Spatially averaged global model of HBr/Cl₂ inductively coupled plasma discharges Open Access

Halogen-based plasmas have primarily been employed in the dry etching process of semiconductor devices to achieve ideal anisotropic profiles. In fact, plasma containing F atoms tends to yield a faster etch rate compared to other halogen atoms, such as Cl or Br. However, controlling the profile and loading is more challenging due to the quick isotropic etching caused by F atoms. Thus, Cl- or Br-based plasmas are frequently preferred for silicon etching processes due to their high etching selectivity for silicon oxide mask structures.^1,2 Cl-based plasma is known to have a higher etch rate for Si compared to Br-based plasmas. On the other hand, Br-based plasma offers higher anisotropy and selectivity. Consequently, Cl- or Br-based plasma etching demands a trade-off among several crucial characteristics.³ To overcome this inherent limitation, a combination of Cl₂ and HBr has proven to be an effective solution for concurrently achieving higher rates, selectivity, and anisotropic etching.

To understand the intricate mechanisms of the HBr/Cl₂ plasma etching process, the competitive halogenation of silicon surfaces was studied for Cl−, Br−, and H-containing species impinging simultaneously. Measurements of the Br and Cl saturation coverage during and after plasma etching revealed that the presence ratio of Br and Cl directly corresponds to the respective halogen ratio in the supply gas, which is directly related to the etching rate of the formed halides on the surface in pure HBr and Cl₂ plasma. Additionally, hydrogen atoms from the dissociation of HBr can affect the etching process in several ways, including hydrogenation of the photoresist enhancing selectivity, formation of volatile etch products influencing the overall etch rate, and surface passivation, which may affect the etch rate.² Moreover, the angular dependence observed during the process of polysilicon etching points toward the likelihood of an ion-enhanced chemical etching mechanism. It is worth noting that distinct discrepancies are observed in the sidewall etching process between pure Cl₂ and HBr.⁴ After extensive research on the HBr/Cl₂ plasma process, further investigation is required due to the complexity of interpreting and optimizing the etching characteristics.

Several numerical studies have been conducted on HBr mixture plasmas. Efremov et al. developed HBr/Ar and HBr/Cl₂ bulk plasma chemistries for dry etching applications.⁵ Using the developed model, the authors in Ref. 6 analyzed the etching characteristics and mechanisms of Pb(Zr,Ti)O₃, Pt, and SiO₂ under external operating conditions. The measured total positive ion density and the electron temperature were used to calculate the charged and radical densities in the bulk. The calculated plasma properties were then used to characterize the dominant etching mechanism by comparing them with the results of etching experiments. Similarly, the characterization of InP thin films in HBr/Cl₂/Ar plasma and Si/SiO₂ etching in HBr/Cl₂/O₂ plasma was carried out by Kim et al.⁷ and Lee et al.⁸ 

Moreover, Gul et al. conducted two-dimensional fluid simulations for capacitively coupled HBr mixture plasmas.^3,9–11 In these simulations, the momentum equation of the charged particles is assumed to follow the drift-diffusion approximation, and the electron mobility and diffusion coefficients are calculated by solving the Boltzmann equation using the BOLSIG+ solver.⁹ Although there is not much comparison with experimental results, their results are significant in that they solve the Boltzmann equation to obtain more accurate rate coefficients for electron collision reactions. In particular, comparing the measured EEPF (electron energy probability function) with the EEPF obtained by solving the Boltzmann equation can be used as a manner to evaluate the reliability of the electron collision cross section. Unfortunately, there are a few EEPF measurements available for HBr/Cl₂ plasmas, and the equipment geometry used in the previous literature is not sufficiently described. Therefore, we were unable to perform calculations that combine the Boltzmann solver with the global model.

Tinck and Bogaerts computationally investigated inductively coupled CF₄/CHF₃/H₂/Cl₂/O₂/HBr plasmas under different gas mixing ratios.¹² They performed fluid-based two-dimensional simulations under various conditions. Among the results, the following are those related to HBr mixing: as the HBr flow rate increased, the chemical etching decreased, because the Cl and F radical densities decreased. However, in contrast, the ion sputtering increased with increasing HBr⁺ and Br⁺ in the plasma.

It is well known that we need a set of chemical reactions and rate coefficients as an input variable for the simulations. Although a reliable set is highly relevant to the accuracy of the simulation, simulations and measurements such as ion and radical densities have not been sufficiently verified for HBr mixture gas discharges. Therefore, in this study, we developed a complete set of chemical reactions for HBr/Cl₂ inductively coupled plasmas. The model was verified by comparing calculated results with previously published experimental measurements. The electron and Cl⁻ densities were verified for pure Cl₂ discharges, HBr and Br₂ densities were compared for pure HBr discharges. Moreover, for HBr/Cl₂ discharges, the experimental and calculated results for the electron temperature, total ion flux, total ion density, and positive ion flux ratio were compared. By using the obtained global model, the characteristics of the HBr and Cl₂ mixture plasmas were analyzed with the calculated results.

The global model is widely employed to analyze the dependence of spatially averaged plasma variables on plasma operating conditions.^13–17 Although the global model cannot compute the spatial distribution of plasma variables, it serves for the development and the verification of chemical reaction sets with reaction coefficients. In our research group, we have conducted numerous studies using our global models and have endeavored to enhance the accuracy of the global model by coupling or including additional physical models.^17–19 The global model comprises a method to calculate electron heating, chemical reactions, transport, and RF sheath modules. The previous global model encompassed a module that calculated the power absorption of plasma from the field generated by the coils (i.e., electron heating); however, it was not considered in this work, because the experimental literature examined in this study lacked detailed coil dimension. Hence, it is assumed that all power supplied to the plasma source is absorbed by the electrons. Consequently, the EEPF was assumed to be Maxwellian, although it can be obtained by solving the Boltzmann²⁰ or Fokker–Planck²¹ equations in a self-consistent manner. Therefore, under low-pressure or low-power conditions, the developed model may introduce errors.

When the unit of $Q in , i$ is sccm, the generation term of the inlet gas $S inlet , i$ is $4.483× 10 17 Q in , i /V$⁠. The loss of species i by the pump $L pump , i$ is $n i Q out /V$⁠, and the pumping speed $Q out$ is controlled by the feedback routine to maintain a certain pressure. The density of the electron $n e$ is calculated as $n e= n +− n −$⁠, where $n +$ is the total density of positive ions, and $n −$ is the density of negative ions based on quasi-neutrality in the bulk plasma region.

The cross sections for vibrational excitations, de-excitations, and dissociative attachment (R1–R12) were obtained from Horáček and Domcke²⁵ and Fedor et al.²⁶ The rate coefficient of dissociative attachment of HBr(v = 3) was assumed to be that of HBr(v = 2), where v is the vibrational excitation level. For the neutral dissociation (R13–R16) and ionization (R17–R20) of HBr, the rate coefficients of the vibrationally excited states were assumed to be those of the ground state.^27,28 The threshold energies of reactions R14–R16 and R18–R20 were calculated by subtracting the excitation energies from the threshold energy of the reactions from the ground state.

The total ionization rate coefficient of Br₂ was calculated from the cross section determined using the binary-encounter-Bethe (BEB) method in Ref. 28. Through the ionization of Br², Br⁺ or Br₂⁺ can be produced, but the partial ionization cross sections that create each ion were not known. In this model, we assumed that the ionization reaction creates Br⁺ and Br₂⁺ at a ratio of 0.7 and 0.3, respectively. The rate coefficients of the electron attachment reactions of Br₂ (R28 and R29) were calculated from the collision cross sections in Ref. 29. When the electron collision energy is lower than 10.41 eV, Br⁻ and Br are produced as R28. When the energy is higher than this value, ion-pair formation of Br⁻ and Br⁺ occurs at R29. The cross sections of other reactions of Br₂ were difficult to obtain from the literature, and the cross sections of Cl₂ were used as assumptions.^5,16,30 We used the electron impact ionization cross-section of Br (R30) calculated using the BEB model.²⁸ Furthermore, the electron collisional excitation cross sections were calculated using the flexible atomic code (FAC) and converted to rate coefficients (R30–R33).³¹ 

The electron collisional cross sections for H₂ and H were obtained from the data collected and evaluated by Yoon et al.³² Cross sections related to the metastable states of H atoms were obtained from the evaluated data by Janev et al.^33,34 In this model, we considered three types of metastable states of H atoms: H^*(2s), H^*(2p), and H^*(n3).

The reactions and rate coefficients (R49–R58, R67, R72, and R73) for Cl₂ plasma modeling were based on the model of Meeks et al.³⁰ The collision cross section of the dissociative ionization producing Cl⁺ (R72) was obtained from Ref. 35. The rate coefficients of the reactions related to the vibrationally excited state of Cl₂ (R61–R66, R68–R71, and R75), and negative ions of Cl (R59 and R60) were obtained from Ref. 16.

The ionization collision cross section of HCl (R76), calculated using the BEB model²⁸ was employed in this model. The dissociative attachment rate coefficient (R77) was derived from the measured collision cross section found in Ref. 26. Additionally, the collision cross section for vibrational excitation (R78) was obtained from the calculations by Fedor et al. using a nonlocal resonance model.³⁶ In this model, we only considered vibrational excitation from the ground state (v = 0) to the first excited state (v = 1) because the cross sections for other excitations are significantly smaller.

Due to limited research on the electron collision reaction mechanism of BrCl, we assumed the collision cross sections based on data from HBr, HCl, and Cl₂. The neutral dissociation rate coefficient (R79) of HBr was adopted.²⁷ Furthermore, the rate coefficients for electron collisional ionization (R80) and dissociative attachment (R81 and R82) of BrCl were based on those of HCl.^26,28 Vibrational excitation (R83–R85) and electron excitation (R86) of BrCl were treated similarly to Cl₂.^16,30

Several reactions between neutral species (R87–R102) were adopted from the model of Efremov et al.⁵ For other reactions (R103 and R104), the rate coefficients were obtained from experiments in Refs. 37 and 38. The rate coefficient of mutual neutralization between a positive ion and a negative ion (R105) was estimated as $1.0× 10 − 6 c m 3 s − 1$⁠. The rate coefficients of charge transfer between a positive ion and a neutral species (R106) were estimated as $1.0× 10 − 10 c m 3 s − 1$ when $E iz , A< E iz , B$⁠. Here, $E iz , A$ and $E iz , B$ are the ionization energies of species A and B, respectively. When $E iz , A> E iz , B$⁠, the charge-transfer reactions were ignored because the rate was significantly lower than that in the opposite cases.

When positive ions collide with a wall, they are neutralized by the electrons on the wall. The reaction rates of the neutral species at the wall are given by Eq. (3), and the wall recombination coefficients of the neutral wall recombination are listed in Table IX. The coefficients of the single-species reactions were adapted from Efremov's model,⁵ and those of the multiple-species reactions were estimated as the average of the sticking coefficients for each species.

The developed model was validated by comparing the calculated results with experimental data. First, the diagnostic results of Fleddermann et al. were used to validate the model for pure Cl₂ plasmas.³⁹ In Ref. 39, the densities of Cl⁻ were obtained from the density measurement of excess electrons by photo detachment from Cl⁻ using the fourth harmonic (266 nm) of an Nd:YAG laser. The electron density was obtained using a microwave interferometer by measuring the phase shift caused by the plasma. The plasma was produced in a modified gaseous electronics conference (GEC) RF reference cell as an inductively coupled plasma (ICP) source. Because the complex geometry of the GEC cell cannot be applied to the global model, the chamber was approximated as a cylinder with a height of 4.05 cm and a radius of 8.26 cm. The applied power was varied from 150 W to 400 W with a gas pressure of 20 mTorr, and the pressure was varied from 6 to 40 mTorr with a power of 300 W.

Figures 1(a) and 1(b) show the comparison of the experimental and numerical results for the Cl₂ discharges. The calculated Cl⁻ and electron densities are in good agreement with the experimental results in terms of tendencies. However, the magnitude of the densities differed by a factor of 2, approximately. These differences may be due to the limitations in the assumed global model for not comparing the densities at multiple positions.

The model was also validated for pure HBr gas plasma by comparing it with the experimental results of Cunge et al.⁴⁰ In the experiment, the densities of HBr and Br₂ molecules in an ICP source were measured using vacuum UV absorption spectroscopy. The plasma reactor had a diameter of 50 cm and a height of 17 cm. The plasma was generated with pure HBr gas at a pressure of 5 mTorr and a power of 100–1000 W. It can be seen that the power dependences of the radical densities are in good agreement, as shown in Fig. 1(c). In particular, the power dependence of Br₂ agrees well with the experimental results quantitatively.

The model was further validated for plasma discharges using HBr diluted in Cl₂. The calculated results were compared with the experimental results conducted by Efremov et al. and Vitale et al.^5,41 In Ref. 5, the plasma was generated in an ICP reactor with a height of 12.8 cm and a radius of 16.0 cm, the applied power was 700 W, and the gas pressure was 6 mTorr. The ratio of HBr gas was varied in the mixture of HBr and Cl₂ gas, and the total flow of the two gases was maintained at 40  sccm. The ion saturation current density and electron temperature were measured using a double Langmuir probe placed at the center of the reactor. A bias power of 300 W was also applied. However, in this study, the power applied by the bias was ignored due to its small power-transfer coefficient compared to the power applied to the antenna. Similarly, in Ref. 41, the authors measured ion saturation current and the electron temperature at the center of the chamber using a Langmuir probe. The reactor had a height of 30.48 cm and a radius of 10.16 cm. The gas pressure was maintained at 5 mTorr, the total gas flow was kept at 40  sccm, and the applied power was 400 W.

Figure 2 shows the comparison between the experimental and numerical results for the ion saturation current density, electron temperature, and total positive ion density. The calculated ion saturation current densities agree well with the experimental results. However, in the model, the electron temperature initially decreased as the HBr flow rate increased from 0 to 0.6, but then increased with a further increase in the HBr ratio. In contrast, during the experiment, the electron temperature monotonically decreased. The difference in results may be because the estimated electron collisional cross sections of BrCl differ from the real cross sections. As mentioned in Sec. II B, there were insufficient cross sections for BrCl in the literature, and several cross sections were estimated based on those of other species. For a more reliable calculation, more accurate cross sections for BrCl are required. However, for the pure Cl₂ plasma and pure HBr plasma, the calculated electron temperatures match well with the experimental results.

The dependence of the ion flux ratio on the HBr flow rate is shown in Fig. 3. Here, the ion flux ratio is defined as the ion flux divided by the total flux of the six ions considered in the experiment. The global model considers H⁺, H₂⁺, and HCl⁺, which were not considered in the experiments. The results show that the HBr ratio dependence of the ion flux ratio is in good agreement with the experimental results. In particular, the peak positions of the Br⁺ and BrCl⁺ flux ratios matched well.

The flux ratio of Br⁺ is highest at the HBr ratio of 0.5 (0.4 in the experiment), which can be explained by the densities of neutral species as shown in Fig. 4. When the HBr ratio is higher than 0.5, the increasing slope of the Br density decreases, while the slope of HBr density becomes steeper. As the ratio of HBr is increased, the flux of Br⁺ increases slowly, while that of HBr⁺ increases rapidly as the densities of Br and HBr molecules increase. Therefore, the flux ratio of Br⁺ decreases at higher HBr ratios greater than 0.5.

The major dissociation reaction of HBr is electron-collisional neutral dissociation (R13). As shown in Fig. 2(d), the electron density decreases as the ratio of HBr increases. The rate of dissociation of each HBr molecule also decreases. Therefore, a larger portion of the supplied HBr remains as the HBr ratio increases, as shown in Fig. 4.

We developed a global model for HBr/Cl₂ in inductively coupled plasma sources. To evaluate the model, we compared the calculated results with experimental data for Cl₂, HBr, and HBr/Cl₂ plasma discharges. For pure Cl₂ discharges, we analyzed the pressure and power dependence of electron and Cl⁻ densities. For pure HBr plasma, we investigated the power dependence of HBr and Br₂ densities. Additionally, for HBr/Cl₂ discharges, we compared various plasma parameters' dependence on the HBr ratio with experimental results. The calculated results showed good agreement with experimental data within the margin of error. We also explained an irregular phenomenon: the ion flux ratio of Br⁺ decreased as the HBr gas flow ratio increased in the HBr/Cl₂ plasma.

However, further theoretical and experimental studies are needed on electron collision reactions in molecules like BrCl. Moreover, for a more accurate evaluation of electron collision cross sections, additional studies should be conducted to solve the Boltzmann or Fokker–Planck equations using collision data to calculate the EEPF and compare it with experimental results. This is an active study and will be reported later.

This work was supported by the National Research Council of Science & Technology (NST) grant by the Korea government (MSIT) (No. CRC-20-01-NFRI). This research was also supported by the MOTIE (Ministry of Trade, Industry & Energy) (No. 1415187722) and KSRC (Korea Semiconductor Research Consortium) (RS-2023-00235950) support program for the development of future semiconductor device, and it was supported by the National R&D Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science and ICT (2021M3H4A6A01048300).

The authors have no conflicts to disclose.

Sang-Young Chung: Conceptualization (equal); Data curation (lead); Formal analysis (lead); Software (supporting); Validation (lead); Visualization (lead); Writing – original draft (lead). Yeong Geun Yook: Investigation (supporting); Validation (supporting); Writing – review & editing (supporting). Won-Seok Chang: Validation (equal); Writing – original draft (supporting). Heechol Choi: Validation (equal); Writing – original draft (supporting). Yeon Ho Im: Conceptualization (equal); Project administration (supporting); Supervision (equal). Deuk-Chul Kwon: Data curation (equal); Project administration (lead); Software (lead); Supervision (lead); Writing – original draft (equal).

The data that support the findings of this study are available within the article.