Physics Letters A. Theoretical model for study of the voltage current curve of a Langmuir-probe used in the hot region of the ECR plasma

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1 Physics Letters A 372 (2008) Contents lists available at ScienceDirect Physics Letters A Theoretical model for study of the voltage current curve of a Langmuir-probe used in the hot region of the ECR plasma L. Kenéz a,,j.karácsony b,a.derzsi b,c,s.biri d a Sapientia Hungarian University of Transylvania, Târgu-Mureş/Corunca, RO , Şos. Sighişoarei Nr. 1C, Romania b Babeş-Bolyai University, Cluj-Napoca, Street M. Kogălniceanu Nr. 1, , Romania c Research Institute for Solid State Physics and Optics of the Hungarian Academy of Sciences, Konkoly-Thege út 29-33, H-1121 Budapest, Hungary d Institute of Nuclear Research (ATOMKI), Bem tér 18/c, H-4026 Debrecen, Hungary article info abstract Article history: Received 10 April 2008 Accepted 25 May 2008 Available online 29 May 2008 Communicated by F. Porcelli PACS: Ni m Keywords: ECRIS Plasma diagnostics Langmuir-probe Simple and emitting probe In the last years the ATOMKI-ECRIS group started a local plasma diagnostics research project, to adapt the probe to the ECR plasma conditions. Until now we made progress in the study of the cold plasma region. The results has been reported in e.g. [L. Kenéz, S. Biri, J. Karácsony, A. Valek, Nucl. Instrum. Methods Phys. Res. B 187 (2) (2002) 249; L. Kenéz, S. Biri, J. Karácsony, A. Valek, T. Nakagawa, K.E. Stiebing, V. Mironov, Rev. Sci. Instrum. 73 (2) (2002) 617]. In this Letter we make a step further report the first experiments carried out in the hot ECR plasma. We used a simple probe inserted in the hot resonant plasma. We point out that this probe works as emitting probe. We developed a theoretical model to explain the unusual shaped voltage current characteristics and tested its validity using computational study of the presented theory Elsevier B.V. All rights reserved. 1. Introduction Among the various diagnostics methods, Langmuir-probes are the most important ones available for local plasma diagnostics. Various experimental configurations were developed, e.g. cylindrical-probes, plane-probes, double-probes, emitting-probes, etc. [1]. Every configuration have its own theory, and depending on the plasma conditions are available to determine different plasma parameters, e.g. electron density, electron temperature (energy), plasma potential, etc. [1]. Due to experimental and theoretical difficulties, there were only few attempts to use Langmuir-probes to explore the complex electron cyclotron resonance (ECR) ion source plasma until the second half of the 90s. However, the ECR ion source has become a very important tool in physics research and many other fields, e.g., medicine, material science, etc., so beside the building high performance ion sources it become important to know the physical process going on in the plasma of the source during tuning the source, which are at the basis of the high performance. Thus, plasma diagnostic research became also important, which can be done in global sense, externally using electromagnetic radiation * Corresponding author. address: l_kenez@ms.sapientia.ro (L. Kenéz). coming out of the plasma, or local sense, using electrical or socalled Langmuir-probes. Each method present advantages and disadvantages and find their place in better understanding the ECR ion source. The interest of researchers working in different fields regarding the local plasma diagnostics method using different types of Langmuir-probes has grown in the last 15 years. Langmuir-probes are used to characterize the ECR plasma, determining different plasma parameters of the different regions of the plasma, e.g. plasma potential, cold electron temperature, plasma density [2 7], but on the other hand they are also used in different experiments to find out information on the plasma during fabrication processes [8], treatments [9], thin film deposition processes [10] and so on. However, there are few information regarding local plasma diagnostics regarding the hot region of the ECR plasma, which is the hart of the ion source, where ionization processes and plasma confinement are going on. A local plasma diagnostics research has been started at the 14.5 GHz room temperature ECRIS in Debrecen, Hungary in The main goal was to adapt the Langmuir-probe method to the ECR conditions filling the lack of information regarding the ECR plasma. At firs we studied the cold plasma region. We used cylindrical tungsten probes, and successfully determined electron densities, parallel electron temperatures and their evolution caused by changes of the macroscopic source parameters [11,12]. Due to the /$ see front matter 2008 Elsevier B.V. All rights reserved. doi: /j.physleta

2 4928 L. Kenéz et al. / Physics Letters A 372 (2008) strong magnetic field, the plasma potential cannot be determined using simple probes [13]. However, emitting probes can be used to determine the plasma potential of the cold plasma [1,14]. Such measurements were performed at the 10 GHz all-permanent room temperature ECRIS in NIRS, Japan. Using the above mentioned probe configurations, we demonstrated explanations for different aspects of the so-called biased-disc effect [15]. Having gained experience in the cold plasma regions of the ECR plasma, we started to explore the hot plasma region of the source. We performed experiments using simple cylindrical probes, inserted in the ECR zone of the source. We measured unusual shaped probe voltage current curves and gave a qualitative model to explain it [15]. In this Letter we present a theoretical model which describes the strange behavior of the voltage current curves and the computational study of this model. 2. Theoretical model To prove the foregoing qualitative explanation we elaborate the following simplified theoretical model. The current collected by the probe in the resonant zone is composed of ion and cold electron currents, hot and secondary electron currents and thermionic current. In case of the probe used in the presented experiments, sheath dimensions were small comparatively with the probe dimensions. Therefore, it is convenient and adequate to use planar probe approximation to build the theoretical model Cold electron current density In terms of Cartesian coordinates, we can write the probe current density of the electrons to the negatively biased probe as j = e f (v x, v y, v z )v z dv x dv y dv z (1) v z min where minimal value of v z from energy conservation is v z min = [ ] 1/2 2e(V pl V pr )/m e (2) and V pl is the plasma potential, V pr is the probe bias-voltage, m e is the electron mass, and e is the magnitude of the electron charge. Assuming loss-cone distribution for the cold electron component [16] ( ) 1/2 me f c (v, v ) = n c exp( m e v 2 ) m e 2π T c 2T c 2 π T c j! ( me v 2 ) j exp( m e v 2 ) (3) 2T c 2T c where T c and T c are the parallel and perpendicular temperatures of the cold electrons, respectively (measured in energetic units), and n c is the cold electron density, we obtain for the cold electron current density the following expression [ j c = j c exp e(v ] pl V pr ), if V pr V pl, T c j e = j e, if V pr V pl, (4) where j e = en c Tc /2πm e is the cold electron saturation current density Incident (primary) hot electron current density Electrons gain high energies in ECR ion sources due to the ECR effect [17]. The B-minimum type magnetic trap of the source, which is formed by the superposition of the axial and the multipole radial magnetic fields, confines these hot electrons. The hot electrons are confined by the multipole magnetic field in the direction perpendicular to the axis of the source; therefore, for simplicity, we consider that the hot electrons are mono-energetic in this direction. However, we assume Maxwell-type distribution in the direction parallel to the source axis. The distribution function obtained this way is the Maxwell-ring distribution, which can be written for the hot electron component as f h (v, v ) = n ( ) 1 h me 2 δ(v w )e mv2 2T h (5) 2π w 2π T h where w is the common perpendicular velocity, n h the electron number density, and T h the parallel temperature of the hot electrons. The expression of the current density of the hot electrons can be obtained substituting the Maxwell-ring distribution function into Eq. (1). [ T h j h = en h exp e(v ] pl V pr ). (6) 2πm e T h 2.3. Secondary electron current density The secondary electron current density can be written as j s = e v z σ (ε w ) f h (v x, v y, v z ) d 3 v (7) where σ is evaluated at ε w, which is the kinetic energy of the incident hot electrons at the probe [1]. The minus sign indicates that the emission current appears as an apparent ion current. The dependence of the coefficient σ on the primary electron energy is described by the semi-empirical formula given by [18]: σ (ε w ) = (2.72) ε [ ( ) 1 ] 2 w εw 2 σ max exp 2 (8) E max E max where E max is the electron energy where the emission coefficient σ has its maximum yield σ max. Using cylindrical system of coordinates, the secondary electron current density can be written as: 2 π j s = e 0 0 v z min v z a(ε + eu) exp( b ε + eu ) f h (v z, v )v dv dv z dϕ. (9) In the above expression we introduced the following notations, a = (2.72) 2 σ max E max, b = 2 Emax (10) and also considered, that the electron kinetic energy on the surface of the probe is given by ε w = ε + eu (11) where U = V pr V pl represents the potential difference between the probe and the plasma, and ε = ε + E is the energy of the hot electrons in the plasma. Inserting the Maxwell-ring distribution (5) in Eq. (9), we obtain j s = en ha 2πme T h eu (ε + E + eu) e b ε +E +eu e ε T h dε (12) where ε = mv 2 z /2 is the electron energy corresponding to the motion parallel to the longitudinal magnetic field and E = mv 2 /2is the perpendicular electron energy.

3 L. Kenéz et al. / Physics Letters A 372 (2008) Ion current density In the ECR plasma (T i T e ) the ion saturation current density is given by the Bohm-current density [19] j i = 0.61ne (13) m i where we used an effective electron temperature taking into account that in the resonant zone there are two types of electrons present with different number density and temperature. The effective temperature can be defined as [20] 1 = n c nt c + n h nt h (14) where n = n c + n h is the total electron plasma density Thermionic current density Fig. 1. Experimental curves. When a probe is heated it emits thermionic electrons, which happens to a probe inserted in the resonant zone. The thermionic current density of a heated probe can be calculated using the Richardson Dushmann formula, which involves the temperature of the probe (T ), the work function of the material of the probe (L) and is given by (15): j T = AT 2 exp( L/kT) (15) where A is the Richardson coefficient and k is the Boltzmann constant. Due to the thermal inertia of the probe, the probe temperature practically does not depend on the probe potential. Hence we neglected it at the above presented qualitative explanation of the probe voltage current curve, while its only effect is to shift the ion part of the curve characteristic to higher negative values Total current density to the probe The total current density to the probe is given by the sum of the cold ( j c ), hot ( j h ), secondary ( j s ) electron current density, the thermionic current density ( j T ), respectively, and the ion current density ( j i ). Substituting the above determined expressions (4), (6), (12), (13) and (15) into the sum j = j c + j h + j s + j i + j T (16) we obtain for the total current density the following expression: ( ) j = AT 2 T c eu exp( L/kT) + en c exp 2πm e T c ( ) T h eu + en h exp 2πm e T h en h a (ε + E + eu)e b ε +E +eu e ε T h dε 2πme T h eu 0.61ne AT 2 exp( L/kT). (17) m i Calculated total current densities j = j(n c,n h, T c, T h, E, V ) are shown in Fig. 1 for various parameter combinations considering tungsten probe (E max = 700 ev and σ max = 1.4). Fig. 2. Calculated probe current densities for different hot electron densities, E = 1000 ev. 3. Computational study of the theoretical model. Conclusions The computational study of the theoretical model presented in Section 2 focuses on the main features of the measured probe voltage current curves presented in Fig. 1. The description of the experimental condition can be found in [15]. After testing many parameter combinations which are realistic from ECR ion source plasma point of view, we concluded that the shape of the curves given by the computational study of the theoretical model correspond to the measured ones presented in Fig On the experimental curves one can observe that the maximum value of the probe current increases when the probe is getting closer to the resonant zone. It is natural to expect (taking into account the mirror effect [17]), that the electron density must increase approaching the resonant zone. To test the validity of the theoretical model we present the analysis of three different sets of curves (Figs. 2, 3 and 4). To make correct conclusions in Fig. 5 we represent the energy dependence of the secondary emission coefficient versus incident electron energy. This coefficient increases monotonously for most metals reaching a maximum value somewhat greater than 1 at incident energies of the order of several hundred evs (E max ). Further increase of the incident electron energy causes the reduction of the secondary emission.

4 4930 L. Kenéz et al. / Physics Letters A 372 (2008) Fig. 3. Calculated probe current densities for different hot electron densities, E = 3000 ev. Fig. 6. Calculated total current density to the probe. Different parallel electron temperatures. nitudes in the different figures, we can conclude that increasing the hot electron temperature (which means that the number of the secondary electron decrease, Fig. 5) the magnitude of the current densities decrease, which is also expectable, because when the same number of electron with higher energies collide to the probe lower number of secondary electrons are released. Fig. 4. Calculated probe current densities for different hot electron densities, E = 5000 ev A series of curves corresponding to different parallel energies of the hot electrons are presented on Fig. 6, all other parameters having constant values. The position on the voltage scale of the minimum of the curve can be associated with the average parallel electron temperature of the hot electrons [15]. One can see that the increase of the parallel temperature causes the minimum position to be shifted toward lower negative voltages. Due to the confinement mechanism of the mirror machines, as the probe departs from the resonant zone, the parallel velocity (temperature) of the hot electrons will decrease and the minimum can be observed at lower negative voltages. The same conclusion can be drawn from the experimental curves (Fig. 1). In case of lower parallel energies, the minimum is obtained again at negative voltages with smaller absolute values, which means this feature is also reproduced by the theoretical model. The computational study of the presented theoretical model proves our assumptions regarding the mechanisms taking place when the simple probe is inserted in the hot resonant zone of the ECR plasma. References Fig. 5. Behavior of the secondary electron emission coefficient. Let us discuss at first separately about Figs. 2, 3 and 4. In all three cases the hot electron density is increased from curve to curve, while all other parameters were kept constant. In each figure can be observed that increasing the hot electron temperature, the maximum value of the probe current also increases. This behavior of the calculated curves matches the behavior of the experimental ones. Let us consider now the magnitudes of the maximum values in every one of the figures. On one hand we can conclude that increasing the hot electron temperature, the difference (denoted by j m, Figs. 2 4) between the minimum values of the current decrease. On the other hand comparing the mag- [1] N. Hershkowitz, Plasma diagnostics, vol. 1, in: O. Aucielo, D.L. Flamm (Eds.), How Langmuir Probes Work, Academic, San Diego, 1989, pp , Chapter 3. [2] H.J. You, K.S. Chung, F.W. Meyer, J. Korean Phys. Soc. 49 (4) (2006) [3]T.Lagarde,Y.Arnal,A.Lacoste,J.Pelletier,PlasmaSourcesSci.Technol.10(2) (2001) 181. [4] S.K. Jain, A. Jain, D. Sharma, P.R. Hannurkar, Indian J. Phys. Proc. Indian Assoc. CultivationSci.80(10)(2006)1011. [5] S.L. Fu, J.F. Chen, J.S. Hu, X.Q. Wu, Y. Lee, S.L. Fan, Plasma Sources Sci. Technol. 15 (2) (2006) 187. [6] A.J. Stoltz, M.J. Sperry, J.D. Benson, J.B. Varesi, M. Martinka, L.A. Almeida, P.R. Boyd, J.H. Dinan, J. Electronic Mater. 34 (6) (2005) 733. [7] J.F. Chen, S.L. Fu, X.Q. Lai, P.F. Xiang, Y. Li, X.Q. Wu, Vacuum 81 (2006) 49. [8] D.H. Thang, K. Sasaki, H. Muta, N. Itagaki, Y. Kawai, Thin Solid Films (2006) 485.

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