Electron Transport Behavior in a Mirror Magnetic Field and a Non-uniform Electric Field

Transcription

1 Commun. Theor. Phys. (Beijing, China) 35 (2001) pp c International Academic Publishers Vol. 35, No. 2, February 15, 2001 Electron Transport Behavior in a Mirror Magnetic Field and a Non-uniform Electric Field LIU Yan-Hong, LIU Zu-Li, YAO Kai-Lun, WEI He-Lin and LIU Hong-Xiang Department of Physics, Huazhong University of Science and Technology, Wuhan , China (Received January 28, 2000) Abstract The behaviors of electrons as they move under the influence of mirror magnetic field and non-uniform electric field in a positive column of helium radio frequency gas discharge are studied by Monte Carlo simulation. Some types of collisions (elastic, excitation and ionization collisions) are considered. Graphs showing how the electron density, electron energy, electron current density, collision rate and the electron-scattering angular distribution are affected by the mirror magnetic field are presented. The results indicate that the mirror magnetic field can control the electron transport behavior in the positive column. In the presence of the mirror magnetic field, the electrons are restricted in the middle part of the positive column, and the electron density is greatly increased. The electron collision rate and the electron current density are enhanced in the middle region, and the electron-scattering angles are extended by the mirror magnetic fields. These results are in good agreement with experimental results. PACS numbers: Ez, Cw Key words: electron transport behavior, plasma, mirror magnetic field 1 Introduction Glow discharges between parallel-plate electrodes are widely used in many microelectronic applications, such as etching and thin film deposition. Some experimental results have shown that in glow discharge plasma the electron transport behaviors can be controlled by a magnetic field, and the parameters of electrons were affected by the magnetic field. [1 7] Heylen and Dargen [2] reported that the cathode sheath parameters were changed by a magnetic field, and the ionization number also depends on the magnetic field. Bonin and Mason et al. [5] found that the fluorescence and ionization processes in glow discharge plasma were enhanced by a magnetic field. These results are useful for the discharge plasma applications, such as the enhancement of the fluorescence light used in lamp industry. Theoretically, the Monte Carlo simulation (MCS) is a tool to study the effect of the magnetic field on the electron transport behavior in discharge plasma. [8,9] Razdan et al. [8] investigated the discharge instability in uniform magnetic field and non-uniform electric field using MCS. Labahn et al. [9] applied a similar model to study the electron energy distribution in helium discharge. In these models, the uniform magnetic fields and the oneor two-dimensional MCS were used. However, experimental results [10] showed that the plasma transport processes could be controlled by a mirror magnetic field, and the conductance of the films deposited by plasma chemical vapor deposition (PCVD) technique is enhanced by the mirror magnetic field. The effects of mirror magnetic field on the electron transport behavior in a non-uniform electric field have not been studied theoretically, which is the aim of this work. In the present paper, the behaviors of electrons as they move under the influence of mirror magnetic field and uniform electric field in a positive column of helium radio frequency (rf) gas discharge are studied by Monte Carlo method. Graphs showing how the electron density, electron energy, electron current density, electron collision rate and the electron-scattering angular distribution are affected by the mirror magnetic fields are presented and discussed. This paper is divided into four sections. The mirror magnetic field, the electric field, and the electron motion will be described in Sec. 2. The electron density, electron energy, electron current density, rate of collision between the electrons and the neutral-particles, and the electron-scattering angular distribution will be presented and discussed in Sec. 3. The conclusions are given in Sec Monte Carlo Simulation The purpose of this paper is mainly to study the effects of mirror magnetic field on the electron transport behavior in a positive column of helium rf gas discharge. Treating electron motion using the three-dimensional MCS method includes three processes: (i) Following each electron s trajectory until it reaches a collision position or the boundary of the positive column. (ii) Treating the electron neutralatom collision process and determining the electron motion parameters at different positions. (iii) Summing the results for all the test electrons. The experimental apparatus in our simulation is shown schematically in Fig. 1. The two coils circle the discharge tube. The midplane between the two coils overlaps the midplane of the positive column as shown in Fig. 1. The The project supported by National Natural Science Foundation of China under Grant No

2 208 LIU Yan-Hong, LIU Zu-Li, YAO Kai-Lun, WEI He-Lin and LIU Hong-Xiang Vol. 35 distance between two coils is L1, and the radius of the coils is R1. The length and the radius of the positive column are L0 and R0, respectively. Fig. 1 Schematics of experimental apparatus in which 1, 2, 3 and 4 denote the discharge tube, the positive column, first coil and second coil, respectively. The a and b denote the a-plane and the b-plane of the boundary of the positive column. 2.1 Mirror Magnetic Field Two galvanized coils produce the mirror magnetic fields. The coil currents are equal, and the current directions are the same; the radius of every coil is 6.0 cm; the coil current is 1.0 A; the distance between two coils are different for different mirror magnetic fields, which is equal to 3.0 cm, 5.0 cm, 9.0 cm and 13.0 cm, respectively. The mirror magnetic field intensities are calculated by using B = (µ 0 /4π) (L 1) (I 1d l 1 r 12 /r 2 12) (where I 1 is the coil current, r 12 is the distance between the current element I 1 d l 1 and the objective point). The magnetic field distributions are similar, but the ratios η = B max /B min (where B max and B min are the largest intensity at the center of the coils and the lowest intensity at the midpoint between two coils, respectively) are different for different distances between two coils. The mirror magnetic field distributions are shown in Fig. 2 with the coil distance 13.0 cm. Figures 2a and 2b are the longitudinal and radial intensities of the mirror magnetic field. Figure 2c gives the general mirror magnetic field intensities. B-axis denotes the intensity, Z-axis and R-axis denote the longitudinal and radial positions between two coils, where z = 0.5 corresponds to the midpoint between the coils. These results are in good agreement with the experimental metrical results. [10] From Fig. 2c one can see that at center of the two coils the intensity of mirror magnetic field is the highest, at the midpoint between the two coils the intensity is the lowest. With increasing distance from midpoint to the coils the magnetic field intensity gradually increases. When the coil distance is 13.0 cm, the ratio η is We also calculate the other ratios η when the coil distances are 3.0 cm, 5.0 cm and 9.0 cm, respectively. It is found that they are equal to 1.11, 1.32 and 2.19, respectively, which shows that the larger value of the coil distance, the larger ratio η of the mirror magnetic field. Fig. 2 Mirror magnetic field distribution for the ratio η = (a), (b) and (c) are the longitudinal, radial and the general intensities of the mirror magnetic field, respectively. 2.2 Electric Field and Electron Motion According to Ref. [11], we assume that the electric field varies with time in the positive column of rf gas discharge and the electric field can be given by E(t) = E 0 sin(2πv rf t), (1) where E 0 is the electric field vector, v rf and t are the rf frequency and time, respectively. The electric field vector E 0 is parallel to the symmetrical axis of the positive column. The symmetrical axis of the mirror magnetic field is parallel to E 0. In the electromagnetic field, the total force acting on an electron is given by F = q E(t) + q ν B, (2) where q is the electron charge, B is the mirror magnetic field evaluated from Fig. 2, and ν is the electron velocity. 2.3 Collision Process Figure 3 shows the phase-space coordinate system used in the calculation. θ 0 and θ 1 are the angles between the

3 No. 2 Electron Transport Behavior in a Mirror Magnetic Field and a Non-uniform Electric Field 209 electron velocity vector and electric field before and after collisions, respectively. γ is the electron scattering angle, and φ is the azimuthal angle. The electron path is determined by θ, γ and φ. Three types of collisions for electrons are taken into account in our model: elastic, excitation and ionization collisions. The cross sections for the elastic, excitation and ionization collisions are σ el (ε), σ ex (ε) and σ j (ε), respectively. The semi-empirical expressions of the cross sections are taken from Ref. [12]. The total collision cross section is σ t (ε) = σ el (ε) + σ ex (ε) + σ i (ε). (3) We can split the total cross section into its component cross sections and the relative probabilities are determined by these components, P k = σ k (ε)/σ t (ε), k = el,ex,i. (4) The sum of these probabilities is unity. One can divide the interval [0,1] into segments of lengths corresponding to these probabilities. We take a random number R (uniformly distributed in [0,1]) to determine the type of collision. The electron neutral-atom collision frequency v(ε) is given by v(ε) = Nσ t (ε)ν, (5) where N is the gas density, ν is the electron velocity, σ t (ε) is the total collision cross section. The flying time between collisions is τ(ε) = 1/v(ε)log(R 1 ), (6) R i (i = 1,2,,n) is a random number in (0,1). After a random collision occurs, the azimuthal angle of the electron is φ = 2πR 2. (7) The electron energy after elastic collision is ε 1 = ε[1 (2m e /M)(1 cos γ)], (8) where ε is the electron energy before collision, m e and M are the electron and neutral-particle masses, respectively. γ is the electron scattering angle, which is given by γ = cos 1 (1 2R 3 ). (9) The electron energy after excitation collision is given by ε 1 = ε ε ex, (10) where ε ex is the excitation potential energy of helium. The scattering angle is γ = πr 4. (11) When the ionization collision occurs, the scatteredelectron energy ε 1 and the second-ejected-electron energy ε 2 are randomly distributed and can be written as ε 1 = (ε ε i )R 5, ε 2 = ε ε i ε 1, (12) where ε i is the ionization potential energy of helium. The scattering angles γ 1, γ 2 are γ 1 = cos 1 {[ε 1/(ε ε i ) 2 ] 1/2 }, γ 2 = cos 1 {[ε 2/(ε ε i ) 2 ] 1/2 }. (13) In Eqs (13) we have made two assumptions: 1) the incident, ejected and scattered electron velocities are coplanar; 2) the scattered and ejected electron velocities are perpendicular. Fig. 3 Electron phase-space coordinate system in collision point where θ is the angle between the electron velocity vector and the symmetrical axis (Z-axis) of the positive column. γ and φ are the electron-scattering angle and azimuthal angle, respectively. The direction of motion for the electron after collision is given by θ 1 = cos 1 (cos θ 0 cos γ + sinθ 0 cos γ sinφ), (14) where θ 1 and θ 0 are the angles made by the electron velocity vector with the electric field after and before collisions, respectively. 3 Simulation Results and Discussions In our model, the electrons start running from different positions at the a-plane of the positive column with different directions and initial energies. The initial directions are chosen randomly by a random number R. According to Ref. [13], we assume that the electron initial energy is uniformly distributed in ev, the neutral particle density is approximately cm 3, the length (L0) and the radius (R0) of the positive column are equal to 4.0 cm and 5.0 cm, the electric field intensity amplitude E 0 is equal to 300 V/m, and the rf frequency is 1.0 MHz. Good statistics has been obtained for 6000 electrons. The electron density, electron energy, electron current density, rate of collision between electron and neutralatoms, and the electron-scattering angular distribution are investigated below for different mirror magnetic fields in a positive column of helium rf gas discharge. 3.1 Spatial Distribution of Electron Density The spatial distributions of electron density are shown in Fig. 4 for different mirror magnetic fields. The ratios η are equal to 1.32 and 3.92, respectively.

4 210 LIU Yan-Hong, LIU Zu-Li, YAO Kai-Lun, WEI He-Lin and LIU Hong-Xiang Vol. 35 From Fig. 4, one can see that when the mirror magnetic field does not exist in the positive column, the electron density is very low, and the distribution of electron density is relatively uniform. When the mirror magnetic field exists in the column, the electron density is increased greatly and a sharp peak is formed near the position z 0.5L0. Especially, when the ratio is 3.92, the peak of electron density is very obvious. These results are due to the effect of the mirror magnetic field on the electron motion process. According to the conservation of electron magnetic moment µ = W /B and electron kinetic energy, when electron moves from the midpoint z = 0.5L0 to the coil, the intensity of mirror magnetic field is gradually increased, and the electron transverse kinetic energy W is also increased, so the electron longitudinal kinetic energy W is continually reduced. When the electron longitudinal velocity ν is zero, the electrons cannot continually go forward, and will be reflected by force F = µ B. Similarly, when the electron longitudinal velocity is reduced to zero again in the strong magnetic field region, the electrons will also be reflected, so the electrons go forth and back in the positive column, and are restricted in the middle region of the positive column. The longer the time of electron motion in the middle part of the positive column is, the higher the electron density is in the region, and a sharp peak of electron density is formed at the position z 0.5L0. Especially, when the ratio is 3.92, the number of electrons that are restricted in the region is increased by the effect of mirror magnetic field, so the electron density is increased greatly and the peak of electron density is very obvious. These results are in good agreement with the experimental results. [10] Fig. 4 Spatial distributions of electron density. (a), (b) and (c) denote the zero of mirror magnetic field and the ratios η of 1.32 and 3.92, respectively. From the above discussions, we can predict that the electrons in a positive column of rf gas discharge can be restricted in a small region of the positive column, and the electron density is also increased by a mirror magnetic field. These results are useful for the plasma applications, such as to restrict the electrons in a small region and to increase the electron density in experiments. 3.2 Rate of Collision Between Electrons and Neutral Particles The rates of collisions (elastic, excitation and ionization collisions) between electrons and neutral-particles are enhanced by the mirror magnetic field. For example, the rates of elastic collision between electrons and neutralparticles are shown in Fig. 5 for different mirror magnetic fields. The ratios η are also equal to 1.32 and 3.92, respectively as in subsection 3.1. Fig. 5 Rate of elastic collision between electrons and neutral-particles. The dashed line, solid line and line with open circles denote the zero of mirror magnetic field and the ratios η of 1.32 and 3.92, respectively.

5 No. 2 Electron Transport Behavior in a Mirror Magnetic Field and a Non-uniform Electric Field 211 It is evident that when the mirror magnetic field does not exist in the positive column, the elastic collision rate is very low. When the mirror magnetic field exists in the region, the collision rates are increased greatly. Especially, when the ratio η is 3.92, the elastic collision rate is very high and a sharp peak is formed at the position z 0.5L0. These results are due to the mirror magnetic field effect on the electron motion process. From the above discussions, when the mirror magnetic field exists in the positive column, the electron density is increased greatly in the region near z = 0.5L0, so the number of collisions between electrons and neutral-particles in the region is also increased, and this is why at the position z 0.5L0 a sharp peak is formed. Fig. 6 Electron current density. The dashed line, solid line and line with open circles denote the zero of mirror magnetic field and the ratios η of 1.32 and 3.92, respectively. 3.3 Distribution of Electron Current Density The distributions of electron current density are shown in Fig. 6 for different mirror magnetic fields. The ratios η are equal to 1.32 and 3.92, respectively as stated above. From Fig. 6, one can see that when the mirror magnetic field does not exist in the positive column, the distribution of electron current density is basically uniform along the Z-axis. When the mirror magnetic field exists in the region, a peak of electron current density is formed at the position z 0.5L0, and in other positions the electron current densities are reduced. Especially, when the ratio is 3.92, the peak of electron current density at position z 0.5L0 is very high, and in other positions the current densities are low. These results are also related to the effects of mirror magnetic field on the electron transport process. We know that when the mirror magnetic field exists in the positive column, most of the electrons are restricted in the middle part of the positive column, and the electron density in the region is increased, so the electron current density near the position z 0.5L0 is enhanced greatly by the mirror magnetic field. 3.4 Spatial Distribution of Average Electron Energy The spatial distributions of electron energy are shown in Fig. 7 for different mirror magnetic fields. The ratios η are equal to 1.32 and 3.92, respectively. Fig. 7 Spatial distributions of electron energy. (a), (b) and (c) denote the zero of mirror magnetic field and the ratios η of 1.32 and 3.92, respectively. It is evident that when the mirror magnetic field does not exist in the positive column, with increasing of distance from the initiative point z = 0.0L0 the electron energy is increased continually. When the mirror magnetic field exists in the region, the energy distribution deviates from the normal distribution. When the ratio η is 1.32, at region z > 0.2L0 and 0.9R0 > R > 0.2R0 the electron energy is increased and a sharp peak of the distribution is formed; when the ratio is 3.92, the energy distribution becomes disordered. These results are related to the effect of mirror magnetic field. When the mirror magnetic field exists in the positive column, the electron density is increased greatly in the region z 0.5L0 and R < 0.2R0, the number of collisions between electrons and neutral particles is increased in the region, and the electron energy can be reduced by the non-elastic collisions between

6 212 LIU Yan-Hong, LIU Zu-Li, YAO Kai-Lun, WEI He-Lin and LIU Hong-Xiang Vol. 35 electrons and neutral atoms, so in the region R < 0.2R0 the electron energy is low. On the other hand, during the electron motion process, the motion direction of electron is changed by the frequent collisions, and the energy of electron can be increased or reduced by the electric field. Therefore, the electron energy distribution deviates from the normal distribution and becomes disordered when the ratio η is Electron-Scattering Angular Distributions at the End of the Positive Column The electron-scattering angular distributions at the end of the positive column z = 1.0L0 are shown in Fig. 8. The ratios η are equal to 1.32 and 3.92, respectively. The angles are made by the electron velocity vector and the symmetrical axis of the positive column (Z-axis). Fig. 8 Electron-scattering angular distributions at the end of the positive column. The dashed line, solid line and line with open circles denote the zero of mirror magnetic field and the ratios η of 1.32 and 3.92, respectively. From Fig. 8 one can see that when the mirror magnetic field does not exist in the positive column, the angles are distributed within 15. When the mirror magnetic fields exist in the region, the angles become larger. When the ratio η is 1.32, a sharp peak is formed at 60 ; when the ratio is 3.92, the sharp peak is formed at 85. As discussed above, when the mirror magnetic field exists in the positive column, the electron motion direction can be continuously changed by the enhanced collisions, and so the electron-scattering angles become larger, and the sharp peak of angular distributions is moved to a larger angle direction. 4 Conclusion The effects of mirror magnetic field on the electron transport behaviors in a positive column of helium rf gas discharge are studied by the three-dimensional MCS technique. Some types of collisions (elastic, excitation and ionization collisions) are considered. The electron free-flying time between collisions is determined by the electron neutral-particle collision frequency. The symmetrical axis of the mirror magnetic field is parallel to the symmetrical axis of the positive column (Z-axis). The electron density, electron energy, electron current density, rate of collision between electrons and the neutral particles, and the electron-scattering angular distribution at the end of the positive column are calculated for different magnetic fields in this paper. The results indicate that the electron transport process can be controlled by the mirror magnetic field in a positive column of rf gas discharge. The electrons are restricted in the middle part of the positive column. The electron density is increased, and the electron current density, the rate of collision between electrons and neutral particles are also enhanced by the mirror magnetic fields in the middle region of the positive column. On the other hand, the electron energy distribution deviates from the normal one, and the electron-scattering angles are extended by the effect of mirror magnetic fields on electron motion process. These results are useful for the plasma applications, such as to restrict the electrons in a small region and to obtain high electron density in experiments by using suitable mirror magnetic fields. References [1] H.N. Fukumasa, H. Nation and S. Kakiyama, J. Appl. Phys. 74 (1993) 848. [2] A.F.D. Heylen and L.L. Dargen, Int. J. Electron 35 (1973) 433. [3] G. Jellison and C.R. Parsons, Phys. Fluids 26 (1983) 117. [4] T.M. Zhou, L.-Z. Wang, D.D. Hollister, et al., J. Illum. Eng. Soc. 16 (1987) 176. [5] K.D. Bonin and T.G. Mason, Phys. Rev. A43 (1990) [6] R.G. Greaves, D.A. Boy, J.A. Antoniades and R.F. Ellis, Phys. Rev. Lett. 64 (1990) 886. [7] R.G. Greaves and D.A. Boy, Phys. Rev. A43 (1990) [8] R. Razdan, C.E. Capjack and H.J.J. Seguin, J. Appl. Phys. 57 (1985) [9] A.H. Labahn, C.E. Capjack and H.J.J. Seguin, J. Appl. Phys. 68 (1990) [10] Y.H. LIU and Z.L. LIU, Transaction of Sensor Technology 10 (1997) 15. [11] M.J. Kushner, IEEE Transactions on Plasma Science 14 (1986) 188. [12] Z.L. LIU and H.L. WEI, Physica A215 (1995) 283. [13] F.F. Yong and C.H.J. Wu, IEEE Transactions on Plasma Science 21 (1993) 312.