Applied Science and Convergence Technology 2020; 29(2): 23-27
Published online March 30, 2020
https://doi.org/10.5757/ASCT.2020.29.2.023
Copyright © The Korean Vacuum Society.
Yejin Shon^{a}, Sora Lee^{a}, Dong-gil Kim^{a}, Deuk-Chul Kwon^{b}, and HeeHwan Choe^{a,*}
^{a}School of Electronics and Information Engineering, Korea Aerospace University, Goyang 10540, Republic of Korea
^{b}Plasma Technology Research Center, National Fusion Research Institute, Gunsan 54004, Republic of Korea
Correspondence to:choehh@kau.ac.kr
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-CommercialLicense (http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution,and reproduction in any medium, provided the original work is properly cited.
This research investigates the plasma characteristics of an inductively coupled plasma discharge which is widely used in semiconductor fabrication processes. Three-dimensional non-symmetric geometry, which contains four discontinuous gas inlet ports and a unique antenna shape with individual power input and grounded output part, is assumed. Fluid-based simulation is performed to reduce the amount of computation that increases with the three-dimensional structure. Ar is used as a reaction gas to simplify the physics. For comparison to increase accuracy in calculating the power induced to the plasma chamber, the capacitive electric field is calculated besides the usual inductive electric field; the impedance of the system is calculated from a field definition based on Poynting’s theorem.
Keywords: Plasma, Asymmetric 3D inductively coupled plasma, Coil analysis, Impedance
An inductively coupled plasma (ICP) discharge is used widely for wafer fabrication processes of microelectronic devices such as semiconductors or displays [1]. Further, an ICP has advantages on plasma-surface interactions such as etching or deposition with an elaborate scale in the manufacturing of micro-circuits because of the high density, low pressure, and good uniformity [2-8].
To increase the efficiency of ultrafine processes, it is necessary to gain an understanding of the physics of plasma through computational modeling, which is an appropriate approach to determine associations between plasma factors and process conditions that are difficult to detect in experiments. The ICP discharge simulations are conducted in a two-dimensional axisymmetric model because of the limited computing resources and time [9]; however, in this study, a three-dimensional asymmetric model is used to investigate the asymmetric effects in plasma properties caused by the geometrical structure, and to simulate the actual process conditions more closely.
In this study, a fluid-based model was used among various plasma analysis methods for the simulation and detailed explanations about this model are present in the literature [10,11]. The fluid model treats plasma particles in the chamber as a fluid, and thus, there are limitations because plasma phenomena occur owing to the movement of individual particles in low-pressure environments. However, fluid model has the benefits of saving resources by using the averaged statistical characteristics and assuming energy distribution of particles to simplify equations [12].
The power flowing through the antenna input generates a magnetic field, and the induced electric field transfers the force to the electrons in the plasma region [13]. The magnitude and distribution of the coil voltage vary based on the impedance of the antenna; further, they are closely related to the magnitude of the transmitted force, and in fact, the plasma characteristics change based on this factor. Among these impedances calculated by field, capacitance and inductance play a considerable role in calculating voltage. Electric energy―the source of capacitance―depends on the electromagnetic field, and therefore, it is important to solve both the inductive and capacitive electric fields to calculate magnetic and electrical physics accurately.
In this paper, a three-dimensional simulation was carried out to consider the asymmetric effect of the plasma characteristics along with the gas inlet and the antenna shape, which could not be considered in the two-dimensional axisymmetric structure. Fluid-based simulations were performed in COMSOL Multiphysics. This study focuses on the characteristics of the plasma depending on the coil impedance variation and the power induced in the chamber according to the method of analyzing the electromagnetic field
In this study, a three-dimensional asymmetric model was used, and the geometric scheme of this model is as shown in Fig. 1. There are four plasma chamber areas: plasma region, ceramic window, antenna, and air cap. The diameter of the plasma region is 35 cm, and each dielectric portion and outlet is located at a thickness of 2.5 cm at the bottom thereof. Four inlet ports, from which the reaction gas flows in, are located at 90° intervals on the side of the chamber. To separate the plasma region from the antenna, a ceramic window is placed on top of the plasma region at a height of 2 cm and they are composed of aluminum oxide. The antenna, which consists of copper, is wound three times with a cross section of 1 cm2 and applied 100 A at a frequency of 13.56 MHz. A shielding cap filled with air is set on the ceramic window to protect the coil; rectangular floating portions around the power supply part of the antenna are designed to avoid the effect of the upper part of the shielding cap that is set to ground.
Argon was used as the reaction gas under the condition of 20 mTorr; Table I summarizes the reactions used in the simulation process. This study was conducted with COMSOL Multiphysics using a fluid model.
Table 1 . Important collision processes in the argon discharge.
Reactio | n Formula | Type | ∆ε(eV) | Ref. |
---|---|---|---|---|
1 | e + Ar → e + Ar | Elastic | 0 | [15] |
2 | e + Ar → e + Ars | Excitation | 11.5 | [15] |
3 | e + Ars → e + Ar | Superelastic | −11.5 | [15] |
4 | e + Ar → 2e + Ar^{+} | Ionization | 15.8 | [15] |
5 | e + Ars → 2e + Ar^{+} | Ionization | 4.24 | [15] |
6 | Ars + Ars → e + Ar + Ar^{+} | Penning ionization | - | [15] |
7 | Ars + Ar → Ar + Ar | Metastable quenching | - | [15] |
The equations for electron density and electron energy density, which follows a drift-diffusion approximation, are provided below.
where
The gas inlets and outlets need to satisfy the conditions of zero charge on the boundary, as indicated in Eq. (6). In addition, to apply the insulation effect, the normal components of the electron and electron energy flux must be zero at the boundary, as indicated in Eqs. (7) and (8).
The lower surface of the ceramic window in contact with the plasma region must meet the surface charge accumulation condition, which builds up charge in the normal direction, as
where
The following equations explain the mixture averaged method for calculating heavy species, which satisfies mass conservation [14].
where
While formulating the electric field, the magnetic field (MF) physics considers the inductive field by considering magnetic vector potential as shown below.
where
Meanwhile, the electric field calculated in magnetic and electric field physics (MEF), which considers both of inductive field and capacitive field by using the electric potential additionally, is given by
where
To prove the physics setting that can calculate the inductance and impedance of the antenna accurately, three cases were considered. Case (1) was simulated with a coil analysis node in MF physics. Cases (2) and (3) were simulated with MEF physics; however, the difference is that coil analysis was used only in Case (2).
The expression of coil inductance and capacitance used in Cases (1) and (2) is given by
and the voltage equation according to the impedance change is defined based on the phasor form of Ohm’s law as
where
Meanwhile the impedance is calculated from a field definition in Case (3) by using Poynting’s theorem, as
where
First, the resistance, inductance, and capacitance of the coil calculated by the electromagnetic field and the coil voltage because of the impedance change for all cases are listed in Table II. Further, the power absorbed by electrons because of inductive discharge is listed for when the current applied to the antenna is 100 A. The absorbed power contributes to computing electron energy density (
Table 2 . Comparison of coil impedance, voltage, and absorbed power between all cases.
Case (1) | Case (2) | Case (3) | |
---|---|---|---|
0.39 | 0.42 | 0.44 | |
3.6 × 10^{-7} | 4.2 × 10^{-7} | 4.1 × 10^{-7} | |
−7.1 × 10^{-15} | −3.83 × 10^{-15} | −5.7 × 10^{-12} | |
39.5 + | 42.1 + | 44.38 + | |
1977 | 2103 | 2195 |
In coil geometry analysis, the voltage out-of-plane to the coil was the source of the current density. This current density was used to calculate the reactance of the coil. Therefore, in this coil geometry analysis, the calculation method was suitable for simple static materials; however, the plasma was very complex for applying this method.
Distribution of coil voltage and electromagnetic fieldFigure 2 shows the real and imaginary parts of the antenna voltage in Cases (2) and (3). It is common for the power injection port to show the highest numerical value (red) and the output part connected to the ground (blue) with the lowest value distribution as shown in Case (3). However, there is an absence of settings for the ground connection to the antenna output part in the coil analysis node. Thus, Case (2) shows a nonzero potential in the output section, which causes a numerical error. In addition, the coil geometry analysis study step, in which the current direction of the antenna is solved, produces optimized results when the coil has a constant cross-section without shape change or bottleneck phenomena. Therefore, the current calculation at the coil bending part could be incorrect.
Figure 3(a) shows the magnetic flux density of the magnetic field generated by the power applied to the coil. Eq. (14) indicates that the magnetic field can be expected to be generated in the rotating direction of the current flow. Further, Fig. 3(b) represents the electric field induced by the magnetic field transmitted to the plasma region. The power generated from the coil is transferred to electrons in the plasma via a time-varying magnetic field, which leads to an azimuthal direction induced electric field that generates high-density plasma.
Figure 4 illustrates normalized plasma characteristics with varying transmitted power caused by coil voltage variations. As shown in Fig. 4, there are slightly differences in the maximum value of electron density, electron temperature, and electric potential between all cases. Since Case (3) received the most power in the plasma region, it has the highest electron density and temperature compared to the other cases. Further, because the time to reach the plasma wall changes based on the speed difference caused by the differences in weight between electrons and ions, the electrons collide into the wall faster than the ions. Thus, the electrons are lost faster than ions, and the inside of the plasma region has a positive charge because it repeats the light emission process of generating electrons and ions. Negative potential near the wall accelerates ions and slows electrons because of the repulsion. The same number of electrons and ions reach the wall, and thus, the wall has a negative charge until the electron current density and ion current density are balanced, which is called the sheath region. Therefore, the electric potential has a negative value at the side of the chamber.
Figures 5 - 8 show the major plasma characteristics by sliced on xy plane at 9 cm where the inlets are located, 5 cm as the center of the chamber, and 1 cm located symmetrically with the inlets relative to the center of the chamber. Since Case (3) has the highest absorbed power, as shown in Table II, exhibits the highest values for collisional power loss and electron density. In addition, due to the structural asymmetry, the distribution of variables is skewed to one side that cannot be observed in the two-dimensional axisymmetric structure.
This study observed the appropriate physics settings required for calculating the correct coil voltage by determining the optimal coil impedance using COMSOL Multiphysics. The simulation is conducted using a three-dimensional asymmetric ICP discharge model for semiconductors manufacturing process, and this work is based on fluid models for plasma. Owing to the difference in the calculation of the electric field, magnetic field physics and magnetic and electric field physics show large differences in the total electric energy that leads to variances in impedance. This shows that accurate electric field calculations are essential because they are closely related to the power induced to the plasma chamber. In addition, due to the absence of a setting to apply ground to the output of the antenna and the diversified geometric characteristics of the antenna structure, the use of coil geometry analysis node results in incorrect current calculations which leads to a numerical error. Therefore, accurate impedance calculations computed from the electromagnetic field based on Poynting’s theorem increase the simulation accuracy.
To obtain more suitable plasma properties, future studies will include heat transfer physics that considers changes in enthalpy caused by collisions and flow physics to consider the change in pressure and velocity fields along the flow of fluid.
This work was supported by the National Research Council of Science & Technology (NST) grant by the Korea government (MSIP) (No. CAP-19-02-NFRI)