A Semi-Empirical Model for the Simulation of Ion Milling in VICTORY Process

Introduction

The three-dimensional process simulator VICTORY Process now includes a semi-empirical model for the simulation of ion milling. During ion milling, the mill (etch) rate depends in a complex way on the local beam’s incidence angle (angle between beam direction and the normal at a particular surface point). The function describing this relationship is called the angle dependent mill rate. A visual representation of such an angle dependent mill rate is shown in Figure 1.

Figure 1: Ion milling rate (sputter rate) as function of the incidence angle of the ion beam.

 

In VICTORY Process this mill rate function can be set (obtained) by two means:

  • An experimentally determined mill rate function can be specified for every material which interacts with the ion beam
  • The mill rate function can be automatically calculated using the model proposed by Yamamura [2][3]

A mixture of both modeling approaches can be used for different materials within a single ion milling simulation:

  • For those materials, where an experimentally determined mill rate function (corresponding to the reactor parameters) is specified, this information is used
  • For all other materials the mill rate function is obtained by means of the Yamamura model, provided that necessary data on reactor conditions and fundamental materials’ properties are supplied.

This article describes the approach based on the Yamamura model, which has been implemented in version 4.3.0 of VICTORY Process. All ion milling models are implemented in the open etching/deposition model library [4] of VICTORY Process, which means that the models are user extendable to new materials and new ions, which are not supported by the default version of the open etching/deposition model library.

 

Ion Milling Simulation in VICTORY Process

In VICTORY Process ion milling simulation is performed by means of the special input deck statement IONMILL. A typical input deck line for ion milling simulation looks like the following :

IONMILL TIME=<time> [<beam parameters>]

[<redeposition parameters>] \

[<equipment parameters>]

The parameters <equipment parameters> are optional as long as experimentally determined mill rate functions are available in the open etching/deposition model library for all materials present in the structure processed by the ion milling step. However, if, at least for one material, experimental data are not found, the Yamamura model is called which requires these data for automatically building the rate function table. Therefore, if you do not have experimental data for some of the materials present in the structure you must provide the <equipment parameters> (see Table 1) .

Parameter
Description
ION Type of the ions used for ion milling (e.g. Ga).
ENERGY Average energy of ions forming the ion beam (in eV).
CURRENTDENSITY Current density of the milling beam
IONIZATION Ionization of the ions in the beam (optional, default value 1)
Table 1: Equipment parameters which can be set by means of the IONMILL statement.

Other data, required by the Yamamura model, are stored in the open material database and in the open etching/deposition library and, therefore, are accessible for modification and extension.

 

The Yamamura Model

Each angle dependent mill rate function is only valid for

  • specific ion/target combination
  • specific ion milling equipment set-up

In order to obtain angle dependent mill rate functions for

  • various ion/target combinations and for
  • various equipment configurations,

the semi-empirical model, proposed by Yamamura in [1] and [2] has been implemented in the open etching/deposition model library of VICTORY Process for the simulation of ion milling. The model is based on the theoretical analysis from Sigmund [3].

• In the model implementation, first the total sputtering yield at normal incidence is calculated →



  • Than the dependence of the total sputtering yield on the incidence angle θ of the ion beam with respect to the surface normal is superimposed →

  • Finally, the angle dependent total sputtering yield is converted into the angle dependent etch rate, which determines the surface velocity of each surface point in each time step of the ion milling process →


Note, that the Yamamura model was derived ([1] and [2]) for single atomic materials only, based on fundamental principles of atom-atom interactions. The implementation within VICTORY Process extends it to compound materials.

At the beginning of an ion milling simulation, mill rate tables, which are a linear piecewise approximation of the angle dependent mill rate functions, are calculated using the Yamamura model, by calling the function

ionmillYamamuraConfiguration

from the etching/deposition model library file

empiricalRateModels.lib

for those materials, present in the structure, for which experimental data are not available. Those tables are used later on within the numerical ion milling calculations.

 

The Yamamura Model for Compound Materials

Although the Yamamura model is originally derived for mono-element materials only ([1] and [2]), VICTORY Process uses it to find the angle dependent total sputtering yield (and hence the angle dependent mill rate) for compound materials as well, by averaging the angle dependent total sputtering yields of each of the material’s elements, taking into account the number of atoms in the molecule as following:

(1)

where

... is the angle dependent total sputtering yield for the compound material,
... is the angle dependent total sputtering yield for an element, included in the materials’ chemical formula,
... is the number of element’s atoms in the material’s chemical formula,
... is the total number of atoms in material’s molecule.

 

For example, the angle dependent total sputtering yield for SiO2 is calculated as

(2)

Usually, the empirical parameters (kept in the open etching/deposition model library) of the Yamamura model are element (atom) specific. This means that, in the example above, and should use different values for those empirical parameters. Nevertheless, for some compound materials like SiO2 a single set of empirical parameters is applied to all elements (atoms) of the molecule. Therefore, in the case of SiO2 , and use the same values for the empirical parameters, allowing you to calibrate the Yamamura model for compound materials and for their components independently.

 

Example

In the following we show the result of applying the new model for ion milling simulation in VICTORY Process to the structure composed of nickel and aluminum shown in Figure 2. For the example, the simulation of ion milling processing with Ga ions with an energy ranging from 50 eV to 300 eV is performed. A static ion beam which is tilted by 20º relative to the wafer normal and rotated by 30º (counter-clockwise) relative to the x-axis of the simulation domain is used. The current density in the beam is 30 and the ions in the beam are single charged.
Such an ion milling process step can be simulated by means of the input deck statement

IONMILL beamAngle=20.0 beamPhi=30 time=2.0 min \

rotation=off ion=Ga energy=<ENERGY> \

currentDensity=30 maxCFL=0.1

whereby <ENERGY> is a placeholder for the energy of the ions within the beam.

Note: Here the numerical parameter maxCFL is set to 0.1 to obtain numerically smoother results (with the penalty of a longer calculation time).

Figure 2: Input structure for ion milling simulation example composed of aluminum and nickel.

 

The default etching/deposition model library, supplied with VICTORY Process, does not contain experimentally determined angle dependent mill rate functions neither for nickel nor for aluminum. Therefore, when executing the above input deck statement, VICTORY Process will use the Yamamura model to calculate the total angle dependent mill rate function for both these materials. During initialization of the simulation step, VICTORY Process reports the calculated total angle dependent mill rate function in its run-time output. The following information is displayed as shown in RTO 1:

  • The Rate for various angles (red in RTO 1)
  • The Mill rate at a plane wafer surface (blue in RTO 1)
  • The mill rate and total sputtering yield at 0º beam incidence (green in RTO 1)

RTO 1: Run-time output generated by the Yamamura model for an ion beam with an energy of 200 eV.

RTO 1 also shows (orange line) that the mill rate functions are stored in an .str file (rate table file), which can be visualized by TonyPlot, in the local working directory. This file can be used to compared the total angle dependent mill rate functions for various processing conditions (various ion energies in this example - see Figure 3). A rate table file also contains:

  • the total angle dependent mill rate function
  • the total angle dependent sputtering yield function
  • relative angle dependent sputtering yield function
Figure 3: Comparison of the total angle dependent mill rate function of aluminum for ion beam energies ranging from 50 eV to 300 eV.

 

A comparison of the total angle dependent sputtering yield functions for various ion beam energies is shown in Figure 4. The maximum of the total sputtering yield function as well as the maximum of the mill rate function shifts towards larger angles with increasing ion energy. This shift is more pronounced for the total sputtering yield function due to a (cos(θ)) coefficient, which takes into account the visible surface element size.

Figure 4: Comparison of the total angle dependent sputtering yield function of aluminum for ion beam energies ranging from 50 eV to 300 eV.

 

The second significant effect when increasing the ion beam energy is the increase of the mill rate. As the run-time output of VICTORY Process shows, the mill rate at a plane aluminum surface increases from 48 A/min at 50 eV to 1137 A/min at 300 eV (see Table 2).

The result of the ion milling step simulation is the 3D shape obtained from the initial structure (see Figure 1) after processing it by the ion milling process step. The structure after ion milling with an energy of 200 eV is shown in Figure 5 and the result obtained by ion milling with an energy of 300 eV is shown Figure 6. Due to the tilted ion beam the structure after ion milling simulation is no longer circular symmetric. On the backward side of the beam the nickel top corner gets faceted. The slope of the facet increases when moving along the top corner towards the forward direction of the beam. At the bottom of the trench a new slanted sidewall is formed due to the shading of the ion beam by the wall of the trench. The depth of the new trench formed at the bottom increases by increasing the ion energy.

(a)


(b)
Figure 5: 3D shape after ion milling by an ion beam with an energy of 200 eV.

 


(a)

(b)
Figure 6: 3D shape after ion milling by an ion beam with an energy of 300 eV.

 

 

Finally Figure 7 compares the two above simulation results by showing the cut plane containing the z-axis of the simulation domain (center of the circular mask) and the ion beam direction. Additionally Figure 7 shows the shape of the structure obtained by ion milling with an energy of 300 eV but with a reduced beam current. The beam current is adjusted such that the rate at a plane surface is identical to the rate of the ion milling process with 200 eV beam energy. Due to the adjustment of the beam current the depth of the trench formed by the 300 eV ion beam can be brought in line with the 200 eV ion milling step. Nevertheless the sidewall slope slightly changes due to a different impact of the beam energy and the beam current on the angle dependent mill rate function.

Figure 7: Comparison of ion milling simulation results with a beam energy of 200 eV and 300 eV within a cut plane containing the z-axis of the simulation domain and the ion beam direction.

 

Conclusion

The ion milling module of VICTORY Process is capable of simulating ion milling for complex three dimensional structures and for a variety of processing conditions which can be set by means of the input deck. If available, you may provide experimental angle dependent mill rates to be used during simulation. Otherwise, they will be generated automatically using the semi-empirical Yamamura model. Since the models for ion milling simulation are implemented in the open etching/deposition model library of VICTORY Process and the material data are stored in the open material database, the ion milling module can be conveniently extended to new materials and new ions. The example shown in this article demonstrates that different processing conditions, with identical mill rates for plane materials, may result in different three-dimensional shapes created by the ion milling processing step.

 

References

  1. Energy dependence of ion-induced sputtering yields from monoatomic solids at normal incidence, Y. Yamamura* and H. Tawara, Atomic data and nuclear data tables 62, pp. 149–253 (1996)
  2. An empirical formula for angular dependence of sputtering yields, Y. Yamamura, Radiation Effects,Vol. 80, Issue 1-2, 1984, pp. 57 - 72 (1984).
  3. Theory of Sputtering. I. Sputtering Yield of Amorphous and Polycrystalline Targets, Peter Sigmund, Phys. Rev. 184, pp. 383 – 416 (1969).
  4. [2] Developing Custom Etching/Deposition Models in VICTORY Process, Simulation Standard,Volume 22, Number 4, October, November, December 2012.
  5. Measurements and calculations of FIB milling yield of bulk metal, J.J.L. Mulders, D.A.M. de Winter, W.J.H.C.P. Duinkerken, Microelectronic Engineering 84 (2007) 1540–1543.
  6. Differential sputtering yields of refractory metals by ion bombardment at normal and oblique incidences, Kirk A. Zoerb, Thesis, Department of Mechanical Engineering, Colorado State University, Fort Collins, Colorado (2007).
  7. Untersuchungen zur Festkörperzerstäubung bei schiefwinkligem Ionenbeschuß polykristalliner Metalloberflächen im Energiebereich um 1 keV, H. Oechsner, Zeitschrift für Physik A Hadrons and Nuclei, Vol. 261, Number 1, pp. 37 – 58 (1973).

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