Optimal Pulse Shaping for Plasma Processing

Tyrone L. Vincent1 and Laxminarayan L. Raja2

1Engineering Division, Colorado School of Mines, Golden, CO 80401tvincent@mines.eduCorresponding Author

2Department of Aerospace Engineering and Engineering Mechanics,The University of Texas at Austin, Austin, Texas 78712

lraja@mail.ae.utexas.edu

August 26, 2002

Submitted to IEEE Transactions on Control Systems Technology

ABSTRACT

Thin Þlm etching and deposition using low pressure plasma reactors is an integral partof the fabrication of Very-Large-Scale Integrated (VLSI) circuits. Standard operation ofplasma reactors uses an RF power source with constant average power to excite a plasma inthe vacuum chamber. Recently, several researchers have shown empirically that operationof plasma reactors with a periodic power input has the promise to increase the ßexibility ofplasma processing, in the sense that a greater range of operating conditions are achievable.This paper presents a numerical analysis of a global model for an argon plasma with the aimof answering the following questions: First, can a periodic input achieve effective operatingconditions that cannot be achieved using steady state inputs? Second, if the answer to theÞrst question is yes, what is the shape of the periodic input required to achieve a particulareffective operating point?

1

1 Introduction

Thin Þlm etching and deposition using low pressure plasma reactors is an integral partof the fabrication of Very-Large-Scale Integrated (VLSI) circuits. As circuit features sizesreduce, aspect ratios increase, and in general, processing requirements become more strin-gent, control of the plasma etching process becomes more difficult. An important aspectof microfabrication using plasma reactors is the possible conßict between different perfor-mance requirements. For example, in plasma etching processes that are ion limited, highcharged species densities and high ion impact energies are required to enhance etch ratesand process throughput, but can compromise etch selectivity and result in device damage.Consequently, novel approaches to the design and control of next generation plasma processreactors must be explored.

Standard operation of plasma reactors uses an RF power source with constant aver-age power to excite a plasma in the vacuum chamber. This mode of operation will becalled continuous wave (CW) processing. Recently, several researchers have shown empir-ically that operation of plasma reactors with a periodic power input has the promise toincrease the ßexibility of plasma processing, in the sense that a greater range of operatingconditions are achievable. Under pulsed operation, the main power to a plasma reactoris deliberately modulated through ON/OFF cycles using pulse width modulated (PWM)square-wave pulses with periods on the order of 10 to 100 µs[1, 2, 3, 4]. This causes theplasma parameters to cycle through periodic trajectories. Of course, even under CW RFpower, the plasma reactor is operated at a periodic steady state. However, the frequencyof the RF source is usually very high, (13.57 MHz for most industrial plasmas,) so that theperiodic behavior is limited to dynamics with very fast time components, such as electronmotion. However, by modulating the RF amplitude using a square wave input with a rel-atively low frequency, dynamic behavior with a slower time constant also is driven along aperiodic steady state, in particular, the gas phase chemistry. This leads to the possibilityof creating average dynamic behavior which is different that what can be achieved using aCW input.

This ßexibility has been demonstrated experimentally through improved etch rates, se-lectivity, process uniformity, and lower notching and charge-induced device damage[1, 2,5]. Additionally, it has been demonstrated that PWM pulses improve quality of plasma-deposited thin Þlms[6] and reduces formation of dust particles in reactors[7]. A number ofmodeling approaches have been used to elucidate the dynamics and chemistry of SWP plas-mas. Zero-dimensional �global� models[8, 9] have revealed the dynamic behavior of theseplasmas, while one-dimensional[10, 11, 12] and two dimensional models[13] have been usedto describe the spatiotemporal behavior of these systems. However, to date, a systematicmethod for selecting the correct modulating waveform to achieve desired objectives has notbeen presented.

The premise of this paper is that a particular etch result is an integration of the plasmaconditions seen during processing. Thus, a given plasma operating condition can be charac-terized by the integral over one period, (i.e. average) of the state trajectory. This averagestate trajectory will be termed an effective operating point. Two questions of are of interest.First, can a periodic input achieve effective operating conditions that cannot be achievedusing steady state inputs? Second, if the answer to the Þrst question is yes, what is theshape of the periodic input required to achieve a particular effective operating point? Notethat if the system is linear, the answer to the Þrst question is no. In this case, it is easy

2

to show that for any given periodic input, the resulting effective operating point can beachieved with a constant input equal to the average of the periodic input. However, fornonlinear systems, there is the possibility that periodic inputs may be able to reach effectiveoperating points off of the manifold of steady state operating points, achieving the increasedßexibility we seek.

In what follows, we describe howmethods for periodic optimization of dynamic systems[14,15, 16, 17, 18] were applied in order to answer the two questions posed above for a particu-lar plasma process. There was a change in focus from the majority of existing literature inthat we considered functions of an average state trajectory, (the effective operating point,)rather than the average of a function of the state trajectory (the average cost). We havenamed this process as applied to plasma processing �Optimal Pulse Shaping� or OPS. TheÞrst results of applying periodic optimal control to the problem of pulsed plasma processinghave been reported in [19]. In this paper, we present the numerical tools in detail, as wellas some new numerical results.

The paper is organized as follows. In Section 2, a dynamic model for an argon plasmais deÞned, and the behavior of the model under CW inputs is shown. In Section 3, thenumerical tools relevant to answering the two questions above are presented. Results ofthese tools applied to the plasma system are given in Section 4.

2 Plasma Dynamic Model

The plasma model describes the bulk plasma region of a pure argon inductively coupledreactor and is developed using a zero-dimensional (global) well-stirred reactor approximation(Figure 1). For low operating pressures (∼ 1-10 mTorr) that are typical of high-densityplasmas, the global model provides a reasonably accurate description of the bulk plasmaphenomena[20]. The reactor is assumed to consist of a single inlet and outlet, and a singlesurface which interacts chemically with the plasma. The reactor is assumed to operate atthe Þxed pressure with a Þxed inlet mass ßow of neutral argon gas.

The plasma is assumed to be weakly ionized as a consequence of relatively low averagepower input to the system. The composition of the plasma can therefore be assumed to bedominated by ground-state argon atoms and the mass fraction of these atoms YAr ≈ 1. Theplasma composition is described in terms of four species; electrons e, ground-state argonatoms Ar, argon ions Ar+, and argon metastables Ar*. Quasi-neutrality in the bulk plasmarestricts the electron number density to equal the argon ion number density, i.e. ne = nAr+.

The time-dependent mass fractions of the ions in the bulk plasma is given by the ioncontinuity equation[20],

dYAr+dt

=1

τßow

¡Y inAr+ − YAr+

¢+

1

ρVAsurf úsAr+WAr+ +

úωAr+WAr+

ρ, (1)

where YAr+ is the mass fraction of argon ions, Y inAr+ is the mass fraction of the argon ions atthe inlet to the reactor (equals zero in this study), úsAr+ is the molar production rate of theargon ions due to surface interactions per unit area of the surface, and úωAr+ is the molarproduction rate of the argon ions due to gas-phase chemical reactions per unit volume ofthe reactor. The argon ion molar mass is given by WAr+ and the reacting surface area isAsurf . The residence time for the ßow through the reactor is given by τßow and is speciÞed

3

Figure 1: Plasma Reactor

based on a time-independent net ßow rate of feed gases through the reactor úm, the reactorvolume V , and the reactor gas mass density ρ as

τßow =ρV

úm. (2)

The mass density ρ = PWAr/RuT , where P is the gas pressure, T is the gas temperature,WAr is the argon molar mass, and Ru is the universal gas constant. The continuity equationfor argon metastable atoms in the bulk plasma is given by

dYAr∗dt

=1

τßow

¡Y inAr∗ − YAr∗

¢+

1

ρVAsurf úsAr∗WAr∗ +

úωAr∗WAr∗ρ

, (3)

where subscript Ar* refers to the appropriate metastable argon quantities. The molar massof the metastables WAr∗ equals that of the ions.

Using the quasi-neutrality condition for the bulk plasma, the electron mass fraction canbe determined as Ye = (We/WAr)YAr+, where We is the electron molar mass. The electrontemperature in the bulk plasma Te is given by

V ddt (ρeue) = úm

¡Y ine h

ine − Yehe

¢+ úseWe (he + hsh)Asurf−

3V RuWAr

ρe (Te − T ) PkBT

³8kBTeπme

´1/2σ(1)en − e V

Pr∆E

eqr + Pinp(t).(4)

Here the electron mass density is given by ρe = NAmeρYAr+/WAr+, where NA is theAvagadro�s number and me is the electron particle mass. The electron internal energyue = 3RuTe/2We and the electron enthalpy he = 5RuTe/2We. The Þrst term on the right

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Table 1: Argon plasma chemistry used in this study[22].Reaction A(cm3/s) Ea(K) ∆Ee(eV)

e + Ar ⇒ e + Ar∗ 3.71×10−8 1.748×105 11.56e + Ar ⇒ 2e + Ar+ 1.23×10−7 2.168×105 15.7e + Ar∗ ⇒ 2e + Ar+ 2.05×10−7 0.575×105 4.16Ar∗ + Ar∗ ⇒ e + Ar+ + Ar 6.2×10−10e + Ar∗ ⇒ e + Ar 2.0×10−7 -11.56

The gas-phase reaction rate coefficients are given in the Arrhenius form:k = A exp(−Ea/Te).

hand side evaluates the net electron energy gain due to ßow into and out of the reac-tor, where the inlet quantities (superscript �in�) equal zero. The second term accountsfor electron energy loss due to surface loss of electrons, where hsh equals the electron en-thalpy corresponding to the potential drops through a grounded sheath and is computedas hsh = kB/2me ln(WAr+/2πWe), while the electron surface production rate úse is assumedto equal the ion surface production rate úsAr+. The third term accounts for energy lossdue to electron elastic collisions with the background gas. σ(1)en is the momentum transfercollision frequency between the electrons and the background neutrals which is assumed tobe 10−19m2. kB is the Boltzmann constant. The fourth terms accounts for loss of energydue to electron inelastic collisions in the gas-phase and the summation is over all electronimpact collision reactions. The term qr is molar rate of progress of each electron impactreaction and ∆Ee is the energy lost per electron due to the inelastic collision. e equals thevalue of unit charge. The last term Pinp is the time-dependent external electromagneticpower supplied to the bulk plasma. This is the control variable, and is a constant underCW processing. We will seek to determine a time varying periodic Pinp to achieve desiredbehavior.

Gas-phase and surface chemistry model: The argon plasma gas-phase chemistry is takenfrom the literature[21] and is listed in Table I. Table I also lists the∆Ee values for importantelectron-impact reactions.

The ion loss rate to the surface is limited by the ion Bohm speed and hence the molarproduction rate of argon ions per unit area of the active surface is given by

úsAr+ = −ξ uB,Ar+µρYAr+WAr+

¶, (5)

where uB,Ar+ = (RuTe/WAr+)1/2 is the Bohm speed of argon ions and ξ = 0.5 is a correction

factor that accounts for the reduction in the bulk plasma ion concentration near the surfacewhen compared to the bulk. Note that the electron surface production rate úse = úsAr+. Thesurface production rate of the metastable atoms is given by

úsAr∗ = −DeffΛ2

µρYAr∗WAr∗

¶, (6)

where Deff is an effective diffusion coefficient for the metastable atoms and Λ is an effectivediffusion length scale for the reactor.

Reactor geometry and problem parameters: A cylindrical inductively coupled plasmareactor geometry is assumed. The top plane of the cylinder deÞnes the location of a dielectric

5

0.2 0.4 0.6 0.8 1 1.2 1.40

1

2

3

4

5

6

Power (kW)

# D

ensi

ty ×

1017

0.2 0.4 0.6 0.8 1 1.2 1.40

1

2

3

elec

tron

tem

pera

ture

(eV)

← nAr

*

← nAr

+

Te →

Figure 2: Steady state plasma variables as a function of power

window through which main RF power is coupled into the plasma. A small fraction of thebottom plane is assumed to correspond to a substrate holder. The reactor diameter isassumed to be 37 cm and the distance between the top and bottom planes is 15 cm. Theeffective total volume of the reactor V = 16, 128 cm3. The active areas of the reactorwhich include the sidewalls and the a substrate holder is assumed to be Asurf = 1871 cm2.A constant reactor pressure P = 5 mTorr and gas temperature T = 300 K is assumed.The inlet conditions correspond to a pure argon ßow rate of 100 sccm with no ionizationor excitation of the feed gas at the inlet. The corresponding mass ßow rate úm = 2.72 ×10−6 kg/s and the ßow residence time τßow = 63.4 milliseconds. The effective length scalefor metastable diffusion in Eq. 6 is taken to be Λ = 4.056 × 10−2m, while the effectivediffusion coefficient is selected as Deff = 1.04 m2/s.

By exercising the model with constant power inputs, the family of operating pointsreachable using constant inputs can be determined. For the model described above, YAr+and YAr∗ are unitless, however, it is more typical to report results in terms of numberdensity (#/m3), where the species number density, ns, is obtained from the correspondingmass fraction, Ys, as ns = YsP/kBT. The results are shown in Figure 2. This Þgure givesan indication of the highly nonlinear nature of the plasma dynamics. While the ion densityis linearly dependent on applied power, the electron temperature is almost independent ofapplied power, and indeed, increases as the power decreases, while the metastable densitysaturates at higher powers. These characteristics are consistent with the classical scaling re-lationship for a high-density plasma reactor as discussed in [21]. Although the dynamics aretightly coupled and complex, the trends of steady state ions density and electron tempera-ture with increased power can be explained as follows. Referring to equation (4), under theassumption that the electron temperature stays fairly constant, increased deposited poweris absorbed by increased electron (and thus ion) generation. The size of the increase isdetermined by the energy balance between deposited power and energy loss due to particlesreaching the walls. Since the energy loss is linearly proportional to particle density, a fairly

6

linear relationship ensues between power and ion density. On the other hand, the steadystate electron temperature is determined mainly from mass balance, rather than energybalance, and is almost independent of applied power.

To explain the relevance of the steady state results, we can consider an industrial plasmaetching process, such as one that would be used in the etching of features to deÞne inte-grated circuits in semiconductor manufacturing. Although industrial plasma etching doesnot normally use only argon as the feed gas, the argon chemistry has the advantage ofsimplicity, while still demonstrating some of the salient features of plasma dynamics. Oneimportant and common example is seen in ßuorocarbon etching of features in SiO2 Þlms[21].A ßuorocarbon plasma creates reactive F and CFx radicals that diffuse to the surface ofthe Þlm. However, these radicals do not spontaneously react with the SiO2 surface. Addi-tional energy is needed to initiate reactions that will produce volatile Si and O containingspecies. This energy is delivered by high velocity ions impinging on the surface, acceleratedby a bias voltage that develops between an RF plasma and the Þlm surface. This etchingmechanism helps to ensure a high Þdelity transfer of a mask pattern to the Þlm surface,since impinging ions have velocities mainly in the vertical direction, and etching occurs onhorizontal surfaces, rather than vertical ones[23]. Thus, in this system, it is imperative thatthe radical ßux rates as well as the ratio of CFx radical-to-ion ßux rates to the surfacebe controlled. To examine this behavior in the context of pulsed power operation, we willuse the argon plasma as a surrogate for a ßuorocarbon plasma with argon ions standing infor ßuorocarbon ions, and argon metastables standing in for ßuorocarbon radicals. Morecomplex chemistries will be the subject of future research.

3 Numerical Analysis of Periodic Inputs

The notion that physical system performance can be improved by using time-varying in-puts is not a new one, and has been studied extensively in the context of optimal controlin chemical reaction engineering[14, 24]. This idea has also been extended to trajectoryplanning for fuel-efficient ßight[25], economic planning[26], and dynamic optimization ofchemical vapor deposition processes[27]. Typically, system performance is measured by anobjective function which depends on the time-dependent variables of the physical model ofthe system, and optimal control theory can be used to determine when a steady state inputis not optimizing.

First order necessary conditions based on weak variations for a periodic input to beoptimal were apparently Þrst presented in [14]. Since an optimizing constant input alsosatisÞes these necessary conditions, second order information is needed to determine if aperiodic input exists that can improve upon a steady state input. When a (weak variation)periodic control exists that improves upon the best available steady state input, the optimalcontrol problem is called (locally) proper. A second order necessary condition for locallyproper, called the π test, was Þrst developed in [16], and corrected in [17] by includingconstraint qualiÞcation conditions.

Although a steepest decent method was introduced in [14] for numerical calculation ofoptimal periodic control, the majority of work has been concentrated on extending the testsfor proper periodic optimal control, e.g. [15, 28, 18, 29]. SigniÞcant exceptions includethe work of Speyer and coworkers [30, 25] who iteratively solve the Þrst order necessaryconditions, except for a transversaility condition associated with the input period. They

7

then travel along the family of trajectories satisfying the Þrst order necessary conditionsuntil the transversaility condition is met. More recently in [26] a comparison between solvingthe Þrst order necessary boundary value problem and utilizing a discretization technique ispresented.

Our approach has focused on gradient methods for the optimization of a Þnite dimen-sional parameterization of the periodic input. A gradient method is chosen because of theeasy availability of numerical packages for constrained optimization based on gradients. Inaddition, utilizing a parameterization of the input can be convenient for implementing atrade-off between computational complexity vs. accuracy, and parameterization may evenbe suggested by implementation constraints.

3.1 DeÞnitions and Notation

In what follows, we will be attempting to Þnd periodic inputs which create desirable trajec-tories in the plasma system. It will be convenient to deÞne notation and some assumptions.The general form of the plasma dynamics is as follows:

úx = f(x(t), u(t)) (7a)

where x ∈ Rn is a length n vector of state variables, and u ∈ Rm is a vector ofm actuatorinputs. In our case, n = 3 and m = 1. We assume that f(x, u) is twice continuouslydifferentiable with respect to both variables, and that df

dx is invertible for all x. Underperiodic excitation, and given appropriate stability assumptions on the system, the statetrajectory will approach a periodic steady state with the same period as the input. Forconvenience, we will deÞne a normalized time s = t/τ where τ is the fundamental periodof the input. Note that in this normalized time variable, the period of the input is alwaysone. Then x(s) = x(t)|t=sτ and

dx

ds= τf(x(s), u(s)) (8)

where u(s) = u(t)|t=sτ . DeÞne fs(x, u) := τf(x(s), u(s)). We will useúx = fs(x, u) (9)

to represent Equation (8).

To deÞne a periodic input, we need only specify the input over one period. Thus, theinput will be given in normalized time by u(s) = uc(smod1) where uc(s) is deÞned fors ∈ [0, 1]. The periodic steady state trajectory can be likewise decomposed as xss(s) =xc(smod1), with xc(s) deÞned for s ∈ [0, 1].

Plasma processing seeks to modify Þlms through deposition or etching. The instanta-neous plasma state is Þltered, or integrated, through the Þlm being processed, which is theÞnal product of the reactor. Thus, it makes sense to speak of an average plasma operatingcondition deÞned by an integral of the plasma trajectory over a given period.

DeÞnition 1 The effective operating point is deÞned as

x(uc(s), τ) =

Z 1

0xc(s)ds. (10)

8

Note that the dependence of the effective operating point on the particular periodicinput was made explicit. With a change in variables back to un-normalized time, this isequivalent to x = 1

τ

R τ0 xc(t)dt, so that this is truly an average of the state trajectory.

What is most interesting and relevant to plasma processing is the possibility that theset of x which are achievable with steady state inputs may be strictly smaller than theset of effective operating conditions achievable with general periodic inputs. As alreadydiscussed, periodic inputs do not increase the size of achievable effective operating pointsif the system is linear. However, as presented in Section 2, the dynamics of plasmas arehighly nonlinear, and as will be shown, effective operating points can indeed be reachedusing periodic inputs that cannot be obtained using steady state inputs. This problem istaken up in the next section. Following this, we discuss methods for determining pulseshapes to achieve particular effective operating point.

3.2 Reachable Effective Operating Points

Consider a constant input u = u0, while assuming exponential stability of the associatedequilibrium point. The effective operating point x = x0 is given by the solution to

0 = f(x0, u0)

No averaging is needed since we are assuming a constant steady state trajectory. If weconÞne ourselves to constant inputs, the implicit function theorem[31, p. 223] characterizesthe directions that other operating points can be reached using small perturbations. In fact,given our assumptions on the function f, there exists ² > 0 and a continuously differentiablefunction g(·) such that for δu < ², the operating point for u = u0 + δu is given by x =

x0+g(δu+u0) anddgdu = −

hdfdx

i−1dfdu . The column space of

dgdu deÞne directions of operating

points locally reachable from x0 using constant inputs. More generally, we deÞne locallyreachable effective operating points for arbitrary inputs

DeÞnition 2 The vector d ∈ Rn is a reachable direction from x0 if there exists uc(s, ²), τand ²0 > 0 such that úx = fτ (x, u0 + uc(s, ²)), 0 < ² < ²0 has a periodic solution x(s, ²) with

period 1, and x(²) = x0 + g(²) where g(²) = 0 anddgd²

¯²=0

= d.

The following result is closely related to the π test, but does not appear elsewhere inthe form we require.

Proposition 3 Consider system (8) with m = 1 and steady state input u0. Select τ > 0,αk ∈ R, αk > 0, for k > 0, and ωk =

2π`kτ , `k ∈ N. Then the following is a reachable

direction:

d = α0

"·∂fs∂x

¸−1 ∂fs∂u

#+

NXk=1

αk

π1(ωk)π2(ωk)...

πn(ωk)

9

where

πi(ω) = G(ω)∗Hi

xxG(ω) +G(ω)∗Hi

xu +HiuxG(ω) +H

iuu (11)

Hi(x, u,λ) = eTi x+ λTi fs(x, u) (12)

λTi = −·∂fs(x, u)

∂x

¸−1eTi (13)

G(ω) =

µωI − df

dx

¶−1 dfdu

(14)

and ei is the unit vector in the ith direction.

The proof is given in Appendix B. We have assumed a single input, but it is straightfor-ward to extend it to multiple inputs. The reachable directions are a linear combination ofthe variation due to changes in the DC value of the input (the Þrst term), with variationsdue to periodic input (the second term). The essential part of this result is the term πi(ω),which relates small signal sinusoidal perturbations of frequency ω to changes in the averageof state i.

3.3 Periodic Optimization

Given a nonlinear model (7) it is of interest to determine a periodic input which can op-timize a desired function of this average operating point, as stated in Problem 4 below.This is a slightly different formulation than standard in optimal periodic control, wheretypically the objective function is a function of the instantaneous state averaged over thestate trajectory[14]. .

Problem 4 Given (9), x deÞned in (10) and g(x) : Rn → R. Determine τ and uc(s),s ∈ [0, 1] to minimize

J(uc(s), τ) = g(x)

A typical choice will be g(x) = 12 kx(uc(s), τ)− xdk2 where kxk =

√xTx and xd is a

desired operating point. By Þnding the gradient of J with respect to uc and τ , a numericalsearch can be performed for a (local) minima. If the minimum is zero, then we haveachieved the desired set-point. Since the input uc(s) is an inÞnite dimensional object, thenumerical algorithm uses a Þnite parameterization of the control uc(s) = µ(s,α), where αis a vector of parameters to be optimized. The available adjustable parameters are thengiven by θ =

£αT τ

¤T ∈ Rp. Gradient methods for numerical solutions of optimal controlproblems are well developed, see for example [32, 33, 34]. Of particular interest here willbe efficient use of gradient methods for achieving desired effective operating conditions forthe plasma system described above.

3.3.1 Second Order Expansion of Cost

In order to apply a numerical parameter optimization, we seek a second order expansion ofthe cost function with respect to the parameters θ in terms of the Jacobian and Hessian. Inaddition to the previously stated differentiability assumptions on the state velocity equa-tions, we will assume throughout that g(x) is twice continuously differentiable, and thatunique periodic solutions exist for θ ∈ Θ where Θ is an open set

10

Proposition 5 Given systemúx = fs(x, θ) (15)

and cost function

J(θ) = g(x) x =

Z 1

0x(s)ds

If θ ∈ Θ and there exists a periodic solution (λ(0) = λ(1)) to the differential equation

úλT= −A(t)Tλ+CT A(t) =

∂fs∂x

C =dg

dx(16)

Then, the second order expansion of the cost function is given by

J(θ) = J(θ0) +∇J(θ0)T δθ + 12δθT∇2J(θ0)δθ + o(δθ2)

where

θ = θ0 + δθ (17)

H(x, θ) = Cx+ λT fτ (x, θ) (18)

∇J(θ0) =·Z 1

0λTB(s)ds

¸T(19)

∇2J(θ0) ="µZ 1

0

£L(s)T I

¤ · Hxx HxθHθx Hθθ

¸·L(s)I

¸ds

¶+

µZ 1

0L(s)ds

¶Td2g

dx2

µZ 1

0L(s)ds

¶#(20)

B(s) = ∂fs∂θ , and

L(s) =£`1(s) `2(s) · · · `p(s)

¤where `i is the periodic solution to

úi = A(s)`i +Bi(s) (21)

and Bi(s) is the ith column of B(s).

The proof is given in Appendix B. Note that Þnding the Jacobian ∇J, requires Þndingthe periodic solution of a linear time-varying system (16). This is a signiÞcant savings overa Þrst difference approximation, which would typically require p + 1 periodic solutions ofthe nonlinear system (15). However, Þnding the Hessian ∇2J requires Þnding p additionalsolutions of the linear time-varying systems (21), while a Þrst difference approximationutilizing the expression for ∇J would typically require p additional periodic solutions of(16). There is thus no computational savings over a Þrst difference approximation, althoughthe accuracy is not dependent on the correct selection of the size of the Þrst difference. TheJacobian and Hessian found above can be used in a Newton or quasi-Newton optimizationmethod in order to Þnd inputs to reach a particular desired effective operating point.

11

3.3.2 Numerical solutions for the periodic steady state

At several instances, a periodic solution of a differential equation is required, for examplewhen Þnding the Jacobian and Hessian, or determining the effective operating point for agiven periodic input. With the input Þxed, and initial condition x0, let the solution of thedifferential equation of interest from time 0 to time 1 be denoted by φ(t, x0). The initialcondition which lies on the periodic solution is obtained when �x = φ(1, �x). Thus we seek toÞnd the solution to the equation x− φ(1, x) = 0. If φ(1, x) is asymptotically stable, in thesense that the discrete time system xk+1 = φ(1, xk) has an asymptotically stable equilibriumpoint at �x, then if x0 is chosen sufficiently close to �x, we can simply simulate the differentialequation from 0 to 1 repeatedly to Þnd xk and �x = limk→∞ xk. This is useful if the mappingφ(1, x) is �sufficiently stable� in the sense that the Jacobian of φ(1, �x) has eigenvalues withmagnitude much smaller than one, so that convergence can occur after only a few iterations.If this is not the case, an alternative is to use Newton�s method. Given an initial guess x0,a simulation results in x01 = φ(1, x0), but in general x01 6= x0. A Taylor series expansion ofφ(1, x) around x0 gives to Þrst order a prediction of how a new initial condition xi wouldresult in Þnal condition xf :

xf = x01 +

∂φ(1, x0)

∂x(xi − x0) .

Note that ∂φ(1,x0)∂x is in fact the state transition matrix from initial time 0 to Þnal time 1 ofthe linearized differential equation evaluated along the state trajectory with initial conditionx0. We will use the notation Φ(t, τ) for the state transition matrix from time τ to time tof the variational system, so that ∂φ(1,x0)

∂x = Φ(1, 0). If the Taylor series approximation isaccurate, the Þxed point such that xi = xf is given by

x = x01 +Φ(1, 0) (x− x0)

so that the next guess for the Þxed point is given by

x1 = (I −Φ(1, 0))−1¡x01 −Φ(1, 0)x0

¢this process is iterated until x0i+1 is sufficiently close to xi.

A key element of Newton�s method is the calculation of the Jacobian of the functionx − φ(1, x), that is, I − Φ(1, 0). To avoid the repeated calculation of the state transitionmatrix, a secant method, such as Broyden�s method, can be used, which generates anapproximation to I −Φ(1, 0) from function evaluations[35].

3.3.3 Numerical Implementation

Once a control parameterization is chosen, using the Jacobian and Hessian expressionsabove, standard parameter optimization methods can be used to Þnd the minimum of theobjective function. In our case, we selected a sequential quadratic programming (SQP)method supplied with the software package MATLAB[36]. At each iteration, the gradientof the objective function with respect to α was found and an approximation to the Hessianwas calculated using a BFGS update method. The Hessian approximation was found tobe more efficient that calculating the Hessian explicitly using (20). A quadratic program-ming subproblem establishes a descent direction based on a quadratic approximation of

12

the Lagrangian. A line search is performed along this descent direction, and the processrepeated until convergence. A detailed discussion of SQP can be found in many textbookson parameter optimization, e.g. [37].

Two different parametrizations have been used. Recall that we use the notation uc(s)for the input deÞned over one (normalized) period. The Þrst parameterization is a pulsewidth modulated signal, which has been used extensively in experimental work. In thiscase, The control is deÞned by the �on� pulse power level, µ0, the fractional pulse width,t1, and the period. A short transition is include to make the input continuous. Thus,

u(s) =

µ0 0 ≤ s < t1 − .01

(t1−s)µ0+(s−t1−.01)µoff.01 t1 − .01 ≤ s < t1µoff t1 ≤ s < .99

(1−s)µoff+(s−.99)µ0.01 .99 ≤ s < 1

For numerical stability when solving the plasma dynamic equations, a small µoff = 0.1 ischosen.

The second parameterization chosen for this work is a piecewise linear control withvariable time intervals, which allows for a much greater array of possible pulse shapes. Thecontrol is parameterized by deÞning the values of the periodic input over one period, uc(s),s ∈ [0, 1], as well as the uc(0) = µ0 and the values of uc(s) at a Þnite set of time points(t1, t1, . . . , tk) within the pulse period, where ti ∈ (0, 1) and k is the last time point in thepulse interval. The parameterized control is determined as,

u(s) =

(t1−s)µ0+sµ1

t10 ≤ s < t1

(ti+1−s)µi+(s−ti)µi+1ti+1−ti ti ≤ s < ti+1

(1−s)µk+(s−tk)µ01−tk tk ≤ s < 1

(22)

The vector of optimization parameters is therefore,

α = [µ0, t1, µ1, t2, µ2, · · · , tk, µk, τ ]T . (23)

Note that u(0) = u(τ) = µ0, so that the control is indeed periodic.

Because we are parameterizing the input with variable time, fs(x, θ) does not satisfythe assumption of two times differentiability. However, in practice, the Hessian still exists,and can be found by using generalized functions (i.e. impulse functions).

Constraints on the control parameters must be enforced to ensure a solution will be foundthat is implementable, as well as to avoid possible singularities in the dynamic model. Abound on the minimum and maximum control magnitude µmin < |µi| < µmax is enforced.In addition, bound on the time interval tmin < |ti − ti−1| < tmax and on the period τmin <τ < τmax is also speciÞed.

For the results presented in this paper, the following parameters were chosen: k = 10,tmin = 0.01, tmax = 1, τmin = 5 × 10−5 s, τmax = 4 × 10−3 s, µmax = 10 kW, andµmin = 0.1 W. In addition, we required that the average power be less than 300 W.

Now, the periodic input which can achieve a speciÞed effective operating point is notunique, since a rotational shift in time will not change the integral over one period of thestate trajectory. (A rotational shift indicates that the part of the signal rotated past either0 or 1 will be appended to the other end) An additional constraint could be imposed to

13

eliminate this redundancy, for example specifying the value of one of the states at time0 [26]. However, this constraint could adversely affect the convergence properties of theparametric optimization. Even if this was done, there would still be no guarantee the inputwaveform to achieve a particular operating point was unique. Instead, 5 different runs withrandom initial conditions were performed to give some insight as to what the �essential�characteristics of the input waveform were, as well as to give some conÞdence as to if aglobal minimum is achieved.

4 Results

Returning the focus to the plasma dynamics discussed in Section 2, we seek to developnumerical answers to questions about the applicability of pulsed plasma operation.

4.1 Local Reachability

To establish the increased ßexibility of pulse plasma processing, the reachable directionsfrom a given DC operating point are found. Let the steady state input be 300 Watts.Recall that πi(ω) is the small signal relative change in the (average of the) ith state due toa sinusoid of frequency ω. In Figure 3, πi(ω)xi

is plotted vs. frequency, giving a visualization ofthe relative changes in each average state. This normalization was used so that the differentscales can be plotted on the same graph. For periodic inputs of low frequency, the average iondensity and electron temperature were both increased by a periodic perturbation, while themetastable density is decreased. This low frequency behavior was caused by the curvature ofthe DC manifold around the steady state operating point, since the system is being run in apseduo-steady state manner. In this case, the input, which is symmetric around the nominaloperating point a0, is mapped to a sinusoidal-like output that is warped in one direction.The resulting change in the effective operating point came from this non-symmetry. Atintermediate frequencies (above 104 rad/s) there was a larger magnitude ion density increaseand metastable decrease, but the electron temperature variation changed sign to a decrease.At very high frequencies, the variation returned to zero, as the perturbation frequencybecomes larger than the fastest time constants of the system. To give an indication ofhow periodic perturbations increase the local reachability over CW operation, in Figure 4,we have re-plotted the information in Figure 2 but in phase space, along with cones thatrepresent possible reachable directions using periodic perturbations only (the second termfrom Proposition 3.) It should be emphasized that since this is a perturbation analysis,these cone merely represent directions, so that the size of the cones have no bearing on theactual set of reachable effective states. Note that effective state variations due to periodicperturbations clearly lie off the DC manifold.

4.2 Optimal Pulse Shaping

Although the analysis in the previous section gives insight into the expected behavior ofperiodic inputs, the analysis is only local, and another method is required to Þnd theperiodic input which can achieve an effective operating point that is signiÞcantly off the DCmanifold. Using the optimization methods discussed above, we present an interesting casewhere the reachable space enlarges as we move from CW to pulsed to OPS operation.

14

103 104 105 106 107-16

-14

-12

-10

-8

-6

-4

-2

0

2x 10-6

Frequency (rad/s)

Rel

ativ

e Va

riatio

n

Ar+

Ar*

Te

Figure 3: Relative variation in average states due to sinusoidal perturbation as a functionof frequency

0.5 1 1.5 2 2.5 3x 1017

5.5

6

6.5

7

7.5x 1016

nAr+

n Ar*

(a)

0.5 1 1.5 2 2.5 3x 1017

2.4

2.5

2.6

2.7

(b)

nAr+

T e

5.5 6 6.5 7x 1016

2.4

2.5

2.6

2.7

nAr*

T e

(c)

Figure 4: CW operation manifold along with incremental directions of effective operatingpoints achievable with periodic operation

15

We examined the case where a high metastable density is desired, while at the sametime allowing the ratio of metastables to ions to be varied. As stated before, we will placea bound on the average power of 300 Watts. From Figure 2, we see that the metastabledensity increases with power, so if only constant inputs are considered, CW operation atthe maximum of 300 Watts should be chosen, resulting in a metastable density of nAr∗ =6.83×1016. At this power level the ion to metastable ratio is 2.6. This ratio can be decreasedby lowering the steady state power, however this also decrease the metastable density. Witha 300 W bound on the average power, there is no way to increase the ion to metastableratio with CW operation. It is of interest to determine if periodic operation can achieve alarger set of effective operating points. The local analysis above indicates it is possible toincrease the ion density using periodic operation, and thus higher ratios will be possible.However, all periodic inputs decrease the metastable density, so a trade-off is expected. Thefollowing objective function was deÞned to quantify this trade-off:

g(x) = γ°°nAr∗ − 6.83× 1016°°2 + α logµ nAr+

nAr∗

¶.

The scalar γ was Þxed at 1× 10−31 in order to provide an appropriate scaling between thetwo terms, while α was varied to obtain different points on the trade-off curve.

Both the pulse width modulated (PWM) and piecewise linear (OPS) parametrizationswere used. The results are shown in Figure 5. This plot shows part of the boundary ofthe reachable effective operating points as determined using the gradient algorithm. Thethick line shows the achievable operating points with steady state (CW) operation. Bychoosing α < 0, lower ion to metastable ratios are selected. In this case, both the CW andOPS solutions converged to the CW case - a DC power level less than 300 W. However,when α was chosen greater than zero, as expected, both PWM and OPS pulse shapes wereable to trade off metastable density for higher ion to metastable ratios. Effective operatingpoints to the left of these lines were also achievable. SigniÞcantly, a more general controlparameterization is able to increase the size of the set of reachable effective operating points,so that at nAr∗ = 5 × 1016/m3 OPS is able to increase the ratio nAr+/nAr∗ by almost afactor of 2.

Viewing the state transient responses gives some insight into the different mechanismsat work. In Figures 6 and 7, the transient response for the PWM input for α = 80, and theOPS input for α = 60 respectively are shown. Examining Figure 6, we see that a PWMinput was able to increase the ion density because of its asymmetric step response. The iondensity increased quickly when the power step came on, but decreased more slowly whenthe power went to the low level. This asymmetric response was due to the dependence ofthe surface generation rate on the electron temperature. When the electron temperatureis high, ions are driven out of the plasma to balance the high electron ßux and maintainquasi-neutrality. However, when the power goes off, the electron temperature decreases andthe ion ßux out of the plasma is reduced. In Figure 7, the OPS input takes advantage of thiseffect as well, but is able to do it more effectively by a combination of pulses. A sharp, highpulse generates a large ion density, which decays slowly due to the effect discussed above.Then, a lower constant voltage beginning at about 7 ms generates metastable species torecover as high a metastable density as possible.

16

3 4 5 6 7 8x 1016

1

2

3

4

5

6

7

8

9

nAr*

nA

r+/ n

Ar*

α= 10

α= 15

α= 30

α= 40

α= 60

α= -10α= -20

α= -40α= -80

Not reachable

OPS/PWM reachable

OPS reachable

α= 100

α= 20α= 40

α= 80

OPSPWMCW

Figure 5:

0 0.02 0.04 0.060

0.5

1

1.5

2

2.5

3

3.5

Time (ms)

Mag

nitu

de

(a)

Power/1000T e

0 0.02 0.04 0.061

2

3

4

5

6

7

8

9

10

11

Time (ms)

Mag

nitu

de

(b)

n Ar+ /1 × 10 17

n Ar* /1 × 10 16

Figure 6: Transient response for α = 80. (a) Input power and electron temperature (b)nAr+ and nAr*

17

0 1 2 30

1

2

3

4

5

6

7

8

9

10

Time (ms)

Mag

nitu

de

(a)

Power/1000T e

0 1 2 30

5

10

15

20

25

30

35

40

Time (ms)

Mag

nitu

de

(b)

n Ar+ /1 × 10 17

n Ar* /1 × 10 16

Figure 7: Transient response for α = 60. (a) Input power and electron temperature (b)nAr+ and nAr*

5 Conclusions and Future Work

This paper investigated the periodic control of low-pressure plasma reactors. In partic-ular, numerical methods to Þnd solutions of periodic controls to achieve desired effectiveoperating conditions were described, utilizing a gradient descent method with an efficientapproximation to the gradient. The resulting periodic controls are able to achieve variableargon ion to metastable argon mass fraction ratio while keeping the argon ion mass fractionconstant. These results give a systematic method for developing pulse shapes to achieve de-sired effective operating points, as well as give a further example of the increased ßexibilitypossible using pulse plasma operation.

Future work will expand the plasma model to include a more complex chemistry as wellas including 1-D and 2-D spatial features. This will require a large increase the number ofstates of the model, and will increase the importance of calculating the gradient withoutrequiring a large number of simulations of the variational equations.

6 Acnowledgements

The Þrst author would like to thank Prof. Thomas Vincent for several useful discussions.

18

7 Appendix A

Proof of Proposition 3: Let uc(s, ²) = u + 2√²ue(s) where ue(s) =

PNk=0Re

£uke

j2π`ks¤,

with `0 = 0 and u0, · · · un−1 to be determined. Under exponential stability, by [38, Theorem7.3], there exists ²0 > 0 such that a periodic solution x(s, ²) with period 1 is well deÞnedfor 0 < ² < ²0. Let xc(s, ²) = x+ �xc(s, ²). It is straightforward to show that the derivativeof xc with respect to ² must satisfy

d

ds

µdxcd²

¶=dfsdx

µdxcd²

¶+dfsdu

1√²ue(s) (24)

The periodic solution of (24) is given by dxcd² (s) =1√²xe(s) where xe(s) =

PNk=0Re

£G(ωk)uke

j2π`ks¤.

Now, given Equations (12) and (13)

eTi (x(²)− x(0)) =Z 1

0Hi(x(s, ²), u(s, ²),λ)−Hi(x0, u0,λ)− λTi úx(s, ²)ds

=

Z 1

0

¡Hi(x(s, ²), u(s, ²),λ)−Hi(x0, u0,λ)

¢ds− λTi (x(1, ²)− x(0, ²))

Since x(s, ²) is periodic with period 1, the second term is zero. Expand Hi around ² = 0using a Taylor Series:

eTi (x(²)− x(0)) =Z 1

0

√²Hi

xxe +√²Hi

uue +1

2²£xe ue

¤ · Hixx Hi

xu

Hiux Hi

uu

¸·xeue

¸ds+ o(²)

By Equation (13) Hx = 0, Thus

eTi (x(²)− x(0)) =Z 1

0

√²Hi

uue +1

2²£xe ue

¤ · Hixx Hi

xu

Hiux Hi

uu

¸·xeue

¸ds+ o(²)

Now

Hu = λT ∂f

∂u= −D

·∂f

∂x

¸−1 ∂f∂u

Using Equation (11), due to orthogonality of sinusoids,

eTi (x(²)− x(0)) = −√²eTi

·∂f

∂x

¸−1 ∂f∂uu0 +

1

2

NXk=0

²πi(ωk) |uk|2 + o(²)

By selecting ei = e1, · · · ,en, we can establish that

x(²)− x(0) = −√²·∂f

∂x

¸−1 ∂f∂uu0 +

1

2²D−1

NXk=0

π1(ωk)π2(ωk)...

πn(ωk)

|uk|2 + o(²)

Select u0 =√²α0, and uk =

√2αk and the result follows.

19

8 Appendix B

Proof of Proposition 5: Consider the cost function

J(θ) = g(x)

We seek to Þnd a second order Taylor series expansion of J around a given θ = θ0. DeÞningδJ = J(θ0 + δθ)− J(θc), we have

δJ =

·dJ

dθ

¸δθ +

1

2δθ

·d2J

dθ2

¸δθ + o(δθ2)

Where dJdθ is the gradient of J with respect to θ and

d2Jdθ2

is the Hessian matrix of J.

Clearly

dJ

dθ=dg

dx

dx

dθ

=

Z 1

0

dg

dx

dx

dθdt

where dxdθ is the change in the periodic trajectory x with respect to changes in the parameterθ. DeÞne C := dg

dx and the related cost function

J 0 =Z 1

0Cx(t)dt

Then we see that dJdθ =dJ 0dθ . If we deÞne

H(x, θ) = Cx+ λT fτ (x, θ)

then

J 0(θ) =Z 1

0(H(x, θ)− λT úx)dt

Replace θ by θ + ²δθ0 and Þnd the derivative with respect to ² :

dJ

dθδθ =

dJ 0(θ0 + ²δθ)d²

¯²=0

=

Z 1

0(∂H

∂x

dx

d²+∂H

∂θδθ − λT d úx

d²)dt

choose λ to be the periodic solution of

úλ = −∂H∂x

T

so that we have

dJ 0(θ0 + ²δθ)d²

¯²=0

=

Z 1

0

∂H

∂θδθ − ( úλT dx

d²− λT d úx

d²)dt

=

Z 1

0

∂H

∂θδθ − d

dt

µλTdx

d²

¶dt

= λT (1)dx

d²(1)− λT (0)dx

d²(0) +

Z 1

0

∂H

∂θdtδθ

20

Since λ is periodic, and x is periodic for any ², the Þrst two terms on the right cancel, andwe have by inspection

dJ

dθ=

Z 1

0

∂H

∂θdt

In summary, substituting for ∂H∂θ and∂H∂x ,

dJ

dθ=

Z 1

0λTB(t)dt

whereúλ = −A(t)Tλ(t)−CT

A(t) = ∂fτ∂x , and B(t) =

∂fτ∂θ .

Now,

d2J

dθ2=d

dθ

·dg

dx

dx

dθ

¸=d

dθ

·Cdx

dθ

¸+

·d

dθ

dg

dx

¸dx

dθ

=d

dθ

·Cdx

dθ

¸+dx

dθ

T ·d2gdx2

¸dx

dθ

where d2gdx2

is the Hessian of g. Consider again the related cost function

J 0 =Z 1

0Cx(t)dt

and note thatd2J 0

dθ2=d

dθ

·Cdx

dθ

¸Let

H(x, θ) = Cx+ λT fτ (x, θ)

andúλ = −A(t)Tλ(t)−CT

so that

J 0(θ) =Z 1

0(H(x, θ)− λT úx)dt

Replace θ by θ + ²δθ and Þnd the second derivative with respect to ² :

δθTd2J 0

dθ2δθ =

d2J 0(θ + ²δθ)d²2

¯²=0

=d2

d²2

Z 1

0(H(x, θ)− λT úx)dt

=

·Z 1

0

µdx

d²

T ∂2H

∂x∂x

dx

d²+dx

d²

T ∂2H

∂x∂θδθ + δθT

∂2H

∂θ∂x

dx

d²+ δθT

∂2H

∂θ2δθ

¶dt

¸²=0

= δθTZ 1

0

£L(t)H I

¤ · Hxx HxθHθx Hθθ

¸·L(t)I

¸dtδθ

where L(t) is deÞned by δx(t) = L(t)δθ and is given by

L(t) =£`1(t) `2(t) · · · `p(t)

¤21

where `i is the periodic solution to

úi = A(t)`i +Bi(t)

with Bi(t) =∂fτ∂θi.

Finally, we note that

dx

dθ=

Z 1

0

dx

dθdt

=

Z 1

0L(t)dt

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24