Estimations of secondary electron emission coefficients of Si, SiO₂, and polyimide electrodes in dual-frequency capacitively coupled discharge

With the miniaturization of semiconductor process, it is required to control the etching shape with high accuracy. ^1,2) In addition, from the viewpoint of improving productivity, an etching process that improves yield and shortens process time is expected. ³⁾ For addressing these demands, much effort has been devoted to increasing the density and controllability of these plasmas for etching ^4,5)

As a plasma source for etching, capacitively coupled plasma equipment has been widely used. To improve the plasma density and controllability, a dual-frequency plasma source in which RF power supplies of different frequencies, connected to the upper and lower electrodes has been proposed. ^6–9) In this method, a high-frequency RF power supply is connected to the upper electrode to generate high-density plasma. On the other hand, a low-frequency RF power supply is connected to the lower electrode to control the dc self-bias V_dc. By adjusting the RF frequency and voltage of the upper and lower electrodes, it is possible to generate the plasma required for etching.

Plasma simulation is effective for improving the efficiency of the development of plasma equipment and the optimization of plasma states. By using plasma simulation, it is possible to visualize the distribution of electron density, ion density, and active species density. ^10–14) Several simulations have been reported so far for dual-frequency plasma. ^15,16) In capacitively coupled plasmas including dual-frequency excitation, secondary electron emission (SEE) due to collisions between positive ions and the electrodes has been focused on, ^17–19) and many papers have been reported. ^20–23) Since the electron density obtained from the simulation largely depends on the SEE coefficient, it is an extremely important parameter. However, the SEE coefficient changes depending on the applied voltage, the electrode material, and the surface state of the electrode.

We performed coupled calculation of plasma and gas flow simulations for dual-frequency excited Ar plasmas and compared them with the experimental results. ^24,25) As a result, the gas phase reaction model of Ar was constructed. In this study, we measured the electron densities by changing the surface material of the lower electrode to Si, SiO₂, and polyimide. Furthermore, the SEE coefficient was changed in the plasma simulation, and the SEE coefficient was fitted so that the electron density in the simulation coincided with that measured in the experiment. From these results, the SEE coefficients of Si, SiO₂, and polyimide are estimated to be 0.06, 0.24, and 0.22, respectively.

A dual-frequency capacitively coupled plasma equipment was used in the experiment. Figures 1 and 2 show the perspective view of the plasma equipment and cross-section of the plasma chamber, respectively. The diameters of the upper and lower electrodes were 50 and 120 mφ, respectively. The gap length between the upper and lower electrodes was set to 119.5 mm. A 4 inch φ wafer can be installed on the lower electrode. The process gas is supplied uniformly from the showerhead of the upper electrode. The gas pressure in the chamber is 4 Pa, and the Ar gas flow rate is 50 sccm. The upper and lower electrodes made of Si are isolated from the GND of the chamber by Teflon and alumina, respectively. The frequencies of the upper and lower electrodes were 60 MHz and 2 MHz, respectively. The RF voltages, when 400 W and 500 W are applied to the upper and lower electrodes are 810 V and 905 V.

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The electron density was measured using an absorption probe. ^26,27) The absorption probe has an antenna diameter of 0.5 mmφ and a length of 10 mm, and the outer diameter of the glass covering the antenna is 4 mmφ. The tip of the probe antenna was located at the center of the chamber and 23 mm above the upper electrode, as shown in Fig. 1.

First, electron densities were measured in chambers of upper and lower electrodes made of Si. Next, 4 inch Si wafers coated with SiO₂ and polyimide were placed on the lower electrode, and the electron densities were measured. The SiO₂ film was deposited on a Si substrate with a thickness of 8 μm using TEOS gas. And a polyimide with a thickness of 100 μm was attached to a Si wafer.

Table I shows the measurement results of the electron density. The electron density for both the upper and lower electrodes made of Si was 14.6 × 10¹⁶ m^–3. On the other hand, the electron densities of the lower electrodes of SiO₂ and polyimide were as high as 26.5 × 10¹⁶ and 20.2 × 10¹⁶ m^–3 We thought that the increase in electron densities in these electrodes were caused due to the SEE generated on the electrode surface and evaluated the effects of the SEE by simulation in Sects. 4 and 6.

Next, we measured the relationship between bias power and electron density in the chamber where the Si wafer covered with SiO₂ was set on the lower electrode. The input power of the upper electrode was kept at 400 W, and the bias power of the lower electrode was changed from 300 to 500 W. The experimental results are shown in Fig. 3. With the increase in bias power, the electron density increased from 22.0 × 10¹⁶ to 25.9 × 10¹⁶ m^–3.

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3.1. Chamber model and boundary conditions

Figure 4 shows the plasma chamber and the simulation model. A 2D fluid model of the cylindrical coordinate system was adopted for the simulation, and one-half of the cross-section of the chamber corresponded to the cylindrical coordinate cross-section. The central axis of the chamber corresponded to the center of the cylindrical coordinates, and the vertical and lateral directions were defined as the z- and r-axes, respectively. In this coordinate system, the position of the absorption probe is (r, z) = (0, 80).

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The boundary conditions for the simulation were set to be the same as the experimental conditions. The gas pressure is 4 Pa and the gas flow rate is 50 sccm. Ar gas was set to be evenly supplied from the upper electrode and exhausted from the space between the alumina and the chamber wall as shown in the lower right of Fig. 4. The relative permittivity of the insulating material Teflon of the upper electrode was 2.5, and the relative permittivity of alumina was 8.5. A frequency of 60 MHz and V = 810 V was applied to the upper electrode, and a frequency of 2 MHz and V = 905 V were applied to the lower electrode. The other metal surfaces were set to the ground potential.

3.2. Plasma simulation

The 2D fluid model PHM (plasma hybrid module) of the Pegasus Software Inc. was used for the plasma simulation. PHM is a simulator of the 2D fluid model, and the continuity equation of electrons is expressed by Eq. (1).

$$\begin{eqnarray}&&\frac{\partial {n}_{{\rm{e}}}}{\partial t}+{\rm{\nabla }}\cdot {\Gamma }_{{\rm{e}}}={R}_{{\rm{e}}},\end{eqnarray} \tag{ 1 }$$

where n_e is the electron density, Γ _e is the electron flux, and R_e is the electron generation rate. Similarly, the ion formation rate is given by Eq. (2).

$$\begin{eqnarray}&&\frac{\partial {n}_{{\rm{i}}}}{\partial t}+{\rm{\nabla }}\cdot {\Gamma }_{{\rm{i}}}={R}_{{\rm{i}}},\end{eqnarray} \tag{ 2 }$$

where n_i is the ion density, Γ _i is the ion flux, and R_i is the ion generation rate. We applied the gas reactions shown in Ref. 24 for the Ar gas reaction model.

The amount of ion flux Γ _{i C} flowing into the electrode from the ion flux Γ _i in Eq. (2) was calculated. Secondary electrons are emitted as shown in Fig. 4 by the collision between the positive ion and the electrode. In the simulation, the SEE was calculated as follows. The amount of ion flux Γ _{i C} flowing into the electrode from the ion flux Γ _i in Eq. (2) was calculated. Assuming that the SEE coefficient is γ, the flux amount Γ _{e E} of the electrons emitted was calculated by Eq. (3).

$$\begin{eqnarray}&&{\Gamma }_{{\rm{e}}E}=\gamma {\Gamma }_{{\rm{i}}C}\end{eqnarray} \tag{ 3 }$$

The Γ _eE calculated by Eq. (3) was added to the electron flux Γ _e of Eq. (1). Γ _eE was added so that the total amount of Γ _e increased without considering the emission direction.

The V_dc generated at the upper and lower electrodes was calculated by integrating the inflowing electron flux and ion flux. ²⁴⁾ The V_dc voltages of the upper and lower electrodes were –691 V and –859 V under the condition that there was no secondary electron emission. ²⁴⁾ The SEE coefficient has been reported in several papers, but it depends on the electrode material and surface condition, and the value is not clearly quantified. ^17–19) The voltage difference between the upper electrode and the lower electrode is 168 V, which is relatively small compared to the absolute value of V_dc. Therefore, we assumed that Ar⁺ ions cause these at the upper and lower electrodes with the same coefficient.

3.3. Coupled calculation of plasma and gas flow simulations

NMEM (neutral momentum equation module) of the PEGSUS software Inc. was used to calculate the diffusion and flow of neutral particles. In NEME, the continuity equation of Eq. (4) is solved for determining the distribution of gas flow velocity. ^28–30)

$$\begin{eqnarray}&&\displaystyle \frac{\partial \rho }{\partial t}+\displaystyle \frac{\partial \left(\rho {v}_{r}\right)}{\partial {\rm{r}}}+\displaystyle \frac{\partial \left(\rho {v}_{z}\right)}{\partial {\rm{z}}}=0,\end{eqnarray} \tag{ 4 }$$

where ρ is density, and t is time. ${v}_{r},$ ${v}_{z}$ are the velocities in the r-axis and z-axis, respectively.

Figure 5 shows the calculation flow for coupled calculation of the plasma and gas flow simulation. The gas reaction is calculated by plasma simulation, and the generated particles and neutral particles are reflected in the flow simulation to calculate the gas state. The continuity equation at the time of coupling is expressed by Eq. (5).

$$\begin{eqnarray}&&\frac{\partial {\rho }_{{\rm{a}}}}{\partial t}+\frac{\partial \left({\rho }_{{\rm{a}}}{v}_{r}\right)}{\partial {\rm{r}}}+\frac{\partial \left({\rho }_{{\rm{a}}}{v}_{z}\right)}{\partial {\rm{z}}}={S}_{{\rm{a}}},\end{eqnarray} \tag{ 5 }$$

where ρ_a is the mass density of the a-type particle and S_a is the amount of a-type particles generated and disappeared per unit time and unit volume. After the movement of various particles in the gas flow simulation was calculated, the density of particles generated by plasma was calculated again. The calculations were repeated until the changes in all particle densities converge.

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First, assuming that there was no SEE, the coupled calculation of plasma and gas flow simulation was performed under the same conditions as the experimental conditions in Sect. 2. The calculation result of the electron density is shown in Fig. 6(a). The region where the electron density is low in the vicinity of the upper and lower electrodes corresponds to the sheath region.

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From the results in Fig. 6(a), we extracted the electron density at the center of the probe position. The probe was set at a height of 23 mm from the lower electrode and the height position corresponded to z = 80 mm in the model shown in Fig. 2. Since r = 0 mm is the center of symmetry of the simulation model and is a computationally singular point, we extracted the electron density at a position 2 mm away from the center. Figure 6(b) shows the density distribution along the z-axis with r = 2 mm. The probe is at a height of 23 mm from the lower electrode, which corresponds to z = 80 mm in the model shown in Fig. 4. The electron density at this point was 9.29 × 10¹⁶ m^–3, which was 36.37 % lower than that in the measurement result.

We assumed that the electron density is low because the influence of the SEE is not applied. Therefore, the value of the secondary electron emission coefficient γ was increased by 0.01 in the range of 0 to 0.10 and compared with the experimental results. From the simulation results, the electron density at (r, z) = (2, 80) was extracted. Figure 7 shows the relationship between the secondary electron emission coefficient γ and the electron density. For comparison, the calculation results of the apparatus where Si wafer covered with SiO₂ or polyimide was placed on the lower electrode were also added to Fig. 7. The calculation method and results are described in detail in Sect 5. The electron density in the case of the Si lower electrode increased with the increase of γ, and at γ = 0.06, it was 13.65 × 10¹⁶ m^–3, which is the closest to the experimental result.

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Hagstrum conducted an experiment in which Ar⁺ ions were accelerated to 1000 eV and collided with a Si target. ³¹⁾ On basis of the experiments, it was determined that the SEE coefficient of Si is 0.039. Since the SEE coefficient is also affected by the surface cleanliness of the electrode, we considered that secondary electron emission 0.06 for Si as valid.

A dielectric layer with a thickness of 0.1 mm was added to the surface on the lower electrode to perform plasma simulation with the lower electrodes SiO₂ and polyimide. In Sect. 2, a Si wafer covered with SiO₂ and polyimide was set on the Si lower electrode and the electron densities were measured. In the experiment, the lower electrode is 0.7 mm higher due to the placement of the wafer. However, 0.7 mm is smaller than the 70 mm space between the upper and lower electrodes. We assumed that the thickness of the lower electrode could be negligible. Although the thickness of SiO₂ is 8 μm, the maximum size of the plasma simulation is 200 mm, the shape of the micro order is difficult to be handled in the simulation. Therefore, we added a 0.1 mm dielectric layer on the lower electrode for both cases of polyimide and SiO₂ without increasing the height of the lower electrode.

The dielectric constant of the electric body layer was 4.0 for SiO₂ and 3.3 for polyimide. Since the upper electrode is Si, the SEE coefficient was set to be 0.06. Figure 8 shows the results calculated without the SEE from the lower electrode. Figure 8(a) shows the result that the relative permittivity of the dielectric layer is 4.0 corresponding to SiO₂ and (b) shows the result that the relative permittivity of the dielectric layer is 3.3 corresponding to the polyimide.

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The electron density at the probe measurement position was 8.40 × 10¹⁶ m^–3 for both the SiO₂ and the polyimide electrode. In the experimental results in Sect. 2, the electron densities of the SiO₂ and the polyimide electrodes were 25.9 × 10¹⁶ m^–3 and 20.2 × 10¹⁶ m^–3, respectively. The electron densities of the simulation results are both lower than those in the experiment due to no SEE effect. Then, under the conditions of the SEE coefficient of the upper electrode at 0.06, the values of the SEE coefficient from the SiO₂ and polyimide surfaces were changed, and the electron densities were calculated.

Figure 7 shows the relationship between the SEE coefficient and the electron density. The electron density increases as the SEE coefficient increases. The SEE coefficients, which are equal to those in the experiment, were 0.24 for SiO₂ and 0.22 for polyimide. Figure 9 summarizes the obtained SEE coefficients. Therefore, we estimated that the SEE coefficient of SiO₂ and polyimide is 0.24 and 0.22, respectively. The value is more than 3 times higher than that of 0.06 for Si.

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Kushner adopted the SEE of 0.15 for metal electrodes in the simulation, ²¹⁾ and Szapiro et al. estimated the SEE on carbon as 0.5. ¹⁷⁾ The SEE coefficients of 0.22 for polyimide obtained in this study are close to these reported values. So far many papers on SEE coefficients of SiO₂ have been reported. ^19,32–34) Lee et al. reported that the SEE coefficient was 0.06 when Ar⁺ collided with SiO₂ at 200 V. ¹⁹⁾ Suppose that the SEE coefficient is proportional to the ion acceleration voltage, the SEE coefficient at 1 keV is estimated to be 0.3. Booth et al. ³²⁾ and Horváth et al. ³³⁾ calculated assuming the SEE coefficient of SiO₂ as 0.4. Sobolewski measured the secondary electron emission coefficient in situ in FR-excited plasma and reported that the secondary electron emission coefficient of SiO₂ is 0.72. ³⁴⁾ In this study, the secondary electron emission coefficient of SiO₂ was estimated to be 0.24. The value is close to the value measured by Lee et. al and is smaller than that reported by Booth et al, Horváth et al, and Sobolewski.

Next, we performed plasma simulation under the condition same as the bias power experiments in Sect. 2. The bias power of the lower electrode was changed to investigate the dependence of the SEE coefficient on bias power. The SEE coefficients were obtained using the experiments and simulations as described in Sects. 4 and 5. Figure 10 shows the calculation results of the SEE coefficient and V_dc. The SEE coefficients were almost the same as 0.23 to 0.24 in the range of 300 to 500 W. The change in V_dc is as small as 200 V in the range of the bias power and the SEE was estimated to be almost constant. It is considered that the ion flux incident on the SiO₂ surface increased with the increase of V_dc and the electron density increase.

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Phelps and Petrovic report the SEE coefficients of dirty and clean metals when Ar⁺ ions collide with them. ³⁵⁾ In their experimental results, the SEE coefficient in dirty metal increases with the acceleration voltage. On the other hand, the SEE coefficient of clean metal is almost constant at an acceleration voltage of 2 kV or less. In our experiment, the SEE coefficient of SiO₂ became almost constant regardless of the increase in V_dc. We consider the experiment result of SiO₂ is similar to the result of clean metal.

In the case that the SiO₂ wafer was set on the lower electrode, SiO₂ was sputtered by the collision of Ar⁺ ions. The released SiO₂ into the plasma may be ionized and thus increase the electron density. We compared the amount of flux of SiO₂ emitted by sputtering with the amount of electron flux emitted as secondary electrons. In the experiment in Sect. 2, the sputtering rate of SiO₂ was 114 nm min⁻¹. Since the diameter of the wafer is 4 inches, the wafer area is 8.11 × 10^–1 cm² and the volume sputtered per second is 1.54 × 10^–7 cm^–3 s^–1. The SiO₂ mass converted at a density of 2.20 g cm⁻³ of TEOS oxide film is 3.39 × 10^–7 g s⁻¹, which corresponds to the mole number of 5.64 × 10^–9 mol s⁻¹. Therefore, SiO₂ molecules are released at a rate of 3.40 × 10¹⁵ s⁻¹ from the entire surface of the wafer.

Next, the amount of Ar⁺ ions incident on the wafer per second was calculated. Figure 11 shows the calculation result of Ar⁺ ion Flux under the condition that the SEE coefficient was 0.24 in the case of the SiO₂ wafer. The average Ar⁺ ion flux in the wafer plane was 1.58 × 10¹⁶ cm^–2 s^–1. Multiplying the value by the wafer area of 8.11 × 10^–1 cm², the number of Ar⁺ ions incident on the wafer was 1.28 × 10¹⁶ s⁻¹. The number of the secondary electrons was estimated to be 3.07 × 10¹⁵ s⁻¹ multiplying the number of Ar⁺ ions incident by the SEE coefficient.

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As described above, the SiO₂ sputtered particles emitted from the entire surface of the wafer are 3.40 × 10¹⁵ s⁻¹. The Ar⁺ ion incident on the wafer is 1.28 × 10¹⁶ s⁻¹, and the sputter yield of SiO₂ is 0.27. Oostra measured the sputter yield of Ar⁺ ions on the SiO₂ film and reported that the sputter yield is 5.2 × 10^–2 at the acceleration voltage of 100 eV and 1.2 × 10^–1 at 250 V. ³⁶⁾ Extrapolating the results to 859 V which corresponds to V_dc of the lower electrode, the sputter yield is estimated to be 0.38. The result in our experiment and simulation was somewhat lower than that in the result of Oostra. Table II summarizes the values of incident Ar⁺ ions, sputtered SiO₂, and secondary electrons calculated in Sect 5.

On other hand, generally, since atoms or molecules collided with an electron are ionized with a probability of 1/1000 to 1/100, the number of electrons generated from the ionized SiO₂ is smaller than that of electrons generated by the SEE. Therefore, it can be estimated that the difference in electron density between the experiment and the simulation is due to the effect of secondary electron emission generated by Ar⁺ ions. In addition, the number of secondary electrons and sputtered SiO₂ molecules are almost the same, and it is estimated that SiO₂ sputter emission and secondary electron emission occur at a rate of about a quarter of the incident Ar⁺ ions.

In a dual-frequency pumped Ar plasma, the SSE coefficients of Si, SiO₂, polyimide, and electrodes were estimated using simulations and experiments. Plasma was generated at the upper and lower electrodes of Si, and the electron density was measured. Plasma simulation was performed under the same conditions as in the experiment, and the electron density was calculated by changing the SEE coefficient. The SEE coefficient was calculated so that the electron density of the simulation was close to that of the experimental result, and the SEE coefficient of the Si electrode was estimated to be 0.06.

Next, we covered the Si surface of the lower electrode, covered with SiO₂ and polyimide, and measured the electron density. While keeping the SEE coefficient of the upper electrode constant at 0.06, the SEE coefficient of the lower electrode was changed to obtain the value at which the electron density in the simulation was equal to that measured in the experiment. As a result, the SEE coefficients of the SiO₂ and the polyimide films were estimated to be 0.24 and 0.22, respectively. Moreover, The SEE coefficients of SiO₂ were almost the same in the range between 300 and 500 W of the lower electrode power.

This work was carried out by the joint usage/research program of the center for Low-temperature Plasma Sciences, Nagoya University.