Calibration of the Ion Enhanced Chemical Etching Model of VICTORY Process Using Virtual Wafer Fab

 

Introduction

In this article we describe a technique to automatically calibrate parameters of an Ion Enhanced Chemical Etching model implemented in the 3D process simulator VICTORY Process to measurements published in [1]. We will briefly describe the model and show how an experiment applying this model can be set up in VWF to automate the calibration by carrying out several individual simulations without user interaction.

 

Virtual Wafer Fab – VWF

Virtual Wafer Fab (VWF) is software that:

  • conveniently sets up simulation experiments
  • automates running an arbitrary number of individual simulations in parallel on a workstation and a cluster of workstations
  • performs comprehensive analysis on the data generated by the set of simulations
  • automatically optimizes simulation parameters with respect to a variety of optimization targets

The fundamental concept of VWF is to define (choose) a strategy to vary parameters within given bounds and to let the software carry out the simulations and obtain the simulation results. Once the virtual experiment has finished, results can be investigated by looking into created structures and electrical characteristics, and by applying statistical methods or visualization techniques.

Experiment Strategies
VWF offers two main strategies for varying parameters. The first strategy is to use one of several supported Designs of Experiments (DOE), whereby a predefined sample of the input parameter set is taken and simulated. The supported DOE types range from Full-Factorial, to Box-Behnken to Latin-Hypercube. In addition, user-defined types are also available which use JavaScript to sample the parameter space.

The second strategy is to use an optimization algorithm to vary the parameter set. Once started, the experiment keeps running and performs simulations with varied parameters until a defined target value is achieved. VWF offers various optimization algorithms ranging from:

  • local, gradient-based methods, like Levenberg-Marquardt
  • global optimization algorithms, like Genetic Optimization, Simulated Annealing or Parallel Tempering

Data Organization
VWF can organize your data either within the file system or in an SQL-92 compliant database. By using the database you can easily share your data with other team members. Permissions similar to that of the UNIX system can be defined for each experiment to allow fine-grained access control. The database system also offers flexible ways to backup and archive the data. The database can be installed on any workstation, and VWF can conveniently connect to remotely running databases. It can even connect to several databases at the same time and migrate experiments between databases.

All experiment data can be held in a directory of the VWF user as an alternative to the database approach. While this lacks the advanced access control features, it is convenient to quickly get started without the need to install a database system. The data is protected by the UNIX access rights of the user who defines and runs the experiments. To offer a smooth transition between the data models, experiments can not only be exchanged between databases, they can also be exported from a database to the file system and vice versa. Experiments can also be stored in TAR files to allow transferring experiments to other sites.

Job Execution Module
VWF contains a powerful job execution module which allows simulation jobs to run on a variety of computer setups. The user can opt to run all simulation jobs on his workstation only. In this case, VWF would typically run as many jobs in parallel as there are cores installed in the system. The behavior can be fine tuned by limiting the number of jobs or by defining a nice level for the jobs. No further installation or setup is necessary.

A more advanced technique is to use a grid computing environment (a cluster) for running simulation jobs. The decision of when (or on what machine) a simulation is executing is no longer taken by VWF but is delegated to the grid environment. Such systems typically offer sophisticated queuing algorithms, allow prioritization of jobs of participating users, or definition of job limits, such as the maximum amount of memory or a maximum CPU time which must not be exceeded.

For grid computing VWF supports all flavors of grid-engine as well as the LSF queuing system.

Visualization Module
Results obtained by VWF can be visualized in several ways. All values extracted from simulation runs are made available in a worksheet. Data in the worksheet can be exported for post-processing via SPAYN or exported as comma separated values files and loaded into spreasheets such as Excel. Data in SPAYN can be statistically analyzed and Response Surface Models (RSM) can be computed and visualized in TonyPlot.

 

Ion Enhanced Chemical Etching Model

Before demonstrating how VWF can be used for calibration purposes, we will first briefly describe the ion enhanced chemical etching (IECE) model which we intend to calibrate. The ion enhanced chemical etching model is implemented in the open etching/deposition model library of the full 3D process simulator VICTORY Process [2]. VICTORY Process performs the etching simulation on the feature scale level. The implementation of the model follows [1], which describes the etching of silicon by fluorine reactants.

Ion enhanced chemical etching is modeled in VICTORY Process as the interaction between two types of plasma particles, namely low energy ions and neutrals with the etched structured. In order to characterize the plasma on the feature scale level, the properties of the ions and neutrals have to be defined as input parameters for the feature scale simulation. These properties are:

  • Momentum distribution and the transport characteristics of the particles
  • Density of the particles in the vicinity of the wafer surface and
  • Reaction characteristic with the various materials of the wafer

The model assumes that the neutral chemicals chemically react with the surface material whereby the low energy ions which are hitting the surface with higher thermal energy can accelerate this etching process because they transfer energy to the surface.

The model does not take into account physical sputtering of surface atoms by the ions. Any removal of surface atoms is just due to a chemical reaction with neutrals. On the basis of this assumption the total etch rate is calculated as:

r = rneutral . fneutral + rion . fion . Θ (1)

where Θ is the surface coverage. The surface reaction model used by the ion enhance chemical etching model assumes that the surface coverage is in steady state. Thereby it also becomes a function of:

  • The partial ion flux fion
  • The partial neutral flux fnuetral
  • The ion related etching rate rion
  • The neutral related etching rnuetral

The partial fluxes are the probabilities to find a specific particle at the surface point.

 

Calibration Task

In [1], experimental results are published, which are used in this article to calibrate the parameters of the ion enhance chemical etching model. For that purpose we focus on obtaining the parameters rion and rnuetral. Those parameters are significantly affected by the equipment set-up, which is not described in [1]. The ion related etch rate rion is not a direct input parameter of the IECE model. Instead, the dimensionless ratio of the SiFx molecules desorption rate to the rate of their formation on a plane surface rationuetral is used as a model parameter. Within a VWF experiment, the two parameters rnuetral and rationuetral shall be varied until the experimental data are reproduced by the simulation. Table 1 summarizes the measurements published in [1]. A trench is etched into silicon through a mask opening. The depth of the trench is provided for various mask openings ranging from 2 to 20 microns.

Width [um]
Trench Depth [Angstrom]
2
51000
5
70000
10
80000
17
86000
20
87000
Table 1: Measured trench depth for various mask openings.

 

VWF Experiment Setup

In VWF, the experiment is set-up as a Levenberg-Marquardt optimization experiment with the following target function to optimize:

(2)

Here mi and si are the measured and simulated trench depths (see Table 1), respectively and N equals the number of mask openings (five in this case).

To compute the target function it is necessary to run several simulations (one per mask width) for each optimizer iteration. Therefore, this type of experiment can be seen as a combination of a DOE and an optimization. The screen-shot in Figure 2 (Parameters Tab) illustrates that. For the parameter size the DOE flag is activated (marked). This allows us to enter an arbitrary amount of target values. Target values are entered in the top frame of the Worksheet Tab (see Figure 1). You can see that five trenchdepth values corresponding to the mask widths of 2-20 microns (according to Table 1) are specified there. On the other hand, the parameters NEU_RATE and NEU_RATIO define the parameters that are varied by the optimizer. It is important to note that low and high bounds must be defined for each parameter (DOE parameters as well as optimization parameters).

Figure 1: Experiments Worksheet Tab which shows how the optimization target is defined.

 

Figure 2: Experiments Parameter Tab which shows that this experiment uses the combined DOE optimization approach.

 

The target function (see Equation 2) itself is implemented as a user-definable JavaScript program, which is specified as part of the experiment setup in the middle frame of the Worksheet Tab (see Figure 1). This script will be executed for every set of 5 (formula in Equation 2) finished simulation runs. The script iterates over:

  • The obtained simulation results (called response within the JavaScript) and
  • The entered target values (called target within the JavaScript)

It then computes ftarget. On return, VWF passes the computed final target value ftarget to the Levenberg-Marquardt optimization algorithm.

While the optimization is running, VWF starts a series of simulations and displays the progress of the optimization. Within the Graphics Tab of the experiment (see Figure 3) you can see how the target value evolves. Starting from a target value (error) of ~10%, the target value reduces to 2.34%. within a total of ~250 simulation runs. Once the optimization is finished, the optimal simulation is marked green within the Graphics Tab. Additionally, you can see in the Results Tab of the experiment (see Figure 4) all the results which are generated while the experiment is running. In the right bottom corner of this tab VWF also reports whether the optimizer has successfully converged and found a result :

Optimizer return code : OPT_RC_CONVERGED

Figure 3: Experiments Graphics Tab showing the progress of the optimization algorithm.

 

For the calibration of the ion enhanced chemical etching model parameters, the Levenberg-Marquardt optimizer has converged and has obtained the results which are marked green in the Results Tab of the experiment (see Figure 4). The optimal values for the calibration parameters NEU_RATE and NEU_RATIO were found to be 12.27 and 1.85, respectively.

Figure 4: Experiments Results Tab showing simulation results and target function.

 

Calibration Results

Table 2 summarizes the simulated trench depths for all five mask openings when the calibrated values for the parameters NEU_RATE and NEU_RATIO are applied. A visual comparison between the measured (Table 1, Figure 5-blue) and simulated (Table 2, Figure 5-red) function of the trench depth over trench width is presented in Figure 5. The overall agreement between simulated and measured data is good. The simulation slightly overestimates the trench depth for mask openings below 15 um but underestimates the trench depth for larger mask widths. The maximum deviation is in the range of ~3%. Better agreement can be achieved by considering more data in the calibration experiment, like for instance the measured under-etch.

Width [um]
Trench Depth [Angstrom]
2
50707.6
5
71302.1
10
82736.5
17
84801.6
20
84233
Table 2: Simulated trench depth for various mask openings when the optimal values for the IECE model parameters NEU_RATE and NEUR_RATIO are used

 

Figure 5: Comparison of the measured and simulated trench depth over trench width function.

 

Conclusion

We have presented a way of using Virtual Wafer Fab (VWF) to perform a fully automated calibration of an Ion Enhanced Chemical Etching model of the 3D process simulator VICTORY Process. The parameters of the model were calibrated by taking advantage of VWFs advanced experiment type which combines DOE and optimization. The optimization task was driven by a Levenberg-Marquardt optimizer, which was used to minimize a given target function ftarget. The obtained values for the model parameters NEU_RATE and NEUR_RATIO result in a good agreement between simulated and measured trench depths over all five mask openings with a relative error of no more than 3%.

 

References

[1] Hauguth 2009, “Integrated plasma processing simulation framework, linking tool scale plasma models with 2D feature scale etch simulator”.

[2] Developing Custom Etching/Deposition Models in VICTORY Process, Simulation Standard,Volume 22, Number 4, October, November, December 2012.

 

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