A Simple Model for the Calculation of Plasma Impedance in Atmospheric Radio Frequency Discharges

Transcription

1 Plasma Science an Technology, Vol.16, No.1, Oct. 214 A Simple Moel for the Calculation of Plasma Impeance in Atmospheric Raio Frequency Discharges GE Lei ( ) an ZHANG Yuantao ( ) Shanong Provincial Key Lab of UHV Technology an Gas Discharge Physics, School of Electrical Engineering, Shanong University, Jinan 2561, China Abstract In atmospheric raio-frequency (rf) ischarges, the plasma parameters, such as electron ensity, sheath thickness an sheath voltage, are not easy to be probe experimentally, while the electrical characteristics, such as impeance, resistance an reactance, are relatively convenient to be measure. In this paper we presente a simple theoretical moel erive from the flui escription of generate plasmas without consiering the circuit moel, to investigate the relationship between the plasma impeance an plasma parameters. By introucing a relaxation frequency, the plasma impeance coul be preicte by formulas presente in this stuy, an the mean electron ensity an sheath thickness can also be calculate from the measure or simulate impeance an reactance, respectively. Keywors: raio frequency ischarges, atmospheric ischarges, electron ensity, plasma simulation PACS: 52.8.Pi, Kj, Fv DOI: 1.188/19-63/16/1/5 (Some figures may appear in colour only in the online journal) 1 Introuction In recent years, atmospheric pressure glow ischarge sources have attracte an increasing attention ue to their many avantages, especially without the nee of expensive vacuum equipment [1,2]. Among them atmospheric raio-frequency (rf) ischarges can provie large-volume, homogenous an low-pressure plasmas, which are well accepte as ieal plasma sources [3 5]. Two istinctive ischarge moes in atmospheric rf ischarges, namely α moe an γ moe, have been observe in experimental an computational stuies [6 8]. However, in the γ moe hot an constricte plasmas coul be generate which are not suitable for many applications. Several effective ways, for example increasing the excitation frequency, reucing the electroe gap an pulse moulate, were propose to enhance the ischarge stability to avoi the moe transition [9 11]. On the other sie, a variety of gas mixtures, such as He+O 2, Ar+O 2, an He+O 2 +H 2 O, were applie as the working gas to prouce plenty of reactive species, especially reactive oxygen species, which are believe to play an important role in the arising plasma meicine [12 16]. Compare to the low pressure plasmas, the high flux of reactive species plays an essential role in the applications of atmospheric plasmas [17], an the plasma ensity an sheath thickness are very important parameters to unerstan the interaction of atmospheric plasma an the substrate [17,18]. But how to get the electron ensity an sheath properties is not very easy at atmospheric pressure, because collisions usually take place within a picosecon timescale an the mean free paths are reuce to nanometers scale in atmospheric rf ischarges [18]. However, usually the electrical characteristics are irectly obtaine by electrical measurements, which appears quite significant to get insight into the operating mechanisms of atmospheric plasmas. In rf ischarges the plasma impeance coul be use to monitor an optimize the plasma processes, an the unerstaning of the plasma impeance is of importance for raising the power coupling efficiency to plasmas, these have been iscusse theoretically in etail at low pressure [19]. Then how to extract useful information of plasma parameters from the measure impeance is very helpful in many applications especially at atmospheric pressure. Although efforts have been mae, the relation between electrical characteristics an plasma parameter is still far from fully unerstoo at atmospheric pressure [2 23]. In this paper a one-imensional flui moel is aopte to investigate the relations between impeance an plasma parameters, by means of the introuce relaxation frequency the plasma impeance can be preicte, an the sheath thickness can also be estimate base on the simulation results, as well as the sheath supporte by National Natural Science Founation of China (No ) an Inepenent Innovation Founation of Shanong University of China (No. 212TS67) 924

2 GE Lei et al.: A Simple Moel for the Calculation of Plasma Impeance in Atmospheric RF Discharges voltage. The analytical results euce from the moel are also applie to gain eep insights into the properties of atmospheric rf plasmas. Section 2 gives a brief escription of the flui moel use in this stuy, followe by the computational results an further iscussion, a summary is obtaine in section 4. 2 Description of physical moel The numerical moel use in this stuy is base on a one-imensional flui escription of atmospheric plasmas, an the high pressure operation usually allows the flui moel to simulate the rf ischarges appropriately especially at a relatively large electroe gap [18]. The governing equations are briefly given below for completeness of this paper, an a more etaile escription has been given in Ref. [24]. The ynamic behaviors of ischarge plasmas are escribe by the continuity equations for electrons, ions an neutral species, as well as the Poisson equation N e,i,n t + Γ e,i,n x = S e,i,n (x, t), (1) E (x, t) = e (N i N e ), (2) x ε where N, Γ an S are the ensity, flux an the source term respectively, the inices e, i an n represent electrons, ions an neutral species, respectively, an E enotes the electric fiel in the electroe gap, ε is the vacuum permittivity. Within the iffusion-rift approximation these fluxes can be written as: Γ e,i = µ e,i EN e,i D e,i N e,i x, (3) Γ n = D n N n x, (4) where µ e an µ i are the electron an ion mobility coefficients respectively, D e an D i enote the iffusion coefficients of electron an ion. For neutral species only the iffusion item shoul be consiere. From Eq. (3), the conuctive current ensity in the ischarge region is given by j g (x, t) = e( i Γ i (x, t) Γ e (x, t)), (5) where e is the elementary charge an the sum is over all the positive ions having ensity N i. In this moel, by introucing the total current ensity I(t), only epening on the time, the current balance equation is obtaine an coul also be applie to calculate the electric fiel in the ischarge region ε E (x, t) t = I(t) j g (x, t). (6) Then the total current ensity obtaine from the current balance equation Eq. (6), is expresse as I(t) = ε V (t) + 1 t j g x, (7) 925 where is the electroe spacing. An the total conuction current ensity I c as a function of time is given by I c (t) = 1 j g (x, t) x. (8) To calculate the electron temperature, the electron energy conservation equation is also solve, which is escribe in etail in Ref. [6]. The numerical scheme use in this paper is mainly taken from Ref. [25], an a sinusoial total current ensity is treate as the input parameter. The backgroun gas is pure helium without any impurities, an the reactions in pure helium plasma an the corresponing reaction coefficients are mostly taken from Ref. [6]. In this stuy the electroe spacing is fixe as 2. mm, an the seconary electron emission is also taken into account as a constant coefficient of.3, since this value has only a minor influence on the ischarge characteristics in the α moe [7]. 3 Results an iscussion In the framework of flui escription, the governing equations can be further simplifie to gain more insights into the rf plasmas accoring to the characteristics of rf ischarges at atmospheric pressure [18]. Due to the ominant effects of rift motion riven by the electric fiel in the ischarge region, at atmospheric pressure the total conuction current I c can be further written by ignoring the iffusion terms I c (t) = 1 1 = 1 eµ en ξ (t) ( ) N i e µ i EN i D i x + µ N e een e D e x x (9) e (µ i EN i + µ e EN e ) x (1) Ex (11) = eµ e N ξ (t) V (t). (12) From Eq. (1) to Eq. (11) the mean value theorem for integration is applie, where N ξ (t) represents a electron ensity at a specific position which can not be clearly ientifie in accorance with the mean value theorem for integration. However, it shoul be mentione that uner some ischarge conitions the iffusion items may play an important role in the sheath region ue to the sharp ensity graient for electrons an ions, but this oes not seriously affect the iscussion given here, as inferre from the computational comparison in Ref. [24]. An the simulation ata presente in this stuy is obtaine by solving the complete control equation set with the iffusion items consiere, the simplifie form without the iffusion terms is use only

3 to investigate the ischarge characteristics analytically an qualitatively. Consiering the final expression Eq. (12) of I c, the equivalent conuctivity σ p of ischarge plasma can be efine as σ p (t) = eµ e N ξ (t), (13) thus the plasma relaxation time τ p (t), which implies the time scale for the electrons generate in the gap to arrive at the electroes, an corresponing relaxation frequency f p shoul be given as τ p (t) = ε σ p (14) = ε, (15) eµ e N ξ 1 f p (t) = (16) 2πτ p (t) = 1 eµ e N ξ, (17) 2πε the physics unerlying these parameters have been explaine in a more etaile way in Ref. [25]. To compute the relaxation frequency, by combining Eqs. (12) an (17) f p can be calculate from f p (t) = I c(t) 2πε V (t), (18) here I c (t) is obtaine from Eq. (9), not Eq. (1). The relaxation frequency f p is epicte in Fig. 1 at a given current ensity of 5 ma/cm 2, which also shows the total current ensity versus rf cycle. In an entire cycle, two separate branches of f p curve can be observe, an the whole curve is iscontinuous at the instant when the applie voltage gets to nearly zero. For a given branch of f p, the relaxation frequency istributes symmetrically in terms of the zero point, an the perio of relaxation frequency is not exactly the same as that of the current ensity. Usually the relaxation frequency is comparable with the riving frequency of MHz, inicate in this figure by a ot line. But it shoul be note that the maximum value of relaxation frequency may not be compute very accurately because it is usually achieve at the instant when the applie voltage is very close to the zero point, as seen in Fig. 1 an Eq. (18). By means of the relaxation frequency f p, from Eq. (7) the total current ensity can be further written as I(t) = ε V (t) + 2πε t f pv (t). (19) Usually the total current ensity is treate as the input parameter with the form of a normal sinusoial function [4,6], namely I(t) = I sin (2πft), where I is the amplitue of total current ensity. This is because before an after the α-γ transition for a constant applie voltage multiple current ensities coul be observe for a given applie voltage [3,7]. Then, the total current ensity can be given symmetrically I(t) = 2πV ε (f cos (2πft) + f p sin (2πft)), (2) Plasma Science an Technology, Vol.16, No.1, Oct which clearly shows that f p represents the characteristic frequency of generate atmospheric plasmas, while f enotes the characteristic frequency of the power source. Fig.1 Relaxation frequency an total current ensity as a function of rf cycle From Eq. (19), the erivative of applie voltage is V (t) t = ε I(t) 2πf p V (t) (21) = ε I sin (2πft) 2πf p V (t), (22) the solution of this ifferential equation, that s the applie voltage V (t), is obtaine an written as V (t) = 2πε I f 2 p + f 2 sin (2πft ϕ) + I 2πf 2πε fp 2 + f 2 exp ( 2πf pt), (23) however, one can notice that the secon item of righthan sie in Eq. (23) ecays very rapily because the relaxation frequency f p is usually very large as clearly shown in Fig. 1. Thus the applie voltage actually shoul be expresse as V (t) = 2πε I f 2 p + f 2 sin (2πft ϕ), (24) with the amplitue of V (t) given by V = 2πε I f 2 p + f 2, (25) where ϕ is the phase angle between the applie voltage an the total current ensity, represente by ( ) f ϕ = arctan, (26) f p which epens on the ratio of the excitation frequency f an the relaxation frequency f p. From Eqs. (24) an (25), the applie voltage is not a normal sinusoial function any more, as its amplitue is not a constant value

4 GE Lei et al.: A Simple Moel for the Calculation of Plasma Impeance in Atmospheric RF Discharges after the ignition. Fig. 2 gives the calculate phase angle in two rf cycles at a current ensity of 5 ma/cm 2. The phase angle is not a constant uring a whole cycle, varying from 9 to 9. The perio of phase angle is not the same as that of the current ensity. Like the relaxation frequency, the phase angle curve in a cycle is separate to two branches at the instant when the relaxation frequency is nearly zero, an it approaches 9 when the conition f p f is satisfie. Usually the values of phase angle can be irectly measure in experiments at the instants when the current ensity or applie voltage reaches the amplitues. peaks. A summary of the plasma impeance with increasing current ensity is shown in Fig. 4, where the resistance an reactance are isplaye as well. Before the ischarge is ignite, the whole ischarge system is capacitive an the impeance is almost equal to the reactance. But after the breakown, at the onset of the α moe a large loa impeance can be observe with only a small current ensity going through the electroe gap. With more power couple into the plasmas plenty of electrons an ions are generate in the ischarge region an a large conuction current ensity forms across the gap, resulting in a ecrease in the plasma impeance, as seen in Fig. 4. In aition, the resistance increases an eventually becomes very close to the impeance, even significantly larger than the reactance as the current ensity rises, inicating that the ischarge has become more resistive. Fig.2 Phase angle an total current ensity as a function of rf cycle As more powers are couple into the ischarge region, both the ischarge current ensity an applie voltage increase. In terms of Eq. (18) the relaxation frequency at the instant when the total current ensity gets to the peak value, increases. This can partly be unerstoo by the enhancement of the electron ensity from the efinition of Eq. (17). From Eq. (26) the phase angle shoul reuce with relaxation frequency, given in Fig. 3, where the phase angle ecreases rapily initially after the ignition, then changes moerately, which well agrees qualitatively with the experimental measurements [5,21]. Fig.3 Phase angle an relaxation frequency as a function of RMS current ensity From the simulation results, the amplitue of applie voltage an current ensity, namely the voltage an current characteristics, can be obtaine, as well as the phase angle at the time when the current ensity Fig.4 Impeance, resistance, reactance an preicte impeance as a function of RMS current ensity Using Eq. (25), the plasma impeance can be irectly given by Z = V I (27) = 2πε 1 f 2 p + f 2 (28) 2πε 1 f p, (29) thus, the plasma impeance coul be preicte from the values of relaxation frequency an frequency base on Eq. (28), especially when the ischarge is ignite with the conition f p f satisfie, the plasma impeance is irectly relate with relaxation frequency. An the preicte impeance is also capture in Fig. 4 as a ash line, where it shows a reasonable agreement with the simulate curve obtaine by solving the whole control equation set. After the ignition with rising electron ensity, Eq. (29) also coul be use to infer the impeances when the conition f p f is satisfie, as given in Fig. 1. The impeance is almost unchange before the breakown, then ecreases with the current ensity, an the increase in relaxation frequency as the current ensity rises shoul be response to the reuction in the plasma impeance. The preicte value is usually smaller than the impeance after the ignition. 927

5 Plasma Science an Technology, Vol.16, No.1, Oct. 214 Usually, the electron ensity generate in atmospheric ischarges is not easy to be measure experimentally, in particular the electron ensity lower than 1 12 cm 3 is ifficult to be irectly measure, compare to the case of low pressure [17]. As we know, the impeance can be probe irectly in experiments, thus how to extract the useful information of electron ensity from impeance becomes an interesting problem, an at low pressure many valuable results have been reveale [19]. As the riving frequency increases uner a constant power ensity, the applie voltage ecreases while the current ensity increases [24], thus the plasma impeance can be inferre to rise significantly. The simulate impeance is given in Fig. 5 at a constant power ensity of 5 W/cm 3 as well as the resistance, an obviously the resistance is a little smaller but very close to the impeance. On the other han, the computational results in Ref. [24] also suggest the relaxation frequency to increase with the excitation frequency. Then, by combination of Eqs. (17) an (29), the electron ensity can be obtaine from N ξ = eµ e Z, (3) that is to say, the electron ensity coul be inferre from the plasma impeance which can be measure in experiments irectly, although here N ξ may not be ientifie clearly ue to its uncertainty nature, it coul be treate as the mean electron ensity [24]. Fig. 5 gives the mean electron ensity from the simulation by solving the whole governing equation set, an the preicte value from Eq. (3) using the simulate plasma impeance shows a reasonable agreement at a constant power of 5 W/cm 3 with increasing the excitation frequency. Fig.6 Reactance, sheath thickness an preicte sheath thickness as a function of frequency at a cosntant power ensity of 5 W/cm 3 In Ref. [3] a capacitive moel is propose to iscuss the sheath characters in atmospheric rf ischarges, the sheath capacitance C is estimate by C = 1.52ε A s, (31) where A is the area of the electroe, here for a oneimensional moel A shoul be an unit area, s is the sheath thickness. The sheath thickness is usually obtaine from the simulation ata by estimating the istance over which the electric fiel reuces from its peak value to less than a given percentage of the peak electric fiel [7], then the sheath voltage coul be compute by integrating the electric fiel within the sheath region. The absolute value of reactance is usually given by X = 1 2πfC, (32) thus, in terms of the reactance X, the sheath thickness s is represente as s = πfε X, (33) consequently, with the measure or simulate reactance X the sheath thickness coul be approximately estimate, which is epicte in Fig. 6 as well as the irect simulate values from the computational coe, an a reasonable agreement can be observe. By means of the simulate values of reactance an total current ensity the sheath voltage can also be inferre, given by V s = XI. (34) Fig.5 Impeance, mean electron ensity an preicte electron ensity as a function of frequency at a constant power ensity of 5 W/cm 3 On the other sie, the reactance usually inicates the properties of sheath region in rf ischarges at low pressure [22,23]. From the simulate impeance an phase angle, the reactance at 5 W/cm 3 as a function of excitation frequency is presente in Fig. 6, where the reactance ecreases rapily initially, then rops moestly, this suggests that the sheath shrinks as the excitation frequency increases. Consequently, with the measure or simulate impeance the sheath properties can be approximately obtaine. 4 Conclusions The plasma impeance plays an important role in monitoring the plasma processes. In this paper we presente a simple analytical moel erive from the flui escription of atmospheric plasmas, without consiering the circuit moel which is always use in the stuy 928

6 GE Lei et al.: A Simple Moel for the Calculation of Plasma Impeance in Atmospheric RF Discharges on plasma impeance, to investigate the relation between electrical characteristics an plasma parameters. The key feature of this analytical moel is the introuction of relaxation frequency f p, which links the electron ensity an excitation frequency. By means of the irect measure or simulate impeance an phase angle, the mean electron ensity, sheath thickness an sheath voltage can be approximately preicte from the present analytical moel, this is very helpful to gain insights into the plasmas characteristics at atmospheric pressure without complicate plasma iagnosis, although the present analytical moel oes not provie an effective solution to accurately calculate the relaxation frequency. References 1 Roth J R. 1995, Inustrial Plasma Engineering, Principles Vol. 1. Institute of Physics, Bristol 2 Friman G, Frieman G, Gutsol A, et al. 28, Plasma Process. Polym., 5: 53 3 Park J, Henins I, Herrmann H W, et al. 21, J. Appl. Phys., 89: 2 4 Balcon N, Hagelaar G J M an Boeuf J P. 28, IEEE Trans. Plasma Sci., 36: Moon S Y, Phee J K, Kim D B, et al. 26, Phys. Plasmas, 13: Yuan X an Raja L L. 23, IEEE Trans. Plasma Sci., 31: Shi J J an Kong M G. 25, J. Appl. Phys., 97: Lou J an Zhang Y T. 213, IEEE Trans. Plasma Sci., 42: Moon S Y, Kim D B, Gweon B, et al. 28, Appl. Phys. Lett., 93: Walsh J L, Iza F, an Kong M G. 28, Appl. Phys. Lett., 93: Zhang Y T, Li Q Q, Lou J, et al. 21, Appl. Phys. Lett., 97: Knake N, Niemi K, Reuter S, et al. 28, Appl. Phys. Lett., 93: Li S Z, Lim J P, Kang J G, et al. 26, Phys. Plasmas, 13: Park G Y, Hong Y J, Lee H W, et al. 21, Plasma Process. Polym, 7: He J an Zhang Y T. 212, Plasma Process. Polym, 9: Zhang Y T an He J. 213, Phys. Plasmas, 2: Iza F, Kim G J, Lee S M, et al. 28, Plasma Process. Polym., 5: Iza F, Lee J K an Kong M G. 27, Phys. Rev. Lett., 99: Lieberman M A an Lichtenberg A J. 25, Principles of Plasma Discharges an Materials Processing. 2n en, Wiley, New York 2 Bera K, Chen C A, an Vitello P. 22, IEEE Trans. Plasma Sci., 3: Zhu W C, Wang B R, Yao Z X, et al. 25, J. Phys. D: Appl. Phys., 38: Byrns B, Wooten D, Linsay A, et al. 212, J. Phys. D: Appl. Phys., 45: Overzet L J, Jung D, Manra M A, et al. 21, Eur. Phys. J. D, 6: Zhang Y T an Cui S Y. 211, Phys. Plasmas., 18: Zhang Y T, Wang D Z an Kong M G J. 26, Appl. Phys., 1: 6334 (Manuscript receive 2 November 213) (Manuscript accepte 17 February 214) aress of corresponing author ZHANG Yuantao: ytzhang@su.eu.cn 929