The role of modeling of non-equilibrium plasmas: Scientific curiosity or industrial tool?

Mark J. Kushner

Dept. Electrical and Computer Engineering, Iowa State University, Ames, Iowa 50011, USA mjk@iastate.edu Abstract: The attributes of models of non-equilibrium plasmas for use as industrial design tools are discussed. The current status of modeling of non-equilibrium plasmas is reviewed with examples of approaches having these desired attributes. A case study of model based design addressing wafer-to-wafer drift in plasma etching is discussed.

Keywords: Modeling and simulation, industrial design, self-aware models, real-time-control. 1. Introduction

The 1991 National Research Council Report Plasma Processing of Materials: Scientific Opportunities and Technological Challenges cited the need for modeling to advance the rate of innovation in the development of plasma equipment and processes in nonequilibrium plas-mas.[1] Since that report, tremendous progress has been made in modeling and simulation, and in our fundamental understanding of plasma processing. Nevertheless, most assessments of the status of plasma materials processing and plasma chemistry cite the need for improvements in modeling and simulation to maintain high levels of inno-vation. Some of these assessments are driven by eco-nomics wherein modeling may be used to shorten the de-sign cycle, while other assessments look to modeling to provide the insights necessary to lead experiments into untested parameter spaces.

There are two differing views of the role of modeling of nonequilibrium plasmas. The first view is the funda-mental perspective in which the role of modeling is to elucidate and investigate fundamental physics and chem-istry issues; and so provide insights to the technologist. As such, validation is of paramount importance and mod-eling is limited to a parameter space in which clean, un-ambiguous experimental data is available. The second view is the technological perspective wherein models are used to address parameter spaces which have not yet been experimentally investigated, and so there is a lack of validating data. Truth is somewhere in between these views. Industrially relevant models must be based on first principles algorithms that are tested and validated; and provide insights to the physics. However to be maximally leveraged by industry, models must also be capable of extending the technologist's reach into untested territory which, by definition, lacks validating data.

In this paper, progress in modeling of industrially rele-vant processes will be reviewed, assessments of the chal-lenges facing modeling will be made, and requirements for further advances will be discussed. 2. Status of Modeling and Simulation

The advances that have been made in modeling and simulation in the context of materials processing are truly

impressive. For example, fully kinetic 2-dimensional simulations of magnetically enhanced low pressure plas-mas can now be applied to the design of processing equipment.[2] (See Fig. 1.) 3-d modeling of induc-tively coupled plasma torches are now able to resolve the injection of powders [3] and chaotic arc attachment to the anode, as shown in Fig. 2 [4]. Plasma surface interac-tions are now well enough understood that the twisting of trenches during plasma etching caused by non-uniform charging can be resolved using modeling. (See Fig. 3.)

Given these existing capabilities and computational in-frastructure, it is surprising that modeling is not more frequently applied in industry to aid in the design of proc-esses and equipment. It is expected that as computing speeds increase, the resolution, dimensionality and chemical complexity of these will models will continue to increase, and perhaps gain greater acceptance by industry. However speed alone is not the only requirement. 3. Expectations for Industrial Relevance

The expectation that models will become industrially relevant tools will be met when advances occur in at least three areas: integration of spatial and time scales by im-proving computational techniques, becoming self-aware of the proper algorithms for the local physics, and trans-parently generating data and reaction mechanisms.

Fig. 1. Particle-in-Cell model of the electron density in a low pressure plasma with multi-polar magnetic confinement. (From Ref. 2.)

Computational Infra-structure: Advances in multi-scale algorithms in computational fluid dy-namics (CFD) and high energy density physics are impressive. Numerical techniques such as adap-tive meshes, automatic mesh refinement (AMR) and multi-gridding have enabled vast differences in spatial scales to be ad-dressed in multi-physics codes. These techniques are only now being ap-plied to modeling of non-equilibrium plasmas. For example, Montijn et al. [5] and Nikandrov et al. [6] developed models for

streamers which resolve the very steep gradients at the head of the stream using adaptive mesh and AMR meth-ods. (See Fig. 4.) These techniques will become even more important as models address multiphase plasmas where, for example, nm to micron sized particles or clus-ters affect plasma transport and so must be resolved si-multaneously with reactor scale phenomena.

"Self-Aware" Models: The integration of vastly different time and space scales often produces conditions which also require different modeling approaches. Although the most accurate models typically use kinetic techniques, in the absence of massively parallel computing, it will be challenging to generalize those techniques to 2-d models addressing geometrically or chemically complex systems or even 3d models in simple systems. As such, models will be required that are able to diagnose the local plasma conditions and select the proper computational technique

(e.g., kinetic or fluid) which resolves the problem with the least computational burden while perhaps using multiple techniques in the same grid. This requires a model that is self-aware with some power to choose.

One example of a self-aware simulation appears in Fig. 5. The model is a fluid simulation using an unstructured mesh in which an electron Monte Carlo simulation (eMCS) is embedded. The fluid modules have sensors that detect regions in which electron transport is suffi-ciently non-local or non-equilibrium that continuum tech-niques may not be valid. Criteria to detect such regions include λ > E/∇E (λ = mean free path, E = electric field or ν < (∂E/∂t)/E (ν = collision frequency). In these re-gions, the eMCS is executed to kinetically resolve the non-equilibrium transport. In the example, breakdown in a cold HID lamp (Ar 30 Torr) is modeled. The sensor looks for regions in which the gradient in electric field exceeds a critical value as the breakdown front crosses the tube from anode to cathode. The eMCS mesh is super-imposed over these regions that move with the ionization front. A fixed eMCS mesh hovers about the large ∇E at

Fig. 2. 3-D Finite-element-model of an argon plasma torch showing chaotic arc attachment to the anode. The heavy parti-cle temperature (K) is shown at different times, noted in the plot of the change in arc voltage. (Adapted from Ref. 4.)

Fig. 4. Use of AMR for streamers. (left) Electron density and potential contours and (right) mesh for simulation of a streamer originating from a spot of plasma. AMR is used to resolve sharp gradients. (Adapted from Ref. 6.)

Fig. 5. A self-aware fluid model of breakdown in a cold HID lamp (30 Torr Ar) detects regions of non-equilibrium and overlays an electron Monte Carlo Simulation (left) ∇Eat different times is used to sense non-equilibrium regions. The eMCS mesh follows those regions. (right) Electron impact ionization sources with and without the eMCS.

Fig. 3. Twisting of trenches due to charging during etching of SiO2 in a fluorocarbon plasma. The polymer (in red) traps negative charge which steers ion trajectories.

the cathode. The resulting electron impact source func-tions show significant differences compared to simula-tions using only the fluid equations.

Reaction Mechanisms: Although the computational infrastructure exists to address chemically complex sys-tems, applying this capability to industrially relevant problems depends on the availability of databases and reaction mechanisms, both volume and surface. The chemical diversity of applications, particularly in plasma materials processing and HID lamps, makes it difficult to develop the databases in advance of their need. As such, the ability to develop databases in real time (days or weeks as opposed to months or years) is required.

Although this data is ideally provided experimentally, from a practical perspective computational approaches will be necessary to provide databases and reaction mechanisms in real time. In this regard, significant pro-gress has been made in the development of user friendly software that is accessible to non-experts to produce elec-tron impact cross sections sets. For example, the ab ini-tio R-Matrix techniques of Tennyson et al. for calculating electron impact cross sections on polyatomic molecules have been captured in a commercial software package Quantemol-N.[7,8] The rapid generation of cross sec-tions in this manner enables rapid screening of processes and identifying which cross section data the result is par-ticularly sensitive to and may require more refinement.

The ultimate test of being industrially relevant is whether the model can be applied to: a) Improving fun-damental understanding of the process so that experimen-tal or empirical development proceeds faster and cheaper; and b) Providing a virtual design capability to either re-duce or eliminate the number of prototyping iterations required. The model must also fit between the two dif-fering perspectives of modeling discussed above: vali-dated to the degree to provide confidence in the algo-rithms but general enough to be applied to untested pa-rameter spaces. An example from a model developed by Iza et al. which fits well between these touch points ap-pears in Fig. 6 [8]. A 2-d particle-in-cell simulation was used to investigate striations in plasma-display-panel cells

operating in 500 Torr Ne. The manner of charging the dielectric was found to be critical to behavior of the stria-tions. 4. Case Study of Model Based Design

To demonstrate the challenges and benefit from devel-oping models for the design of industrially relevant proc-esses, a case study will be discussed. The example case is poly-Si etching in an Ar/Cl2 inductively coupled plasma having a biased substrate. (See Fig. 7.) In this process, ion activated etching of the wafer produces a SiClx prod-uct that diffuses off the wafer and deposits on the interior surfaces of the reactor. The SiClx also deposits back onto the wafer. The passivation of the walls of the reac-tor by the etch product reduces the sticking coefficients of additional etch products which in turn increases the rede-position of SiClx onto the wafer. This occurs while sput-tering of the quartz window by capacitive coupling from the coils injects oxygen into the plasma which forms etch blocking SiOxCly deposits on the wafer. The wall pas-sivation increases over time. The end result is that the etch rate decreases both during a single etch sequence and wafer-to-wafer as the reactor walls are further passivated. This process drift is usually addressed by cleaning the reactor after each wafer. The industrial perspective is: Can modeling be used to propose inexpensive design changes in the reactor or real-time-control (RTC) tech-niques to minimize or eliminate process drift?

The requirements for a model capable of addressing this design issue include: 2-d (or 3-d) capability to model different surfaces, neutral and charged particle hydrody-namics, kinetic electron, ion and neutral (e.g., sputtered atoms) transport to resolve energy distributions in the plasma and onto surfaces, time resolution of ns to tens-of-seconds, electric circuit model, gas phase and surface chemistry (including sputtering) and RTC algo-rithms (sensors and actuators).

A model having many of these attributes was used to suggest design changes and RTC strategies to eliminate

Fig. 7. Plasma properties for an ICP sustained in 15 mTorr Ar/Cl2 = 90/10. (left) Total ion density and SiCl2 etch product diffusing off the wafer. (right) O atom density and fluxes originating from sputtering of the quartz window.

Fig. 6. Predictions of plasma emissions (left) and experi-ments (right) revealing striations in a plasma-display-panel cell. Design capable models must be validated in well characterized parameter spaces but capable of being extended beyond that space. (Adapted from Refs. 9 and 10.)

process drift. The predicted total ion density and SiCl2 etch product for an Ar/Cl2=90/10, 15 mTorr plasma etching a p-Si wafer are shown in Fig. 7 (100 sccm, 500 W ICP, 75 V substrate bias). A total ion density of 3 × 1011 cm-3 produces an etch rate of 0.1 µm/min and an etch product density 1012 cm-3 above the wafer. With an rf 75 V sub-strate bias and a com-

mensurate dc bias, ion energies as large as 150 eV are incident onto the substrate, as shown in Fig. 8. Ion en-ergies incident onto the side wall do not appreciably ex-ceed 60 eV. The end result is that etch products deposit on the side walls without there being significant sputter-ing to remove them. This passivation reduces the stick-ing of etch products to the side wall which in turn in-creases the flux of etch products back onto the wafer. These fluxes produce etch blocks which, both during a single process and wafer-to-wafer, reduce the etch rate, as shown in Fig. 9. Capacitive coupling from the outer coil creates a sufficiently large sheath voltage to produce 130 eV ions onto the quartz window which sputters O-atoms into the plasma. (See Fig. 7). This produces additional SiOxCly etch blocks which further reduce the etch rate.

Inexpensive design changes and RTC strategies were derived from the model to stabilize etch rates.

Optimizing Reactor Design: Geometrical flexibility and coupling of circuit parameters to plasma properties en-abled model based optimization of the coil. By lifting the coil off the quartz window by about 1 cm, the ion en-

ergies onto the quartz were reduced to below 100 eV (see Fig. 8) which significantly reduced the sputtering of the quartz. This reduced sputtering nearly eliminated the detrimental effects of SiOxCly etch blocks. (See Fig. 9.)

Model Based RTC Strategies: The model was also used to propose RTC strategies. For example, an etch rate sensor based on a laser interferometer was interfaced through a PID controller to the bias voltage. Without RTC, the etch rate progressively decreases during an etch and as well as wafer-to-wafer. With RTC, the desired etch rate is recouped by adjusting the bias voltage to compensate for the etch blocks. 5. Concluding Remarks

Impressive progress has been made in the development of models capable of industrially relevant design. These models are in large part used by modeling experts. This is not an unusual situation in other disciplines. For ex-ample, it is rare that CFD models are used in the aero-space industry by non-experts. Having said that, the arguably additional complexities of plasma materials processing compared to CFD calls for higher degrees of expert intervention to choose data (e.g., cross sections) and the proper physical models to resolve the problem. As such, the acceptance and broader use of models for industrial design may require more self-aware models that reduce this high degree of intervention. Acknowledgments The author thanks A. Agarwal, A. Bhoj, P. Trellus, W. Brok, J. K. Lee, M. Kong and V. Kolobov for their con-tributions. This work was supported by the National Science Foundation, Semiconductor Research Corp., Ap-plied Materials, Inc., and Micron Technology, Inc. References [1] Plasma Processing of Materials: Scientific Opportu-

nities and Technological Challenges (National Academy Press, Washington DC, 1991).

[2] H. Takehida and K. Nanbu, Trans. Plasma Sci. 33, 345 (2005).

[3] D. Bernardi, et al. Plasma Sci. 33, 424 (2005). [4] J P Trelles, J V R Heberlein, and E Pfender, submit-

ted to J. Phys. D. [5] C. Montijn, W. Hundsdorfer and U. Ebert, J. Comp.

Phys. 219, 801 (2006). [6] D. Nikandrov, L. Tsendin, V. Kolobov and R.

Arslanbekov, Proc. 28th ICPIG, Prague, 2007. [7] I. Rozum, N. J. Mason and J. Tennyson J. New J.

Phys. 5, 155 (2003). [8] http://www.quantemol.com/products.html. [9] F. Iza, S. S. Yang, H. C. Kim, and J. K. Lee, J. Appl.

Phys. 98, 043302 (2005). [10] R. Ganter, J. Ouyang, Th. Callegari and J. P. Boeuf, J.

Appl. Phys. 91, 992 (2002).

Fig. 8. Ion energy and angular distributions incident on the wafer, side wall, quartz window under the outer coil and onto the quartz with the coils offset from the window.

Fig. 9. Etch rates in the ICP reactor. (left) At the end of the process for successive wafers without sputtering of the quartz, with sputtering and with offset coils. (right) Etch rates with and without RTC, and bias voltage provided by the controller.