jkl are costats obtaied from the respose surfaces. The basic goal i the problems we studied was to adjust the actuators X = [x 1, x 2,, x ] T so that the sesor sigal Y = [y 1, y 2,, y ] T ca be made to approach a desired target Τ = [t 1, t 2,, t ] T. To determie how much the actuators eed to be adjusted i a give situatio, we cosider a small chage δx k i actuators i Eq. (1). This will modify the sesor outputs to y j + δy j. Assumig that δx k << x k, oe ca differetiate Eq. (1), liearize the resultig equatio ad write it i matrix form as δ X = A 1 δy, (2) where δy = [δy 1, δy 2,, δy ] T, δx = [δx 1, δx 2,, δx ] T ad A is a matrix with elemets Ajk = a jk + ( bjkl + bjlk)( xl xl0). (3) l= 1 Settig δy = T - Y m ad δx = X m+1 - X m, where the subscript m deotes the curret settigs ad m+1 deotes the ew values, we ca write Eq. (2) as X = X + B A 1 ( T Y ). (4) m+ 1 m m We multiplied A 1 with a diagoal matrix B so that the actuator gais ca be idividually chaged to improve stability. cotrollers. We used Eq. (4) for implemetig the

4 FEEDBACK CONTROL OF PLASMAS I this sectio, we describe two problems i which feedback cotrollers have bee used to compesate for exteral disturbaces. Both problems have bee studied i the iductively coupled Gaseous Electroics Coferece (GEC) referece cell [7]. I the first problem, we cosider a Ar discharge (20 mtorr, 400 W), ad desig a 2-iput 2-output cotroller which is meat to keep sesor outputs at give values. Gas pressure ad iductive power depositio are the two actuators. Average electro desity (as might be measured by a microwave iterferometer) ad total Ar + io flux to the wafer (to emulate etch rate of a io drive etch process) are the sesors. It was ascertaied that this cotroller ca stably cotrol drifts i actuators. The problem we cosider here examies the behavior of this cotroller i the presece of a uwated leak of N 2 ito the reactor (emulatig a air leak). The results for this case are show i Fig. 2. Whe N 2 is added ito the reactor, both plasma desity ad Ar + io flux to the wafer decrease. This is a cosequece of the fact that some of the iductive power is ow diverted towards o-ioizig collisios of electros with N 2 molecules. For costat power depositio, less power is therefore available for ioizatio. To compesate for the decrease i sesor sigals, the cotroller icreased the iductive power ad slightly decreased the gas pressure. This brought the sesor sigals back to their origial values. The small oscillatio i the steady-state is a cosequece of the fact that the cotroller is operatig i a slightly differet system (Ar/N 2 ) tha the oe it was desiged for (Ar). Whe reactive gases are used i plasma processig reactors, wall coditios may chage over time due to passivatio or polymer buildup. This chage i wall coditios ca appreciably modify the plasma characteristics by chagig the stickig coefficiet of radical species. A useful applicatio of feedback cotrollers would be to compesate for these log term drifts. I the ext example, whose results are show i Fig. 3, we cosider a 10% mixture of Cl 2 i argo (27 mtorr, 380 W). To desig the 2- iput 2-output cotroller, we chose iductive power ad rf bias voltage as the actuators. The sesors were average electro desity ad total flux of Cl + ios to the wafer. To simulate a chage i wall coditios, we modified the stickig coefficiet of Cl Cl 2 at the reactor walls. Whe the stickig coefficiet was icreased, first from to 0.01 ad the from 0.01 to 0.04, Cl 2 cocetratio i the reactor icreased, which reduced the plasma desity due to attachmet ad reduced the Cl atom desity. With fewer electros ad Cl atoms available, Cl + desity decreased. The cotroller respoded to the chage i sesor sigals by icreasig the iductive power ad decreasig the rf bias voltage. This brought the sesor sigals back to their origial values.

5 CONCLUSIONS I this paper, we described a geeral plasma equipmet simulatio tool, the Virtual Plasma Equipmet Model (VPEM), that was used to study feedback cotrol problems i ICP reactors. The VPEM uses the HPEM for simulatig the plasma. The iput ad output of the HPEM are liked through a sesor, actuator ad a programmable cotroller module. I the feedback cotrol problems that were discussed, cotrollers were desiged usig a respose surface based techique. These results demostrate the viability of computatioally desigig cotrollers to stabilize drifts i process parameters such as pressure ad power, compesate for gas leaks ad ullify the effect of log term chages i wall coditios. ACKNOWLEDGMENTS This research has bee supported by Advaced Research Projects Agecy/Air Force Office of Scietific Research (F ), Natioal Istitute of Stadards ad Techology, Natioal Sciece Foudatio (ECS ), Semicoductor Research Corporatio ad the Uiversity of Wiscosi ERC for Plasma Aided Maufacturig. REFERENCES 1. B. A. Rashap et al., IEEE Tras. Semicoduct. Maufact. 8, 286 (199). 2. P. K. Mozumder, ad G. G. Bara, IEEE Tras. Semicoduct. Maufact. 7, 1 (1994). 3. P. L. G. Vetzek et al., J. Vac. Sci. Techol. A, 13, 246 (199). 4. P. L. G. Vetzek, R. J. Hoekstra, ad M. J. Kusher, J. Vac. Sci. Techol. B 12, 461 (1994).. M. J. Kusher et al., J. Appl. Phys. 80, 1337 (1996). 6. W. Z. Colliso, ad M. J. Kusher, Appl. Phys. Lett. 68, 903 (199). 7. P. A. Miller et al., J. Res. Natl. Stad. Techol. 100, 427 (199).

6 Start Module HPEM Module Cotroller Module No Termiate Yes Stop FIG. 1: A schematic of the Virtual Plasma Equipmet Model (VPEM).

7 2 0.0 Disturbace 1 0 a) 9.20 Disturbace 0.00 a) b) b) c) c) d) 43 3 d) e) Time (cotroller time steps) FIG. 2: Time history of a cotrol case i which the cotroller compesates for a N 2 leak ito the reactor e) Time (cotroller time steps) FIG. 3: Time history of a cotrol case i which the cotroller compesates for a chage i the stickig coefficiet of Cl Cl 2 at the reactor walls.

ECE-S352 Introduction to Digital Signal Processing Lecture 3A Direct Solution of Difference Equations

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