Plasma fluid modeling with transient to stochastic transformation

The invention relates to the simulation method and apparatus used in plasma modeling. It includes a method to transform transient formulations of the phenomenological plasma model into a quasi-stochastic spatial formulation. Specifically, the invention aids in decreasing computational time for the modeling of plasma in a plasma processing system, particularly those involving two different time-based parameters. The invention is particularly described in connection with plasma simulations used for the optimization dual-frequency capacitively-coupled plasma etching systems.

FIELD OF THE INVENTION

The invention relates to the modeling of plasma distribution properties in the design and optimization of plasma processing systems. More particularly, the invention relates to a system and a method for modeling the distribution within the chamber of a plasma processing system, of plasma density, plasma species, or other plasma properties. The invention is particularly useful in the design and optimization of dual-frequency CCP etching systems, IPVD deposition systems, and other plasma processing or plasma enhanced processing systems.

BACKGROUND

Advanced plasma tools have been using very high frequency power sources or combination radiofrequency (RF) and direct current (DC) power supplies to advance greater control over the plasma formation process and performance. Recently, for example, capacitively coupled plasma (CCP) reactors have employed dual frequency (DF) power delivery to control the energy and density of the ions and radicals at the surface of a wafer being processed. In such reactors, the effectiveness and the quality of the process being performed depends on the distributions of this energy and of the various particle species in the processing chamber, particularly near the wafer surface. Inductively coupled plasma (ICP) reactors, such as ionized physical vapor deposition (iPVD) reactors, employ combinations of gas and coating material species plus energy introduced from a number of different sources within the geometry of a processing chamber. In these systems, the material and energy distributions that affect the process being performed are reached after initial parameters are established, transient behavior has subsided, and the system has stabilized. However, the physics of this process has not been fully understood and has been the subject of continued theoretical and experimental investigations.

Plasma modeling and numerical simulations play in an important role in understanding plasma behavior in these systems. Modeling and simulation methods can play important roles in the development and optimization of plasma equipment and related plasma processes.

While, in the last decade, computing systems have greatly increased capabilities for numerically modeling complex and robust plasma-chemical systems like etching systems (for example central processing unit (CPU) speed, multi-core architecture, memory resources, and large size data storage capacities), the computational time required for a full numerical simulation is often on the scale of many hours, and in some specific complex cases, can be several days. Most computing systems and software are not yet ready and/or capable for parallel computing for a variety and specific tasks.

For example, even with the simplest plasma model, e.g. inert argon gas with a basic geometry etching system, a computer simulation of the plasma density distribution can take several hours (2-6 hours) to converge if the model includes mutually coupled mass transport equations, heat and energy balance equations, and electromagnetic (EM) wave propagation equations of 100 MHz (i.e. RF cycle on a 10 ns time scale). To generate a full transient solution on a system excited by a lower frequency EM wave would require yet longer CPU times.

Computational time becomes even longer when more complex chemistry (number of reactant species and/or number of participating reactions) and higher resolution reactor geometry are modeled. Computing time increases yet again if a transient solution is desired, when a pulsed process is investigated by simulation, or when a process is operating on multiple frequency scales, such as in the DF-CCP etching system. Multiple frequency scales or other multiple time based factors increase computational time because electron and ion transport properties within the plasma must be solved at both time scales. For example, the first frequency of the DF-CCP etching system can be 100 MHz (corresponding to a 10 ns time scale); and the second frequency can be 1-2 MHz (corresponding to a 1 μs time scale). Thus, the DF-CCP etching system must be solved for both the 10 ns and the 500 ns (or 1 μs) time scales. The computational time is related to the product of the number of iterations that each solution requires.

If the interactions between the ion flux and the processed wafer are considered, which can be over the time scale of several thousand RF cycles, the number of iterations and computational times can increase drastically.

Thus, there is a need to develop fast methods that would reduce computational real time in complex chemistry and geometry plasma systems. The fast algorithm and/or method could apply to solving numerically by finite elements modeling (FEM) methods, multiple frequency applications, pulsed plasma processes, sequential processing, contamination studies, feature profile evolutions, or developing industrial applications for etching or deposition technologies in plasmas.

Further, there is a need to develop and invent new, fast algorithms for technological plasma simulations and modeling. The simulation procedure should be practical and efficient for aiding in process and/or hardware development, for serving as an operational and predictive flexible tool, and for generating outputs as close as possible to the development needs and workflow in real time. Thousands of wafers are processed by the technological process on a daily basis, thus a large amount of experimental data related to hardware and software performance is readily available. Simulation procedures should be competitive in the rate of generating results to provide the ability to correlate, correct, or change the technological process in real time. This can represent large material and technical resource savings.

SUMMARY OF THE INVENTION

According to principles of the present invention, a computer method is provided for simulating plasma density distribution within a plasma reactor that is responsive to at least two variables of different time bases. The computer method includes executing computer program codes according to a process of approximating transient sheath behavior of the plasma occurring in the plasma reactor in response to a variable having a first time basis and performing, on a spatial character that is based upon the probability function, a time-domain simulation of the transient plasma sheath behavior occurring in the plasma reactor in response to a variable having a second time basis. The behavior is approximated by defining a model domain that is characteristic of the plasma reactor, constructing a probability function for a time dependent variable within the model domain, and transferring the time dependent variable into the spatial character. In certain embodiments, the performing is carried out to arrive at a steady state plasma density distribution.

According to an exemplary embodiment of the invention, a computer method is provided for simulating plasma density distribution within a capacitively coupled plasma reactor having a dual frequency power source coupled to the plasma, where transient sheath behavior of the plasma occurring in the plasma reactor at a first frequency of the dual frequency power source is approximated and a time-domain simulation of the transient plasma sheath behavior occurring in the plasma reactor is performed on the spatial character at a second frequency of the dual frequency power source.

In certain embodiments of the invention, the constructing includes random time sampling of the time dependent variable and constructing a probability function of a predetermined plasma parameter response to an incremental unit of a time dependent variable within the model domain. In some embodiments of the invention, the defining includes establishing at least one initial value and at least one boundary condition for the model domain.

A plasma reactor optimized in accordance with the simulation performed by the computer method of the invention is also provided. For example, a dual-frequency plasma etching reactor optimized in accordance with the simulation performed by the computer method of the invention is provided.

A computer program product is also provided having computer readable program codes embodied in the medium that when executed causes a computer to execute the method of the invention. A simulation apparatus is also provided having a calculating unit programmed with the computer program product.

DETAILED DESCRIPTION

Turning to the drawings, wherein like numbers denote like parts throughout the several views,FIG. 1illustrates a hardware and software environment for a computing system10that may include an integrated circuit device (hereinafter “ICD”) consistent with embodiments of the invention. The computing system10, for purposes of this invention, may represent any type of computer, computer system, computing system, server, disk array, or programmable device such as multi-user computers, single-user computers, handheld devices, networked devices, etc. The computing system10may be implemented using one or more networked computers, e.g., in a cluster or other distributed computing system. The computing system10will be referred to as “computer” for brevity sake, although it should be appreciated that the term “computing system” may also include other suitable programmable electronic devices consistent with embodiments of the invention.

The computer10typically includes at least one processing unit12(illustrated as “CPU”) coupled to a memory14along with several different types of peripheral devices, e.g., a mass storage device16, a user interface18(including, for example, user input devices and a display), and a network interface20. The memory14may be comprised of dynamic random access memory (DRAM), static random access memory (SRAM), non-volatile random access memory (NVRAM), persistent memory, flash memory, at least one hard disk drive, and/or another digital storage medium. The mass storage device16is typically at least one hard disk drive and may be located externally to the computer10, such as in a separate enclosure or in one or more networked computers21, one or more networked storage devices22(including, for example, a tape drive), and/or one or more other networked devices23(including, for example, a server). The computer10may communicate with the networked computer21, networked storage device22, and/or networked device23through a network24. As illustrated inFIG. 1, the computer10includes one processing unit12, which, in various embodiments, may be a single-thread, multithreaded, multi-core, and/or multi-element processing unit as is well known in the art. In alternative embodiments, the computer10may include a plurality of processing units12that may include single-thread processing units, multithreaded processing units, multi-core processing units, multi-element processing units, and/or combinations thereof as is well known in the art. Similarly, memory14may include one or more levels of data, instruction and/or combination caches, with caches serving an individual processing unit or multiple processing units as is well known in the art. In some embodiments, the computer10may also be configured as a member of a distributed computing environment and communicate with other members of that distributed computing environment through the network24.

The memory14of the computer10may include an operating system26to control the primary operation of the computer10in a manner that is well known in the art. In a specific embodiment, the operating system26may be a Unix-like operating system, such as Linux. The memory14may also include at least one application28, or other software program, configured to execute in combination with the operating system26and perform a task. It will be appreciated by one having ordinary skill in the art that other operating systems may be used, such as Windows, MacOS, or Unix-based operating systems, for example, Red Hat, Debian, Debian GNU/Linux, etc.

In general, the routines executed to implement the embodiments of the invention, whether implemented as part of an operating system26or a specific application28, component, algorithm, program, object, module or sequence of instructions, or even a subset thereof, will be referred to herein as “computer program code” or simply “program code.” Program code typically comprises one or more instructions that are resident at various times in memory and storage devices in a computer, and that, when read and executed by at least one processor in a computer, cause that computer to perform the steps necessary to execute steps or elements embodying the various aspects of the invention. Moreover, while the invention has been, and hereinafter will be, described in the context of fully functioning computers and computer systems, those skilled in the art will appreciate that the various embodiments of the invention are capable of being distributed as a program product in a variety of forms, and that the invention applies regardless of the particular type of computer readable media used to actually carry out the invention. Examples of computer readable media include, but are not limited to, recordable type media such as volatile and non-volatile memory devices, floppy and other removable disks, hard disk drives, tape drives, optical disks (e.g., CD-ROM's, DVD's, HD-DVD's, Blu-Ray Discs), among others, and transmission-type media such as digital and analog communications links.

Those skilled in the art will recognize that the environment illustrated inFIG. 1is not intended to limit the present invention. Indeed, those skilled in the art will recognize that other alternative hardware and/or software environments may be used without departing from the scope of the invention.

With the computing system described in some detail, an exemplary physical system to be modeled is described. The system to be modeled and simulated includes an etching apparatus described in the plasma model and can be a DF-CCP etching system such as those taught in U.S. Pat. Nos. 5,997,962; 6,426,477; and 6,576,860 each incorporated herein by reference in its entirety; however, the necessary details will be described specifically herein. This system is selected because its nominal complexity renders it suitable for illustration purposes. It should be understood that the principles described herein are applicable to a wide variety of other plasma processing systems. Other examples of plasma processing systems and plasma sources that are described with the aid of computer simulations are described in U.S. Pat. Nos. 7,273,533 and 6,523,493 and U.S. Patent Application Publication No. US2005/0160985, each expressly incorporated by reference herein in its entirety.

FIG. 2schematically illustrates the basic components of a representative DF-CCP system30, which includes a process chamber32formed within a container34constructed from conductive material, such as aluminum having an oxidized alumite surface. The container34is grounded through a ground line36. A support plate38formed of an insulating material is arranged on the bottom of the process chamber32. A wafer support40is arranged on the insulating support plate38and is operable to hold the wafer42to be etched. The wafer support40also acts as a lower electrode.

The wafer support40is made of aluminum having an oxide surface and can include several features, such as heating and chilling elements (not shown).

The wafer support40is surrounded by a baffle plate44arranged around the wafer support40. An inner portion of the baffle plate44is affixed to the wafer support40. The baffle plate44serves to facilitate the uniform flow of process gas from a processing volume46of the process chamber32above the wafer42to an exhaust volume48in the lower portion of the process chamber32from which it is exhausted by an exhaust pump50.

A shower head52is arranged at the top of the process chamber32for introducing a gas from a gas source54. The shower head52is attached to the ceiling56of the process chamber32through an insulating support58made of alumina and a cooling member60made of aluminum. An upper electrode62is fixed to the bottom of the shower head52and is made of a conductive material, such as single crystalline silicon. The upper electrode62serves as the outlet plate of the shower head52and includes a plurality of delivery holes64through which a gas within the shower head52can be delivered into the process chamber32above the wafer42on the wafer support40. The upper electrode62should be approximately the same in size as the lower electrode (represented by the wafer support40).

The process chamber32further includes a shield ring66near the upper electrode62and made of an insulating material such as quartz. A second insulating ring68can be made of a fluoride-based resin and positioned between the top of the shield ring66and ceiling56of the process chamber32. Generally, the frequency of the RF for the upper electrode62should be much greater than the frequency of the RF to the lower electrode (represented by the wafer support40).

The upper electrode62and the lower electrode (represented by the wafer support40) are each respectively connected to individual power sources or RF power supplies70,72. Each RF power supply is respectively connected through a matching network74,76to the upper electrode62and the lower electrode (represented by the wafer support40). Generally, the RF power supply72for the lower electrode will operate at a lower frequency, such as 2 MHz (from about 1 MHz to about 5 MHz). The RF power supply70for the upper electrode62will operate at a higher frequency, such as 100 MHz (from about 10 MHz to about 200 MHz).

The DF-CCP system30includes a control system (not shown) operable to govern the RF power supplies70,72and to operate the other sub-systems of the system30to perform a plasma etch process on the wafer42.

In operation, the DF-CCP system30will generate a capacitively coupled plasma in the processing volume46for etching the wafer42upon the wafer support40according to a first embodiment of the present invention by which, for example, an oxide film such as SiO2is etched on the wafer42.

In performing an etching process in the DF-CCP system30, a wafer42is positioned upon the wafer support40and the processing volume46is set to a predetermined vacuum. One or more process gases, including Ar, N2, CF4and others, is supplied from the gas source54at a predetermined flow rate. The one or more process gases are exhausted at the exhaust volume48by the vacuum exhaust pump50. Typical pressures are in the range of from 1 to 20 mTorr (up to about 300 mTorr).

After the vacuum is established, the upper electrode62is supplied with an RF energy having a frequency of 100 MHz and a power of, e.g., 2 kW from the RF power supply70. The lower electrode (represented by the wafer support40) is supplied with an RF energy having a frequency of 2 MHz and a power of, e.g., 1 kW from the RF power supply72. The RF power applied to the upper electrode62generates a plasma between the upper electrode62and the wafer support40, while the RF power to the lower electrode (represented by the wafer support40) provides a self-bias on the wafer42for directing the etchant ions within the plasma onto the wafer42. This combination of lower and higher frequency RF energies creates an immediate and stable plasma.

Within the plasma, electrons are energized by the high frequency RF field. As the high energy electrons diffuse through the plasma, the high energy electrons will undergo non-elastic collisions with molecules of the process gas. These collisions can result in the generation of radicals, cations, and/or anions by the dissociation of the process gas wherein the generating of these species will depend on the particular gas, the energy of the high energy electrons, and the number of collisions occurring between electrons and a particular gas molecule. Additionally, a portion of the dissociated radicals, anions, and cations may undergo recombination into the process gas molecules.

Cation etchant ions of gas from the plasma are attracted by a negative bias on the wafer support40such that the cation etchant ions are incident onto the wafer42. As a result, the silicon oxide (SiO2) film on the surface of the wafer42can be etched at a uniform etching rate that is determined in part by the composition, density, and distribution of the plasma in the processing volume46, and physical parameters such as the geometric, electric, and chemical parameters of the process chamber32and the DF-CCP system30.

When the etching is complete, the controller stops the supply of RF energy to the upper electrode62and the wafer support40, and stops the supply of the process gas into the process chamber32, whereupon the wafer42can be removed from the process chamber32.

Existing between the plasma and the upper and lower electrode there is a plasma sheath. The plasma sheath properties are usually considered to be a vacuum.

While the basic components of the DF-CCP system30have been described in some detail, it would be understood than additional features or geometries could be included within the simulation model.

With the DF-CCP system30now described with some detail, the simulation according to an embodiment of the present invention for modeling of the distribution and other parameters of the plasma, and the computer method for such modeling, will now be described. The plasma to be simulated is specified in a mathematical coordinate system called the simulation domain. The simulation domain used herein is two-dimensional and is constructed with axial symmetry; however, multiple dimension systems are also possible. The plasma, for example, can be modeled in a manner described by Lee et al., “Modeling electromagnetic effects in capacitive discharges,”Plasma Sources Sci. Technol.17, 1-16 (2008), incorporated herein by reference in its entirety.

The program code to simulate the plasma may be executed as part of, or executed on behalf of, a software suite, application, command, or request. In some embodiments, the program code may be incorporated with, or executed on behalf of, device simulation software. In a specific embodiment, the program code may be incorporated with, or executed on behalf of, a version of COMSOL application/software suite as distributed by The COMSOL Group of Burlington, Mass. In alternative embodiments, the program code may be incorporated with, or executed on behalf of, mathematical software, such as a version of Fluent by ANSYS Corp or Mathematica by Wolfram Research, Inc.

The conventional transient method of modeling the plasma is shown inFIG. 3. This method can, for example, model transient behavior that leads to steady state solutions of various plasma properties. To begin the method, and in accordance with steps80,82,84and86, configuration settings, initial conditions, boundary conditions, and parameters of the etching system are entered into the start of the simulation program. These initial parameters and settings are utilized within various subsequent steps as shown.

After the simulation domain and initial parameters are set, the simulation begins at step88, wherein the simulation domain is entered. This simulation domain involves solving the transient domain behavior for the time-based variables to arrive at a steady state solution of the plasma. In the DC-CCP system30(FIG. 2), this involves solving equations that describe the system for each of the time-based variables, which in the case of the system30(FIG. 2) are CCP power that has a first frequency of, for example, 100 MHz and the substrate bias power that has a second frequency of, for example, 2 MHz, in a series of nested loops.

The transient behavior based on the CCP power at the first frequency is iterated in a first and inner loop87over a time interval sufficient for convergence to a steady state solution, which for the 100 MHz frequency is found to be a time interval of from 0 to approximately 1-10 microseconds, in 0.1 nanosecond increments, for a total of about 10,000 to 100,000 iterations. The transient behavior based on the bias power at the 2 MHz frequency is iterated in a second and outer loop89over a time interval also deemed sufficient for convergence. Where the first frequency is much larger than the second frequency, this is found to be one complete cycle of 500 nanoseconds, calculated in 10 or 15 degree increments, for example, for twenty iterations of 25 nanosecond. The solution for the system30(FIG. 2) with the two time bases thus requires about 250,000 to 2,500,000 iterations that each involves solving of all of the equations describing the system.

Execution of the inner loop87begins with calculating the RF power deposition within the plasma as in step90, using the configuration settings (Block80), the initial settings (Block82), and the boundary conditions (Block84). One manner of calculating the RF power deposition is by an Electro-Magnetic solver for Maxwell Equations in a simple format having 2-D axial symmetry (i.e. electric field with components in the r-z plane and a magnetic field having only an azimuthal component). As such, the RF power deposition can be computed from:

∇×((ɛr-j⁢⁢σωɛ0)-1⁢∇×Hφ)-μr⁢k0⁢Hφ=0EQUATION⁢⁢1
where εris the relative permittivity in the sheath, j is the complex number index, σ is the conductivity, ω is the RF excitation angular frequency, ε0is the permittivity of free space, Hφis the magnetizing field intensity, μris the permeability, and k0is the wave number in vacuum given by k0=ω√{square root over (ε0μ0)}=ω/c0, where ω is the angular frequency and c0is the speed of light. Solving for Hφresults in the electric field:
Hφ(Er,Ez)  EQUATION 2
and the RF power that is deposited into the plasma is calculated as:

Qabs=12⁢σp⁢E2EQUATION⁢⁢3
where Qabsis the heat absorbed from the RF field and σpis the conductivity of the plasma.

Once the iteration step on the calculation of the radiofrequency field and power deposition is complete, the gas flow is calculated in step92. This can be accomplished from steady state Navier-Stokes equations assuming a low Reynolds number and laminar flow in 2-D axial symmetry geometry. The total stress tensor formulation is given by:

ρ⁢∂u→∂t-∇·[-p⁢⁢I→+η(∇u→+(∇u→)T)]+ρ⁡(u→·∇)⁢u→=F→EQUATION⁢⁢4
where ρ is the gas density, {right arrow over (u)} is the gas velocity, p is the process chamber pressure, {right arrow over (I)} is the unit diagonal matrix, η is the gas viscosity, and {right arrow over (F)} are the body forces. The mass continuity equation under imcompressible gas assumption (that is with density being constant) will simplify to:
∇·{right arrow over (u)}=0  EQUATION 5

Turning now to iteration step94, the temperature of the gas is calculated in accordance with energy conservation. As noted above, the gas molecules are heated (source term Q) due to the non-elastic collisions between gas molecules and RF field energized electrons. Thus, the gas temperature, Tg, is computed through a heat transport by a gas convection and heat conduction module:
∇(−k∇Tg)=Q−ρcp{right arrow over (u)}∇TgEQUATION 6
where k is Boltzmann constant and cp, ρ, and k are gas heat capacity, density, and thermal conductivity of the gas mixture, respectively.

With the gas flow and temperature calculated, the local time iterative scheme can begin at step96. As noted above, the upper electrode is powered by a 100 MHz RF frequency, which corresponds to a 10 ns time scale. As a result, the iterations to simultaneously solve the equations of step96through step100occur at 0.1 ns intervals until a transient convergence of the intermediate values of the electron temperature, electron density, and radical densities is achieved. This iteration process can typically continue at about 10−4second (or 100 μs) intervals within the internal time of the model. Specifically, the iterative calculations include the following steps and equations.

In step96, the electron temperature (Te) is computed in accordance with the conservation of energy:

∂∂t⁢(32⁢ne⁢Te)+∇(52⁢Te⁢Γe-k⁢⁢∇Te)=Qabs-e⁢⁢Γe⁢Ea-3⁢mM⁢vm⁢ne⁢Te-∑i⁢Vi⁢υi⁢⁢neEQUATION⁢⁢7
where Eais an ambipolar electric field; Γeis electron flux; viis the collision frequency of the “i”-th reaction and Viis its threshold; e is the elementary charge; m is electron mass; and M is total mass of two particles in elastic collision.

In step98, the electron (or ion) density is calculated by an ambipolar-drift diffusion approach, with a transient mass balance equation for “i” type of ions:

∂ni∂t+∇(-Da⁢∇ni+ni⁢u→)=RiEQUATION⁢⁢8
Where niis the “i”-th ion and Dais the ambipolar diffusion coefficient.

Finally, in step100, the neutral species transport within the plasma is calculated from the mass balance equations:
∇(−Dk∇nk+nk{right arrow over (u)})=RkEQUATION 9
where Dkis the diffusion coefficient of the “k”-th particle and nkis the “k” particle, and where k=all neutral species of the reaction. The generation and recombination of the positive etchant ions is given by:

where neis the electron, njis the “j” particle, σjkand σkjare the recombination rates due to collisions with electrons, Rjkl(forward) and Rkjl(backward) are the rate constants due to collision with species nkor nj, and veis the electron velocity.

As indicated above, iteration steps90-100are repeated at approximately 0.1 ns intervals until the transient solutions converge. Typically, the iterations involve calculations of steps96-100for electron temperature, electron density, and radical densities to converge. Convergence usually occurs within an internal model time of between 1 μs and 10 μs, at which time the iteration scheme computation proceeds to loop switch102, which involves recalculation of all steps90-100.

Throughout step96to step100, the local constraints and convergence criteria of step104can be changed or adjusted to aid in the convergence of the mentioned parameters.

If convergence does not occur at step102, the new iteration sequence will start again at Step90through Step100under the satisfaction of the global constraints and convergence criteria at step105. The process begins again at step90with these new criteria.

When the electron density for the first frequency has converged, then the RF cycle at the second frequency is computed in the outer loop89at phase steps that are less than 5% of the RF cycle duration, that is phase modulated by 10° to 15° steps, with the process of steps88through106are repeated for each iteration of loop89. Typically, steps90through102can take about 4 hours to about 6 hours to complete the coupled iteration scheme computation using a dual core PC at 2-3 GHz. Thus, a full transient solution for a dual frequency system can take nearly 100+ hours.

To decrease the computational time needed for achieving a solution, the transient model is converted to a stochastic model as generally shown inFIG. 4Aand in accordance with the present invention.

Turning now toFIG. 4A, the method108begins with defining the simulation domain according to one embodiment of the present invention. The simulation domain defining the characteristics of the etching system will include time dependent variables that are dependent upon the chemicals, materials, or geometries of the plasma. Often, the simulation domain may contain at least one or more time dependent variables, or sub-domains for differing chemicals, materials, and/or geometries. That is, for each sub-domain, the time dependence of the variable in each point within the sub-domain should be a homogenous transient formulation, i.e. have similar time dependence differing only in phase for various points. Otherwise, the simulation domain is a heterogeneous transient solution and should be divided into two or more sub-domains of homogeneous transient formulations.

For example, as illustrated inFIG. 5, the simulation domain D2110can include points (x1,z1) and (x2,z2) shown to differ in their time dependence of a particular variable. Thus, points (x1,z1) and (x2,z2) will exist in first and second sub-domains (D21, D22)112,114, respectively. While only two sub-domains D21and D22112,114within the simulation domain D2110are shown, it would be understood that more sub-domains are possible.

A time dependent function, Ψ, for the time dependent variable of each sub-domain can be determined by empirical data or by analytical derivation and are represented by the graphs Ψ versus time withinFIG. 5.

Returning again toFIG. 4A, and specifically to Step116where a probability function Θ for each time dependent function, Ψ, is determined, wherein the probability function Θ is a statistical distribution of the values of Ψ. The probability function can be determined from the known analytical expression of the transient dependence of the particular variable by measured or empirically established transient dependence, hypothetically derived, or measured directly. As seen inFIG. 5, each Ψ of a separate sub-domain may have a different statistical distribution shown in graphs Θ versus Ψ.

Continuing withFIG. 4A, and step118, the probability function Θ is normalized to ensure that an occurrence of all transient values corresponds to the probability equal to 1. In other words, the probability function is multiplied by a normalizing constant, κ according to:
∫ΨminΨmaxκΘ(Ψ)dΨ=1  EQUATION 11

This will ensure that the area under the Θ versus Ψ curve (seeFIG. 6) should be equal to 1. The normalized probability function will then be given by:

With the probability function now normalized, the method proceeds to step120ofFIG. 4Awith reference made toFIG. 6. In step120, the interval of the function Ψ values, that is <Ψmin,Ψmax> is subdivided into “n” subintervals <Ψmin:Ψmax>nks, wherein the size of each subinterval will be given by:

ΔΨ⁡(x,z)=Ψmax⁡(x,z)-Ψmin⁡(x,z)nksEQUATION⁢⁢13
And the characteristic value,Ψb, is determined for each nkssub-interval. TheΨbcan be mean value of Ψ values between <Ψmin:Ψmax>; however, other values may also be appropriate. Additionally, the subintervals should be sufficiently small to minimize sampling error. This is illustrated inFIG. 6, wherein the characteristic valueΨfor subintervalsΨb−1,Ψb, andΨb+1are shown.

Turning again toFIG. 4A, and specifically to Step122, wherein the characteristic valuesΨbare arranged into corresponding strings, Bn, wherein the length of Bn is given by:
Δb=INT[nNSnMSΘnorm[Ψb(x,z)]]  EQUATION 14
where nNSand nMSare the number of points within the simulation domain110in the x- and z-axes, respectively, and each member within the string will equal the value ofΨb.

In step124, all strings of Bnare arranged into an array Ω corresponding to each sub-domain, as shown inFIG. 7.

In step126ofFIG. 4Aand further inFIG. 7, the values within each Ω are reassigned randomly into the points of the corresponding sub-domain. Once all of the points of the original simulation domain are randomly reassigned values from the respective Ω, then the resultant array is a stochastic simulation array128and is ready to undergo simulation, as described inFIG. 4B.

FIG. 4Billustrates the simulation of the DF-CCP device with the stochastic simulation array128created from step130(FIGS. 4A and 4B). While the majority of the simulation method is similar to what was detailed previously withFIG. 3, distinct differences will be noted. Most notably is the use of the stochastic simulation array eliminates the need for the super-loop or outer loop89(FIG. 3) with a phase-modulation at the second frequency in the iterative scheme

Described herein is a method and an algorithm for converting a transient to spatial stochastic formulation for enhancing the computation of plasma fluid models. The computer method may be encoded on a computer usable medium and provided in the form of a computer program product. The system ofFIG. 1, when programmed with the computer program product, is a simulation apparatus that includes a calculating unit programmed to calculate a probability function as described in connection withFIG. 4Aand a simulation unit programmed to perform a simulation of transient plasma sheath behavior as described in connection withFIG. 4B.