Hybrid Monte Carlo Fluid Model Research Articles

Elucidating the effects of gas flow rate on an SF6 inductively coupled plasma and on the silicon etch rate, by a combined experimental and theoretical investigation

Experiments show that the etch rate of Si with SF6 inductively coupled plasma (ICP) is significantly influenced by the absolute gas flow rate in the range of 50–600 sccm, with a maximum at around 200 sccm. Therefore, we numerically investigate the effects of the gas flow rate on the bulk plasma properties and on the etch rate, to obtain more insight in the underlying reasons of this effect. A hybrid Monte Carlo—fluid model is applied to simulate an SF6 ICP. It is found that the etch rate is influenced by two simultaneous effects: (i) the residence time of the gas and (ii) the temperature profile of the plasma in the ICP volume, resulting indeed in a maximum etch rate at 200 sccm.

Cryogenic etching of silicon with SF6 inductively coupled plasmas: a combined modelling and experimental study

A hybrid Monte Carlo—fluid model is applied to simulate the wafer-temperature-dependent etching of silicon with SF6 inductively coupled plasmas (ICP). The bulk plasma within the ICP reactor volume as well as the surface reactions occurring at the wafer are self-consistently described. The calculated etch rates are validated by experiments. The calculations and experiments are performed at two different wafer temperatures, i.e. 300 and 173 K, resembling conventional etching and cryoetching, respectively. In the case of cryoetching, a physisorbed SFx layer (x = 0–6) is formed on the wafer, which is negligible at room temperature, because of fast thermal desorption, However, even in the case of cryoetching, this layer can easily be disintegrated by low-energy ions, so it does not affect the etch rates. In the investigated pressure range of 1–9 Pa, the etch rate is always slightly higher at cryogenic conditions, both in the experiments and in the model, and this could be explained in the model due to a local cooling of the gas above the wafer, making the gas denser and increasing the flux of reactive neutrals, like F and F2, towards the wafer.

Modeling and experimental investigation of the plasma uniformity in CF4/O2 capacitively coupled plasmas, operating in single frequency and dual frequency regime

A two-dimensional hybrid Monte Carlo–fluid model, incorporating a full-wave solution of Maxwell's equations, is employed to describe the behavior of high frequency (HF) and very high frequency capacitively coupled plasmas (CCPs), operating both at single frequency (SF) and dual frequency (DF) in a CF4/O2 gas mixture. First, the authors investigate the plasma composition, and the simulations reveal that besides CF4 and O2, also COF2, CF3, and CO2 are important neutral species, and CF3+ and F− are the most important positive and negative ions. Second, by comparing the results of the model with and without taking into account the electromagnetic effects for a SF CCP, it is clear that the electromagnetic effects are important, both at 27 and 60 MHz, because they affect the absolute values of the calculation results and also (to some extent) the spatial profiles, which accordingly affects the uniformity in plasma processing. In order to improve the plasma radial uniformity, which is important for the etch process, a low frequency (LF) source is added to the discharge. Therefore, in the major part of the paper, the plasma uniformity is investigated for both SF and DF CCPs, operating at a HF of 27 and 60 MHz and a LF of 2 MHz. For this purpose, the authors measure the etch rates as a function of position on the wafer in a wide range of LF powers, and the authors compare them with the calculated fluxes toward the wafer of the plasma species playing a role in the etch process, to explain the trends in the measured etch rate profiles. It is found that at a HF of 60 MHz, the uniformity of the etch rate is effectively improved by adding a LF power of 2 MHz and 300 W, while its absolute value increases by about 50%, thus a high etch rate with a uniform distribution is observed under this condition.

Numerical investigation of HBr/He transformer coupled plasmas used for silicon etching

A two-dimensional hybrid Monte Carlo—fluid model is applied to study HBr/He inductively coupled plasmas used for etching of Si. Complete sets of gas-phase and surface reactions are presented and the effects of the gas mixing ratio on the plasma characteristics and on the etch rates are discussed. A comparison with experimentally measured etch rates is made to validate the modelling results. The etch rate in the HBr plasma is found to be quite low under the investigated conditions compared to typical etch rates of Si with F- or Cl-containing gases. This allows for a higher control and fine-tuning of the etch rate when creating ultra-small features. Our calculations predict a higher electron temperature at higher He fraction, because the electrons do not lose their energy so efficiently in vibrational and rotational excitations. As a consequence, electron impact ionization and dissociation become more important, yielding higher densities of ions, electrons and H atoms. This results in more pronounced sputtering of the surface. Nevertheless, the overall etch rate decreases upon increasing He fraction, suggesting that chemical etching is still the determining factor for the overall etch rate.

Effect of small amounts of hydrogen added to argon glow discharges: hybrid Monte Carlo-fluid model.

A hybrid Monte Carlo-fluid modeling network is developed for an argon-hydrogen mixture, to predict the effect of small amounts of hydrogen added to a dc argon glow discharge. The species considered in the model include the Ar gas atoms, electrons, Ar(+) ions and fast Ar atoms, ArH(+), H(+), H(2)(+) and H(3)(+) ions, and H atoms and H(2) molecules, as well as Ar metastable atoms, sputtered Cu atoms, and the corresponding Cu(+) ions. Sixty-five reactions between these species are incorporated in the model. The effect of hydrogen on various calculation results is investigated, such as the species densities, the relative role of different production and loss processes for the various species, the cathode sputtering rate and contributions by different bombarding species, and the dissociation degree of H(2) and the ionization degree of Ar and Cu. The calculation results are presented and discussed for 1% H(2) addition, and comparison is also made with a pure argon discharge and with only 0.1% H(2) addition.

Self-Consistent Description of Nitrogen dc Glow Discharge

A self-consistent hybrid Monte Carlo fluid model is presented to describethe nitrogen dc glow discharge. The movement of fast electrons is simulated bythe Monte Carlo method while the dynamics of slow electrons and ions is byfluid equations. The spatial features of the charged species and thecorresponding electric field throughout the discharge have been calculated,which include the creation rates of ions and slow electrons, densities ofthe charged species, the electric field and the potential distribution.These closely related results can give a self-consistent explanation of thedischarge characteristics throughout the space of nitrogen dc glowdischarge. The calculated ion density is also compared with thecorresponding experimental result.

Hybrid model for a cylindrical hollow cathode glow discharge and comparison with experiments

Some of the fundamental characteristics of the hollow cathode glow discharge are presented for different discharge conditions, based on a hybrid Monte Carlo fluid model, and on electrical and spectroscopic measurements. The Monte Carlo model describes the movement of the fast electrons as particles, while in the fluid model, the slow electrons and positive ions are treated as a continuum. The transient continuity equations are solved together with the Poisson equation in order to obtain a self-consistent electric field. The source terms of the continuity equations and the electron multiplication coefficient (used for the determination of the secondary electron emission coefficient) are obtained from the Monte Carlo simulation. These two models are run iteratively until convergence is reached. Typical results are, among others, the charged particles densities, the fluxes, the electric field and potential distribution. It is found that the influence of the bottom of the cylindrical hollow cathode cannot be neglected. A very good agreement between calculations and experimental data was obtained.

Improved hybrid Monte Carlo–fluid model for the electrical characteristics in an analytical radio-frequency glow discharge in argon

An improved hybrid Monte Carlo–fluid model for electrons, argon ions and fast argon atoms, is presented for the rf Grimm-type glow discharge. In this new approach, all electrons, including the large slow electron group in the bulk plasma, are treated with the Monte Carlo model. The calculation results presented here are the electrical characteristics (voltage, current and power as a function of time in the rf cycle, as well as the electrical potential and field distribution in the discharge), the electron and argon ion densities, and the electron, fast argon ion and atom impact ionization rates. In particular, the newly calculated electron impact ionization rate is more reliable now, because it is explicitly calculated in the electron Monte Carlo model instead of using an approximate formula, as in the earlier fluid model. Consequently, the major difference with our previous calculation results is found in the electron impact ionization rate. The calculated electrical characteristics (voltage, current and power) are, however, very similar to the results of our previous model. The new model confirms that the plasma displacement current is lower than the ion and electron conduction currents at the typical analytical rf Grimm-type glow discharge conditions, and therefore that the plasma current and voltage are in phase with each other. This is in contrast to other modeling results published recently, but in agreement with experimental observations where the capacitive current of the measuring circuit had also been subtracted from the total current.

Modeling of a microsecond pulsed glow discharge: behavior of the argon excited levels and of the sputtered copper atoms and ions

A set of models is developed for a microsecond pulsed glow discharge, to describe the behavior of argon excited levels, including the metastables, as well as the cathodic sputtering and the behavior of sputtered copper atoms and ions. These models are coupled to a hybrid Monte Carlo–fluid model for electrons, argon ions and atoms, which was developed previously, to obtain an overall picture of the pulsed glow discharge. Typical results of the present model are the densities of copper atoms and ions, the level populations of argon and copper excited levels, the erosion rate due to cathode sputtering and the contributions due to argon ion, atom and copper ion bombardment, the rates of various collision processes in the plasma, as well as the optical emission intensities. The results are presented as a function of distance from the cathode and as a function of time during and after the pulse.

Modeling of ionization of argon in an analytical capacitively coupled radio-frequency glow discharge

A hybrid Monte Carlo-fluid model has been developed for the description of electrons, argon ions, and fast argon atoms in a capacitively coupled radio-frequency (rf) glow discharge used in analytical spectroscopy. Typical operating conditions are about 6 Torr pressure and 10 W electrical power. The discharge cell is rather small and is characterized by a much smaller rf-powered electrode than grounded electrode, which yields a high dc bias voltage. The electron density at these conditions is in the order of 1013 cm−3. The computation time to simulate all these electrons with a Monte Carlo or a particle-in-cell method was found to be too long. Therefore, the electrons are subdivided in two groups. The fast electrons emitted from the rf electrode, as well as the ones formed by ionization with sufficiently high total (=kinetic+potential) energy for further ionization, give rise to so-called γ ionization; these are described with a Monte Carlo method. The slow electrons, which can, however, be heated again by the fluctuating electric field, give rise to so-called α ionization; they are described with a fluid approach, which also treats the argon ions. Moreover, the fast argon ions and atoms are treated with a Monte Carlo model in the rf sheath. Typical results of this model include the electrical characteristics (i.e., dc bias and rf amplitude voltages, electrical current, potential, and electric field distributions), the electron densities and mean energies, the ionization rates due to the electron impact α and γ ionization and fast argon ion and atom impact ionization, and the relative contributions of these ionization mechanisms to the overall ionization.

Comparison between a radio-frequency and direct current glow discharge in argon by a hybrid Monte Carlo–fluid model for electrons, argon ions and fast argon atoms

A hybrid Monte Carlo–fluid model has been developed for the electrons, argon ions and fast argon atoms in an argon glow discharge, either operated in the dc mode or the capacitively coupled rf mode. Typical working conditions for rf GD-OES are considered, i.e. approximately 6 torr argon gas pressure and approximately 10 W power. Typical results of the model, like the potential distributions, densities, fluxes and ionization rates, will be presented and compared between the two operation modes. It will be demonstrated that the rf discharge yields more efficient ionization than the dc discharge, and hence the rf discharge requires lower voltages to obtain the same amount of power, which is in good correspondence to experimental observations.

Monte Carlo model for the argon ions and fast argon atoms in a radio-frequency discharge

A three-dimensional Monte Carlo model has been developed for the argon ions and fast argon atoms in the RF sheath of a capacitively coupled RF glow discharge in argon. Our interest in the argon ions and fast argon atoms in the RF sheath arises from the fact that the glow discharge under study is used as sputtering source for analytical chemistry. This source operates at typical working conditions of a few torr pressure and about 10 W incoming power. The argon ion and atom Monte Carlo model has been coupled to a hybrid Monte Carlo-fluid model for electrons and argon ions developed before to obtain fully self-consistent results. Typical results of this model include, among others, the densities, fluxes, collision rates, mean energies, and energy distributions of the argon ions and atoms. Moreover, we have investigated how many RF cycles have to be followed before periodic steady state is reached, and the effects of all previous RF cycles are correctly accounted for. This is found to be the case for about 20-25 RF cycles.

The role of fast argon ions and atoms in the ionization of argon in a direct-current glow discharge: A mathematical simulation

A model is developed for a direct-current glow discharge in argon by a combination of a hybrid Monte Carlo fluid model of electrons and ions in the entire discharge and a Monte Carlo model of ions and fast atoms in the cathode dark space, in which fast ion and atom impact ionization are incorporated. The relative importance of these processes, compared to electron impact ionization is investigated, as a function of distance from the cathode and at different discharge conditions. It is found that they are dominant close to the cathode, and that they gain importance with increasing voltages. With the incorporation of these processes it was possible to predict current–voltage relations which are in excellent agreement with experiment. Also, the length of the cathode dark space, as a function of pressure and voltage, is calculated; the results agree with Aston’s empirical formula.